#! /bin/sh
# This is a shell archive.  Remove anything before this line, then unpack
# it by saving it into a file and typing "sh file".  To overwrite existing
# files, type "sh file -c".  You can also feed this as standard input via
# unshar, or by typing "sh <file", e.g..  If this archive is complete, you
# will see the following message at the end:
#		"End of shell archive."
# Contents:  AREADME.1ST Manual.ps Contents.m app_hh.m baart.m bidiag.m
#   bsvd.m cgls.m csdecomp.m csvd.m deriv2.m discrep.m dsvd.m
#   fil_fac.m fnder.m foxgood.m gcv.m gen_form.m gen_hh.m get_l.m
#   gsvd.m heat.m heb_new.m ilaplace.m l_corner.m l_curve.m lagrange.m
#   lanc_b.m lsolve.m lsqi.m lsqr.m ltsolve.m maxent.m mgs.m mtsvd.m
#   newton.m nu.m parallax.m pcgls.m phillips.m picard.m pinit.m
#   plot_lc.m plsqr.m pnu.m ppbrk.m ppcut.m ppmak.m ppual.m pythag.m
#   quasiopt.m regudemo.m shaw.m sorted.m sp2pp.m spbrk.m spikes.m
#   spleval.m spmak.m sprpp.m std_form.m tgsvd.m tikhonov.m tsvd.m
#   ttls.m ursell.m wing.m
# Wrapped by michela@aurora on Sat Oct 31 12:10:42 1998
PATH=/bin:/usr/bin:/usr/ucb ; export PATH
if test -f 'AREADME.1ST' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'AREADME.1ST'\"
else
echo shar: Extracting \"'AREADME.1ST'\" \(3303 characters\)
sed "s/^X//" >'AREADME.1ST' <<'END_OF_FILE'
X  ***************************************************************************
X  * All the software  contained in this library  is protected by copyright. *
X  * Permission  to use, copy, modify, and  distribute this software for any *
X  * purpose without fee is hereby granted, provided that this entire notice *
X  * is included  in all copies  of any software which is or includes a copy *
X  * or modification  of this software  and in all copies  of the supporting *
X  * documentation for such software.                                        *
X  ***************************************************************************
X  * THIS SOFTWARE IS BEING PROVIDED "AS IS", WITHOUT ANY EXPRESS OR IMPLIED *
X  * WARRANTY. IN NO EVENT, NEITHER  THE AUTHORS, NOR THE PUBLISHER, NOR ANY *
X  * MEMBER  OF THE EDITORIAL BOARD OF  THE JOURNAL  "NUMERICAL ALGORITHMS", *
X  * NOR ITS EDITOR-IN-CHIEF, BE  LIABLE FOR ANY ERROR  IN THE SOFTWARE, ANY *
X  * MISUSE  OF IT  OR ANY DAMAGE ARISING OUT OF ITS USE. THE ENTIRE RISK OF *
X  * USING THE SOFTWARE LIES WITH THE PARTY DOING SO.                        *
X  ***************************************************************************
X  * ANY USE  OF THE SOFTWARE  CONSTITUTES  ACCEPTANCE  OF THE TERMS  OF THE *
X  * ABOVE STATEMENT.                                                        *
X  ***************************************************************************
X
X   AUTHOR:
X
X       P. C. HANSEN
X          DEPT. OF MATHEMATICAL MODELLLING
X          TECHNICAL UNIVERSITY OF DENMARK
X
X   REFERENCE:
X
X       REGULARIZATION TOOLS: A MATLAB PACKAGE FOR ANALYSIS AND SOLUTION OF
X       DISCRETE ILL-POSED PROBLEMS,
X       NUMERICAL ALGORITHMS, 6 (1994), PP. 1-35
X
X   SOFTWARE REVISION:
X
X       Ver 2.1  MARCH 31, 1998
X
X   SOFTWARE LANGUAGE:
X
X       MATLAB 4
X
X**************************************************************************
X
XRegularization Tools.
XVersion 2.1  31-March-98.
XCopyright (c) 1998 by Per Christian Hansen and UNI-C.
X
XThe installation of Regularization Tools is very simple:
X
X  1.  Unpack the shell archive na4-matlab4 by executing the command
X         /bin/sh na4-matlab4
X
X  2.  Remove the file na4-matlab4
X
X  3.  If the Spline Toolbox is installed on your computer, then remove
X      the following 10 m-files
X         fnder.m   sp2pp.m
X         ppbrk.m   sorted.m
X         ppcut.m   spbrk.m
X         ppmak.m   spmak.m
X         ppual.m   sprpp.m
X
X  4.  The file Manual.ps contains the related manual in PostScript form
X      (not revised)
X
X***************************************************************
X* This is Version 2.1 of Regularization Tools for Matlab 4.2c *
X*-------------------------------------------------------------*
X*                 Per Christian Hansen, IMM                   *
X***************************************************************
X
XRevisions in Version 2.1
X
X02/01/94:
XFixed bug in cgls (s -> s2).
X
X08/03/94:
XExpanded stopping criterion in newton.
X
X11/01/94:
XModified get_l slightly such that the sign of L*x is correct.
X
X02/09/95:
XRenamed csd to csdecomp (csd is now a function in the Signal Proc. Toolbox).
XRevised gsvd to call csdecomp.
X
X07/02/97:
XFixed bug in pcgls when computing filter factors.
X
X11/11/97:
XModified gen_hh to compensate for Matlab's signum function.
X
X12/29/97:
XCorrected bugs in discrep and lsqi.
X
END_OF_FILE
if test 3303 -ne `wc -c <'AREADME.1ST'`; then
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fi
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fi
if test -f 'Manual.ps' -a "${1}" != "-c" ; then 
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else
echo shar: Extracting \"'Manual.ps'\" \(893990 characters\)
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X%%EndProlog
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X%%Feature: *Resolution 300dpi
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X/Helvetica 14 FMS
X
X
X/Helvetica 10 FMS
X
X
X/Helvetica 14 FMS
X
X
X/Helvetica 10 FMS
X
X
X/Helvetica 14 FMS
X
X
X/Helvetica 10 FMS
X
X
X/Helvetica 14 FMS
X
X
X/Helvetica 10 FMS
X
X
X/Helvetica 14 FMS
X
X
X/Helvetica 10 FMS
X
X
X/Helvetica 14 FMS
X
X
X/Helvetica 10 FMS
X
X
X/Helvetica 14 FMS
X
X
X/Helvetica 10 FMS
X
X
X/Helvetica 14 FMS
X
X
X/Helvetica 10 FMS
X
X
X/Helvetica 14 FMS
X
X
X/Helvetica 10 FMS
X
X
X/Helvetica 14 FMS
X
X
X/Helvetica 10 FMS
X
X
X/Helvetica 14 FMS
X
X
X/Helvetica 10 FMS
X
X
X/Helvetica 14 FMS
X
Xc1
X   0    0 6912 5184 PR
XDO
XSO
Xc0
X 898 2165 mt 6221 2165 L
X 898  414 mt 6221  414 L
X 898 2165 mt  898  414 L
X6221 2165 mt 6221  414 L
X 898 2165 mt  898 2165 L
X6221 2165 mt 6221 2165 L
X 898 2165 mt 6221 2165 L
X 898 2165 mt  898  414 L
X 898 2165 mt  898 2165 L
X 898 2165 mt  898 2112 L
X 898  414 mt  898  467 L
X 852 2348 mt (0) s
X1658 2165 mt 1658 2112 L
X1658  414 mt 1658  467 L
X1612 2348 mt (5) s
X2419 2165 mt 2419 2112 L
X2419  414 mt 2419  467 L
X2326 2348 mt (10) s
X3179 2165 mt 3179 2112 L
X3179  414 mt 3179  467 L
X3086 2348 mt (15) s
X3940 2165 mt 3940 2112 L
X3940  414 mt 3940  467 L
X3847 2348 mt (20) s
X4700 2165 mt 4700 2112 L
X4700  414 mt 4700  467 L
X4607 2348 mt (25) s
X5461 2165 mt 5461 2112 L
X5461  414 mt 5461  467 L
X5368 2348 mt (30) s
X6221 2165 mt 6221 2112 L
X6221  414 mt 6221  467 L
X6128 2348 mt (35) s
X 898 2165 mt  951 2165 L
X6221 2165 mt 6168 2165 L
X
X/Helvetica 10 FMS
X
X
X/Helvetica 14 FMS
X
X 512 2227 mt (10) s
X
X/Helvetica 10 FMS
X
X 698 2116 mt (-20) s
X
X/Helvetica 14 FMS
X
X 898 2095 mt  925 2095 L
X6221 2095 mt 6194 2095 L
X 898 2025 mt  925 2025 L
X6221 2025 mt 6194 2025 L
X 898 1955 mt  925 1955 L
X6221 1955 mt 6194 1955 L
X 898 1885 mt  925 1885 L
X6221 1885 mt 6194 1885 L
X 898 1815 mt  925 1815 L
X6221 1815 mt 6194 1815 L
X 898 1745 mt  925 1745 L
X6221 1745 mt 6194 1745 L
X 898 1675 mt  925 1675 L
X6221 1675 mt 6194 1675 L
X 898 1605 mt  925 1605 L
X6221 1605 mt 6194 1605 L
X 898 1535 mt  925 1535 L
X6221 1535 mt 6194 1535 L
X 898 1815 mt  951 1815 L
X6221 1815 mt 6168 1815 L
X
X/Helvetica 10 FMS
X
X
X/Helvetica 14 FMS
X
X 512 1877 mt (10) s
X
X/Helvetica 10 FMS
X
X 698 1766 mt (-15) s
X
X/Helvetica 14 FMS
X
X 898 1745 mt  925 1745 L
X6221 1745 mt 6194 1745 L
X 898 1675 mt  925 1675 L
X6221 1675 mt 6194 1675 L
X 898 1605 mt  925 1605 L
X6221 1605 mt 6194 1605 L
X 898 1535 mt  925 1535 L
X6221 1535 mt 6194 1535 L
X 898 1465 mt  925 1465 L
X6221 1465 mt 6194 1465 L
X 898 1395 mt  925 1395 L
X6221 1395 mt 6194 1395 L
X 898 1325 mt  925 1325 L
X6221 1325 mt 6194 1325 L
X 898 1254 mt  925 1254 L
X6221 1254 mt 6194 1254 L
X 898 1184 mt  925 1184 L
X6221 1184 mt 6194 1184 L
X 898 1465 mt  951 1465 L
X6221 1465 mt 6168 1465 L
X
X/Helvetica 10 FMS
X
X
X/Helvetica 14 FMS
X
X 512 1527 mt (10) s
X
X/Helvetica 10 FMS
X
X 698 1416 mt (-10) s
X
X/Helvetica 14 FMS
X
X 898 1395 mt  925 1395 L
X6221 1395 mt 6194 1395 L
X 898 1325 mt  925 1325 L
X6221 1325 mt 6194 1325 L
X 898 1254 mt  925 1254 L
X6221 1254 mt 6194 1254 L
X 898 1184 mt  925 1184 L
X6221 1184 mt 6194 1184 L
X 898 1114 mt  925 1114 L
X6221 1114 mt 6194 1114 L
X 898 1044 mt  925 1044 L
X6221 1044 mt 6194 1044 L
X 898  974 mt  925  974 L
X6221  974 mt 6194  974 L
X 898  904 mt  925  904 L
X6221  904 mt 6194  904 L
X 898  834 mt  925  834 L
X6221  834 mt 6194  834 L
X 898 1114 mt  951 1114 L
X6221 1114 mt 6168 1114 L
X
X/Helvetica 10 FMS
X
X
X/Helvetica 14 FMS
X
X 512 1176 mt (10) s
X
X/Helvetica 10 FMS
X
X 698 1065 mt (-5) s
X
X/Helvetica 14 FMS
X
X 898 1044 mt  925 1044 L
X6221 1044 mt 6194 1044 L
X 898  974 mt  925  974 L
X6221  974 mt 6194  974 L
X 898  904 mt  925  904 L
X6221  904 mt 6194  904 L
X 898  834 mt  925  834 L
X6221  834 mt 6194  834 L
X 898  764 mt  925  764 L
X6221  764 mt 6194  764 L
X 898  694 mt  925  694 L
X6221  694 mt 6194  694 L
X 898  624 mt  925  624 L
X6221  624 mt 6194  624 L
X 898  554 mt  925  554 L
X6221  554 mt 6194  554 L
X 898  484 mt  925  484 L
X6221  484 mt 6194  484 L
X 898  764 mt  951  764 L
X6221  764 mt 6168  764 L
X
X/Helvetica 10 FMS
X
X
X/Helvetica 14 FMS
X
X 512  826 mt (10) s
X
X/Helvetica 10 FMS
X
X 698  715 mt (0) s
X
X/Helvetica 14 FMS
X
X 898  694 mt  925  694 L
X6221  694 mt 6194  694 L
X 898  624 mt  925  624 L
X6221  624 mt 6194  624 L
X 898  554 mt  925  554 L
X6221  554 mt 6194  554 L
X 898  484 mt  925  484 L
X6221  484 mt 6194  484 L
X 898  414 mt  925  414 L
X6221  414 mt 6194  414 L
X 898  414 mt  951  414 L
X6221  414 mt 6168  414 L
X
X/Helvetica 10 FMS
X
X
X/Helvetica 14 FMS
X
X 512  476 mt (10) s
X
X/Helvetica 10 FMS
X
X 698  365 mt (5) s
X
X/Helvetica 14 FMS
X
Xgs 898 414 5324 1752 MR c np
X152 23 152 21 153 6 152 12 152 4 152 17 152 5 152 5 
X152 9 152 5 152 21 152 145 152 38 152 37 153 113 152 66 
X152 82 152 78 152 67 152 50 152 41 152 65 152 88 152 38 
X152 53 152 10 153 17 152 57 152 30 152 18 152 14 1050 731 32 MP stroke
Xgr 
X1014  651 mt 1086  723 L
X1086  651 mt 1014  723 L
Xgs 898 414 5324 1752 MR c np
Xgr 
X1166  706 mt 1238  778 L
X1238  706 mt 1166  778 L
Xgs 898 414 5324 1752 MR c np
Xgr 
X1318  691 mt 1390  763 L
X1390  691 mt 1318  763 L
Xgs 898 414 5324 1752 MR c np
Xgr 
X1470  750 mt 1542  822 L
X1542  750 mt 1470  822 L
Xgs 898 414 5324 1752 MR c np
Xgr 
X1622  836 mt 1694  908 L
X1694  836 mt 1622  908 L
Xgs 898 414 5324 1752 MR c np
Xgr 
X1775  849 mt 1847  921 L
X1847  849 mt 1775  921 L
Xgs 898 414 5324 1752 MR c np
Xgr 
X1927  859 mt 1999  931 L
X1999  859 mt 1927  931 L
Xgs 898 414 5324 1752 MR c np
Xgr 
X2079 1001 mt 2151 1073 L
X2151 1001 mt 2079 1073 L
Xgs 898 414 5324 1752 MR c np
Xgr 
X2231  981 mt 2303 1053 L
X2303  981 mt 2231 1053 L
Xgs 898 414 5324 1752 MR c np
Xgr 
X2383 1078 mt 2455 1150 L
X2455 1078 mt 2383 1150 L
Xgs 898 414 5324 1752 MR c np
Xgr 
X2535 1197 mt 2607 1269 L
X2607 1197 mt 2535 1269 L
Xgs 898 414 5324 1752 MR c np
Xgr 
X2687 1200 mt 2759 1272 L
X2759 1200 mt 2687 1272 L
Xgs 898 414 5324 1752 MR c np
Xgr 
X2839 1276 mt 2911 1348 L
X2911 1276 mt 2839 1348 L
Xgs 898 414 5324 1752 MR c np
Xgr 
X2991 1450 mt 3063 1522 L
X3063 1450 mt 2991 1522 L
Xgs 898 414 5324 1752 MR c np
Xgr 
X3143 1445 mt 3215 1517 L
X3215 1445 mt 3143 1517 L
Xgs 898 414 5324 1752 MR c np
Xgr 
X3295 1562 mt 3367 1634 L
X3367 1562 mt 3295 1634 L
Xgs 898 414 5324 1752 MR c np
Xgr 
X3447 1657 mt 3519 1729 L
X3519 1657 mt 3447 1729 L
Xgs 898 414 5324 1752 MR c np
Xgr 
X3600 1779 mt 3672 1851 L
X3672 1779 mt 3600 1851 L
Xgs 898 414 5324 1752 MR c np
Xgr 
X3752 1770 mt 3824 1842 L
X3824 1770 mt 3752 1842 L
Xgs 898 414 5324 1752 MR c np
Xgr 
X3904 1780 mt 3976 1852 L
X3976 1780 mt 3904 1852 L
Xgs 898 414 5324 1752 MR c np
Xgr 
X4056 1775 mt 4128 1847 L
X4128 1775 mt 4056 1847 L
Xgs 898 414 5324 1752 MR c np
Xgr 
X4208 1779 mt 4280 1851 L
X4280 1779 mt 4208 1851 L
Xgs 898 414 5324 1752 MR c np
Xgr 
X4360 1779 mt 4432 1851 L
X4432 1779 mt 4360 1851 L
Xgs 898 414 5324 1752 MR c np
Xgr 
X4512 1770 mt 4584 1842 L
X4584 1770 mt 4512 1842 L
Xgs 898 414 5324 1752 MR c np
Xgr 
X4664 1778 mt 4736 1850 L
X4736 1778 mt 4664 1850 L
Xgs 898 414 5324 1752 MR c np
Xgr 
X4816 1824 mt 4888 1896 L
X4888 1824 mt 4816 1896 L
Xgs 898 414 5324 1752 MR c np
Xgr 
X4968 1782 mt 5040 1854 L
X5040 1782 mt 4968 1854 L
Xgs 898 414 5324 1752 MR c np
Xgr 
X5120 1783 mt 5192 1855 L
X5192 1783 mt 5120 1855 L
Xgs 898 414 5324 1752 MR c np
Xgr 
X5272 1782 mt 5344 1854 L
X5344 1782 mt 5272 1854 L
Xgs 898 414 5324 1752 MR c np
Xgr 
X5425 1791 mt 5497 1863 L
X5497 1791 mt 5425 1863 L
Xgs 898 414 5324 1752 MR c np
Xgr 
X5577 1799 mt 5649 1871 L
X5649 1799 mt 5577 1871 L
Xgs 898 414 5324 1752 MR c np
Xgr 
X5729 1764 mt 5801 1836 L
X5801 1764 mt 5729 1836 L
Xgs 898 414 5324 1752 MR c np
Xgr 
X1014  685 1086  757 FO
Xgs 898 414 5324 1752 MR c np
Xgr 
X1166  725 1238  797 FO
Xgs 898 414 5324 1752 MR c np
Xgr 
X1318  692 1390  764 FO
Xgs 898 414 5324 1752 MR c np
Xgr 
X1470  721 1542  793 FO
Xgs 898 414 5324 1752 MR c np
Xgr 
X1622  750 1694  822 FO
Xgs 898 414 5324 1752 MR c np
Xgr 
X1775  746 1847  818 FO
Xgs 898 414 5324 1752 MR c np
Xgr 
X1927  746 1999  818 FO
Xgs 898 414 5324 1752 MR c np
Xgr 
X2079  835 2151  907 FO
Xgs 898 414 5324 1752 MR c np
Xgr 
X2231  778 2303  850 FO
Xgs 898 414 5324 1752 MR c np
Xgr 
X2383  786 2455  858 FO
Xgs 898 414 5324 1752 MR c np
Xgr 
X2535  840 2607  912 FO
Xgs 898 414 5324 1752 MR c np
Xgr 
X2687  802 2759  874 FO
Xgs 898 414 5324 1752 MR c np
Xgr 
X2839  829 2911  901 FO
Xgs 898 414 5324 1752 MR c np
Xgr 
X2991  936 3063 1008 FO
Xgs 898 414 5324 1752 MR c np
Xgr 
X3143  853 3215  925 FO
Xgs 898 414 5324 1752 MR c np
Xgr 
X3295  887 3367  959 FO
Xgs 898 414 5324 1752 MR c np
Xgr 
X3447  916 3519  988 FO
Xgs 898 414 5324 1752 MR c np
Xgr 
X3600  925 3672  997 FO
Xgs 898 414 5324 1752 MR c np
Xgr 
X3752  879 3824  951 FO
Xgs 898 414 5324 1752 MR c np
Xgr 
X3904  851 3976  923 FO
Xgs 898 414 5324 1752 MR c np
Xgr 
X4056  701 4128  773 FO
Xgs 898 414 5324 1752 MR c np
Xgr 
X4208  684 4280  756 FO
Xgs 898 414 5324 1752 MR c np
Xgr 
X4360  678 4432  750 FO
Xgs 898 414 5324 1752 MR c np
Xgr 
X4512  661 4584  733 FO
Xgs 898 414 5324 1752 MR c np
Xgr 
X4664  664 4736  736 FO
Xgs 898 414 5324 1752 MR c np
Xgr 
X4816  705 4888  777 FO
Xgs 898 414 5324 1752 MR c np
Xgr 
X4968  645 5040  717 FO
Xgs 898 414 5324 1752 MR c np
Xgr 
X5120  643 5192  715 FO
Xgs 898 414 5324 1752 MR c np
Xgr 
X5272  630 5344  702 FO
Xgs 898 414 5324 1752 MR c np
Xgr 
X5425  633 5497  705 FO
Xgs 898 414 5324 1752 MR c np
Xgr 
X5577  621 5649  693 FO
Xgs 898 414 5324 1752 MR c np
Xgr 
X5729  562 5801  634 FO
Xgs 898 414 5324 1752 MR c np
Xgr 
X3542 2525 mt (i) s
X1606  281 mt (--- = sing. values     x = rhs. coef.     o = solution coef.) s
X 898 2165 mt 6221 2165 L
X 898  414 mt 6221  414 L
X 898 2165 mt  898  414 L
X6221 2165 mt 6221  414 L
X 898  414 mt  898  414 L
X6221  414 mt 6221  414 L
XDO
XSO
X 898 4614 mt 6221 4614 L
X 898 2864 mt 6221 2864 L
X 898 4614 mt  898 2864 L
X6221 4614 mt 6221 2864 L
X 898 4614 mt  898 4614 L
X6221 4614 mt 6221 4614 L
X 898 4614 mt 6221 4614 L
X 898 4614 mt  898 2864 L
X 898 4614 mt  898 4614 L
X 898 4614 mt  898 4561 L
X 898 2864 mt  898 2917 L
X 852 4797 mt (0) s
X1658 4614 mt 1658 4561 L
X1658 2864 mt 1658 2917 L
X1612 4797 mt (5) s
X2419 4614 mt 2419 4561 L
X2419 2864 mt 2419 2917 L
X2326 4797 mt (10) s
X3179 4614 mt 3179 4561 L
X3179 2864 mt 3179 2917 L
X3086 4797 mt (15) s
X3940 4614 mt 3940 4561 L
X3940 2864 mt 3940 2917 L
X3847 4797 mt (20) s
X4700 4614 mt 4700 4561 L
X4700 2864 mt 4700 2917 L
X4607 4797 mt (25) s
X5461 4614 mt 5461 4561 L
X5461 2864 mt 5461 2917 L
X5368 4797 mt (30) s
X6221 4614 mt 6221 4561 L
X6221 2864 mt 6221 2917 L
X6128 4797 mt (35) s
X 898 4614 mt  951 4614 L
X6221 4614 mt 6168 4614 L
X
X/Helvetica 10 FMS
X
X
X/Helvetica 14 FMS
X
X 512 4676 mt (10) s
X
X/Helvetica 10 FMS
X
X 698 4565 mt (-20) s
X
X/Helvetica 14 FMS
X
X 898 4395 mt  925 4395 L
X6221 4395 mt 6194 4395 L
X 898 4177 mt  925 4177 L
X6221 4177 mt 6194 4177 L
X 898 3958 mt  925 3958 L
X6221 3958 mt 6194 3958 L
X 898 3739 mt  925 3739 L
X6221 3739 mt 6194 3739 L
X 898 3520 mt  925 3520 L
X6221 3520 mt 6194 3520 L
X 898 3302 mt  925 3302 L
X6221 3302 mt 6194 3302 L
X 898 3083 mt  925 3083 L
X6221 3083 mt 6194 3083 L
X 898 2864 mt  925 2864 L
X6221 2864 mt 6194 2864 L
X 898 4177 mt  951 4177 L
X6221 4177 mt 6168 4177 L
X
X/Helvetica 10 FMS
X
X
X/Helvetica 14 FMS
X
X 512 4239 mt (10) s
X
X/Helvetica 10 FMS
X
X 698 4128 mt (-10) s
X
X/Helvetica 14 FMS
X
X 898 3958 mt  925 3958 L
X6221 3958 mt 6194 3958 L
X 898 3739 mt  925 3739 L
X6221 3739 mt 6194 3739 L
X 898 3520 mt  925 3520 L
X6221 3520 mt 6194 3520 L
X 898 3302 mt  925 3302 L
X6221 3302 mt 6194 3302 L
X 898 3083 mt  925 3083 L
X6221 3083 mt 6194 3083 L
X 898 2864 mt  925 2864 L
X6221 2864 mt 6194 2864 L
X 898 3739 mt  951 3739 L
X6221 3739 mt 6168 3739 L
X
X/Helvetica 10 FMS
X
X
X/Helvetica 14 FMS
X
X 512 3801 mt (10) s
X
X/Helvetica 10 FMS
X
X 698 3690 mt (0) s
X
X/Helvetica 14 FMS
X
X 898 3520 mt  925 3520 L
X6221 3520 mt 6194 3520 L
X 898 3302 mt  925 3302 L
X6221 3302 mt 6194 3302 L
X 898 3083 mt  925 3083 L
X6221 3083 mt 6194 3083 L
X 898 2864 mt  925 2864 L
X6221 2864 mt 6194 2864 L
X 898 3302 mt  951 3302 L
X6221 3302 mt 6168 3302 L
X
X/Helvetica 10 FMS
X
X
X/Helvetica 14 FMS
X
X 512 3364 mt (10) s
X
X/Helvetica 10 FMS
X
X 698 3253 mt (10) s
X
X/Helvetica 14 FMS
X
X 898 3083 mt  925 3083 L
X6221 3083 mt 6194 3083 L
X 898 2864 mt  925 2864 L
X6221 2864 mt 6194 2864 L
X 898 2864 mt  951 2864 L
X6221 2864 mt 6168 2864 L
X
X/Helvetica 10 FMS
X
X
X/Helvetica 14 FMS
X
X 512 2926 mt (10) s
X
X/Helvetica 10 FMS
X
X 698 2815 mt (20) s
X
X/Helvetica 14 FMS
X
Xgs 898 2864 5324 1751 MR c np
X152 15 152 13 153 4 152 7 152 2 152 11 152 3 152 3 
X152 6 152 3 152 13 152 91 152 23 152 24 153 70 152 41 
X152 52 152 49 152 41 152 31 152 26 152 41 152 55 152 23 
X152 34 152 6 153 10 152 36 152 19 152 11 152 9 1050 3718 32 MP stroke
Xgr 
X1014 3655 mt 1086 3727 L
X1086 3655 mt 1014 3727 L
Xgs 898 2864 5324 1751 MR c np
Xgr 
X1166 3689 mt 1238 3761 L
X1238 3689 mt 1166 3761 L
Xgs 898 2864 5324 1751 MR c np
Xgr 
X1318 3680 mt 1390 3752 L
X1390 3680 mt 1318 3752 L
Xgs 898 2864 5324 1751 MR c np
Xgr 
X1470 3716 mt 1542 3788 L
X1542 3716 mt 1470 3788 L
Xgs 898 2864 5324 1751 MR c np
Xgr 
X1622 3771 mt 1694 3843 L
X1694 3771 mt 1622 3843 L
Xgs 898 2864 5324 1751 MR c np
Xgr 
X1775 3779 mt 1847 3851 L
X1847 3779 mt 1775 3851 L
Xgs 898 2864 5324 1751 MR c np
Xgr 
X1927 3785 mt 1999 3857 L
X1999 3785 mt 1927 3857 L
Xgs 898 2864 5324 1751 MR c np
Xgr 
X2079 3846 mt 2151 3918 L
X2151 3846 mt 2079 3918 L
Xgs 898 2864 5324 1751 MR c np
Xgr 
X2231 3848 mt 2303 3920 L
X2303 3848 mt 2231 3920 L
Xgs 898 2864 5324 1751 MR c np
Xgr 
X2383 3835 mt 2455 3907 L
X2455 3835 mt 2383 3907 L
Xgs 898 2864 5324 1751 MR c np
Xgr 
X2535 3829 mt 2607 3901 L
X2607 3829 mt 2535 3901 L
Xgs 898 2864 5324 1751 MR c np
Xgr 
X2687 3846 mt 2759 3918 L
X2759 3846 mt 2687 3918 L
Xgs 898 2864 5324 1751 MR c np
Xgr 
X2839 3841 mt 2911 3913 L
X2911 3841 mt 2839 3913 L
Xgs 898 2864 5324 1751 MR c np
Xgr 
X2991 3841 mt 3063 3913 L
X3063 3841 mt 2991 3913 L
Xgs 898 2864 5324 1751 MR c np
Xgr 
X3143 3836 mt 3215 3908 L
X3215 3836 mt 3143 3908 L
Xgs 898 2864 5324 1751 MR c np
Xgr 
X3295 3838 mt 3367 3910 L
X3367 3838 mt 3295 3910 L
Xgs 898 2864 5324 1751 MR c np
Xgr 
X3447 3826 mt 3519 3898 L
X3519 3826 mt 3447 3898 L
Xgs 898 2864 5324 1751 MR c np
Xgr 
X3600 3834 mt 3672 3906 L
X3672 3834 mt 3600 3906 L
Xgs 898 2864 5324 1751 MR c np
Xgr 
X3752 3823 mt 3824 3895 L
X3824 3823 mt 3752 3895 L
Xgs 898 2864 5324 1751 MR c np
Xgr 
X3904 3826 mt 3976 3898 L
X3976 3826 mt 3904 3898 L
Xgs 898 2864 5324 1751 MR c np
Xgr 
X4056 3842 mt 4128 3914 L
X4128 3842 mt 4056 3914 L
Xgs 898 2864 5324 1751 MR c np
Xgr 
X4208 3855 mt 4280 3927 L
X4280 3855 mt 4208 3927 L
Xgs 898 2864 5324 1751 MR c np
Xgr 
X4360 3836 mt 4432 3908 L
X4432 3836 mt 4360 3908 L
Xgs 898 2864 5324 1751 MR c np
Xgr 
X4512 3846 mt 4584 3918 L
X4584 3846 mt 4512 3918 L
Xgs 898 2864 5324 1751 MR c np
Xgr 
X4664 3835 mt 4736 3907 L
X4736 3835 mt 4664 3907 L
Xgs 898 2864 5324 1751 MR c np
Xgr 
X4816 3841 mt 4888 3913 L
X4888 3841 mt 4816 3913 L
Xgs 898 2864 5324 1751 MR c np
Xgr 
X4968 3839 mt 5040 3911 L
X5040 3839 mt 4968 3911 L
Xgs 898 2864 5324 1751 MR c np
Xgr 
X5120 3856 mt 5192 3928 L
X5192 3856 mt 5120 3928 L
Xgs 898 2864 5324 1751 MR c np
Xgr 
X5272 3823 mt 5344 3895 L
X5344 3823 mt 5272 3895 L
Xgs 898 2864 5324 1751 MR c np
Xgr 
X5425 3851 mt 5497 3923 L
X5497 3851 mt 5425 3923 L
Xgs 898 2864 5324 1751 MR c np
Xgr 
X5577 3837 mt 5649 3909 L
X5649 3837 mt 5577 3909 L
Xgs 898 2864 5324 1751 MR c np
Xgr 
X5729 3862 mt 5801 3934 L
X5801 3862 mt 5729 3934 L
Xgs 898 2864 5324 1751 MR c np
Xgr 
X1014 3676 1086 3748 FO
Xgs 898 2864 5324 1751 MR c np
Xgr 
X1166 3701 1238 3773 FO
Xgs 898 2864 5324 1751 MR c np
Xgr 
X1318 3681 1390 3753 FO
Xgs 898 2864 5324 1751 MR c np
Xgr 
X1470 3699 1542 3771 FO
Xgs 898 2864 5324 1751 MR c np
Xgr 
X1622 3717 1694 3789 FO
Xgs 898 2864 5324 1751 MR c np
Xgr 
X1775 3715 1847 3787 FO
Xgs 898 2864 5324 1751 MR c np
Xgr 
X1927 3714 1999 3786 FO
Xgs 898 2864 5324 1751 MR c np
Xgr 
X2079 3742 2151 3814 FO
Xgs 898 2864 5324 1751 MR c np
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X2231 3721 2303 3793 FO
Xgs 898 2864 5324 1751 MR c np
Xgr 
X2383 3653 2455 3725 FO
Xgs 898 2864 5324 1751 MR c np
Xgr 
X2535 3606 2607 3678 FO
Xgs 898 2864 5324 1751 MR c np
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X2687 3598 2759 3670 FO
Xgs 898 2864 5324 1751 MR c np
Xgr 
X2839 3561 2911 3633 FO
Xgs 898 2864 5324 1751 MR c np
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X2991 3520 3063 3592 FO
Xgs 898 2864 5324 1751 MR c np
Xgr 
X3143 3466 3215 3538 FO
Xgs 898 2864 5324 1751 MR c np
Xgr 
X3295 3417 3367 3489 FO
Xgs 898 2864 5324 1751 MR c np
Xgr 
X3447 3363 3519 3435 FO
Xgs 898 2864 5324 1751 MR c np
Xgr 
X3600 3301 3672 3373 FO
Xgs 898 2864 5324 1751 MR c np
Xgr 
X3752 3266 3824 3338 FO
Xgs 898 2864 5324 1751 MR c np
Xgr 
X3904 3246 3976 3318 FO
Xgs 898 2864 5324 1751 MR c np
Xgr 
X4056 3171 4128 3243 FO
Xgs 898 2864 5324 1751 MR c np
Xgr 
X4208 3171 4280 3243 FO
Xgs 898 2864 5324 1751 MR c np
Xgr 
X4360 3149 4432 3221 FO
Xgs 898 2864 5324 1751 MR c np
Xgr 
X4512 3153 4584 3225 FO
Xgs 898 2864 5324 1751 MR c np
Xgr 
X4664 3139 4736 3211 FO
Xgs 898 2864 5324 1751 MR c np
Xgr 
X4816 3142 4888 3214 FO
Xgs 898 2864 5324 1751 MR c np
Xgr 
X4968 3130 5040 3202 FO
Xgs 898 2864 5324 1751 MR c np
Xgr 
X5120 3144 5192 3216 FO
Xgs 898 2864 5324 1751 MR c np
Xgr 
X5272 3104 5344 3176 FO
Xgs 898 2864 5324 1751 MR c np
Xgr 
X5425 3128 5497 3200 FO
Xgs 898 2864 5324 1751 MR c np
Xgr 
X5577 3100 5649 3172 FO
Xgs 898 2864 5324 1751 MR c np
Xgr 
X5729 3111 5801 3183 FO
Xgs 898 2864 5324 1751 MR c np
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X6221 4614 mt 6221 2864 L
X 898 2864 mt  898 2864 L
X6221 2864 mt 6221 2864 L
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X-30 910 124 -351 31 -185 2016 1008 4 MP  PP
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X
X-30 910 124 -351 31 -185 2016 1008 4 MP  DP
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X-30 -135 124 214 31 50 2139 948 4 MP  PP
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X
X-30 -135 124 214 31 50 2139 948 4 MP  DP
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X
X-31 72 124 135 31 7 2170 998 4 MP  PP
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X
X-31 72 124 135 31 7 2170 998 4 MP  DP
Xc9
X
X-31 185 124 -97 31 -8 1892 928 4 MP  PP
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X
X-31 185 124 -97 31 -8 1892 928 4 MP  DP
Xcolortable /c11 { 0.000000 0.380952 1.000000 sc} put
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X
X-31 -316 125 129 30 78 2109 870 4 MP  PP
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X
X-31 -316 125 129 30 78 2109 870 4 MP  DP
Xcolortable /c12 { 0.000000 0.571429 1.000000 sc} put
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X
X-31 16 124 -123 31 10 1923 920 4 MP  PP
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X
X-31 16 124 -123 31 10 1923 920 4 MP  DP
Xc10
X
X-31 196 124 -29 31 -32 2201 1005 4 MP  PP
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X
X-31 196 124 -29 31 -32 2201 1005 4 MP  DP
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X
X-31 8 124 -44 31 16 1768 948 4 MP  PP
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X
X-31 8 124 -44 31 16 1768 948 4 MP  DP
Xc12
X
X-31 -63 125 -78 30 18 1954 930 4 MP  PP
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X-31 -63 125 -78 30 18 1954 930 4 MP  DP
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X-30 -78 124 -15 31 15 1984 948 4 MP  PP
Xc0
X
X-30 -78 124 -15 31 15 1984 948 4 MP  DP
Xc13
X
X-31 -10 125 -47 30 13 1799 964 4 MP  PP
Xc0
X
X-31 -10 125 -47 30 13 1799 964 4 MP  DP
Xc13
X
X-31 -50 124 24 31 11 2015 963 4 MP  PP
Xc0
X
X-31 -50 124 24 31 11 2015 963 4 MP  DP
Xc4
X
X-31 -318 124 -109 31 63 2078 807 4 MP  PP
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X-31 -318 124 -109 31 63 2078 807 4 MP  DP
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X-31 29 124 -364 31 -16 2047 823 4 MP  PP
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X
X-31 29 124 -364 31 -16 2047 823 4 MP  DP
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X
X-30 -18 124 -39 31 10 1829 977 4 MP  PP
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X
X-30 -18 124 -39 31 10 1829 977 4 MP  DP
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X
X-31 -7 124 28 31 3 2046 974 4 MP  PP
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X
X-31 -7 124 28 31 3 2046 974 4 MP  DP
Xc13
X
X-31 -16 125 -35 30 12 1644 987 4 MP  PP
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X-31 -16 125 -35 30 12 1644 987 4 MP  DP
Xc13
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X-31 -15 124 -30 31 6 1860 987 4 MP  PP
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X-31 -15 124 -30 31 6 1860 987 4 MP  DP
Xc13
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X-30 -13 124 -33 31 11 1674 999 4 MP  PP
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X-30 -13 124 -33 31 11 1674 999 4 MP  DP
Xc12
X
X-31 32 125 -3 30 -1 2077 977 4 MP  PP
Xc0
X
X-31 32 125 -3 30 -1 2077 977 4 MP  DP
Xc13
X
X-31 191 125 -175 30 -45 2232 973 4 MP  PP
Xc0
X
X-31 191 125 -175 30 -45 2232 973 4 MP  DP
Xc12
X
X-31 -11 124 -24 31 5 1891 993 4 MP  PP
Xc0
X
X-31 -11 124 -24 31 5 1891 993 4 MP  DP
Xc13
X
X-31 -10 124 -31 31 8 1705 1010 4 MP  PP
Xc0
X
X-31 -10 124 -31 31 8 1705 1010 4 MP  DP
Xc9
X
X-30 -12 124 -35 31 11 1519 1023 4 MP  PP
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X
X-30 -12 124 -35 31 11 1519 1023 4 MP  DP
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X
X-31 -3 125 -25 30 4 1922 998 4 MP  PP
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X
X-31 -3 125 -25 30 4 1922 998 4 MP  DP
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X
X-30 45 124 -48 31 0 2107 976 4 MP  PP
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X
X-30 45 124 -48 31 0 2107 976 4 MP  DP
Xc13
X
X-31 -6 124 -31 31 6 1736 1018 4 MP  PP
Xc0
X
X-31 -6 124 -31 31 6 1736 1018 4 MP  DP
Xc13
X
X-31 -11 124 -32 31 8 1550 1034 4 MP  PP
Xc0
X
X-31 -11 124 -32 31 8 1550 1034 4 MP  DP
Xc15
X
X-30 -62 124 106 31 53 2385 1026 4 MP  PP
Xc0
X
X-30 -62 124 106 31 53 2385 1026 4 MP  DP
Xc14
X
X-31 -173 125 97 30 67 2355 959 4 MP  PP
Xc0
X
X-31 -173 125 97 30 67 2355 959 4 MP  DP
Xc15
X
X-30 1 124 -30 31 4 1952 1002 4 MP  PP
Xc0
X
X-30 1 124 -30 31 4 1952 1002 4 MP  DP
Xc12
X
X-31 -5 125 -32 30 6 1767 1024 4 MP  PP
Xc0
X
X-31 -5 125 -32 30 6 1767 1024 4 MP  DP
Xc9
X
X-31 72 124 19 31 15 2416 1079 4 MP  PP
Xc0
X
X-31 72 124 19 31 15 2416 1079 4 MP  DP
Xc13
X
X-31 -8 124 -32 31 8 1581 1042 4 MP  PP
Xc0
X
X-31 -8 124 -32 31 8 1581 1042 4 MP  DP
Xc13
X
X-31 -11 124 -32 31 10 1395 1056 4 MP  PP
Xc0
X
X-31 -11 124 -32 31 10 1395 1056 4 MP  DP
Xcolortable /c16 { 0.000000 0.285714 1.000000 sc} put
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X
X-30 71 124 -217 31 -29 2262 928 4 MP  PP
Xc0
X
X-30 71 124 -217 31 -29 2262 928 4 MP  DP
Xc11
X
X-31 29 124 -83 31 6 2138 976 4 MP  PP
Xc0
X
X-31 29 124 -83 31 6 2138 976 4 MP  DP
Xc4
X
X-31 -185 124 -9 31 50 2324 909 4 MP  PP
Xc0
X
X-31 -185 124 -9 31 50 2324 909 4 MP  DP
Xc12
X
X-30 -4 124 -33 31 5 1797 1030 4 MP  PP
Xc0
X
X-30 -4 124 -33 31 5 1797 1030 4 MP  DP
Xc12
X
X-31 -6 125 -33 30 7 1612 1050 4 MP  PP
Xc0
X
X-31 -6 125 -33 30 7 1612 1050 4 MP  DP
Xc11
X
X-31 0 124 -37 31 7 1983 1006 4 MP  PP
Xc0
X
X-31 0 124 -37 31 7 1983 1006 4 MP  DP
Xc13
X
X-31 -8 124 -34 31 10 1426 1066 4 MP  PP
Xc0
X
X-31 -8 124 -34 31 10 1426 1066 4 MP  DP
Xc4
X
X-31 -83 124 -144 31 10 2293 899 4 MP  PP
Xc0
X
X-31 -83 124 -144 31 10 2293 899 4 MP  DP
Xc16
X
X-31 -10 124 -87 31 14 2169 982 4 MP  PP
Xc0
X
X-31 -10 124 -87 31 14 2169 982 4 MP  DP
Xc12
X
X-30 -6 124 -33 31 6 1642 1057 4 MP  PP
Xc0
X
X-30 -6 124 -33 31 6 1642 1057 4 MP  DP
Xc15
X
X-31 -4 124 -36 31 7 1828 1035 4 MP  PP
Xc0
X
X-31 -4 124 -36 31 7 1828 1035 4 MP  DP
Xc11
X
X-31 -6 124 -42 31 11 2014 1013 4 MP  PP
Xc0
X
X-31 -6 124 -42 31 11 2014 1013 4 MP  DP
Xc13
X
X-31 -8 125 -34 30 8 1457 1076 4 MP  PP
Xc0
X
X-31 -8 125 -34 30 8 1457 1076 4 MP  DP
Xc9
X
X-31 137 124 -98 31 -20 2447 1094 4 MP  PP
Xc0
X
X-31 137 124 -98 31 -20 2447 1094 4 MP  DP
Xc9
X
X-31 -10 124 -36 31 11 1271 1091 4 MP  PP
Xc0
X
X-31 -10 124 -36 31 11 1271 1091 4 MP  DP
Xc16
X
X-31 -50 125 -58 30 21 2200 996 4 MP  PP
Xc0
X
X-31 -50 125 -58 30 21 2200 996 4 MP  DP
Xc11
X
X-30 -67 124 -16 31 25 2230 1017 4 MP  PP
Xc0
X
X-30 -67 124 -16 31 25 2230 1017 4 MP  DP
Xc11
X
X-31 -53 124 14 31 23 2261 1042 4 MP  PP
Xc0
X
X-31 -53 124 14 31 23 2261 1042 4 MP  DP
Xc11
X
X-31 -14 125 -42 30 14 2045 1024 4 MP  PP
Xc0
X
X-31 -14 125 -42 30 14 2045 1024 4 MP  DP
Xc11
X
X-31 -7 124 -37 31 8 1859 1042 4 MP  PP
Xc0
X
X-31 -7 124 -37 31 8 1859 1042 4 MP  DP
Xc15
X
X-31 -5 124 -35 31 7 1673 1063 4 MP  PP
Xc0
X
X-31 -5 124 -35 31 7 1673 1063 4 MP  DP
Xc12
X
X-30 -7 124 -35 31 8 1487 1084 4 MP  PP
Xc0
X
X-30 -7 124 -35 31 8 1487 1084 4 MP  DP
Xc13
X
X-31 -10 125 -37 30 11 1302 1102 4 MP  PP
Xc0
X
X-31 -10 125 -37 30 11 1302 1102 4 MP  DP
Xc15
X
X-31 -15 124 11 31 18 2292 1065 4 MP  PP
Xc0
X
X-31 -15 124 11 31 18 2292 1065 4 MP  DP
Xc11
X
X-30 -21 124 -38 31 17 2075 1038 4 MP  PP
Xc0
X
X-30 -21 124 -38 31 17 2075 1038 4 MP  DP
Xc11
X
X-31 -11 125 -36 30 10 1890 1050 4 MP  PP
Xc0
X
X-31 -11 125 -36 30 10 1890 1050 4 MP  DP
Xc12
X
X-31 -6 124 -36 31 7 1518 1092 4 MP  PP
Xc0
X
X-31 -6 124 -36 31 7 1518 1092 4 MP  DP
Xc15
X
X-31 -7 124 -35 31 7 1704 1070 4 MP  PP
Xc0
X
X-31 -7 124 -35 31 7 1704 1070 4 MP  DP
Xc13
X
X-30 -8 124 -38 31 9 1332 1113 4 MP  PP
Xc0
X
X-30 -8 124 -38 31 9 1332 1113 4 MP  DP
Xc13
X
X-31 -11 125 -33 30 11 1147 1124 4 MP  PP
Xc0
X
X-31 -11 125 -33 30 11 1147 1124 4 MP  DP
Xc15
X
X-31 20 125 -22 30 13 2323 1083 4 MP  PP
Xc0
X
X-31 20 125 -22 30 13 2323 1083 4 MP  DP
Xc11
X
X-31 -25 124 -30 31 17 2106 1055 4 MP  PP
Xc0
X
X-31 -25 124 -30 31 17 2106 1055 4 MP  DP
Xc12
X
X-31 93 125 -163 30 -28 2478 1074 4 MP  PP
Xc0
X
X-31 93 125 -163 30 -28 2478 1074 4 MP  DP
Xc11
X
X-30 -14 124 -35 31 13 1920 1060 4 MP  PP
Xc0
X
X-30 -14 124 -35 31 13 1920 1060 4 MP  DP
Xc13
X
X-31 -8 124 -39 31 9 1363 1122 4 MP  PP
Xc0
X
X-31 -8 124 -39 31 9 1363 1122 4 MP  DP
Xc11
X
X-31 -8 125 -36 30 9 1735 1077 4 MP  PP
Xc0
X
X-31 -8 125 -36 30 9 1735 1077 4 MP  DP
Xc12
X
X-31 -7 124 -36 31 7 1549 1099 4 MP  PP
Xc0
X
X-31 -7 124 -36 31 7 1549 1099 4 MP  DP
Xc13
X
X-30 -11 124 -33 31 11 1177 1135 4 MP  PP
Xc0
X
X-30 -11 124 -33 31 11 1177 1135 4 MP  DP
Xc11
X
X-31 -23 124 -25 31 18 2137 1072 4 MP  PP
Xc0
X
X-31 -23 124 -25 31 18 2137 1072 4 MP  DP
Xc15
X
X-30 28 124 -63 31 13 2353 1096 4 MP  PP
Xc0
X
X-30 28 124 -63 31 13 2353 1096 4 MP  DP
Xc11
X
X-31 -17 124 -32 31 14 1951 1073 4 MP  PP
Xc0
X
X-31 -17 124 -32 31 14 1951 1073 4 MP  DP
Xc15
X
X-31 -62 125 85 30 36 2601 1115 4 MP  PP
Xc0
X
X-31 -62 125 85 30 36 2601 1115 4 MP  DP
Xc11
X
X-30 -10 124 -36 31 10 1765 1086 4 MP  PP
Xc0
X
X-30 -10 124 -36 31 10 1765 1086 4 MP  DP
Xc11
X
X-31 -18 125 -26 30 19 2168 1090 4 MP  PP
Xc0
X
X-31 -18 125 -26 30 19 2168 1090 4 MP  DP
Xc12
X
X-31 -7 124 -38 31 6 1394 1131 4 MP  PP
Xc0
X
X-31 -7 124 -38 31 6 1394 1131 4 MP  DP
Xc15
X
X-31 -7 125 -36 30 7 1580 1106 4 MP  PP
Xc0
X
X-31 -7 125 -36 30 7 1580 1106 4 MP  DP
Xc14
X
X-31 -142 124 59 31 47 2570 1068 4 MP  PP
Xc0
X
X-31 -142 124 59 31 47 2570 1068 4 MP  DP
Xc13
X
X-31 -9 124 -33 31 9 1208 1146 4 MP  PP
Xc0
X
X-31 -9 124 -33 31 9 1208 1146 4 MP  DP
Xc9
X
X-30 -11 124 -36 31 12 1022 1159 4 MP  PP
Xc0
X
X-30 -11 124 -36 31 12 1022 1159 4 MP  DP
Xc16
X
X-30 -24 124 -133 31 -6 2508 1046 4 MP  PP
Xc0
X
X-30 -24 124 -133 31 -6 2508 1046 4 MP  DP
Xc11
X
X-31 -17 124 -32 31 17 1982 1087 4 MP  PP
Xc0
X
X-31 -17 124 -32 31 17 1982 1087 4 MP  DP
Xc14
X
X-31 -125 124 -36 31 28 2539 1040 4 MP  PP
Xc0
X
X-31 -125 124 -36 31 28 2539 1040 4 MP  DP
Xc12
X
X-30 54 124 28 31 3 2631 1151 4 MP  PP
Xc0
X
X-30 54 124 28 31 3 2631 1151 4 MP  DP
Xc11
X
X-30 -13 124 -33 31 20 2198 1109 4 MP  PP
Xc0
X
X-30 -13 124 -33 31 20 2198 1109 4 MP  DP
Xc15
X
X-31 6 124 -84 31 15 2384 1109 4 MP  PP
Xc0
X
X-31 6 124 -84 31 15 2384 1109 4 MP  DP
Xc11
X
X-31 -13 124 -34 31 11 1796 1096 4 MP  PP
Xc0
X
X-31 -13 124 -34 31 11 1796 1096 4 MP  DP
Xc12
X
X-31 -7 125 -37 30 6 1425 1137 4 MP  PP
Xc0
X
X-31 -7 125 -37 30 6 1425 1137 4 MP  DP
Xc13
X
X-31 -9 124 -32 31 8 1239 1155 4 MP  PP
Xc0
X
X-31 -9 124 -32 31 8 1239 1155 4 MP  DP
Xc15
X
X-30 -9 124 -34 31 7 1610 1113 4 MP  PP
Xc0
X
X-30 -9 124 -34 31 7 1610 1113 4 MP  DP
Xc13
X
X-31 -11 124 -37 31 12 1053 1171 4 MP  PP
Xc0
X
X-31 -11 124 -37 31 12 1053 1171 4 MP  DP
Xc11
X
X-31 -18 125 -32 30 18 2013 1104 4 MP  PP
Xc0
X
X-31 -18 125 -32 30 18 2013 1104 4 MP  DP
Xc15
X
X-31 -13 124 -40 31 20 2229 1129 4 MP  PP
Xc0
X
X-31 -13 124 -40 31 20 2229 1129 4 MP  DP
Xc15
X
X-31 -28 124 -69 31 13 2415 1124 4 MP  PP
Xc0
X
X-31 -28 124 -69 31 13 2415 1124 4 MP  DP
Xc11
X
X-31 -14 124 -33 31 13 1827 1107 4 MP  PP
Xc0
X
X-31 -14 124 -33 31 13 1827 1107 4 MP  DP
Xc11
X
X-31 -10 124 -32 31 8 1641 1120 4 MP  PP
Xc0
X
X-31 -10 124 -32 31 8 1641 1120 4 MP  DP
Xc12
X
X-31 -6 125 -33 30 7 1270 1163 4 MP  PP
Xc0
X
X-31 -6 125 -33 30 7 1270 1163 4 MP  DP
Xc15
X
X-30 -7 124 -35 31 5 1455 1143 4 MP  PP
Xc0
X
X-30 -7 124 -35 31 5 1455 1143 4 MP  DP
Xc13
X
X-31 -9 124 -38 31 10 1084 1183 4 MP  PP
Xc0
X
X-31 -9 124 -38 31 10 1084 1183 4 MP  DP
Xc11
X
X-30 -19 124 -34 31 21 2043 1122 4 MP  PP
Xc0
X
X-30 -19 124 -34 31 21 2043 1122 4 MP  DP
Xc15
X
X-31 -47 125 -31 30 9 2446 1137 4 MP  PP
Xc0
X
X-31 -47 125 -31 30 9 2446 1137 4 MP  DP
Xc15
X
X-31 -15 124 -42 31 17 2260 1149 4 MP  PP
Xc0
X
X-31 -15 124 -42 31 17 2260 1149 4 MP  DP
Xc11
X
X-31 -17 125 -32 30 16 1858 1120 4 MP  PP
Xc0
X
X-31 -17 125 -32 30 16 1858 1120 4 MP  DP
Xc11
X
X-31 -20 124 -36 31 22 2074 1143 4 MP  PP
Xc0
X
X-31 -20 124 -36 31 22 2074 1143 4 MP  DP
Xc15
X
X-31 129 124 -69 31 -32 2662 1154 4 MP  PP
Xc0
X
X-31 129 124 -69 31 -32 2662 1154 4 MP  DP
Xc11
X
X-31 -11 124 -32 31 11 1672 1128 4 MP  PP
Xc0
X
X-31 -11 124 -32 31 11 1672 1128 4 MP  DP
Xc13
X
X-31 -8 125 -39 30 9 1115 1193 4 MP  PP
Xc0
X
X-31 -8 125 -39 30 9 1115 1193 4 MP  DP
Xc12
X
X-30 -6 124 -32 31 5 1300 1170 4 MP  PP
Xc0
X
X-30 -6 124 -32 31 5 1300 1170 4 MP  DP
Xc11
X
X-30 -36 124 4 31 1 2476 1146 4 MP  PP
Xc0
X
X-30 -36 124 4 31 1 2476 1146 4 MP  DP
Xc15
X
X-31 -7 124 -32 31 4 1486 1148 4 MP  PP
Xc0
X
X-31 -7 124 -32 31 4 1486 1148 4 MP  DP
Xc15
X
X-31 -13 125 -40 30 11 2291 1166 4 MP  PP
Xc0
X
X-31 -13 125 -40 30 11 2291 1166 4 MP  DP
Xc11
X
X-30 -18 124 -33 31 19 1888 1136 4 MP  PP
Xc0
X
X-30 -18 124 -33 31 19 1888 1136 4 MP  DP
Xc15
X
X-31 -20 124 -37 31 21 2105 1165 4 MP  PP
Xc0
X
X-31 -20 124 -37 31 21 2105 1165 4 MP  DP
Xc16
X
X-31 -13 125 -32 30 13 1703 1139 4 MP  PP
Xc0
X
X-31 -13 125 -32 30 13 1703 1139 4 MP  DP
Xc13
X
X-30 -7 124 -41 31 9 1145 1202 4 MP  PP
Xc0
X
X-30 -7 124 -41 31 9 1145 1202 4 MP  DP
Xc11
X
X-31 -3 124 15 31 -8 2507 1147 4 MP  PP
Xc0
X
X-31 -3 124 15 31 -8 2507 1147 4 MP  DP
Xc15
X
X-31 -5 124 -32 31 5 1331 1175 4 MP  PP
Xc0
X
X-31 -5 124 -32 31 5 1331 1175 4 MP  DP
Xc11
X
X-31 -8 124 -30 31 6 1517 1152 4 MP  PP
Xc0
X
X-31 -8 124 -30 31 6 1517 1152 4 MP  DP
Xc11
X
X-31 -21 124 -34 31 22 1919 1155 4 MP  PP
Xc0
X
X-31 -21 124 -34 31 22 1919 1155 4 MP  DP
Xc15
X
X-30 -9 124 -33 31 2 2321 1177 4 MP  PP
Xc0
X
X-30 -9 124 -33 31 2 2321 1177 4 MP  DP
Xc15
X
X-31 -17 125 -36 30 16 2136 1186 4 MP  PP
Xc0
X
X-31 -17 125 -36 30 16 2136 1186 4 MP  DP
Xc16
X
X-30 -16 124 -33 31 17 1733 1152 4 MP  PP
Xc0
X
X-30 -16 124 -33 31 17 1733 1152 4 MP  DP
Xc11
X
X-31 -22 124 -35 31 23 1950 1177 4 MP  PP
Xc0
X
X-31 -22 124 -35 31 23 1950 1177 4 MP  DP
Xc12
X
X-31 -5 124 -44 31 8 1176 1211 4 MP  PP
Xc0
X
X-31 -5 124 -44 31 8 1176 1211 4 MP  DP
Xc16
X
X-31 -11 125 -29 30 10 1548 1158 4 MP  PP
Xc0
X
X-31 -11 125 -29 30 10 1548 1158 4 MP  DP
Xc15
X
X-31 -4 124 -33 31 5 1362 1180 4 MP  PP
Xc0
X
X-31 -4 124 -33 31 5 1362 1180 4 MP  DP
Xc12
X
X-30 -11 124 -34 31 9 2166 1202 4 MP  PP
Xc0
X
X-30 -11 124 -34 31 9 2166 1202 4 MP  DP
Xc11
X
X-31 -1 124 -26 31 -6 2352 1179 4 MP  PP
Xc0
X
X-31 -1 124 -26 31 -6 2352 1179 4 MP  DP
Xc14
X
X-31 32 124 -6 31 -11 2538 1139 4 MP  PP
Xc0
X
X-31 32 124 -6 31 -11 2538 1139 4 MP  DP
Xc11
X
X-31 -19 124 -35 31 21 1764 1169 4 MP  PP
Xc0
X
X-31 -19 124 -35 31 21 1764 1169 4 MP  DP
Xc15
X
X-31 -21 125 -35 30 21 1981 1200 4 MP  PP
Xc0
X
X-31 -21 125 -35 30 21 1981 1200 4 MP  DP
Xc14
X
X-31 114 124 -140 31 -43 2693 1122 4 MP  PP
Xc0
X
X-31 114 124 -140 31 -43 2693 1122 4 MP  DP
Xc16
X
X-30 -13 124 -30 31 14 1578 1168 4 MP  PP
Xc0
X
X-30 -13 124 -30 31 14 1578 1168 4 MP  DP
Xc12
X
X-31 -5 124 -47 31 8 1207 1219 4 MP  PP
Xc0
X
X-31 -5 124 -47 31 8 1207 1219 4 MP  DP
Xc11
X
X-31 -6 125 -35 30 8 1393 1185 4 MP  PP
Xc0
X
X-31 -6 125 -35 30 8 1393 1185 4 MP  DP
Xc11
X
X-31 -22 124 -36 31 23 1795 1190 4 MP  PP
Xc0
X
X-31 -22 124 -36 31 23 1795 1190 4 MP  DP
Xc15
X
X-31 -2 124 -32 31 0 2197 1211 4 MP  PP
Xc0
X
X-31 -2 124 -32 31 0 2197 1211 4 MP  DP
Xc12
X
X-30 -16 124 -35 31 16 2011 1221 4 MP  PP
Xc0
X
X-30 -16 124 -35 31 16 2011 1221 4 MP  DP
Xc11
X
X-31 -47 125 43 30 84 2847 1216 4 MP  PP
Xc0
X
X-31 -47 125 43 30 84 2847 1216 4 MP  DP
Xc16
X
X-31 8 124 -25 31 -9 2383 1173 4 MP  PP
Xc0
X
X-31 8 124 -25 31 -9 2383 1173 4 MP  DP
Xc16
X
X-31 -17 124 -32 31 19 1609 1182 4 MP  PP
Xc0
X
X-31 -17 124 -32 31 19 1609 1182 4 MP  DP
Xc15
X
X-31 -23 125 -37 30 24 1826 1213 4 MP  PP
Xc0
X
X-31 -23 125 -37 30 24 1826 1213 4 MP  DP
Xc4
X
X-31 43 125 -43 30 -6 2569 1128 4 MP  PP
Xc0
X
X-31 43 125 -43 30 -6 2569 1128 4 MP  DP
Xc15
X
X-31 -5 125 -50 30 8 1238 1227 4 MP  PP
Xc0
X
X-31 -5 125 -50 30 8 1238 1227 4 MP  DP
Xc9
X
X-30 62 124 -69 31 50 2877 1300 4 MP  PP
Xc0
X
X-30 62 124 -69 31 50 2877 1300 4 MP  DP
Xc16
X
X-30 -10 124 -36 31 11 1423 1193 4 MP  PP
Xc0
X
X-30 -10 124 -36 31 11 1423 1193 4 MP  DP
Xcolortable /c17 { 0.095238 0.000000 1.000000 sc} put
Xc17
X
X-31 -138 124 80 31 85 2816 1131 4 MP  PP
Xc0
X
X-31 -138 124 80 31 85 2816 1131 4 MP  DP
Xc12
X
X-31 -9 124 -34 31 8 2042 1237 4 MP  PP
Xc0
X
X-31 -9 124 -34 31 8 2042 1237 4 MP  DP
Xc11
X
X-31 6 124 -32 31 -6 2228 1211 4 MP  PP
Xc0
X
X-31 6 124 -32 31 -6 2228 1211 4 MP  DP
Xc16
X
X-31 -21 124 -36 31 25 1640 1201 4 MP  PP
Xc0
X
X-31 -21 124 -36 31 25 1640 1201 4 MP  DP
Xc15
X
X-30 -21 124 -36 31 20 1856 1237 4 MP  PP
Xc0
X
X-30 -21 124 -36 31 20 1856 1237 4 MP  DP
Xcolortable /c18 { 0.190476 0.000000 1.000000 sc} put
Xc18
X
X-31 23 125 -139 30 -24 2724 1079 4 MP  PP
Xc0
X
X-31 23 125 -139 30 -24 2724 1079 4 MP  DP
Xcolortable /c19 { 0.000000 0.095238 1.000000 sc} put
Xc19
X
X-31 11 125 -29 30 -7 2414 1164 4 MP  PP
Xc0
X
X-31 11 125 -29 30 -7 2414 1164 4 MP  DP
Xc16
X
X-31 -14 124 -38 31 16 1454 1204 4 MP  PP
Xc0
X
X-31 -14 124 -38 31 16 1454 1204 4 MP  DP
Xc15
X
X-30 -8 124 -52 31 10 1268 1235 4 MP  PP
Xc0
X
X-30 -8 124 -52 31 10 1268 1235 4 MP  DP
Xcolortable /c20 { 0.380952 0.000000 1.000000 sc} put
Xc20
X
X-31 -154 124 27 31 60 2785 1071 4 MP  PP
Xc0
X
X-31 -154 124 27 31 60 2785 1071 4 MP  DP
Xc11
X
X-31 -23 125 -39 30 26 1671 1226 4 MP  PP
Xc0
X
X-31 -23 125 -39 30 26 1671 1226 4 MP  DP
Xc15
X
X-31 0 124 -33 31 -1 2073 1245 4 MP  PP
Xc0
X
X-31 0 124 -33 31 -1 2073 1245 4 MP  DP
Xc12
X
X-31 -16 124 -34 31 14 1887 1257 4 MP  PP
Xc0
X
X-31 -16 124 -34 31 14 1887 1257 4 MP  DP
Xc17
X
X-30 24 124 -75 31 8 2599 1122 4 MP  PP
Xc0
X
X-30 24 124 -75 31 8 2599 1122 4 MP  DP
Xc20
X
X-30 -88 124 -67 31 16 2754 1055 4 MP  PP
Xc0
X
X-30 -88 124 -67 31 16 2754 1055 4 MP  DP
Xc16
X
X-31 9 125 -32 30 -9 2259 1205 4 MP  PP
Xc0
X
X-31 9 125 -32 30 -9 2259 1205 4 MP  DP
Xcolortable /c21 { 0.000000 1.000000 0.952381 sc} put
Xc21
X
X-31 97 124 -154 31 -12 2908 1350 4 MP  PP
Xc0
X
X-31 97 124 -154 31 -12 2908 1350 4 MP  DP
Xc16
X
X-31 -19 124 -39 31 20 1485 1220 4 MP  PP
Xc0
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X-31 -19 124 -39 31 20 1485 1220 4 MP  DP
Xc15
X
X-30 -24 124 -40 31 25 1701 1252 4 MP  PP
Xc0
X
X-30 -24 124 -40 31 25 1701 1252 4 MP  DP
Xc15
X
X-31 -11 124 -52 31 11 1299 1245 4 MP  PP
Xc0
X
X-31 -11 124 -52 31 11 1299 1245 4 MP  DP
Xc4
X
X-30 6 124 -37 31 2 2444 1157 4 MP  PP
Xc0
X
X-30 6 124 -37 31 2 2444 1157 4 MP  DP
Xc12
X
X-31 -8 124 -32 31 6 1918 1271 4 MP  PP
Xc0
X
X-31 -8 124 -32 31 6 1918 1271 4 MP  DP
Xc11
X
X-31 -25 125 -39 30 25 1516 1240 4 MP  PP
Xc0
X
X-31 -25 125 -39 30 25 1516 1240 4 MP  DP
Xc17
X
X-31 -16 124 -84 31 25 2630 1130 4 MP  PP
Xc0
X
X-31 -16 124 -84 31 25 2630 1130 4 MP  DP
Xc11
X
X-31 6 125 -32 30 -7 2104 1244 4 MP  PP
Xc0
X
X-31 6 125 -32 30 -7 2104 1244 4 MP  DP
Xc12
X
X-31 -20 124 -38 31 18 1732 1277 4 MP  PP
Xc0
X
X-31 -20 124 -38 31 18 1732 1277 4 MP  DP
Xc16
X
X-30 -84 124 11 31 47 2722 1242 4 MP  PP
Xc0
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X-30 -84 124 11 31 47 2722 1242 4 MP  DP
Xc15
X
X-31 -50 124 22 31 39 2753 1289 4 MP  PP
Xc0
X
X-31 -50 124 22 31 39 2753 1289 4 MP  DP
Xc15
X
X-31 -16 124 -50 31 14 1330 1256 4 MP  PP
Xc0
X
X-31 -16 124 -50 31 14 1330 1256 4 MP  DP
Xc19
X
X-31 -85 125 -26 30 48 2692 1194 4 MP  PP
Xc0
X
X-31 -85 125 -26 30 48 2692 1194 4 MP  DP
Xc17
X
X-31 -60 124 -63 31 39 2661 1155 4 MP  PP
Xc0
X
X-31 -60 124 -63 31 39 2661 1155 4 MP  DP
Xc19
X
X-30 7 124 -34 31 -5 2289 1196 4 MP  PP
Xc0
X
X-30 7 124 -34 31 -5 2289 1196 4 MP  DP
Xc11
X
X-30 -26 124 -38 31 25 1546 1265 4 MP  PP
Xc0
X
X-30 -26 124 -38 31 25 1546 1265 4 MP  DP
Xc17
X
X-31 -8 124 -42 31 13 2475 1159 4 MP  PP
Xc0
X
X-31 -8 124 -42 31 13 2475 1159 4 MP  DP
Xc12
X
X-31 -14 124 -34 31 10 1763 1295 4 MP  PP
Xc0
X
X-31 -14 124 -34 31 10 1763 1295 4 MP  DP
Xc15
X
X-31 1 125 -30 30 -3 1949 1277 4 MP  PP
Xc0
X
X-31 1 125 -30 30 -3 1949 1277 4 MP  DP
Xc13
X
X-31 12 124 -13 31 23 2784 1328 4 MP  PP
Xc0
X
X-31 12 124 -13 31 23 2784 1328 4 MP  DP
Xc11
X
X-31 -20 125 -46 30 16 1361 1270 4 MP  PP
Xc0
X
X-31 -20 125 -46 30 16 1361 1270 4 MP  DP
Xcolortable /c22 { 0.000000 0.857143 1.000000 sc} put
Xc22
X
X-31 -48 124 -10 31 -96 2939 1338 4 MP  PP
Xc0
X
X-31 -48 124 -10 31 -96 2939 1338 4 MP  DP
Xc15
X
X-31 -25 124 -34 31 21 1577 1290 4 MP  PP
Xc0
X
X-31 -25 124 -34 31 21 1577 1290 4 MP  DP
Xc16
X
X-30 9 124 -33 31 -8 2134 1237 4 MP  PP
Xc0
X
X-30 9 124 -33 31 -8 2134 1237 4 MP  DP
Xc17
X
X-31 -25 124 -43 31 26 2506 1172 4 MP  PP
Xc0
X
X-31 -25 124 -43 31 26 2506 1172 4 MP  DP
Xc4
X
X-31 -2 124 -36 31 4 2320 1191 4 MP  PP
Xc0
X
X-31 -2 124 -36 31 4 2320 1191 4 MP  DP
Xc12
X
X-31 -6 125 -29 30 1 1794 1305 4 MP  PP
Xc0
X
X-31 -6 125 -29 30 1 1794 1305 4 MP  DP
Xc15
X
X-30 -25 124 -38 31 17 1391 1286 4 MP  PP
Xc0
X
X-30 -25 124 -38 31 17 1391 1286 4 MP  DP
Xc4
X
X-31 -39 125 -39 30 35 2537 1198 4 MP  PP
Xc0
X
X-31 -39 125 -39 30 35 2537 1198 4 MP  DP
Xc12
X
X-31 -18 124 -31 31 15 1608 1311 4 MP  PP
Xc0
X
X-31 -18 124 -31 31 15 1608 1311 4 MP  DP
Xc11
X
X-30 7 124 -31 31 -6 1979 1274 4 MP  PP
Xc0
X
X-30 7 124 -31 31 -6 1979 1274 4 MP  DP
Xc19
X
X-30 -48 124 -31 31 40 2567 1233 4 MP  PP
Xc0
X
X-30 -48 124 -31 31 40 2567 1233 4 MP  DP
Xc9
X
X-31 96 125 -109 30 0 2815 1351 4 MP  PP
Xc0
X
X-31 96 125 -109 30 0 2815 1351 4 MP  DP
Xc19
X
X-31 5 124 -34 31 -4 2165 1229 4 MP  PP
Xc0
X
X-31 5 124 -34 31 -4 2165 1229 4 MP  DP
Xc16
X
X-31 -47 124 -25 31 41 2598 1273 4 MP  PP
Xc0
X
X-31 -47 124 -25 31 41 2598 1273 4 MP  DP
Xc15
X
X-31 -25 124 -28 31 15 1422 1303 4 MP  PP
Xc0
X
X-31 -25 124 -28 31 15 1422 1303 4 MP  DP
Xc17
X
X-31 -13 124 -37 31 14 2351 1195 4 MP  PP
Xc0
X
X-31 -13 124 -37 31 14 2351 1195 4 MP  DP
Xc12
X
X-31 -10 125 -28 30 7 1639 1326 4 MP  PP
Xc0
X
X-31 -10 125 -28 30 7 1639 1326 4 MP  DP
Xc15
X
X-31 -39 124 -23 31 37 2629 1314 4 MP  PP
Xc0
X
X-31 -39 124 -23 31 37 2629 1314 4 MP  DP
Xc15
X
X-30 3 124 -28 31 -4 1824 1306 4 MP  PP
Xc0
X
X-30 3 124 -28 31 -4 1824 1306 4 MP  DP
Xc16
X
X-31 8 124 -33 31 -6 2010 1268 4 MP  PP
Xc0
X
X-31 8 124 -33 31 -6 2010 1268 4 MP  DP
Xc15
X
X-31 -21 124 -21 31 14 1453 1318 4 MP  PP
Xc0
X
X-31 -21 124 -21 31 14 1453 1318 4 MP  DP
Xc17
X
X-31 -26 125 -36 30 25 2382 1209 4 MP  PP
Xc0
X
X-31 -26 125 -36 30 25 2382 1209 4 MP  DP
Xc12
X
X-31 -23 125 -30 30 30 2660 1351 4 MP  PP
Xc0
X
X-31 -23 125 -30 30 30 2660 1351 4 MP  DP
Xc4
X
X-31 -4 124 -35 31 5 2196 1225 4 MP  PP
Xc0
X
X-31 -4 124 -35 31 5 2196 1225 4 MP  DP
Xc4
X
X-30 -35 124 -35 31 34 2412 1234 4 MP  PP
Xc0
X
X-30 -35 124 -35 31 34 2412 1234 4 MP  DP
Xc15
X
X-30 -1 124 -26 31 -1 1669 1333 4 MP  PP
Xc0
X
X-30 -1 124 -26 31 -1 1669 1333 4 MP  DP
Xc13
X
X-30 0 124 -52 31 22 2690 1381 4 MP  PP
Xc0
X
X-30 0 124 -52 31 22 2690 1381 4 MP  DP
Xc16
X
X-31 6 124 -27 31 -7 1855 1302 4 MP  PP
Xc0
X
X-31 6 124 -27 31 -7 1855 1302 4 MP  DP
Xc15
X
X-31 -15 125 -16 30 10 1484 1332 4 MP  PP
Xc0
X
X-31 -15 125 -16 30 10 1484 1332 4 MP  DP
Xc19
X
X-31 -40 124 -32 31 37 2443 1268 4 MP  PP
Xc0
X
X-31 -40 124 -32 31 37 2443 1268 4 MP  DP
Xc19
X
X-31 4 124 -36 31 -1 2041 1262 4 MP  PP
Xc0
X
X-31 4 124 -36 31 -1 2041 1262 4 MP  DP
Xc17
X
X-31 -14 125 -36 30 15 2227 1230 4 MP  PP
Xc0
X
X-31 -14 125 -36 30 15 2227 1230 4 MP  DP
Xc16
X
X-31 -41 124 -31 31 40 2474 1305 4 MP  PP
Xc0
X
X-31 -41 124 -31 31 40 2474 1305 4 MP  DP
Xc15
X
X-31 -37 125 -31 30 37 2505 1345 4 MP  PP
Xc0
X
X-31 -37 125 -31 30 37 2505 1345 4 MP  DP
Xc11
X
X-31 4 124 -27 31 -3 1700 1332 4 MP  PP
Xc0
X
X-31 4 124 -27 31 -3 1700 1332 4 MP  DP
Xc15
X
X-30 -7 124 -17 31 8 1514 1342 4 MP  PP
Xc0
X
X-30 -7 124 -17 31 8 1514 1342 4 MP  DP
Xc4
X
X-30 -25 124 -36 31 25 2257 1245 4 MP  PP
Xc0
X
X-30 -25 124 -36 31 25 2257 1245 4 MP  DP
Xc14
X
X-31 6 124 -30 31 -3 1886 1295 4 MP  PP
Xc0
X
X-31 6 124 -30 31 -3 1886 1295 4 MP  DP
Xc12
X
X-30 -30 124 -34 31 33 2535 1382 4 MP  PP
Xc0
X
X-30 -30 124 -34 31 33 2535 1382 4 MP  DP
Xc4
X
X-31 -5 125 -38 30 7 2072 1261 4 MP  PP
Xc0
X
X-31 -5 125 -38 30 7 2072 1261 4 MP  DP
Xc4
X
X-31 -34 124 -35 31 33 2288 1270 4 MP  PP
Xc0
X
X-31 -34 124 -35 31 33 2288 1270 4 MP  DP
Xc13
X
X-31 -22 124 -39 31 27 2566 1415 4 MP  PP
Xc0
X
X-31 -22 124 -39 31 27 2566 1415 4 MP  DP
Xc11
X
X-31 1 124 -23 31 5 1545 1350 4 MP  PP
Xc0
X
X-31 1 124 -23 31 5 1545 1350 4 MP  DP
Xc16
X
X-31 7 124 -30 31 -4 1731 1329 4 MP  PP
Xc0
X
X-31 7 124 -30 31 -4 1731 1329 4 MP  DP
Xc19
X
X-31 -37 124 -35 31 37 2319 1303 4 MP  PP
Xc0
X
X-31 -37 124 -35 31 37 2319 1303 4 MP  DP
Xc4
X
X-30 -15 124 -40 31 17 2102 1268 4 MP  PP
Xc0
X
X-30 -15 124 -40 31 17 2102 1268 4 MP  DP
Xc19
X
X-31 1 125 -35 30 4 1917 1292 4 MP  PP
Xc0
X
X-31 1 125 -35 30 4 1917 1292 4 MP  DP
Xc16
X
X-31 -40 125 -32 30 37 2350 1340 4 MP  PP
Xc0
X
X-31 -40 125 -32 30 37 2350 1340 4 MP  DP
Xc11
X
X-30 -37 124 -31 31 36 2380 1377 4 MP  PP
Xc0
X
X-30 -37 124 -31 31 36 2380 1377 4 MP  DP
Xc16
X
X-31 3 124 -33 31 7 1576 1355 4 MP  PP
Xc0
X
X-31 3 124 -33 31 7 1576 1355 4 MP  DP
Xc4
X
X-31 -25 124 -39 31 24 2133 1285 4 MP  PP
Xc0
X
X-31 -25 124 -39 31 24 2133 1285 4 MP  DP
Xc14
X
X-31 3 125 -35 30 2 1762 1325 4 MP  PP
Xc0
X
X-31 3 125 -35 30 2 1762 1325 4 MP  DP
Xc12
X
X-31 -33 124 -31 31 33 2411 1413 4 MP  PP
Xc0
X
X-31 -33 124 -31 31 33 2411 1413 4 MP  DP
Xc4
X
X-30 -7 124 -39 31 11 1947 1296 4 MP  PP
Xc0
X
X-30 -7 124 -39 31 11 1947 1296 4 MP  DP
Xc19
X
X-31 -33 124 -37 31 31 2164 1309 4 MP  PP
Xc0
X
X-31 -33 124 -37 31 31 2164 1309 4 MP  DP
Xc13
X
X-31 -27 124 -32 31 28 2442 1446 4 MP  PP
Xc0
X
X-31 -27 124 -32 31 28 2442 1446 4 MP  DP
Xc14
X
X-31 -37 125 -33 30 33 2195 1340 4 MP  PP
Xc0
X
X-31 -37 125 -33 30 33 2195 1340 4 MP  DP
Xc16
X
X-31 4 125 -45 30 8 1607 1362 4 MP  PP
Xc0
X
X-31 4 125 -45 30 8 1607 1362 4 MP  DP
Xc4
X
X-31 -17 124 -42 31 20 1978 1307 4 MP  PP
Xc0
X
X-31 -17 124 -42 31 20 1978 1307 4 MP  DP
Xc19
X
X-30 -4 124 -39 31 8 1792 1327 4 MP  PP
Xc0
X
X-30 -4 124 -39 31 8 1792 1327 4 MP  DP
Xc16
X
X-30 -37 124 -30 31 34 2225 1373 4 MP  PP
Xc0
X
X-30 -37 124 -30 31 34 2225 1373 4 MP  DP
Xc4
X
X-31 -24 124 -43 31 25 2009 1327 4 MP  PP
Xc0
X
X-31 -24 124 -43 31 25 2009 1327 4 MP  DP
Xc11
X
X-31 -36 124 -27 31 33 2256 1407 4 MP  PP
Xc0
X
X-31 -36 124 -27 31 33 2256 1407 4 MP  DP
Xc16
X
X-30 -2 124 -56 31 13 1637 1370 4 MP  PP
Xc0
X
X-30 -2 124 -56 31 13 1637 1370 4 MP  DP
Xc19
X
X-31 -11 124 -43 31 15 1823 1335 4 MP  PP
Xc0
X
X-31 -11 124 -43 31 15 1823 1335 4 MP  DP
Xc15
X
X-31 -33 124 -24 31 30 2287 1440 4 MP  PP
Xc0
X
X-31 -33 124 -24 31 30 2287 1440 4 MP  DP
Xc19
X
X-31 -31 125 -41 30 29 2040 1352 4 MP  PP
Xc0
X
X-31 -31 125 -41 30 29 2040 1352 4 MP  DP
Xc12
X
X-31 -28 125 -22 30 26 2318 1470 4 MP  PP
Xc0
X
X-31 -28 125 -22 30 26 2318 1470 4 MP  DP
Xc14
X
X-30 -33 124 -38 31 30 2070 1381 4 MP  PP
Xc0
X
X-30 -33 124 -38 31 30 2070 1381 4 MP  DP
Xc14
X
X-31 -8 124 -65 31 17 1668 1383 4 MP  PP
Xc0
X
X-31 -8 124 -65 31 17 1668 1383 4 MP  DP
Xc19
X
X-31 -20 124 -45 31 22 1854 1350 4 MP  PP
Xc0
X
X-31 -20 124 -45 31 22 1854 1350 4 MP  DP
Xc16
X
X-31 -34 124 -34 31 30 2101 1411 4 MP  PP
Xc0
X
X-31 -34 124 -34 31 30 2101 1411 4 MP  DP
Xc19
X
X-31 -25 125 -46 30 26 1885 1372 4 MP  PP
Xc0
X
X-31 -25 125 -46 30 26 1885 1372 4 MP  DP
Xc16
X
X-31 -15 124 -70 31 20 1699 1400 4 MP  PP
Xc0
X
X-31 -15 124 -70 31 20 1699 1400 4 MP  DP
Xc11
X
X-31 -33 124 -29 31 28 2132 1441 4 MP  PP
Xc0
X
X-31 -33 124 -29 31 28 2132 1441 4 MP  DP
Xc14
X
X-30 -29 124 -45 31 28 1915 1398 4 MP  PP
Xc0
X
X-30 -29 124 -45 31 28 1915 1398 4 MP  DP
Xc15
X
X-31 -30 125 -25 30 26 2163 1469 4 MP  PP
Xc0
X
X-31 -30 125 -25 30 26 2163 1469 4 MP  DP
Xc16
X
X-31 -22 125 -71 30 23 1730 1420 4 MP  PP
Xc0
X
X-31 -22 125 -71 30 23 1730 1420 4 MP  DP
Xc16
X
X-31 -30 124 -44 31 29 1946 1426 4 MP  PP
Xc0
X
X-31 -30 124 -44 31 29 1946 1426 4 MP  DP
Xc12
X
X-30 -26 124 -23 31 24 2193 1495 4 MP  PP
Xc0
X
X-30 -26 124 -23 31 24 2193 1495 4 MP  DP
Xc11
X
X-30 -26 124 -69 31 24 1760 1443 4 MP  PP
Xc0
X
X-30 -26 124 -69 31 24 1760 1443 4 MP  DP
Xc11
X
X-31 -30 124 -41 31 27 1977 1455 4 MP  PP
Xc0
X
X-31 -30 124 -41 31 27 1977 1455 4 MP  DP
Xc11
X
X-31 -28 124 -66 31 25 1791 1467 4 MP  PP
Xc0
X
X-31 -28 124 -66 31 25 1791 1467 4 MP  DP
Xc15
X
X-31 -28 125 -39 30 26 2008 1482 4 MP  PP
Xc0
X
X-31 -28 125 -39 30 26 2008 1482 4 MP  DP
Xc15
X
X-31 -29 124 -60 31 23 1822 1492 4 MP  PP
Xc0
X
X-31 -29 124 -60 31 23 1822 1492 4 MP  DP
Xc15
X
X-30 -26 124 -37 31 24 2038 1508 4 MP  PP
Xc0
X
X-30 -26 124 -37 31 24 2038 1508 4 MP  DP
Xc12
X
X-31 -27 125 -56 30 23 1853 1515 4 MP  PP
Xc0
X
X-31 -27 125 -56 30 23 1853 1515 4 MP  DP
Xc12
X
X-31 -24 124 -35 31 22 2069 1532 4 MP  PP
Xc0
X
X-31 -24 124 -35 31 22 2069 1532 4 MP  DP
Xc12
X
X-30 -26 124 -51 31 21 1883 1538 4 MP  PP
Xc0
X
X-30 -26 124 -51 31 21 1883 1538 4 MP  DP
Xc13
X
X-31 -24 124 -47 31 20 1914 1559 4 MP  PP
Xc0
X
X-31 -24 124 -47 31 20 1914 1559 4 MP  DP
Xc13
X
X-31 -22 124 -43 31 18 1945 1579 4 MP  PP
Xc0
X
X-31 -22 124 -43 31 18 1945 1579 4 MP  DP
Xgr 
XDO
XSO
X4979 2165 mt 6221 1825 L
X4026 1722 mt 4979 2165 L
X4026 1722 mt 4026  754 L
X4979 2165 mt 5008 2179 L
X5033 2346 mt (0) s
X5600 1995 mt 5629 2009 L
X5654 2176 mt (5) s
X6221 1825 mt 6250 1838 L
X6275 2006 mt (10) s
X4303 1850 mt 4272 1859 L
X4060 2022 mt (10) s
X4610 1993 mt 4579 2002 L
X4367 2165 mt (20) s
X4918 2136 mt 4886 2145 L
X4674 2308 mt (30) s
X4026 1600 mt 3997 1586 L
X3730 1637 mt (-30) s
X4026 1318 mt 3997 1304 L
X3730 1355 mt (-20) s
X4026 1036 mt 3997 1022 L
X3730 1073 mt (-10) s
X4133  281 mt (Tikh filter factors, log scale) s
Xgs 4026 414 2196 1752 MR c np
Xc2
X
X-31 -14 124 -34 31 14 5144 448 4 MP  PP
Xc0
X
X-31 -14 124 -34 31 14 5144 448 4 MP  DP
Xc2
X
X-30 -15 124 -34 30 15 5175 462 4 MP  PP
Xc0
X
X-30 -15 124 -34 30 15 5175 462 4 MP  DP
Xc2
X
X-31 -14 124 -34 31 14 5205 477 4 MP  PP
Xc0
X
X-31 -14 124 -34 31 14 5205 477 4 MP  DP
Xc2
X
X-31 -14 125 -34 30 14 5020 482 4 MP  PP
Xc0
X
X-31 -14 125 -34 30 14 5020 482 4 MP  DP
Xc2
X
X-31 -14 124 -34 31 14 5236 491 4 MP  PP
Xc0
X
X-31 -14 124 -34 31 14 5236 491 4 MP  DP
Xc2
X
X-30 -15 124 -34 31 15 5050 496 4 MP  PP
Xc0
X
X-30 -15 124 -34 31 15 5050 496 4 MP  DP
Xc2
X
X-31 -14 124 -34 31 14 5267 505 4 MP  PP
Xc0
X
X-31 -14 124 -34 31 14 5267 505 4 MP  DP
Xc2
X
X-31 -14 124 -34 31 14 5081 511 4 MP  PP
Xc0
X
X-31 -14 124 -34 31 14 5081 511 4 MP  DP
Xc2
X
X-30 -14 124 -34 31 14 4895 516 4 MP  PP
Xc0
X
X-30 -14 124 -34 31 14 4895 516 4 MP  DP
Xc2
X
X-30 -15 124 -34 30 15 5298 519 4 MP  PP
Xc0
X
X-30 -15 124 -34 30 15 5298 519 4 MP  DP
Xc3
X
X-31 -32 125 -63 30 31 5759 1373 4 MP  PP
Xc0
X
X-31 -32 125 -63 30 31 5759 1373 4 MP  DP
Xc2
X
X-31 -14 124 -34 31 14 5112 525 4 MP  PP
Xc0
X
X-31 -14 124 -34 31 14 5112 525 4 MP  DP
Xc2
X
X-31 -15 124 -34 31 15 4926 530 4 MP  PP
Xc0
X
X-31 -15 124 -34 31 15 4926 530 4 MP  DP
Xcolortable /c23 { 0.000000 1.000000 0.761905 sc} put
Xc23
X
X-31 -131 124 -64 31 131 5728 1242 4 MP  PP
Xc0
X
X-31 -131 124 -64 31 131 5728 1242 4 MP  DP
Xc2
X
X-31 -14 124 -34 31 14 5328 534 4 MP  PP
Xc0
X
X-31 -14 124 -34 31 14 5328 534 4 MP  DP
Xcolortable /c24 { 0.095238 1.000000 0.000000 sc} put
Xc24
X
X-30 -18 124 -64 31 19 5789 1404 4 MP  PP
Xc0
X
X-30 -18 124 -64 31 19 5789 1404 4 MP  DP
Xc22
X
X-30 -45 124 -63 31 44 5666 1153 4 MP  PP
Xc0
X
X-30 -45 124 -63 31 44 5666 1153 4 MP  DP
Xc2
X
X-31 -14 125 -35 30 15 5143 539 4 MP  PP
Xc0
X
X-31 -14 125 -35 30 15 5143 539 4 MP  DP
Xc21
X
X-31 -44 124 -64 31 45 5697 1197 4 MP  PP
Xc0
X
X-31 -44 124 -64 31 45 5697 1197 4 MP  DP
Xc16
X
X-31 -105 125 -64 30 105 5636 1048 4 MP  PP
Xc0
X
X-31 -105 125 -64 30 105 5636 1048 4 MP  DP
Xc2
X
X-31 -14 124 -34 31 14 4957 545 4 MP  PP
Xc0
X
X-31 -14 124 -34 31 14 4957 545 4 MP  DP
Xc24
X
X-31 -22 124 -63 31 21 5820 1423 4 MP  PP
Xc0
X
X-31 -22 124 -63 31 21 5820 1423 4 MP  DP
Xcolortable /c25 { 0.190476 1.000000 0.000000 sc} put
Xc25
X
X-30 -31 124 -61 31 31 5634 1434 4 MP  PP
Xc0
X
X-30 -31 124 -61 31 31 5634 1434 4 MP  DP
Xc2
X
X-31 -14 124 -34 31 14 5359 548 4 MP  PP
Xc0
X
X-31 -14 124 -34 31 14 5359 548 4 MP  DP
Xc2
X
X-31 -14 124 -34 31 14 4771 550 4 MP  PP
Xc0
X
X-31 -14 124 -34 31 14 4771 550 4 MP  DP
Xc4
X
X-31 -67 124 -64 31 68 5605 980 4 MP  PP
Xc0
X
X-31 -67 124 -64 31 68 5605 980 4 MP  DP
Xc2
X
X-30 -15 124 -34 31 14 5173 554 4 MP  PP
Xc0
X
X-30 -15 124 -34 31 14 5173 554 4 MP  DP
Xcolortable /c26 { 0.000000 1.000000 0.571429 sc} put
Xc26
X
X-31 -131 125 -61 30 131 5604 1303 4 MP  PP
Xc0
X
X-31 -131 125 -61 30 131 5604 1303 4 MP  DP
Xcolortable /c27 { 0.476190 0.000000 1.000000 sc} put
Xc27
X
X-31 -81 124 -63 31 80 5574 900 4 MP  PP
Xc0
X
X-31 -81 124 -63 31 80 5574 900 4 MP  DP
Xcolortable /c28 { 0.285714 1.000000 0.000000 sc} put
Xc28
X
X-31 -19 124 -61 31 19 5665 1465 4 MP  PP
Xc0
X
X-31 -19 124 -61 31 19 5665 1465 4 MP  DP
Xc25
X
X-31 -18 124 -63 31 18 5851 1444 4 MP  PP
Xc0
X
X-31 -18 124 -63 31 18 5851 1444 4 MP  DP
Xc2
X
X-31 -14 125 -34 30 14 4988 559 4 MP  PP
Xc0
X
X-31 -14 125 -34 30 14 4988 559 4 MP  DP
Xc2
X
X-31 -17 125 -45 30 28 5390 562 4 MP  PP
Xc0
X
X-31 -17 125 -45 30 28 5390 562 4 MP  DP
Xc21
X
X-31 -44 124 -61 31 44 5542 1214 4 MP  PP
Xc0
X
X-31 -44 124 -61 31 44 5542 1214 4 MP  DP
Xc2
X
X-31 -15 124 -34 31 15 4802 564 4 MP  PP
Xc0
X
X-31 -15 124 -34 31 15 4802 564 4 MP  DP
Xcolortable /c29 { 0.857143 0.000000 1.000000 sc} put
Xc29
X
X-30 -77 124 -64 31 77 5543 823 4 MP  PP
Xc0
X
X-30 -77 124 -64 31 77 5543 823 4 MP  DP
Xc23
X
X-31 -45 124 -61 31 45 5573 1258 4 MP  PP
Xc0
X
X-31 -45 124 -61 31 45 5573 1258 4 MP  DP
Xc2
X
X-30 -45 124 -63 31 63 5420 590 4 MP  PP
Xc0
X
X-30 -45 124 -63 31 63 5420 590 4 MP  DP
Xc15
X
X-30 -105 124 -61 31 105 5511 1109 4 MP  PP
Xc0
X
X-30 -105 124 -61 31 105 5511 1109 4 MP  DP
Xc2
X
X-31 -14 124 -34 31 14 5204 568 4 MP  PP
Xc0
X
X-31 -14 124 -34 31 14 5204 568 4 MP  DP
Xcolortable /c30 { 1.000000 0.000000 0.380952 sc} put
Xc30
X
X-31 -47 124 -63 31 47 5451 653 4 MP  PP
Xc0
X
X-31 -47 124 -63 31 47 5451 653 4 MP  DP
Xcolortable /c31 { 1.000000 0.000000 0.761905 sc} put
Xc31
X
X-31 -68 125 -64 30 69 5513 754 4 MP  PP
Xc0
X
X-31 -68 125 -64 30 69 5513 754 4 MP  DP
Xc28
X
X-31 -21 124 -61 31 21 5696 1484 4 MP  PP
Xc0
X
X-31 -21 124 -61 31 21 5696 1484 4 MP  DP
Xc25
X
X-31 -18 125 -64 30 19 5882 1462 4 MP  PP
Xc0
X
X-31 -18 125 -64 30 19 5882 1462 4 MP  DP
Xcolortable /c32 { 1.000000 0.000000 0.571429 sc} put
Xc32
X
X-31 -54 124 -63 31 54 5482 700 4 MP  PP
Xc0
X
X-31 -54 124 -63 31 54 5482 700 4 MP  DP
Xc28
X
X-31 -31 124 -64 31 32 5510 1497 4 MP  PP
Xc0
X
X-31 -31 124 -64 31 32 5510 1497 4 MP  DP
Xc2
X
X-30 -15 124 -34 31 15 5018 573 4 MP  PP
Xc0
X
X-30 -15 124 -34 31 15 5018 573 4 MP  DP
Xc19
X
X-31 -68 125 -61 30 68 5481 1041 4 MP  PP
Xc0
X
X-31 -68 125 -61 30 68 5481 1041 4 MP  DP
Xc2
X
X-31 -14 125 -34 30 14 4833 579 4 MP  PP
Xc0
X
X-31 -14 125 -34 30 14 4833 579 4 MP  DP
Xcolortable /c33 { 0.000000 1.000000 0.380952 sc} put
Xc33
X
X-30 -131 124 -63 31 131 5479 1366 4 MP  PP
Xc0
X
X-30 -131 124 -63 31 131 5479 1366 4 MP  DP
Xc2
X
X-31 -14 124 -35 31 15 5235 582 4 MP  PP
Xc0
X
X-31 -14 124 -35 31 15 5235 582 4 MP  DP
Xcolortable /c34 { 0.285714 0.000000 1.000000 sc} put
Xc34
X
X-31 -80 124 -61 31 80 5450 961 4 MP  PP
Xc0
X
X-31 -80 124 -61 31 80 5450 961 4 MP  DP
Xc25
X
X-30 -28 124 -64 31 28 5912 1481 4 MP  PP
Xc0
X
X-30 -28 124 -64 31 28 5912 1481 4 MP  DP
Xcolortable /c35 { 0.476190 1.000000 0.000000 sc} put
Xc35
X
X-31 -19 124 -63 31 18 5541 1529 4 MP  PP
Xc0
X
X-31 -19 124 -63 31 18 5541 1529 4 MP  DP
Xc28
X
X-31 -18 125 -61 30 18 5727 1505 4 MP  PP
Xc0
X
X-31 -18 125 -61 30 18 5727 1505 4 MP  DP
Xc2
X
X-31 -14 124 -34 31 14 4647 584 4 MP  PP
Xc0
X
X-31 -14 124 -34 31 14 4647 584 4 MP  DP
Xc23
X
X-31 -44 124 -64 31 45 5418 1277 4 MP  PP
Xc0
X
X-31 -44 124 -64 31 45 5418 1277 4 MP  DP
Xc2
X
X-31 -14 124 -34 31 14 5049 588 4 MP  PP
Xc0
X
X-31 -14 124 -34 31 14 5049 588 4 MP  DP
Xcolortable /c36 { 0.666667 0.000000 1.000000 sc} put
Xc36
X
X-31 -77 124 -61 31 77 5419 884 4 MP  PP
Xc0
X
X-31 -77 124 -61 31 77 5419 884 4 MP  DP
Xc26
X
X-31 -45 125 -63 30 44 5449 1322 4 MP  PP
Xc0
X
X-31 -45 125 -63 30 44 5449 1322 4 MP  DP
Xc2
X
X-30 -28 124 -57 30 50 5266 597 4 MP  PP
Xc0
X
X-30 -28 124 -57 30 50 5266 597 4 MP  DP
Xc13
X
X-31 -105 124 -63 31 105 5387 1172 4 MP  PP
Xc0
X
X-31 -105 124 -63 31 105 5387 1172 4 MP  DP
Xc2
X
X-30 -14 124 -34 31 14 4863 593 4 MP  PP
Xc0
X
X-30 -14 124 -34 31 14 4863 593 4 MP  DP
Xcolortable /c37 { 1.000000 0.000000 0.190476 sc} put
Xc37
X
X-31 -63 124 -61 31 67 5296 647 4 MP  PP
Xc0
X
X-31 -63 124 -61 31 67 5296 647 4 MP  DP
Xcolortable /c38 { 1.000000 0.000000 0.476190 sc} put
Xc38
X
X-31 -47 124 -61 31 47 5327 714 4 MP  PP
Xc0
X
X-31 -47 124 -61 31 47 5327 714 4 MP  DP
Xc28
X
X-31 -18 124 -63 31 17 5943 1509 4 MP  PP
Xc0
X
X-31 -18 124 -63 31 17 5943 1509 4 MP  DP
Xcolortable /c39 { 1.000000 0.000000 0.952381 sc} put
Xc39
X
X-30 -69 124 -61 31 69 5388 815 4 MP  PP
Xc0
X
X-30 -69 124 -61 31 69 5388 815 4 MP  DP
Xc35
X
X-31 -21 125 -64 30 22 5572 1547 4 MP  PP
Xc0
X
X-31 -21 125 -64 30 22 5572 1547 4 MP  DP
Xcolortable /c40 { 0.380952 1.000000 0.000000 sc} put
Xc40
X
X-30 -19 124 -61 31 19 5757 1523 4 MP  PP
Xc0
X
X-30 -19 124 -61 31 19 5757 1523 4 MP  DP
Xc31
X
X-31 -54 125 -61 30 54 5358 761 4 MP  PP
Xc0
X
X-31 -54 125 -61 30 54 5358 761 4 MP  DP
Xc35
X
X-31 -32 124 -61 31 32 5386 1558 4 MP  PP
Xc0
X
X-31 -32 124 -61 31 32 5386 1558 4 MP  DP
Xc2
X
X-31 -15 125 -34 30 15 4678 598 4 MP  PP
Xc0
X
X-31 -15 125 -34 30 15 4678 598 4 MP  DP
Xc2
X
X-31 -14 124 -35 31 15 5080 602 4 MP  PP
Xc0
X
X-31 -14 124 -35 31 15 5080 602 4 MP  DP
Xc16
X
X-30 -68 124 -63 30 67 5357 1105 4 MP  PP
Xc0
X
X-30 -68 124 -63 30 67 5357 1105 4 MP  DP
Xcolortable /c41 { 0.000000 1.000000 0.190476 sc} put
Xc41
X
X-31 -131 124 -61 31 131 5355 1427 4 MP  PP
Xc0
X
X-31 -131 124 -61 31 131 5355 1427 4 MP  DP
Xc28
X
X-31 -23 124 -64 31 24 5974 1526 4 MP  PP
Xc0
X
X-31 -23 124 -64 31 24 5974 1526 4 MP  DP
Xc2
X
X-31 -15 124 -34 31 15 4894 607 4 MP  PP
Xc0
X
X-31 -15 124 -34 31 15 4894 607 4 MP  DP
Xc17
X
X-31 -80 124 -64 31 81 5326 1024 4 MP  PP
Xc0
X
X-31 -80 124 -64 31 81 5326 1024 4 MP  DP
Xc40
X
X-31 -28 124 -61 31 28 5788 1542 4 MP  PP
Xc0
X
X-31 -28 124 -61 31 28 5788 1542 4 MP  DP
Xcolortable /c42 { 0.571429 1.000000 0.000000 sc} put
Xc42
X
X-31 -18 124 -61 31 18 5417 1590 4 MP  PP
Xc0
X
X-31 -18 124 -61 31 18 5417 1590 4 MP  DP
Xc35
X
X-30 -18 124 -63 31 17 5602 1569 4 MP  PP
Xc0
X
X-30 -18 124 -63 31 17 5602 1569 4 MP  DP
Xc26
X
X-31 -45 124 -60 31 44 5294 1338 4 MP  PP
Xc0
X
X-31 -45 124 -60 31 44 5294 1338 4 MP  DP
Xc2
X
X-30 -14 124 -34 31 14 4708 613 4 MP  PP
Xc0
X
X-30 -14 124 -34 31 14 4708 613 4 MP  DP
Xc27
X
X-31 -77 124 -63 31 77 5295 947 4 MP  PP
Xc0
X
X-31 -77 124 -63 31 77 5295 947 4 MP  DP
Xc33
X
X-30 -44 124 -61 30 45 5325 1382 4 MP  PP
Xc0
X
X-30 -44 124 -61 30 45 5325 1382 4 MP  DP
Xc2
X
X-31 -15 125 -40 30 20 5111 617 4 MP  PP
Xc0
X
X-31 -15 125 -40 30 20 5111 617 4 MP  DP
Xc22
X
X-31 -105 124 -61 31 105 5263 1233 4 MP  PP
Xc0
X
X-31 -105 124 -61 31 105 5263 1233 4 MP  DP
Xc2
X
X-31 -14 124 -34 31 14 4523 618 4 MP  PP
Xc0
X
X-31 -14 124 -34 31 14 4523 618 4 MP  DP
Xc40
X
X-31 -19 125 -64 30 19 6005 1550 4 MP  PP
Xc0
X
X-31 -19 125 -64 30 19 6005 1550 4 MP  DP
Xc30
X
X-31 -67 124 -63 31 67 5172 710 4 MP  PP
Xc0
X
X-31 -67 124 -63 31 67 5172 710 4 MP  DP
Xcolortable /c43 { 1.000000 0.000000 0.666667 sc} put
Xc43
X
X-31 -47 124 -64 31 48 5203 777 4 MP  PP
Xc0
X
X-31 -47 124 -64 31 48 5203 777 4 MP  DP
Xc35
X
X-31 -17 124 -61 31 17 5819 1570 4 MP  PP
Xc0
X
X-31 -17 124 -61 31 17 5819 1570 4 MP  DP
Xc29
X
X-31 -69 124 -63 31 68 5264 879 4 MP  PP
Xc0
X
X-31 -69 124 -63 31 68 5264 879 4 MP  DP
Xcolortable /c44 { 0.666667 1.000000 0.000000 sc} put
Xc44
X
X-30 -22 124 -61 30 22 5448 1608 4 MP  PP
Xc0
X
X-30 -22 124 -61 30 22 5448 1608 4 MP  DP
Xc42
X
X-31 -19 124 -63 31 19 5633 1586 4 MP  PP
Xc0
X
X-31 -19 124 -63 31 19 5633 1586 4 MP  DP
Xc39
X
X-30 -54 124 -64 30 54 5234 825 4 MP  PP
Xc0
X
X-30 -54 124 -64 30 54 5234 825 4 MP  DP
Xc2
X
X-30 -50 124 -63 31 73 5141 637 4 MP  PP
Xc0
X
X-30 -50 124 -63 31 73 5141 637 4 MP  DP
Xc2
X
X-31 -14 124 -34 31 14 4925 622 4 MP  PP
Xc0
X
X-31 -14 124 -34 31 14 4925 622 4 MP  DP
Xc44
X
X-31 -32 124 -63 31 31 5262 1622 4 MP  PP
Xc0
X
X-31 -32 124 -63 31 31 5262 1622 4 MP  DP
Xc15
X
X-30 -67 124 -61 31 67 5232 1166 4 MP  PP
Xc0
X
X-30 -67 124 -61 31 67 5232 1166 4 MP  DP
Xc2
X
X-31 -14 124 -35 31 15 4739 627 4 MP  PP
Xc0
X
X-31 -14 124 -35 31 15 4739 627 4 MP  DP
Xc40
X
X-30 -32 124 -63 31 31 6035 1569 4 MP  PP
Xc0
X
X-30 -32 124 -63 31 31 6035 1569 4 MP  DP
Xcolortable /c45 { 0.000000 1.000000 0.095238 sc} put
Xc45
X
X-31 -131 124 -64 31 131 5231 1491 4 MP  PP
Xc0
X
X-31 -131 124 -64 31 131 5231 1491 4 MP  DP
Xc35
X
X-31 -24 125 -61 30 24 5850 1587 4 MP  PP
Xc0
X
X-31 -24 125 -61 30 24 5850 1587 4 MP  DP
Xc19
X
X-31 -81 125 -61 30 81 5202 1085 4 MP  PP
Xc0
X
X-31 -81 125 -61 30 81 5202 1085 4 MP  DP
Xc42
X
X-31 -28 124 -63 31 28 5664 1605 4 MP  PP
Xc0
X
X-31 -28 124 -63 31 28 5664 1605 4 MP  DP
Xcolortable /c46 { 0.761905 1.000000 0.000000 sc} put
Xc46
X
X-31 -18 125 -64 30 19 5293 1653 4 MP  PP
Xc0
X
X-31 -18 125 -64 30 19 5293 1653 4 MP  DP
Xc2
X
X-30 -15 124 -34 30 15 4554 632 4 MP  PP
Xc0
X
X-30 -15 124 -34 30 15 4554 632 4 MP  DP
Xc44
X
X-31 -17 124 -61 31 17 5478 1630 4 MP  PP
Xc0
X
X-31 -17 124 -61 31 17 5478 1630 4 MP  DP
Xc2
X
X-31 -15 125 -38 30 19 4956 636 4 MP  PP
Xc0
X
X-31 -15 125 -38 30 19 4956 636 4 MP  DP
Xc8
X
X-31 -44 125 -64 30 45 5170 1401 4 MP  PP
Xc0
X
X-31 -44 125 -64 30 45 5170 1401 4 MP  DP
Xc35
X
X-31 -33 124 -63 31 33 6066 1600 4 MP  PP
Xc0
X
X-31 -33 124 -63 31 33 6066 1600 4 MP  DP
Xc20
X
X-31 -77 124 -61 31 77 5171 1008 4 MP  PP
Xc0
X
X-31 -77 124 -61 31 77 5171 1008 4 MP  DP
Xcolortable /c47 { 0.000000 1.000000 0.285714 sc} put
Xc47
X
X-30 -45 124 -64 31 45 5200 1446 4 MP  PP
Xc0
X
X-30 -45 124 -64 31 45 5200 1446 4 MP  DP
Xc2
X
X-31 -15 124 -35 31 15 4770 642 4 MP  PP
Xc0
X
X-31 -15 124 -35 31 15 4770 642 4 MP  DP
Xc10
X
X-31 -105 124 -63 31 104 5139 1297 4 MP  PP
Xc0
X
X-31 -105 124 -63 31 104 5139 1297 4 MP  DP
Xc42
X
X-30 -19 124 -61 31 19 5880 1611 4 MP  PP
Xc0
X
X-30 -19 124 -61 31 19 5880 1611 4 MP  DP
Xcolortable /c48 { 1.000000 0.000000 0.857143 sc} put
Xc48
X
X-31 -48 125 -61 30 48 5079 838 4 MP  PP
Xc0
X
X-31 -48 125 -61 30 48 5079 838 4 MP  DP
Xc32
X
X-31 -67 124 -61 31 67 5048 771 4 MP  PP
Xc0
X
X-31 -67 124 -61 31 67 5048 771 4 MP  DP
Xc44
X
X-31 -17 125 -64 30 18 5695 1633 4 MP  PP
Xc0
X
X-31 -17 125 -64 30 18 5695 1633 4 MP  DP
Xc36
X
X-31 -68 124 -61 31 68 5140 940 4 MP  PP
Xc0
X
X-31 -68 124 -61 31 68 5140 940 4 MP  DP
Xcolortable /c49 { 0.857143 1.000000 0.000000 sc} put
Xc49
X
X-30 -22 124 -63 31 21 5323 1672 4 MP  PP
Xc0
X
X-30 -22 124 -63 31 21 5323 1672 4 MP  DP
Xc44
X
X-31 -19 124 -61 31 19 5509 1647 4 MP  PP
Xc0
X
X-31 -19 124 -61 31 19 5509 1647 4 MP  DP
Xcolortable /c50 { 0.952381 0.000000 1.000000 sc} put
Xc50
X
X-30 -54 124 -61 31 54 5109 886 4 MP  PP
Xc0
X
X-30 -54 124 -61 31 54 5109 886 4 MP  DP
Xc49
X
X-31 -31 125 -61 30 32 5138 1682 4 MP  PP
Xc0
X
X-31 -31 125 -61 30 32 5138 1682 4 MP  DP
Xc2
X
X-31 -14 124 -34 31 14 4584 647 4 MP  PP
Xc0
X
X-31 -14 124 -34 31 14 4584 647 4 MP  DP
Xc2
X
X-30 -20 124 -51 31 33 4986 655 4 MP  PP
Xc0
X
X-30 -20 124 -51 31 33 4986 655 4 MP  DP
Xcolortable /c51 { 1.000000 0.000000 0.095238 sc} put
Xc51
X
X-31 -73 124 -61 31 83 5017 688 4 MP  PP
Xc0
X
X-31 -73 124 -61 31 83 5017 688 4 MP  DP
Xc13
X
X-31 -67 124 -64 31 68 5108 1229 4 MP  PP
Xc0
X
X-31 -67 124 -64 31 68 5108 1229 4 MP  DP
Xc2
X
X-31 -14 125 -34 30 14 4399 652 4 MP  PP
Xc0
X
X-31 -14 125 -34 30 14 4399 652 4 MP  DP
Xc42
X
X-31 -31 124 -61 31 31 5911 1630 4 MP  PP
Xc0
X
X-31 -31 124 -61 31 31 5911 1630 4 MP  DP
Xc24
X
X-31 -131 124 -60 31 130 5107 1552 4 MP  PP
Xc0
X
X-31 -131 124 -60 31 130 5107 1552 4 MP  DP
Xc2
X
X-31 -14 125 -36 30 15 4801 657 4 MP  PP
Xc0
X
X-31 -14 125 -36 30 15 4801 657 4 MP  DP
Xc44
X
X-30 -24 124 -63 31 23 5725 1651 4 MP  PP
Xc0
X
X-30 -24 124 -63 31 23 5725 1651 4 MP  DP
Xc16
X
X-30 -81 124 -63 31 80 5077 1149 4 MP  PP
Xc0
X
X-30 -81 124 -63 31 80 5077 1149 4 MP  DP
Xc46
X
X-31 -28 125 -61 30 28 5540 1666 4 MP  PP
Xc0
X
X-31 -28 125 -61 30 28 5540 1666 4 MP  DP
Xcolortable /c52 { 0.952381 1.000000 0.000000 sc} put
Xc52
X
X-30 -19 124 -61 31 19 5168 1714 4 MP  PP
Xc0
X
X-30 -19 124 -61 31 19 5168 1714 4 MP  DP
Xc49
X
X-31 -17 124 -64 31 18 5354 1693 4 MP  PP
Xc0
X
X-31 -17 124 -64 31 18 5354 1693 4 MP  DP
Xc47
X
X-30 -45 124 -61 31 45 5045 1462 4 MP  PP
Xc0
X
X-30 -45 124 -61 31 45 5045 1462 4 MP  DP
Xc2
X
X-31 -15 124 -36 31 17 4615 661 4 MP  PP
Xc0
X
X-31 -15 124 -36 31 17 4615 661 4 MP  DP
Xc44
X
X-31 -33 124 -61 31 33 5942 1661 4 MP  PP
Xc0
X
X-31 -33 124 -61 31 33 5942 1661 4 MP  DP
Xc18
X
X-31 -77 125 -64 30 78 5047 1071 4 MP  PP
Xc0
X
X-31 -77 125 -64 30 78 5047 1071 4 MP  DP
Xc45
X
X-31 -45 124 -61 31 45 5076 1507 4 MP  PP
Xc0
X
X-31 -45 124 -61 31 45 5076 1507 4 MP  DP
Xcolortable /c53 { 0.000000 1.000000 0.857143 sc} put
Xc53
X
X-31 -104 125 -61 30 105 5015 1357 4 MP  PP
Xc0
X
X-31 -104 125 -61 30 105 5015 1357 4 MP  DP
Xc2
X
X-30 -15 124 -34 31 15 4429 666 4 MP  PP
Xc0
X
X-30 -15 124 -34 31 15 4429 666 4 MP  DP
Xc2
X
X-30 -19 124 -52 31 35 4831 672 4 MP  PP
Xc0
X
X-30 -19 124 -52 31 35 4831 672 4 MP  DP
Xc46
X
X-31 -19 124 -63 31 19 5756 1674 4 MP  PP
Xc0
X
X-31 -19 124 -63 31 19 5756 1674 4 MP  DP
Xc50
X
X-30 -48 124 -63 31 47 4954 902 4 MP  PP
Xc0
X
X-30 -48 124 -63 31 47 4954 902 4 MP  DP
Xc31
X
X-31 -67 125 -64 30 68 4924 834 4 MP  PP
Xc0
X
X-31 -67 125 -64 30 68 4924 834 4 MP  DP
Xc49
X
X-30 -18 124 -61 31 18 5570 1694 4 MP  PP
Xc0
X
X-30 -18 124 -61 31 18 5570 1694 4 MP  DP
Xc27
X
X-31 -68 124 -63 31 68 5016 1003 4 MP  PP
Xc0
X
X-31 -68 124 -63 31 68 5016 1003 4 MP  DP
Xc52
X
X-31 -21 124 -61 31 21 5199 1733 4 MP  PP
Xc0
X
X-31 -21 124 -61 31 21 5199 1733 4 MP  DP
Xc49
X
X-31 -19 124 -64 31 19 5385 1711 4 MP  PP
Xc0
X
X-31 -19 124 -64 31 19 5385 1711 4 MP  DP
Xcolortable /c54 { 0.761905 0.000000 1.000000 sc} put
Xc54
X
X-31 -54 124 -63 31 54 4985 949 4 MP  PP
Xc0
X
X-31 -54 124 -63 31 54 4985 949 4 MP  DP
Xcolortable /c55 { 1.000000 0.952381 0.000000 sc} put
Xc55
X
X-30 -32 124 -64 31 32 5013 1746 4 MP  PP
Xc0
X
X-30 -32 124 -64 31 32 5013 1746 4 MP  DP
Xc2
X
X-31 -15 124 -39 31 18 4646 678 4 MP  PP
Xc0
X
X-31 -15 124 -39 31 18 4646 678 4 MP  DP
Xcolortable /c56 { 1.000000 0.000000 0.285714 sc} put
Xc56
X
X-31 -83 124 -63 31 84 4893 750 4 MP  PP
Xc0
X
X-31 -83 124 -63 31 84 4893 750 4 MP  DP
Xc51
X
X-31 -33 124 -62 31 43 4862 707 4 MP  PP
Xc0
X
X-31 -33 124 -62 31 43 4862 707 4 MP  DP
Xc22
X
X-31 -68 124 -60 31 67 4984 1290 4 MP  PP
Xc0
X
X-31 -68 124 -60 31 67 4984 1290 4 MP  DP
Xc46
X
X-31 -31 124 -64 31 32 5787 1693 4 MP  PP
Xc0
X
X-31 -31 124 -64 31 32 5787 1693 4 MP  DP
Xc28
X
X-31 -130 125 -64 30 131 4983 1615 4 MP  PP
Xc0
X
X-31 -130 125 -64 30 131 4983 1615 4 MP  DP
Xc49
X
X-31 -23 124 -61 31 23 5601 1712 4 MP  PP
Xc0
X
X-31 -23 124 -61 31 23 5601 1712 4 MP  DP
Xc2
X
X-31 -14 124 -35 31 15 4460 681 4 MP  PP
Xc0
X
X-31 -14 124 -35 31 15 4460 681 4 MP  DP
Xc11
X
X-31 -80 124 -61 31 81 4953 1209 4 MP  PP
Xc0
X
X-31 -80 124 -61 31 81 4953 1209 4 MP  DP
Xc52
X
X-30 -28 124 -64 30 28 5416 1730 4 MP  PP
Xc0
X
X-30 -28 124 -64 30 28 5416 1730 4 MP  DP
Xcolortable /c57 { 1.000000 0.857143 0.000000 sc} put
Xc57
X
X-31 -19 124 -63 31 18 5044 1778 4 MP  PP
Xc0
X
X-31 -19 124 -63 31 18 5044 1778 4 MP  DP
Xc55
X
X-31 -18 124 -61 31 18 5230 1754 4 MP  PP
Xc0
X
X-31 -18 124 -61 31 18 5230 1754 4 MP  DP
Xc45
X
X-31 -45 124 -63 31 44 4921 1526 4 MP  PP
Xc0
X
X-31 -45 124 -63 31 44 4921 1526 4 MP  DP
Xc2
X
X-30 -14 124 -35 31 15 4274 686 4 MP  PP
Xc0
X
X-30 -14 124 -35 31 15 4274 686 4 MP  DP
Xc49
X
X-31 -33 125 -64 30 33 5818 1725 4 MP  PP
Xc0
X
X-31 -33 125 -64 30 33 5818 1725 4 MP  DP
Xc2
X
X-30 -15 124 -43 30 19 4677 696 4 MP  PP
Xc0
X
X-30 -15 124 -43 30 19 4677 696 4 MP  DP
Xc4
X
X-30 -78 124 -60 31 77 4922 1132 4 MP  PP
Xc0
X
X-30 -78 124 -60 31 77 4922 1132 4 MP  DP
Xc24
X
X-31 -45 124 -63 31 45 4952 1570 4 MP  PP
Xc0
X
X-31 -45 124 -63 31 45 4952 1570 4 MP  DP
Xcolortable /c58 { 0.000000 1.000000 0.666667 sc} put
Xc58
X
X-30 -105 124 -64 31 105 4890 1421 4 MP  PP
Xc0
X
X-30 -105 124 -64 31 105 4890 1421 4 MP  DP
Xc49
X
X-31 -19 124 -61 31 19 5632 1735 4 MP  PP
Xc0
X
X-31 -19 124 -61 31 19 5632 1735 4 MP  DP
Xc54
X
X-31 -47 124 -61 31 48 4830 962 4 MP  PP
Xc0
X
X-31 -47 124 -61 31 48 4830 962 4 MP  DP
Xc48
X
X-30 -68 124 -60 31 67 4799 895 4 MP  PP
Xc0
X
X-30 -68 124 -60 31 67 4799 895 4 MP  DP
Xc55
X
X-31 -18 124 -63 31 17 5446 1758 4 MP  PP
Xc0
X
X-31 -18 124 -63 31 17 5446 1758 4 MP  DP
Xc2
X
X-31 -17 124 -48 31 30 4491 696 4 MP  PP
Xc0
X
X-31 -17 124 -48 31 30 4491 696 4 MP  DP
Xc34
X
X-31 -68 125 -61 30 68 4892 1064 4 MP  PP
Xc0
X
X-31 -68 125 -61 30 68 4892 1064 4 MP  DP
Xc57
X
X-31 -21 124 -63 31 21 5075 1796 4 MP  PP
Xc0
X
X-31 -21 124 -63 31 21 5075 1796 4 MP  DP
Xc55
X
X-31 -19 125 -61 30 19 5261 1772 4 MP  PP
Xc0
X
X-31 -19 125 -61 30 19 5261 1772 4 MP  DP
Xcolortable /c59 { 0.571429 0.000000 1.000000 sc} put
Xc59
X
X-31 -54 124 -61 31 54 4861 1010 4 MP  PP
Xc0
X
X-31 -54 124 -61 31 54 4861 1010 4 MP  DP
Xcolortable /c60 { 1.000000 0.761905 0.000000 sc} put
Xc60
X
X-31 -32 124 -60 31 31 4889 1807 4 MP  PP
Xc0
X
X-31 -32 124 -60 31 31 4889 1807 4 MP  DP
Xc2
X
X-31 -35 124 -59 31 51 4707 715 4 MP  PP
Xc0
X
X-31 -35 124 -59 31 51 4707 715 4 MP  DP
Xc38
X
X-31 -84 125 -61 30 85 4769 810 4 MP  PP
Xc0
X
X-31 -84 125 -61 30 85 4769 810 4 MP  DP
Xc21
X
X-31 -67 125 -64 30 67 4860 1354 4 MP  PP
Xc0
X
X-31 -67 125 -64 30 67 4860 1354 4 MP  DP
Xc2
X
X-31 -15 124 -35 31 15 4305 701 4 MP  PP
Xc0
X
X-31 -15 124 -35 31 15 4305 701 4 MP  DP
Xc56
X
X-31 -43 124 -60 31 44 4738 766 4 MP  PP
Xc0
X
X-31 -43 124 -60 31 44 4738 766 4 MP  DP
Xc52
X
X-31 -32 125 -60 30 31 5663 1754 4 MP  PP
Xc0
X
X-31 -32 125 -60 30 31 5663 1754 4 MP  DP
Xc35
X
X-30 -131 124 -61 30 131 4859 1676 4 MP  PP
Xc0
X
X-30 -131 124 -61 30 131 4859 1676 4 MP  DP
Xc51
X
X-31 -18 125 -55 30 25 4522 726 4 MP  PP
Xc0
X
X-31 -18 125 -55 30 25 4522 726 4 MP  DP
Xc55
X
X-31 -23 124 -64 31 24 5477 1775 4 MP  PP
Xc0
X
X-31 -23 124 -64 31 24 5477 1775 4 MP  DP
Xc12
X
X-31 -81 124 -64 31 81 4829 1273 4 MP  PP
Xc0
X
X-31 -81 124 -64 31 81 4829 1273 4 MP  DP
Xc55
X
X-30 -28 124 -61 31 28 5291 1791 4 MP  PP
Xc0
X
X-30 -28 124 -61 31 28 5291 1791 4 MP  DP
Xcolortable /c61 { 1.000000 0.666667 0.000000 sc} put
Xc61
X
X-31 -18 124 -61 31 19 4920 1838 4 MP  PP
Xc0
X
X-31 -18 124 -61 31 19 4920 1838 4 MP  DP
Xc60
X
X-31 -18 125 -63 30 18 5106 1817 4 MP  PP
Xc0
X
X-31 -18 125 -63 30 18 5106 1817 4 MP  DP
Xc24
X
X-31 -44 124 -61 31 44 4797 1587 4 MP  PP
Xc0
X
X-31 -44 124 -61 31 44 4797 1587 4 MP  DP
Xc55
X
X-30 -33 124 -61 31 34 5693 1785 4 MP  PP
Xc0
X
X-30 -33 124 -61 31 34 5693 1785 4 MP  DP
Xc14
X
X-31 -77 124 -64 31 77 4798 1196 4 MP  PP
Xc0
X
X-31 -77 124 -64 31 77 4798 1196 4 MP  DP
Xc28
X
X-31 -45 124 -61 31 45 4828 1631 4 MP  PP
Xc0
X
X-31 -45 124 -61 31 45 4828 1631 4 MP  DP
Xc2
X
X-31 -15 124 -39 31 19 4336 716 4 MP  PP
Xc0
X
X-31 -15 124 -39 31 19 4336 716 4 MP  DP
Xc8
X
X-31 -105 124 -61 31 105 4766 1482 4 MP  PP
Xc0
X
X-31 -105 124 -61 31 105 4766 1482 4 MP  DP
Xc51
X
X-30 -19 124 -58 31 22 4552 751 4 MP  PP
Xc0
X
X-30 -19 124 -58 31 22 4552 751 4 MP  DP
Xc55
X
X-31 -19 124 -64 31 19 5508 1799 4 MP  PP
Xc0
X
X-31 -19 124 -64 31 19 5508 1799 4 MP  DP
Xc59
X
X-31 -48 124 -63 31 47 4706 1026 4 MP  PP
Xc0
X
X-31 -48 124 -63 31 47 4706 1026 4 MP  DP
Xc50
X
X-31 -67 124 -64 31 67 4675 959 4 MP  PP
Xc0
X
X-31 -67 124 -64 31 67 4675 959 4 MP  DP
Xc57
X
X-31 -17 124 -61 31 17 5322 1819 4 MP  PP
Xc0
X
X-31 -17 124 -61 31 17 5322 1819 4 MP  DP
Xc17
X
X-30 -68 124 -64 30 68 4768 1128 4 MP  PP
Xc0
X
X-30 -68 124 -64 30 68 4768 1128 4 MP  DP
Xc61
X
X-31 -21 125 -61 30 21 4951 1857 4 MP  PP
Xc0
X
X-31 -21 125 -61 30 21 4951 1857 4 MP  DP
Xc60
X
X-30 -19 124 -63 31 19 5136 1835 4 MP  PP
Xc0
X
X-30 -19 124 -63 31 19 5136 1835 4 MP  DP
Xc20
X
X-31 -54 124 -64 31 55 4737 1073 4 MP  PP
Xc0
X
X-31 -54 124 -64 31 55 4737 1073 4 MP  DP
Xc2
X
X-31 -15 124 -37 31 17 4150 721 4 MP  PP
Xc0
X
X-31 -15 124 -37 31 17 4150 721 4 MP  DP
Xcolortable /c62 { 1.000000 0.571429 0.000000 sc} put
Xc62
X
X-31 -31 124 -64 31 32 4765 1870 4 MP  PP
Xc0
X
X-31 -31 124 -64 31 32 4765 1870 4 MP  DP
Xc37
X
X-31 -51 124 -64 31 57 4583 773 4 MP  PP
Xc0
X
X-31 -51 124 -64 31 57 4583 773 4 MP  DP
Xc2
X
X-31 -30 125 -58 30 49 4367 735 4 MP  PP
Xc0
X
X-31 -30 125 -58 30 49 4367 735 4 MP  DP
Xc43
X
X-30 -85 124 -64 30 85 4645 874 4 MP  PP
Xc0
X
X-30 -85 124 -64 30 85 4645 874 4 MP  DP
Xc23
X
X-30 -67 124 -61 30 67 4736 1415 4 MP  PP
Xc0
X
X-30 -67 124 -61 30 67 4736 1415 4 MP  DP
Xc38
X
X-31 -44 124 -64 31 44 4614 830 4 MP  PP
Xc0
X
X-31 -44 124 -64 31 44 4614 830 4 MP  DP
Xc57
X
X-30 -31 124 -64 30 31 5539 1818 4 MP  PP
Xc0
X
X-30 -31 124 -64 30 31 5539 1818 4 MP  DP
Xc44
X
X-30 -131 124 -63 31 131 4734 1739 4 MP  PP
Xc0
X
X-30 -131 124 -63 31 131 4734 1739 4 MP  DP
Xc60
X
X-31 -24 124 -61 31 24 5353 1836 4 MP  PP
Xc0
X
X-31 -24 124 -61 31 24 5353 1836 4 MP  DP
Xc9
X
X-31 -81 124 -61 31 81 4705 1334 4 MP  PP
Xc0
X
X-31 -81 124 -61 31 81 4705 1334 4 MP  DP
Xc60
X
X-31 -28 124 -63 31 28 5167 1854 4 MP  PP
Xc0
X
X-31 -28 124 -63 31 28 5167 1854 4 MP  DP
Xcolortable /c63 { 1.000000 0.476190 0.000000 sc} put
Xc63
X
X-31 -19 124 -63 31 18 4796 1902 4 MP  PP
Xc0
X
X-31 -19 124 -63 31 18 4796 1902 4 MP  DP
Xc62
X
X-30 -18 124 -61 31 18 4981 1878 4 MP  PP
Xc0
X
X-30 -18 124 -61 31 18 4981 1878 4 MP  DP
Xc37
X
X-30 -25 124 -60 31 27 4397 784 4 MP  PP
Xc0
X
X-30 -25 124 -60 31 27 4397 784 4 MP  DP
Xc2
X
X-31 -15 124 -41 31 19 4181 738 4 MP  PP
Xc0
X
X-31 -15 124 -41 31 19 4181 738 4 MP  DP
Xc28
X
X-31 -44 124 -64 31 45 4673 1650 4 MP  PP
Xc0
X
X-31 -44 124 -64 31 45 4673 1650 4 MP  DP
Xc60
X
X-31 -34 124 -63 31 33 5569 1849 4 MP  PP
Xc0
X
X-31 -34 124 -63 31 33 5569 1849 4 MP  DP
Xc11
X
X-31 -77 124 -61 31 77 4674 1257 4 MP  PP
Xc0
X
X-31 -77 124 -61 31 77 4674 1257 4 MP  DP
Xc35
X
X-31 -45 125 -63 30 44 4704 1695 4 MP  PP
Xc0
X
X-31 -45 125 -63 30 44 4704 1695 4 MP  DP
Xc47
X
X-31 -105 124 -63 31 105 4642 1545 4 MP  PP
Xc0
X
X-31 -105 124 -63 31 105 4642 1545 4 MP  DP
Xc60
X
X-31 -19 125 -61 30 19 5384 1860 4 MP  PP
Xc0
X
X-31 -19 125 -61 30 19 5384 1860 4 MP  DP
Xc20
X
X-31 -47 124 -61 31 47 4582 1087 4 MP  PP
Xc0
X
X-31 -47 124 -61 31 47 4582 1087 4 MP  DP
Xc54
X
X-31 -67 124 -61 31 67 4551 1020 4 MP  PP
Xc0
X
X-31 -67 124 -61 31 67 4551 1020 4 MP  DP
Xc61
X
X-31 -17 124 -63 31 17 5198 1882 4 MP  PP
Xc0
X
X-31 -17 124 -63 31 17 5198 1882 4 MP  DP
Xc56
X
X-31 -22 124 -60 31 22 4428 811 4 MP  PP
Xc0
X
X-31 -22 124 -60 31 22 4428 811 4 MP  DP
Xc4
X
X-30 -68 124 -61 31 69 4643 1188 4 MP  PP
Xc0
X
X-30 -68 124 -61 31 69 4643 1188 4 MP  DP
Xc63
X
X-30 -21 124 -64 30 22 4827 1920 4 MP  PP
Xc0
X
X-30 -21 124 -64 30 22 4827 1920 4 MP  DP
Xc62
X
X-31 -19 124 -61 31 19 5012 1896 4 MP  PP
Xc0
X
X-31 -19 124 -61 31 19 5012 1896 4 MP  DP
Xc18
X
X-31 -55 125 -60 30 54 4613 1134 4 MP  PP
Xc0
X
X-31 -55 125 -60 30 54 4613 1134 4 MP  DP
Xc2
X
X-31 -19 125 -53 30 31 4212 757 4 MP  PP
Xc0
X
X-31 -19 125 -53 30 31 4212 757 4 MP  DP
Xc31
X
X-30 -85 124 -61 31 85 4520 935 4 MP  PP
Xc0
X
X-30 -85 124 -61 31 85 4520 935 4 MP  DP
Xc30
X
X-31 -57 124 -61 31 58 4459 833 4 MP  PP
Xc0
X
X-31 -57 124 -61 31 58 4459 833 4 MP  DP
Xc58
X
X-30 -67 124 -63 31 67 4611 1478 4 MP  PP
Xc0
X
X-30 -67 124 -63 31 67 4611 1478 4 MP  DP
Xc32
X
X-31 -44 125 -61 30 44 4490 891 4 MP  PP
Xc0
X
X-31 -44 125 -61 30 44 4490 891 4 MP  DP
Xc51
X
X-30 -49 124 -63 31 59 4242 788 4 MP  PP
Xc0
X
X-30 -49 124 -63 31 59 4242 788 4 MP  DP
Xc60
X
X-30 -31 124 -61 31 31 5414 1879 4 MP  PP
Xc0
X
X-30 -31 124 -61 31 31 5414 1879 4 MP  DP
Xc61
X
X-31 -24 125 -63 30 24 5229 1899 4 MP  PP
Xc0
X
X-31 -24 125 -63 30 24 5229 1899 4 MP  DP
Xc10
X
X-31 -81 125 -63 30 81 4581 1397 4 MP  PP
Xc0
X
X-31 -81 125 -63 30 81 4581 1397 4 MP  DP
Xc62
X
X-31 -28 124 -61 31 28 5043 1915 4 MP  PP
Xc0
X
X-31 -28 124 -61 31 28 5043 1915 4 MP  DP
Xcolortable /c64 { 1.000000 0.380952 0.000000 sc} put
Xc64
X
X-31 -18 124 -64 31 18 4857 1942 4 MP  PP
Xc0
X
X-31 -18 124 -64 31 18 4857 1942 4 MP  DP
Xc30
X
X-31 -27 124 -63 31 27 4273 847 4 MP  PP
Xc0
X
X-31 -27 124 -63 31 27 4273 847 4 MP  DP
Xc62
X
X-31 -33 124 -61 31 33 5445 1910 4 MP  PP
Xc0
X
X-31 -33 124 -61 31 33 5445 1910 4 MP  DP
Xc12
X
X-31 -77 124 -63 31 77 4550 1320 4 MP  PP
Xc0
X
X-31 -77 124 -63 31 77 4550 1320 4 MP  DP
Xc62
X
X-30 -19 124 -63 31 19 5259 1923 4 MP  PP
Xc0
X
X-30 -19 124 -63 31 19 5259 1923 4 MP  DP
Xc18
X
X-31 -47 125 -64 30 48 4458 1150 4 MP  PP
Xc0
X
X-31 -47 125 -64 30 48 4458 1150 4 MP  DP
Xc59
X
X-31 -67 124 -63 31 67 4427 1083 4 MP  PP
Xc0
X
X-31 -67 124 -63 31 67 4427 1083 4 MP  DP
Xc63
X
X-31 -17 125 -61 30 17 5074 1943 4 MP  PP
Xc0
X
X-31 -17 125 -61 30 17 5074 1943 4 MP  DP
Xc14
X
X-31 -69 124 -63 31 68 4519 1252 4 MP  PP
Xc0
X
X-31 -69 124 -63 31 68 4519 1252 4 MP  DP
Xc38
X
X-31 -22 124 -64 31 23 4304 874 4 MP  PP
Xc0
X
X-31 -22 124 -64 31 23 4304 874 4 MP  DP
Xc64
X
X-31 -19 124 -63 31 18 4888 1960 4 MP  PP
Xc0
X
X-31 -19 124 -63 31 18 4888 1960 4 MP  DP
Xc4
X
X-30 -54 124 -64 31 54 4488 1198 4 MP  PP
Xc0
X
X-30 -54 124 -64 31 54 4488 1198 4 MP  DP
Xc39
X
X-31 -85 124 -63 31 85 4396 998 4 MP  PP
Xc0
X
X-31 -85 124 -63 31 85 4396 998 4 MP  DP
Xc32
X
X-31 -58 125 -63 30 57 4335 897 4 MP  PP
Xc0
X
X-31 -58 125 -63 30 57 4335 897 4 MP  DP
Xc31
X
X-30 -44 124 -63 31 44 4365 954 4 MP  PP
Xc0
X
X-30 -44 124 -63 31 44 4365 954 4 MP  DP
Xc62
X
X-31 -31 124 -63 31 31 5290 1942 4 MP  PP
Xc0
X
X-31 -31 124 -63 31 31 5290 1942 4 MP  DP
Xc63
X
X-30 -24 124 -61 31 24 5104 1960 4 MP  PP
Xc0
X
X-30 -24 124 -61 31 24 5104 1960 4 MP  DP
Xc64
X
X-31 -28 125 -64 30 29 4919 1978 4 MP  PP
Xc0
X
X-31 -28 125 -64 30 29 4919 1978 4 MP  DP
Xc64
X
X-31 -33 124 -64 31 34 5321 1973 4 MP  PP
Xc0
X
X-31 -33 124 -64 31 34 5321 1973 4 MP  DP
Xc64
X
X-31 -19 124 -61 31 19 5135 1984 4 MP  PP
Xc0
X
X-31 -19 124 -61 31 19 5135 1984 4 MP  DP
Xcolortable /c65 { 1.000000 0.285714 0.000000 sc} put
Xc65
X
X-30 -17 124 -64 31 17 4949 2007 4 MP  PP
Xc0
X
X-30 -17 124 -64 31 17 4949 2007 4 MP  DP
Xc64
X
X-31 -31 124 -61 31 31 5166 2003 4 MP  PP
Xc0
X
X-31 -31 124 -61 31 31 5166 2003 4 MP  DP
Xc65
X
X-31 -24 124 -63 31 23 4980 2024 4 MP  PP
Xc0
X
X-31 -24 124 -63 31 23 4980 2024 4 MP  DP
Xc65
X
X-31 -34 125 -60 30 33 5197 2034 4 MP  PP
Xc0
X
X-31 -34 125 -60 30 33 5197 2034 4 MP  DP
Xcolortable /c66 { 1.000000 0.190476 0.000000 sc} put
Xc66
X
X-31 -19 124 -63 31 19 5011 2047 4 MP  PP
Xc0
X
X-31 -19 124 -63 31 19 5011 2047 4 MP  DP
Xc66
X
X-31 -31 125 -64 30 32 5042 2066 4 MP  PP
Xc0
X
X-31 -31 125 -64 30 32 5042 2066 4 MP  DP
Xcolortable /c67 { 1.000000 0.095238 0.000000 sc} put
Xc67
X
X-30 -33 124 -64 31 33 5072 2098 4 MP  PP
Xc0
X
X-30 -33 124 -64 31 33 5072 2098 4 MP  DP
Xgr 
XDO
XSO
X1851 4614 mt 3094 4274 L
X 898 4171 mt 1851 4614 L
X 898 4171 mt  898 3204 L
X2162 4529 mt 2191 4543 L
X2216 4710 mt (5) s
X2550 4423 mt 2580 4436 L
X2604 4604 mt (10) s
X2939 4317 mt 2968 4330 L
X2992 4498 mt (15) s
X1175 4300 mt 1144 4308 L
X 932 4471 mt (10) s
X1482 4443 mt 1451 4451 L
X1239 4614 mt (20) s
X1790 4585 mt 1759 4594 L
X1547 4757 mt (30) s
X 898 4023 mt  869 4010 L
X 695 4060 mt (-4) s
X 898 3836 mt  869 3823 L
X 695 3873 mt (-2) s
X 898 3649 mt  869 3635 L
X 751 3686 mt (0) s
X 898 3462 mt  869 3448 L
X 751 3499 mt (2) s
X 898 3274 mt  869 3261 L
X 751 3311 mt (4) s
X1413 2731 mt (LSQR solutions) s
Xgs 898 2864 2197 1751 MR c np
Xc13
X
X-30 808 77 -295 31 4 2063 3314 4 MP  PP
Xc0
X
X-30 808 77 -295 31 4 2063 3314 4 MP  DP
Xc12
X
X-31 -4 78 -28 31 23 1985 3323 4 MP  PP
Xc0
X
X-31 -4 78 -28 31 23 1985 3323 4 MP  DP
Xc15
X
X-31 -32 78 279 30 -2 2217 3358 4 MP  PP
Xc0
X
X-31 -32 78 279 30 -2 2217 3358 4 MP  DP
Xc12
X
X-30 -206 77 245 31 3 2186 3355 4 MP  PP
Xc0
X
X-30 -206 77 245 31 3 2186 3355 4 MP  DP
Xc13
X
X-30 -13 77 -31 31 16 2016 3346 4 MP  PP
Xc0
X
X-30 -13 77 -31 31 16 2016 3346 4 MP  DP
Xc12
X
X-31 -23 78 -21 30 23 1908 3344 4 MP  PP
Xc0
X
X-31 -23 78 -21 30 23 1908 3344 4 MP  DP
Xc13
X
X-31 -14 77 -27 31 10 2047 3362 4 MP  PP
Xc0
X
X-31 -14 77 -27 31 10 2047 3362 4 MP  DP
Xc13
X
X-31 -16 78 -22 31 17 1938 3367 4 MP  PP
Xc0
X
X-31 -16 78 -22 31 17 1938 3367 4 MP  DP
Xc12
X
X-31 -294 78 42 31 10 2155 3345 4 MP  PP
Xc0
X
X-31 -294 78 42 31 10 2155 3345 4 MP  DP
Xc11
X
X-31 136 78 146 31 -3 2247 3356 4 MP  PP
Xc0
X
X-31 136 78 146 31 -3 2247 3356 4 MP  DP
Xc13
X
X-30 -23 77 -24 31 12 1830 3379 4 MP  PP
Xc0
X
X-30 -23 77 -24 31 12 1830 3379 4 MP  DP
Xc12
X
X-31 -10 78 -20 30 3 2078 3372 4 MP  PP
Xc0
X
X-31 -10 78 -20 30 3 2078 3372 4 MP  DP
Xc13
X
X-31 -10 78 -21 31 9 1969 3384 4 MP  PP
Xc0
X
X-31 -10 78 -21 31 9 1969 3384 4 MP  DP
Xc12
X
X-31 130 78 -438 30 13 2094 3318 4 MP  PP
Xc0
X
X-31 130 78 -438 30 13 2094 3318 4 MP  DP
Xc13
X
X-31 -17 78 -17 30 10 1861 3391 4 MP  PP
Xc0
X
X-31 -17 78 -17 30 10 1861 3391 4 MP  DP
Xc13
X
X-31 -12 78 -22 31 13 1752 3400 4 MP  PP
Xc0
X
X-31 -12 78 -22 31 13 1752 3400 4 MP  DP
Xc12
X
X-31 -210 78 -242 31 14 2124 3331 4 MP  PP
Xc0
X
X-31 -210 78 -242 31 14 2124 3331 4 MP  DP
Xc12
X
X-31 -3 78 -15 31 -2 2108 3375 4 MP  PP
Xc0
X
X-31 -3 78 -15 31 -2 2108 3375 4 MP  DP
Xc12
X
X-30 -3 77 -21 31 3 2000 3393 4 MP  PP
Xc0
X
X-30 -3 77 -21 31 3 2000 3393 4 MP  DP
Xc12
X
X-31 -9 78 -14 31 6 1891 3401 4 MP  PP
Xc0
X
X-31 -9 78 -14 31 6 1891 3401 4 MP  DP
Xc13
X
X-30 -10 77 -21 31 9 1783 3413 4 MP  PP
Xc0
X
X-30 -10 77 -21 31 9 1783 3413 4 MP  DP
Xc15
X
X-30 2 77 -15 31 -2 2139 3373 4 MP  PP
Xc0
X
X-30 2 77 -15 31 -2 2139 3373 4 MP  DP
Xc13
X
X-31 -13 78 -21 30 13 1675 3421 4 MP  PP
Xc0
X
X-31 -13 78 -21 30 13 1675 3421 4 MP  DP
Xc12
X
X-31 2 78 -21 30 -2 2031 3396 4 MP  PP
Xc0
X
X-31 2 78 -21 30 -2 2031 3396 4 MP  DP
Xc12
X
X-31 -3 78 -15 31 4 1922 3407 4 MP  PP
Xc0
X
X-31 -3 78 -15 31 4 1922 3407 4 MP  DP
Xc16
X
X-31 216 78 -70 31 0 2278 3353 4 MP  PP
Xc0
X
X-31 216 78 -70 31 0 2278 3353 4 MP  DP
Xc12
X
X-31 -6 77 -21 31 6 1814 3422 4 MP  PP
Xc0
X
X-31 -6 77 -21 31 6 1814 3422 4 MP  DP
Xc13
X
X-31 -9 78 -21 31 9 1705 3434 4 MP  PP
Xc0
X
X-31 -9 78 -21 31 9 1705 3434 4 MP  DP
Xc11
X
X-31 3 77 -17 31 -1 2170 3371 4 MP  PP
Xc0
X
X-31 3 77 -17 31 -1 2170 3371 4 MP  DP
Xc13
X
X-30 -13 77 -23 31 8 1597 3449 4 MP  PP
Xc0
X
X-30 -13 77 -23 31 8 1597 3449 4 MP  DP
Xc15
X
X-31 2 78 -21 31 -2 2061 3394 4 MP  PP
Xc0
X
X-31 2 78 -21 31 -2 2061 3394 4 MP  DP
Xc15
X
X-30 2 77 -19 31 2 1953 3411 4 MP  PP
Xc0
X
X-30 2 77 -19 31 2 1953 3411 4 MP  DP
Xc12
X
X-31 -4 78 -21 30 4 1845 3428 4 MP  PP
Xc0
X
X-31 -4 78 -21 30 4 1845 3428 4 MP  DP
Xc12
X
X-31 -6 78 -21 31 6 1736 3443 4 MP  PP
Xc0
X
X-31 -6 78 -21 31 6 1736 3443 4 MP  DP
Xc13
X
X-31 -9 78 -20 30 6 1628 3457 4 MP  PP
Xc0
X
X-31 -9 78 -20 30 6 1628 3457 4 MP  DP
Xc16
X
X-31 0 78 -22 30 5 2201 3370 4 MP  PP
Xc0
X
X-31 0 78 -22 30 5 2201 3370 4 MP  DP
Xc13
X
X-31 -8 78 -21 31 8 1519 3470 4 MP  PP
Xc0
X
X-31 -8 78 -21 31 8 1519 3470 4 MP  DP
Xc11
X
X-31 1 78 -22 31 0 2092 3392 4 MP  PP
Xc0
X
X-31 1 78 -22 31 0 2092 3392 4 MP  DP
Xc11
X
X-31 2 78 -23 30 2 1984 3413 4 MP  PP
Xc0
X
X-31 2 78 -23 30 2 1984 3413 4 MP  DP
Xc15
X
X-31 -2 78 -21 31 2 1875 3432 4 MP  PP
Xc0
X
X-31 -2 78 -21 31 2 1875 3432 4 MP  DP
Xc12
X
X-30 -4 77 -21 31 4 1767 3449 4 MP  PP
Xc0
X
X-30 -4 77 -21 31 4 1767 3449 4 MP  DP
Xc12
X
X-31 -6 78 -18 31 4 1658 3463 4 MP  PP
Xc0
X
X-31 -6 78 -18 31 4 1658 3463 4 MP  DP
Xc14
X
X-31 -6 78 -27 31 11 2231 3375 4 MP  PP
Xc0
X
X-31 -6 78 -27 31 11 2231 3375 4 MP  DP
Xc14
X
X-30 172 77 -248 31 6 2309 3353 4 MP  PP
Xc0
X
X-30 172 77 -248 31 6 2309 3353 4 MP  DP
Xc13
X
X-30 -6 77 -21 31 6 1550 3478 4 MP  PP
Xc0
X
X-30 -6 77 -21 31 6 1550 3478 4 MP  DP
Xc16
X
X-31 -20 78 212 30 19 2463 3441 4 MP  PP
Xc0
X
X-31 -20 78 212 30 19 2463 3441 4 MP  DP
Xc16
X
X-30 -5 77 -21 31 4 2123 3392 4 MP  PP
Xc0
X
X-30 -5 77 -21 31 4 2123 3392 4 MP  DP
Xc16
X
X-30 -182 77 211 31 21 2432 3420 4 MP  PP
Xc0
X
X-30 -182 77 211 31 21 2432 3420 4 MP  DP
Xc11
X
X-31 0 78 -26 31 3 2014 3415 4 MP  PP
Xc0
X
X-31 0 78 -26 31 3 2014 3415 4 MP  DP
Xc13
X
X-31 -8 78 -21 30 7 1442 3492 4 MP  PP
Xc0
X
X-31 -8 78 -21 30 7 1442 3492 4 MP  DP
Xc11
X
X-30 -2 77 -21 31 2 1906 3434 4 MP  PP
Xc0
X
X-30 -2 77 -21 31 2 1906 3434 4 MP  DP
Xc15
X
X-31 -2 78 -21 30 2 1798 3453 4 MP  PP
Xc0
X
X-31 -2 78 -21 30 2 1798 3453 4 MP  DP
Xc14
X
X-30 -16 77 -27 31 16 2262 3386 4 MP  PP
Xc0
X
X-30 -16 77 -27 31 16 2262 3386 4 MP  DP
Xc12
X
X-31 -4 78 -18 31 4 1689 3467 4 MP  PP
Xc0
X
X-31 -4 78 -18 31 4 1689 3467 4 MP  DP
Xc12
X
X-31 -4 77 -21 31 4 1581 3484 4 MP  PP
Xc0
X
X-31 -4 77 -21 31 4 1581 3484 4 MP  DP
Xc14
X
X-31 -11 78 -21 30 11 2154 3396 4 MP  PP
Xc0
X
X-31 -11 78 -21 30 11 2154 3396 4 MP  DP
Xc13
X
X-31 -6 78 -21 31 6 1472 3499 4 MP  PP
Xc0
X
X-31 -6 78 -21 31 6 1472 3499 4 MP  DP
Xc16
X
X-31 -4 78 -27 31 5 2045 3418 4 MP  PP
Xc0
X
X-31 -4 78 -27 31 5 2045 3418 4 MP  DP
Xc14
X
X-31 -234 78 50 31 24 2401 3396 4 MP  PP
Xc0
X
X-31 -234 78 50 31 24 2401 3396 4 MP  DP
Xc11
X
X-31 -3 77 -21 31 3 1937 3436 4 MP  PP
Xc0
X
X-31 -3 77 -21 31 3 1937 3436 4 MP  DP
Xc14
X
X-31 -21 77 -25 31 19 2293 3402 4 MP  PP
Xc0
X
X-31 -21 77 -25 31 19 2293 3402 4 MP  DP
Xc13
X
X-30 -7 77 -14 31 9 1364 3504 4 MP  PP
Xc0
X
X-30 -7 77 -14 31 9 1364 3504 4 MP  DP
Xc16
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Xc11
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Xc11
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X-31 -27 78 -34 31 30 1869 3769 4 MP  DP
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X-30 -31 77 -35 31 28 1806 3949 4 MP  DP
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X-30 -24 77 -38 31 19 1790 4046 4 MP  DP
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X-31 -21 78 -34 30 17 1821 4065 4 MP  PP
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X-31 -21 78 -34 30 17 1821 4065 4 MP  DP
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XDO
XSO
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X4026 4171 mt 4026 3204 L
X5290 4529 mt 5319 4543 L
X5343 4710 mt (5) s
X5678 4423 mt 5707 4436 L
X5731 4604 mt (10) s
X6066 4317 mt 6095 4330 L
X6119 4498 mt (15) s
X4303 4300 mt 4272 4308 L
X4060 4471 mt (10) s
X4610 4443 mt 4579 4451 L
X4367 4614 mt (20) s
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X4674 4757 mt (30) s
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X3730 4064 mt (-30) s
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X3730 3791 mt (-20) s
X4026 3481 mt 3997 3468 L
X3730 3518 mt (-10) s
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X59 178 1772 2 v 59 306 a Fo([48])21 b(P)l(.)d(C.)f(Hansen)h(&)g(D.)f(P)
Xl(.)g(O'Leary)l(,)i Fg(The)f(use)g(of)h(the)g(L-curve)f(in)g(the)h(r)n
X(e)n(gularization)f(of)h(dis-)152 362 y(cr)n(ete)h(il)r(l-p)n(ose)n(d)f
X(pr)n(oblems)p Fo(,)g(Rep)q(ort)g(CS-TR-2781,)g(Dept.)f(of)h(Computer)f
X(Science,)k(Univ.)d(of)152 419 y(Maryland,)c(1991;)f(to)h(app)q(ear)g
X(in)h(SIAM)g(J.)f(Sci.)h(Comput.)59 512 y([49])21 b(S.)12
Xb(C.)g(Eisenstat,)g(P)l(.)g(C.)f(Hansen,)i(D.)f(P)l(.)f(O'Leary)i(&)f
X(G.)f(W.)h(Stew)o(art,)f Fg(R)n(e)n(gularizing)h(c)n(onjugate)152
X569 y(gr)n(adient)k(iter)n(ations)g(for)h(solving)e(discr)n(ete)g(il)r
X(l-p)n(ose)n(d)h(pr)n(oblems)p Fo(,)e(w)o(ork)h(in)h(progress.)59
X663 y([50])21 b(P)l(.)g(C.)f(Hansen,)i(T.)e(Sekii)i(&)f(H.)f
X(Shibahashi,)j Fg(The)e(mo)n(di\014e)n(d)g(trunc)n(ate)n(d)g(SVD)g
X(metho)n(d)g(for)152 719 y(r)n(e)n(gularization)16 b(is)g(gener)n(al)f
X(form)p Fo(,)g(SIAM)h(J.)f(Sci.)h(Stat.)e(Comput.)g Fp(13)i
XFo(\(1992\),)d(1142-1150.)59 813 y([51])21 b(B.)g(Hofmann,)h
XFg(R)n(e)n(gularization)f(for)h(Applie)n(d)f(Inverse)f(and)i(Il)r
X(l-Pose)n(d)e(Pr)n(oblems)p Fo(,)h(T)l(eubner,)152 869
Xy(Leipzig,)c(1986.)59 963 y([52])k(T.)d(Kitaga)o(w)o(a,)f
XFg(A)i(deterministic)f(appr)n(o)n(ach)h(to)h(optimal)f(r)n(e)n
X(gularization|the)g(\014nite)f(dimen-)152 1019 y(sional)e(c)n(ase)p
XFo(,)e(Japan)h(J.)h(Appl.)g(Math.)e Fp(4)h Fo(\(1987\),)e(371{391.)59
X1113 y([53])21 b(R.)16 b(Kress,)f Fg(Line)n(ar)g(Inte)n(gr)n(al)g
X(Equations)p Fo(,)f(Springer,)i(Berlin,)g(1989.)59 1207
Xy([54])21 b(C.)14 b(L.)h(La)o(wson)f(&)h(R.)g(J.)f(Hanson,)g
XFg(Solving)h(L)n(e)n(ast)f(Squar)n(es)h(Pr)n(oblems)p
XFo(,)f(Pren)o(tice-Hall,)i(Engle-)152 1263 y(w)o(o)q(o)q(d)f(Cli\013s,)
Xh(1974.)59 1357 y([55])21 b(P)l(.)h(Linz,)j Fg(Unc)n(ertainty)d(in)g
X(the)g(solution)h(of)f(line)n(ar)g(op)n(er)n(ator)h(e)n(quations)p
XFo(,)g(BIT)g Fp(24)f Fo(\(1984\),)152 1413 y(92{101.)59
X1507 y([56])f Fg(Matlab)c(R)n(efer)n(enc)n(e)d(Guide)p
XFo(,)i(The)f(MathW)l(orks,)f(Mass.,)f(1992.)59 1601 y([57])21
Xb(K.)13 b(Miller,)h Fg(L)n(e)n(ast)e(squar)n(es)i(metho)n(ds)f(for)i
X(il)r(l-p)n(ose)n(d)e(pr)n(oblems)g(with)h(a)g(pr)n(escrib)n(e)n(d)f(b)
Xn(ound)p Fo(,)f(SIAM)152 1657 y(J.)k(Math.)e(Anal.)h
XFp(1)g Fo(\(1970\),)f(52{74.)59 1751 y([58])21 b(V.)g(A.)g(Morozo)o(v,)
Xg Fg(Metho)n(ds)h(for)g(Solving)f(Inc)n(orr)n(e)n(ctly)f(Pose)n(d)h(Pr)
Xn(oblems)p Fo(,)h(Springer)g(V)l(erlag,)152 1807 y(New)16
Xb(Y)l(ork,)e(1984.)59 1901 y([59])21 b(F.)14 b(Natterer,)g
XFg(The)h(Mathematics)h(of)g(Computerize)n(d)g(T)m(omo)n(gr)n(aphy)p
XFo(,)e(John)h(Wiley)l(,)h(New)e(Y)l(ork,)152 1958 y(1986.)59
X2051 y([60])21 b(F.)15 b(Natterer,)f Fg(Numeric)n(al)i(tr)n(e)n(atment)
Xg(of)g(il)r(l-p)n(ose)n(d)g(pr)n(oblems)p Fo(,)e(in)i([67)o(].)59
X2145 y([61])21 b(D.)12 b(P)l(.)f(O'Leary)h(&)g(J.)f(A.)h(Simmons,)g
XFg(A)h(bidiagonalization-r)n(e)n(gularization)f(pr)n(o)n(c)n(e)n(dur)n
X(e)h(for)g(lar)n(ge)152 2202 y(sc)n(ale)h(discr)n(etizations)g(of)h(il)
Xr(l-p)n(ose)n(d)g(pr)n(oblems)p Fo(,)e(SIAM)h(J.)g(Sci.)g(Stat.)f
X(Comput.)g Fp(2)h Fo(\(1981\),)e(474{)152 2258 y(489.)59
X2352 y([62])21 b(C.)16 b(C.)g(P)o(aige)g(&)g(M.)g(A.)f(Saunders,)i
XFg(LSQR:)f(an)h(algorithm)h(for)f(sp)n(arse)g(line)n(ar)f(e)n(quations)
Xh(and)152 2408 y(sp)n(arse)f(le)n(ast)g(squar)n(es)p
XFo(,)e(A)o(CM)g(T)l(rans.)h(Math.)f(Soft)o(w)o(are)g
XFp(8)h Fo(\(1982\),)e(43{71.)59 2502 y([63])21 b(R.)d(L.)g(P)o(ark)o
X(er,)g Fg(Understanding)f(inverse)h(the)n(ory)p Fo(,)g(Ann.)g(Rev.)h
X(Earth)e(Planet)h(Sci.)h Fp(5)f Fo(\(1977\),)152 2558
Xy(35{64.)59 2652 y([64])j(D.)15 b(L.)g(Phillips,)i Fg(A)f(te)n(chnique)
Xf(for)h(the)h(numeric)n(al)e(solution)h(of)g(c)n(ertain)f(inte)n(gr)n
X(al)g(e)n(quations)h(of)152 2708 y(the)h(\014rst)f(kind)p
XFo(,)e(J.)h(A)o(CM)g Fp(9)g Fo(\(1962\),)e(84{97.)59
X2802 y([65])21 b(F.)h(San)o(tosa,)h(Y.-H.)f(P)o(ao,)h(W.)f(W.)g(Symes)g
X(&)h(C.)f(Holland)i(\(Eds.\),)f Fg(Inverse)e(Pr)n(oblems)i(of)152
X2859 y(A)n(c)n(oustic)16 b(and)g(Elastic)g(Waves)p Fo(,)f(SIAM,)g
X(Philadelphia,)j(1984.)p eop
X%%Page: 109 110
X109 109 bop 59 166 a Fm(BIBLIOGRAPHY)1343 b Fo(109)p
X59 178 1772 2 v 59 306 a([66])21 b(C.)13 b(Ra)o(y)g(Smith)h(&)f(W.)g
X(T.)f(Grandy)l(,)h(Jr.)g(\(Eds.\),)f Fg(Maximum-Entr)n(opy)k(and)e
X(Bayesian)g(Metho)n(ds)152 362 y(in)i(Inverse)f(Pr)n(oblems)p
XFo(,)f(Reidel,)j(Boston,)d(1985.)59 457 y([67])21 b(G.)c(T)l(alen)o
X(ti,)h Fg(Inverse)f(Pr)n(oblems)p Fo(,)g(Lecture)h(Notes)f(in)h
X(Mathematics)f(1225,)f(Springer)j(V)l(erlag,)152 514
Xy(Berlin,)e(1986.)59 609 y([68])k(H.)12 b(J.)f(J.)h(te)f(Riele,)j
XFg(A)f(pr)n(o)n(gr)n(am)g(for)h(solving)d(\014rst)i(kind)f(F)m(r)n(e)n
X(dholm)g(inte)n(gr)n(al)g(e)n(quations)h(by)g(me)n(ans)152
X665 y(of)k(r)n(e)n(gularization)p Fo(,)d(Computer)h(Ph)o(ysics)h(Comm.)
Xe Fp(36)h Fo(\(1985\),)e(423{432.)59 760 y([69])21 b(A.)e(N.)g(Tikhono)
Xo(v,)h Fg(Solution)f(of)h(inc)n(orr)n(e)n(ctly)f(formulate)n(d)h(pr)n
X(oblems)f(and)h(the)g(r)n(e)n(gularization)152 816 y(metho)n(d)p
XFo(,)13 b(Dokl.)e(Ak)m(ad.)g(Nauk.)h(SSSR)g Fp(151)g
XFo(\(1963\),)e(501{504)f(=)j(So)o(viet)g(Math.)e(Dokl.)h
XFp(4)h Fo(\(1963\),)152 873 y(1035{1038.)59 968 y([70])21
Xb(A.)16 b(N.)g(Tikhono)o(v)g(&)g(V.)f(Y.)h(Arsenin,)h
XFg(Solutions)f(of)h(Il)r(l-Pose)n(d)f(Pr)n(oblems)p Fo(,)f(Winston)h(&)
Xg(Sons,)152 1024 y(W)l(ashington,)f(D.C.,)f(1977.)59
X1119 y([71])21 b(A.)f(N.)g(Tikhono)o(v)g(&)h(A.)f(V.)g(Gonc)o(harsky)l
X(,)g Fg(Il)r(l-Pose)n(d)g(Pr)n(oblems)g(in)g(the)h(Natur)n(al)g(Scienc)
Xn(es)p Fo(,)152 1176 y(MIR)16 b(Publishers,)h(Mosco)o(w,)c(1987.)59
X1271 y([72])21 b(A.)15 b(v)m(an)h(der)g(Sluis,)g Fg(The)g(c)n(onver)n
X(genc)n(e)e(b)n(ehavior)j(of)f(c)n(onjugate)g(gr)n(adients)g(and)g(R)o
X(itz)g(values)g(in)152 1327 y(various)k(cir)n(cumstanc)n(es)p
XFo(;)e(in)h(R.)g(Beau)o(w)o(ens)f(&)h(P)l(.)f(de)h(Gro)q(en)f
X(\(Eds.\),)g Fg(Iter)n(ative)g(Metho)n(ds)h(in)152 1383
Xy(Line)n(ar)d(A)o(lgebr)n(a)p Fo(,)e(North-Holland,)h(Amsterdam,)g
X(1992.)59 1478 y([73])21 b(A.)e(v)m(an)h(der)f(Sluis)i(&)f(H.)e(A.)h(v)
Xm(an)h(der)f(V)l(orst,)g Fg(SIR)m(T-)g(and)h(CG-typ)n(e)g(metho)n(ds)g
X(for)g(iter)n(ative)152 1535 y(solution)15 b(of)h(sp)n(arse)e(line)n
X(ar)g(le)n(ast-squar)n(es)g(pr)n(oblems)p Fo(,)g(Lin.)h(Alg.)f(Appl.)h
XFp(130)f Fo(\(1990\),)e(257{302.)59 1630 y([74])21 b(J.)c(M.)g(V)l
X(arah,)f Fg(On)i(the)g(numeric)n(al)f(solution)h(of)g(il)r(l-c)n
X(onditione)n(d)f(line)n(ar)g(systems)g(with)h(appli-)152
X1686 y(c)n(ations)e(to)h(il)r(l-p)n(ose)n(d)e(pr)n(oblems)p
XFo(,)f(SIAM)i(J.)f(Numer.)g(Anal.)h Fp(10)f Fo(\(1973\),)e(257{267.)59
X1781 y([75])21 b(J.)c(M.)e(V)l(arah,)h Fg(A)h(pr)n(actic)n(al)g
X(examination)g(of)g(some)g(numeric)n(al)g(metho)n(ds)g(for)h(line)n(ar)
Xe(discr)n(ete)152 1838 y(il)r(l-p)n(ose)n(d)g(pr)n(oblems)p
XFo(,)e(SIAM)i(Rev.)f Fp(21)h Fo(\(1979\),)d(100{111.)59
X1932 y([76])21 b(J.)c(M.)e(V)l(arah,)h Fg(Pitfal)r(ls)g(in)h(the)g
X(numeric)n(al)g(solution)g(of)g(line)n(ar)f(il)r(l-p)n(ose)n(d)h(pr)n
X(oblems)p Fo(,)e(SIAM)i(J.)152 1989 y(Sci.)f(Stat.)e(Comput.)h
XFp(4)g Fo(\(1983\),)e(164{176.)59 2084 y([77])21 b(C.)c(R.)g(V)l(ogel,)
Xh Fg(Optimal)g(choic)n(e)g(of)g(a)h(trunc)n(ation)e(level)g(for)i(the)f
X(trunc)n(ate)n(d)g(SVD)g(solution)g(of)152 2140 y(line)n(ar)h(\014rst)g
X(kind)h(inte)n(gr)n(al)e(e)n(quations)h(when)h(data)g(ar)n(e)g(noisy)p
XFo(,)f(SIAM)g(J.)g(Numer.)f(Anal.)h Fp(23)152 2197 y
XFo(\(1986\),)14 b(109{117.)59 2292 y([78])21 b(C.)27
Xb(R.)g(V)l(ogel,)j Fg(Solving)25 b(il)r(l-c)n(onditione)n(d)h(line)n
X(ar)g(systems)g(using)h(the)g(c)n(onjugate)g(gr)n(adient)152
X2348 y(metho)n(d)p Fo(,)16 b(Rep)q(ort,)f(Dept.)g(of)f(Mathematical)i
X(Sciences,)g(Mon)o(tana)e(State)h(Univ)o(ersit)o(y)l(,)g(1987.)59
X2443 y([79])21 b(G.)c(W)l(ah)o(ba,)g Fg(Spline)g(Mo)n(dels)g(for)i
X(Observational)e(Data)p Fo(,)h(CBMS-NSF)g(Regional)g(Conference)152
X2499 y(Series)f(in)f(Applied)h(Mathematics,)d(V)l(ol.)i(59,)e(SIAM,)h
X(Philadelphi)q(a,)i(1990.)59 2594 y([80])k(G.)c(M.)g(Wing,)h
XFg(Condition)g(numb)n(ers)f(of)i(matric)n(es)f(arising)f(fr)n(om)i(the)
Xg(numeric)n(al)f(solution)g(of)152 2651 y(line)n(ar)j(inte)n(gr)n(al)e
X(e)n(quations)i(of)g(the)h(\014rst)e(kind)p Fo(,)h(J.)f(In)o(tegral)h
X(Equations)f Fp(9)h Fo(\(Suppl.\))g(\(1985\),)152 2707
Xy(191{204.)59 2802 y([81])g(G.)13 b(M.)g(Wing)g(&)h(J.)f(D.)g(Zahrt,)f
XFg(A)j(Primer)g(on)f(Inte)n(gr)n(al)f(Equations)h(of)h(the)g(First)f
X(Kind)p Fo(,)f(SIAM,)152 2859 y(Philadelphi)q(a,)k(1991.)p
Xeop
X%%Page: 110 111
X110 110 bop 59 166 a Fo(110)1339 b Fm(BIBLIOGRAPHY)p
X59 178 1772 2 v 59 306 a Fo([82])21 b(H.)16 b(Zha)g(&)h(P)l(.)f(C.)f
X(Hansen,)i Fg(R)n(e)n(gularization)f(and)i(the)f(gener)n(al)f
X(Gauss-Markov)i(line)n(ar)e(mo)n(del)p Fo(,)152 362 y(Math.)f(Comp.)f
XFp(55)h Fo(\(1990\),)f(613{624.)p eop
X%%Trailer
Xend
Xuserdict /end-hook known{end-hook}if
X%%EOF
END_OF_FILE
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sed "s/^X//" >'Contents.m' <<'END_OF_FILE'
X% Regularization Tools.
X% Version 2.1  16-March-93, Revised 31-March-98.
X% Copyright (c) 1993 by Per Christian Hansen and UNI-C.
X%
X% Demonstration.
X%   regudemo  - Tutorial introduction to Regularization Tools.
X%
X% Test examples.
X%   baart     - Fredholm integral equation of the first kind.
X%   deriv2    - Computation of the second derivative.
X%   foxgood   - Severely ill-posed problem.
X%   heat      - Inverse heat equation.
X%   ilaplace  - Inverse Laplace transformation.
X%   parallax  - Stellar parallax problem with 28 fixed observations.
X%   phillips  - Philips' "famous" test problem.
X%   shaw      - One-dimensional image restoration problem.
X%   spikes    - Test problem with a "spiky" solution.
X%   ursell    - Integral equation with no square integrable solution.
X%   wing      - Test problem with a discontinuous solution.
X%
X% Regularization routines.
X%   cgls      - Computes the least squares solution based on k steps
X%               of the conjugate gradient algorithm.
X%   discrep   - Minimizes the solution (semi-)norm subject to an upper
X%               bound on the residual norm (discrepancy principle).
X%   dsvd      - Computes a damped SVD/GSVD solution.
X%   lsqi      - Minimizes the residual norm subject to an upper bound
X%               on the (semi-)norm of the solution.
X%   lsqr      - Computes the least squares solution based on k steps
X%               of the LSQR algorithm (Lanczos bidiagonalization).
X%   maxent    - Computes the maximum entropy regularized solution.
X%   mtsvd     - Computes the modified TSVD solution.
X%   nu        - Computes the solution based on k steps of Brakhage's
X%               iterative nu-method.
X%   pcgls     - Same as cgls, but for general-form regularization.
X%   plsqr     - Same as lsqr, but for general-form regularization.
X%   pnu       - Same as nu, but for general-form regularization.
X%   tgsvd     - Computes the truncated GSVD solution.
X%   tikhonov  - Computes the Tikhonov regularized solution.
X%   tsvd      - Computes the truncated SVD solution.
X%   ttls      - Computes the truncated TLS solution.
X%
X% Analysis routines.
X%   fil_fac   - Computes filter factors for some regularization methods.
X%   gcv       - Plots the GCV function and computes its minimum.
X%   l_corner  - Locates the L-shaped corner of the L-curve.
X%   l_curve   - Computes the L-curve, plots it, and computes its corner.
X%   lagrange  - Plots the Lagrange function ||Ax-b||^2 + lambda^2*||Lx||^,
X%               and its derivative.
X%   picard    - Plots the (generalized) singular values,
X%               the Fourier coefficient for the right-hand side, and a
X%               (smoothed curve of) the solution Fourier-coefficients.
X%   plot_lc   - Plots an L-curve.
X%   quasiopt  - Plots the quasi-optimality function and computes its minimum.
X%
X% Routines for transforming a problem in general form into one in
X% standard form, and back again.
X%   gen_form  - Transforms a standard-form solution back into the
X%               general-form setting.
X%   std_form  - Transforms a general-form problem into one in
X%               standard form.
X%
X% Utility routines.
X%   bidiag    - Bidiagonalization of a matrix by Householder transformations.
X%   bsvd      - Computes the singular values, or the compact SVD,
X%               of a bidiagonal matrix stored in compact form.
X%   csdecomp  - Computes the CS decomposition.
X%   csvd      - Computes the compact SVD of an m-by-n matrix.
X%   get_l     - Produces a p-by-n matrix which is the discrete
X%               approximation to the d'th order derivative operator.
X%   gsvd      - Computes the generalized SVD of a matrix pair.
X%   lanc_b    - Performs k steps of the Lanczos bidiagonalization
X%               process with/without reorthogonalization.
X%
X% Auxiliary routines required by some of the above routines.
X%   app_hh_l  - Applies a Householder transformation from the left.
X%   gen_hh    - Generates a Householder transformation.
X%   heb_new   - Newton-Raphson iteration with Hebden's rational
X%               approximation, used in lsqi.
X%   heb_new2  - Ditto, used in discrep.
X%   lsolve    - Inversion with A-weighted generalized inverse of L.
X%   ltsolve   - Inversion with transposed A-weighted inverse of L.
X%   mgs       - Modified Gram-Schmidt orthonormalization.
X%   newton    - Newton-Raphson iteration, used in discrep.
X%   pinit     - Initialization for treating general-form problems.
X%   pythag    - Computes sqrt(a^2 + b^2).
X%   spleval   - Computes points on a spline or spline curve.
X%
X% The routine l_corner requires the following routines from the Spline
X% Toolbox:
X%   fnder, ppbrk, ppcut, ppmak, sp2pp, spbrk, spmak.
X% If the Spline Toolbox is not available, then dummy functions with these
X% names should reside in the same directory as Regularization Tools.
END_OF_FILE
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if test -f 'app_hh.m' -a "${1}" != "-c" ; then 
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echo shar: Extracting \"'app_hh.m'\" \(304 characters\)
sed "s/^X//" >'app_hh.m' <<'END_OF_FILE'
Xfunction A = app_hh(A,beta,v)
X%APP_HH Apply a Householder transformation.
X%
X% A = app_hh(A,beta,v)
X%
X% Applies the Householder transformation, defined by
X% vector v and scaler beta, to the matrix A; i.e.
X%     A = (eye - beta*v*v')*A .
X
X% Per Christian Hansen, UNI-C, 03/11/92.
X
XA = A - (beta*v)*(v'*A);
END_OF_FILE
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if test -f 'baart.m' -a "${1}" != "-c" ; then 
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echo shar: Extracting \"'baart.m'\" \(1577 characters\)
sed "s/^X//" >'baart.m' <<'END_OF_FILE'
Xfunction [A,b,x] = baart(n)
X%BAART Test problem: Fredholm integral equation of the first kind.
X%
X% [A,b,x] = baart(n)
X%
X% Discretization of a first kind Fredholm integral equation with
X% kernel K and right-hand side g given by
X%    K(s,t) = exp(s*cos(t)) ,  g(s) = 2*sinh(s)/s ,
X% and with integration intervals  s in [0,pi/2] ,  t in [0,pi] .
X% The solution is given by
X%    f(t) = sin(t) .
X%
X% The order n must be even.
X 
X% Reference: M. L. Baart, "The use of auto-correlation for pseudo-
X% rank determination in noisy ill-conditioned linear least-squares
X% problems", IMA J. Numer. Anal. 2 (1982), 241-247.
X
X% Discretized by Galerkin method with orthonormal box functions;
X% one integration is exact, the other is done by Simpson's rule.
X
X% Per Christian Hansen, UNI-C, 09/16/92.
X
X% Check input.
Xif (rem(n,2)~=0), error('The order n must be even'), end
X
X% Generate the matrix.
Xhs = pi/(2*n); ht = pi/n; c = 1/(3*sqrt(2));
XA = zeros(n,n); ihs = [0:n]'*hs; n1 = n+1; nh = n/2;
Xf3 = exp(ihs(2:n1)) - exp(ihs(1:n));
Xfor j=1:n
X  f1 = f3; co2 = cos((j-.5)*ht); co3 = cos(j*ht);
X  f2 = (exp(ihs(2:n1)*co2) - exp(ihs(1:n)*co2))/co2;
X  if (j==nh)
X    f3 = hs*ones(n,1);
X  else
X    f3 = (exp(ihs(2:n1)*co3) - exp(ihs(1:n)*co3))/co3;
X  end
X  A(:,j) = c*(f1 + 4*f2 + f3);
Xend
X
X% Generate the right-hand side.
Xif (nargout>1)
X  si(1:2*n) = [.5:.5:n]'*hs; si = sinh(si)./si;
X  b = zeros(n,1);
X  b(1) = 1 + 4*si(1) + si(2);
X  b(2:n) = si(2:2:2*n-2) + 4*si(3:2:2*n-1) + si(4:2:2*n);
X  b = b*sqrt(hs)/3;
Xend
X
X% Generate the solution.
Xif (nargout==3)
X  x = -diff(cos([0:n]'*ht))/sqrt(ht);
Xend
END_OF_FILE
if test 1577 -ne `wc -c <'baart.m'`; then
    echo shar: \"'baart.m'\" unpacked with wrong size!
fi
# end of 'baart.m'
fi
if test -f 'bidiag.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'bidiag.m'\"
else
echo shar: Extracting \"'bidiag.m'\" \(2059 characters\)
sed "s/^X//" >'bidiag.m' <<'END_OF_FILE'
Xfunction [U,B_n,V] = bidiag(A)
X%BIDIAG Bidiagonalization of an m-times-n matrix with m >= n.
X%
X% B_n = bidiag(A)
X% [U,B_n,V] = bidiag(A)
X% 
X% Computes the bidiagonalization of the m-times-n matrix A
X% with m >= n:
X%     A = U*B*V' ,
X% where B is an upper bidiagonal n-times-n matrix, and U and
X% V have orthogonal columns.
X%
X% The matrix B is stored in compact form in B_n as follows:
X%         [b_11 b_12            ]         [b_11    b_12]
X%     B = [     b_22 b_23       ]   B_n = [b_22    b_23] .
X%         [            .    .   ]         [ .       .  ] 
X%         [                b_nn ]         [b_nn    NaN ]
X% The NaN in B_n(n,2) is used to distinguish a "compact" upper
X% bidiagonal matrix from a "compact" lower bidiagonal one.
X
X% Reference: L. Elden, "Algorithms for regularization of ill-
X% conditioned least-squares problems", BIT 17 (1977), 134-145.
X
X% Per Christian Hansen, UNI-C, 03/11/92.
X
X% Initialization.
X[m,n] = size(A);
Xif (m<n), error('Illegal dimensions of A'), end
XB_n = zeros(n,2);
Xif (nargout> 1), U = [eye(n);zeros(m-n,n)]; betaU = zeros(n,1); end
Xif (nargout==3), V = eye(n); betaV = zeros(n,1); end
X
X% Bidiagonalization; save Householder quantities.
Xif (m > n), k_last = n; else k_last = n-1; end
Xfor k=1:k_last
X
X  [B_n(k,1),beta,A(k:m,k)] = gen_hh(A(k:m,k));
X  if (k < n), A(k:m,k+1:n) = app_hh(A(k:m,k+1:n),beta,A(k:m,k)); end
X  if (nargout>1), betaU(k) = beta; end
X
X  if (k < n-1)
X    [B_n(k,2),beta,v] = gen_hh(A(k,k+1:n)'); A(k,k+1:n) = v';
X    A(k+1:m,k+1:n) = app_hh(A(k+1:m,k+1:n)',beta,A(k,k+1:n)')';
X    if (nargout==3), betaV(k) = beta;, end
X  elseif (k == n-1)
X    B_n(n-1,2) = A(n-1,n);
X  end
X
Xend
X
X% Save bottom element if A is square.
Xif (k_last < n), B_n(n,1) = A(n,n); end
X
X% Put a NaN in bottom element of B_n.
XB_n(n,2) = NaN;
X
X% Compute U if wanted.
Xif (nargout>1)
X  for k=k_last:-1:1
X    U(k:m,k:n) = app_hh(U(k:m,k:n),betaU(k),A(k:m,k));
X  end
Xend
X
X% Compute V if wanted.
Xif (nargout==3)
X  for k=n-2:-1:1
X    V(k+1:n,k:n) = app_hh(V(k+1:n,k:n),betaV(k),A(k,k+1:n)');
X  end
Xend
X
Xif (nargout==1), U = B_n; end
END_OF_FILE
if test 2059 -ne `wc -c <'bidiag.m'`; then
    echo shar: \"'bidiag.m'\" unpacked with wrong size!
fi
# end of 'bidiag.m'
fi
if test -f 'bsvd.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'bsvd.m'\"
else
echo shar: Extracting \"'bsvd.m'\" \(1091 characters\)
sed "s/^X//" >'bsvd.m' <<'END_OF_FILE'
Xfunction [U,s,V] = bsvd(B_k)
X%BSVD SVD of a bidiagonal matrix stored in "compact form".
X%
X% s = bsvd(B_k)
X% [U,s,V] = bsvd(B_k)
X%
X% Computes the singular values, or the compact SVD, of the
X% bidiagonal matrix B stored in compact form in B_k.
X%
X% If the bottom right element of B_k is a NAN, then B_k repre-
X% sents an upper bidiagonal matrix (such as produced by bidiag),
X% stored with its diagonal and upper bidiagonal in the first and
X% second columns of B_k, repsectively.
X%
X% Otherwise, B_k represents a lower bidiagonal matrix (such as
X% produced by lanc_b), stored with its lower bidiagonal and its
X% diagonal in the first and second columns of B_k, respectively.
X
X% Per Christian Hansen, UNI-C, 03/11/92.
X
X% Initialization.
X[k,l] = size(B_k);
Xif (l~=2), error('B_k does not represent a bidiagonal matrix'), end
X
X% Determine which bidiagonal form.
Xif (B_k(k,2)==NaN)
X  B = diag(B_k(:,1)) + diag(B_k(1:k-1,2),1);
Xelse
X  B = diag(B_k(:,1),-1) + diag([B_k(:,2);0]);
X  [k1,k1] = size(B); B = B(:,1:k1-1);
Xend
X
X% Compute the SVD.
Xif (nargout<=1)
X  U = svd(B);
Xelse
X  [U,s,V] = csvd(B);
Xend
END_OF_FILE
if test 1091 -ne `wc -c <'bsvd.m'`; then
    echo shar: \"'bsvd.m'\" unpacked with wrong size!
fi
# end of 'bsvd.m'
fi
if test -f 'cgls.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'cgls.m'\"
else
echo shar: Extracting \"'cgls.m'\" \(2084 characters\)
sed "s/^X//" >'cgls.m' <<'END_OF_FILE'
Xfunction [X,rho,eta,F] = cgls(A,b,k,s)
X%CGLS Conjugate gradient algorithm applied implicitly to the normal equations.
X%
X% [X,rho,eta,F] = cgls(A,b,k,s)
X%
X% Performs k steps of the conjugate gradient algorithm applied
X% implicitly to the normal equations A'*A*x = A'*b.
X%
X% The routine returns all k solutions, stored as columns of
X% the matrix X.  The solution norm and residual norm are returned
X% in eta and rho, respectively.
X%
X% If the singular values s are also provided, cgls computes the
X% filter factors associated with each step and stores them
X% columnwise in the matrix F.
X
X% References: A. Bjorck, "Least Squares Methods", in P. G.
X% Ciarlet & J. L Lions (Eds.), "Handbook of  Numerical Analysis,
X% Vol. I", Elsevier, Amsterdam, 1990; p. 560.
X% C. R. Vogel, "Solving ill-conditioned linear systems using the
X% conjugate gradient method", Report, Dept. of Mathematical
X% Sciences, Montana State University, 1987.
X 
X% Per Christian Hansen, UNI-C, 03/31/98.
X
X% The fudge threshold is used to prevent filter factors from exploding.
Xfudge_thr = 1e-4;
X 
X% Initialization.
Xif (k < 1), error('Number of steps k must be positive'), end
X[m,n] = size(A); X = zeros(n,k);
Xif (nargout > 1)
X  eta = zeros(k,1); rho = eta;
Xend
Xif (nargout==4 & nargin==3), error('Too few imput arguments'), end
Xif (nargin==4)
X  F = zeros(n,k); Fd = zeros(n,1); s2 = s.^2;
Xend
X
X% Prepare for CG iteration.
Xx = zeros(n,1);
Xd = A'*b;
Xr = b;
Xnormr2 = d'*d;
X
X% Iterate.
Xfor j=1:k
X
X  Ad = A*d; alpha = normr2/(Ad'*Ad);
X  x  = x + alpha*d;
X  r  = r - alpha*Ad;
X  s  = A'*r;
X  normr2_new = s'*s;
X  beta = normr2_new/normr2;
X  normr2 = normr2_new;
X  d = s + beta*d;
X  X(:,j) = x;
X  if (nargout>1), rho(j) = norm(r); end
X  if (nargout>2), eta(j) = norm(x); end
X
X  if (nargin==4)
X    if (j==1)
X      F(:,1) = alpha*s2;
X      Fd = s2 - s2.*F(:,1) + beta*s2;
X    else
X      F(:,j) = F(:,j-1) + alpha*Fd;
X      Fd = s2 - s2.*F(:,j) + beta*Fd;
X    end
X    if (j > 2)
X      f = find(abs(F(:,j-1)-1) < fudge_thr & abs(F(:,j-2)-1) < fudge_thr);
X      if (length(f) > 0), F(f,j) = ones(length(f),1); end
X    end
X  end
X
Xend
END_OF_FILE
if test 2084 -ne `wc -c <'cgls.m'`; then
    echo shar: \"'cgls.m'\" unpacked with wrong size!
fi
# end of 'cgls.m'
fi
if test -f 'csdecomp.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'csdecomp.m'\"
else
echo shar: Extracting \"'csdecomp.m'\" \(1794 characters\)
sed "s/^X//" >'csdecomp.m' <<'END_OF_FILE'
Xfunction [U1,U2,cs,V] = csdecomp(Q1,Q2)
X%CSD CS decomposition.
X% cs = csdecomp(Q1,Q2)
X% [U1,U2,cs,V] = csdecomp(Q1,Q2) ,  cs = [c,s]
X%
X% Computes the CS decomposition
X%    [ Q1 ] = [ U1  0 ]*[ diag(c)     0     ]*V'
X%    [ Q2 ]   [ 0  U2 ] [    0     eye(n-p) ]
X%                       [ diag(s)     0     ]
X% of a matrix with orthonormal columns.  The number of rows in Q1
X% must be greater than or equal to the number of rows in Q2.
X%    U1  is  m-by-n ,   c  is  p-by-1
X%    U2  is  p-by-p ,   s  is  p-by-1
X%    V   is  n-by-n .
X
X% Reference: C. F. Van Loan, "Computing the CS and the generalized
X% singular value decomposition", Numer. Math. 46 (1985), 479-491.
X
X% Per Christian Hansen, IMM, 04/17/97.
X
X% Initialization.
X[m,n1] = size(Q1); [p,n] = size(Q2);
Xif (m<n | n<p | n1~=n)
X  error('Incorrect dimensions of Q1 and Q2')
Xend
Xc = zeros(p,1); s = c; thr = .99;
X
X% Compute SVD of Q2.
X[U2,s,V] = svd(Q2); s = diag(s(1:p,1:p));
Xk = length(find(s <= thr)); pk = p-k;
X
X% Compute U1.
X[U1,R] = qr(Q1*V(:,n:-1:1),0); U1 = U1(:,n:-1:1);
Xfor i=1:n
X  if (R(i,i) < 0)
X    if (i > n-p+k), R(i,:) = -R(i,:); end
X    U1(:,n+1-i) = -U1(:,n+1-i);
X  end
Xend
Xc(p:-1:pk+1) = abs(diag(R(n-p+1:n-pk,n-p+1:n-pk)));
XR = R(n:-1:n-pk+1,n:-1:n-pk+1);
X
X% Compute c, U1 and V.
X[U1t,gamma,Vt] = svd(R);
Xc(pk:-1:1) = diag(gamma); clear gamma;
XU1(:,1:pk) = U1(:,1:pk)*U1t(:,pk:-1:1);
XV(:,1:pk) = V(:,1:pk)*Vt(:,pk:-1:1);
XR = Vt; for i=1:pk, R(i,:) = s(i)*R(i,pk:-1:1); end
X
X% Compute s and U2.
XU2t = zeros(pk,pk);
Xfor i=1:pk,
X  s(i) = norm(R(:,i));
X  U2t(:,i) = R(:,i)/s(i);
Xend
XU2(:,1:pk) = U2(:,1:pk)*U2t;
X
X% Make sure that c and s do not exceed one.
Xix = find(c>1); c(ix) = ones(length(ix),1);
Xix = find(s>1); s(ix) = ones(length(ix),1);
X
X% Return the desired quantities.
Xcs = [c,s];
Xif (nargout < 2), U1 = cs; end
END_OF_FILE
if test 1794 -ne `wc -c <'csdecomp.m'`; then
    echo shar: \"'csdecomp.m'\" unpacked with wrong size!
fi
# end of 'csdecomp.m'
fi
if test -f 'csvd.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'csvd.m'\"
else
echo shar: Extracting \"'csvd.m'\" \(724 characters\)
sed "s/^X//" >'csvd.m' <<'END_OF_FILE'
Xfunction [U,s,V] = csvd(A,tst)
X%CSVD Compact singular value decomposition.
X%
X% s = csvd(A)
X% [U,s,V] = csvd(A)
X% [U,s,V] = csvd(A,'full')
X%
X% Computes the compact form of the SVD of A:
X%    A = U*diag(s)*V',
X% where
X%    U  is  m-by-min(m,n)
X%    s  is  min(m,n)-by-1
X%    V  is  n-by-min(m,n).
X%
X% If a second argument is present, the full U and V are returned.
X
X% Per Christian Hansen, UNI-C, 06/22/93.
X
Xif (nargin==1)
X  if (nargout > 1)
X    [m,n] = size(A);
X    if (m >= n)
X      [U,s,V] = svd(full(A),0); s = diag(s);
X    else
X      [V,s,U] = svd(full(A)',0); s = diag(s);
X    end
X  else
X    U = svd(full(A));
X  end
Xelse
X  if (nargout > 1)
X    [U,s,V] = svd(full(A)); s = diag(s);
X  else
X    U = svd(full(A));
X  end
Xend
END_OF_FILE
if test 724 -ne `wc -c <'csvd.m'`; then
    echo shar: \"'csvd.m'\" unpacked with wrong size!
fi
# end of 'csvd.m'
fi
if test -f 'deriv2.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'deriv2.m'\"
else
echo shar: Extracting \"'deriv2.m'\" \(2671 characters\)
sed "s/^X//" >'deriv2.m' <<'END_OF_FILE'
Xfunction [A,b,x] = deriv2(n,case)
X%DERIV2 Test problem: computation of the second derivative.
X%
X% [A,b,x] = deriv2(n,case)
X%
X% This is a mildly ill-posed problem.  It is a discretization of a
X% first kind Fredholm integral equation whose kernel K is the
X% Green's function for the second derivative:
X%    K(s,t) = | s(t-1)  ,  s <  t .
X%             | t(s-1)  ,  s >= t
X% Both integration intervals are [0,1], and as right-hand side g
X% and correspond solution f one can choose between the following:
X%    case = 1 : g(s) = (s^3 - s)/6          ,  f(t) = t
X%    case = 2 : g(s) = exp(s) + (1-e)s - 1  ,  f(t) = exp(t)
X%    case = 3 : g(s) = | (4s^3 - 3s)/24               ,  s <  0.5
X%                      | (-4s^3 + 12s^2 - 9s + 1)/24  ,  s >= 0.5
X%               f(t) = | t    ,  t <  0.5
X%                      | 1-t  ,  t >= 0.5
X
X% References.  The first two examples are from L. M. Delves & J. L.
X% Mohamed, "Computational Methods for Integral Equations", Cambridge
X% University Press, 1985; p. 310.  The third example is from A. K.
X% Louis & P. Maass, "A mollifier method for linear operator equations
X% of the first kind", Inverse Problems 6 (1990), 427-440.
X
X% Discretized by Galerkin method with orthonormal box functions.
X
X% Per Christian Hansen, UNI-C, 05/28/93.
X
X% Initialization.
Xif (nargin==1), case = 1; end
Xh = 1/n; sqh = sqrt(h); h32 = h*sqh; h2 = h^2; sqhi = 1/sqh;
Xt = 2/3; A = zeros(n,n);
X
X% Compute the matrix A.
Xfor i=1:n
X  A(i,i) = h2*((i^2 - i + 0.25)*h - (i - t));
X  for j=1:i-1
X    A(i,j) = h2*(j-0.5)*((i-0.5)*h-1);
X  end
Xend
XA = A + tril(A,-1)';
X
X% Compute the right-hand side vector b.
Xif (nargout>1)
X  b = zeros(n,1);
X  if (case==1)
X    for i=1:n
X      b(i) = h32*(i-0.5)*((i^2 + (i-1)^2)*h2/2 - 1)/6;
X    end
X  elseif (case==2)
X    ee = 1 - exp(1);
X    for i=1:n
X      b(i) = sqhi*(exp(i*h) - exp((i-1)*h) + ee*(i-0.5)*h2 - h);
X    end
X  elseif (case==3)
X    if (rem(n,2)~=0), error('Order n must be even'), else
X      for i=1:n/2
X        s12 = (i*h)^2; s22 = ((i-1)*h)^2;
X        b(i) = sqhi*(s12 + s22 - 1.5)*(s12 - s22)/24;
X      end
X      for i=n/2+1:n
X        s1 = i*h; s12 = s1^2; s2 = (i-1)*h; s22 = s2^2;
X        b(i) = sqhi*(-(s12+s22)*(s12-s22) + 4*(s1^3 - s2^3) - ...
X                    4.5*(s12 - s22) + h)/24;
X      end
X    end
X  else
X    error('Illegal value of case')
X  end
Xend
X
X% Compute the solution vector x.
Xif (nargout==3)
X  x = zeros(n,1);
X  if (case==1)
X    for i=1:n, x(i) = h32*(i-0.5); end
X  elseif(case==2)
X    for i=1:n, x(i) = sqhi*(exp(i*h) - exp((i-1)*h)); end
X  else
X    for i=1:n/2,   x(i) = sqhi*((i*h)^2 - ((i-1)*h)^2)/2; end
X    for i=n/2+1:n, x(i) = sqhi*(h - ((i*h)^2 - ((i-1)*h)^2)/2); end
X  end
Xend
END_OF_FILE
if test 2671 -ne `wc -c <'deriv2.m'`; then
    echo shar: \"'deriv2.m'\" unpacked with wrong size!
fi
# end of 'deriv2.m'
fi
if test -f 'discrep.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'discrep.m'\"
else
echo shar: Extracting \"'discrep.m'\" \(2756 characters\)
sed "s/^X//" >'discrep.m' <<'END_OF_FILE'
Xfunction [x_delta,lambda] = discrep(U,s,V,b,delta,x_0)
X%DISCREP Discrepancy principle criterion for choosing the reg. parameter.
X%
X% [x_delta,lambda] = discrep(U,s,V,b,delta,x_0)
X% [x_delta,lambda] = discrep(U,sm,X,b,delta,x_0)  ,  sm = [sigma,mu]
X%
X% Least squares minimization with a quadratic inequality constraint:
X%    min || x - x_0 ||       subject to   || A x - b || <= delta
X%    min || L (x - x_0) ||   subject to   || A x - b || <= delta
X% where x_0 is an initial guess of the solution, and delta is a
X% positive constant.  Requires either the compact SVD of A saved as
X% U, s, and V, or part of the GSVD of (A,L) saved as U, sm, and X.
X% The regularization parameter lambda is also returned.
X%
X% If delta is a vector, then x_delta is a matrix such that
X%    x_delta = [ x_delta(1), x_delta(2), ... ] .
X%
X% If x_0 is not specified, x_0 = 0 is used.
X
X% Reference: V. A. Morozov, "Methods for Solving Incorrectly Posed
X% Problems", Springer, 1984; Chapter 26.
X
X% Per Christian Hansen, UNI-C, 02/20/92.
X
X% Initialization.
X[n,p] = size(V);    [p,ps] = size(s);      ld  = length(delta);
Xx_k = zeros(n,ld);  lambda = zeros(ld,1);  rho = zeros(p,1);
Xif (min(delta)<0)
X  error('Illegal inequality constraint delta')
Xend
Xif (nargin==5), x_0 = zeros(n,1); end
Xif (ps == 1), omega = V'*x_0; else, omega = V\x_0; end
X
X% Compute residual norms corresponding to TSVD/TGSVD.
Xbeta = U'*b;
Xnb   = norm(b);
Xsnz  = length(find(s(:,1)>0));
Xif (ps == 1)
X  delta_0 = norm(b - U*beta);
X  rho(n) = delta_0^2;
X  for i=n:-1:2
X    rho(i-1) = rho(i) + (beta(i) - s(i)*omega(i))^2;
X  end
Xelse
X  delta_0 = norm(b - U*beta);
X  rho(1) = delta_0^2;
X  for i=1:p-1
X    rho(i+1) = rho(i) + (beta(i) - s(i,1)*omega(i))^2;
X  end
Xend
X
X% Check input.
Xif (min(delta) < delta_0)
X  error('Irrelevant delta < || (I - U*U'')*b ||')
Xend
X
Xif (ps == 1)
X  s2 = s.^2;
X  for k=1:ld
X    if (delta(k)^2 >= norm(beta - s.*omega)^2 + delta_0^2)
X      x_delta(:,k) = x_0;
X    else
X      [dummy,kmin] = min(abs(rho - delta(k)^2));
X      lambda_0 = s(kmin);
X      lambda(k) = newton(lambda_0,delta(k),s,beta,omega,delta_0);
X      e = s./(s2 + lambda(k)^2); f = s.*e;
X      x_delta(:,k) = V(:,1:p)*(e.*beta + (1-f).*omega);
X    end
X  end
Xelse
X  omega = omega(1:p); gamma = s(:,1)./s(:,2);
X  x_u   = V(:,p+1:n)*beta(p+1:n);
X  for k=1:ld
X    if (delta(k)^2 >= norm(beta(1:p) - s(:,1).*omega)^2 + delta_0^2)
X      x_delta(:,k) = V*[omega;U(:,p+1:n)'*b];
X    else
X      [dummy,kmin] = min(abs(rho - delta(k)^2));
X      lambda_0 = gamma(kmin);
X      lambda(k) = newton(lambda_0,delta(k),s,beta(1:p),omega,delta_0);
X      e = gamma./(gamma.^2 + lambda(k)^2); f = gamma.*e;
X      x_delta(:,k) = V(:,1:p)*(e.*beta(1:p)./s(:,2) + ...
X                               (1-f).*s(:,2).*omega) + x_u;
X    end
X  end
Xend
END_OF_FILE
if test 2756 -ne `wc -c <'discrep.m'`; then
    echo shar: \"'discrep.m'\" unpacked with wrong size!
fi
# end of 'discrep.m'
fi
if test -f 'dsvd.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'dsvd.m'\"
else
echo shar: Extracting \"'dsvd.m'\" \(1192 characters\)
sed "s/^X//" >'dsvd.m' <<'END_OF_FILE'
Xfunction x_lambda = dsvd(U,s,V,b,lambda)
X%DSVD Damped SVD regularization.
X%
X% x_lambda = dsvd(U,s,V,b,lambda)
X% x_lambda = dsvd(U,sm,X,b,lambda) ,  sm = [sigma,mu]
X%
X% Computes the damped SVD solution defined as
X%    x_lambda = V*inv(diag(s + lambda))*U'*b .
X% If lambda is a vector, then x_lambda is a matrix such that
X%    x_lambda = [ x_lambda(1), x_lambda(2), ... ] .
X%
X% If sm and X are specified, then the damped GSVD solution:
X%    x_lambda = X*[ inv(diag(sigma + lambda*mu)) 0 ]*U'*b
X%                 [            0                 I ]
X% is computed.
X
X% Reference: M. P. Ekstrom & R. L. Rhoads, "On the application of
X% eigenvector expansions to numerical deconvolution", J. Comp.
X% Phys. 14 (1974), 319-340.
X
X% Per Christian Hansen, UNI-C, 07/21/90.
X
X% Initialization.
Xif (min(lambda)<0)
X  error('Illegal regularization parameter lambda')
Xend
X[n,pv] = size(V); [p,ps] = size(s);
X
X% Compute x_lambda.
Xbeta = U(:,1:p)'*b;
Xll = length(lambda); x_lambda = zeros(n,ll);
Xif (ps==1)
X  for i=1:ll
X    x_lambda(:,i) = V(:,1:p)*(beta./(s + lambda(i)));
X  end
Xelse
X  x0 = V(:,p+1:n)*U(:,p+1:n)'*b; 
X  for i=1:ll
X    x_lambda(:,i) = V(:,1:p)*(beta./(s(:,1) + lambda(i)*s(:,2))) + x0;
X  end
Xend
END_OF_FILE
if test 1192 -ne `wc -c <'dsvd.m'`; then
    echo shar: \"'dsvd.m'\" unpacked with wrong size!
fi
# end of 'dsvd.m'
fi
if test -f 'fil_fac.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'fil_fac.m'\"
else
echo shar: Extracting \"'fil_fac.m'\" \(2527 characters\)
sed "s/^X//" >'fil_fac.m' <<'END_OF_FILE'
Xfunction f = fil_fac(s,reg_param,method,s1,V1)
X%FIL_FAC Filter factors for some regularization methods.
X%
X% f = fil_fac(s,reg_param,method)
X% f = fil_fac(sm,reg_param,method)  ,  sm = [sigma,mu]
X% f = fil_fac(s1,k,'ttls',s1,V1)
X%
X% Computes all the filter factors corresponding to the
X% singular values in s and the regularization parameter
X% reg_param, for the following methods:
X%    method = 'dsvd' : damped SVD
X%    method = 'tsvd' : truncated SVD
X%    method = 'Tikh' : Tikhonov regularization
X%    method = 'ttls' : truncated TLS.
X% If sm = [sigma,mu] is specified, then the filter factors
X% for the corresponding generalized methods are computed.
X%
X% If method = 'ttls' then the singular values s1 and the
X% right singular matrix V1 of [A,b] must also be supplied.
X%
X% If method is not specified, 'Tikh' is default.
X
X% Per Christian Hansen, UNI-C, 06/23/93.
X
X% Initialization.
X[p,ps] = size(s); lr = length(reg_param);
Xif (nargin==2), method = 'Tikh'; end
Xf = zeros(p,lr);
X
X% Check input data.
Xif (min(reg_param) <= 0)
X  error('Regularization parameter must be positive')
Xend
Xif (method ~= 'Tikh' & min(reg_param) > p)
X  error('Truncation parameter too large')
Xend
X
X% Compute the filter factors.
Xfor j=1:lr
X  if (method(1:2)=='cg' | method(1:2)=='nu' | method(1:2)=='ls')
X    error('Filter factors for iterative methods are not supported')
X  elseif (method(1:4)=='dsvd')
X    if (ps==1)
X      f(:,j) = s./(s + reg_param(j));
X    else
X      f(:,j) = s(:,1)./(s(:,1) + reg_param(j)*s(:,2));
X    end
X  elseif (method(1:4)=='Tikh' | method(1:4)=='tikh')
X    if (ps==1)
X      f(:,j) = (s.^2)./(s.^2 + reg_param(j)^2);
X    else
X      f(:,j) = (s(:,1).^2)./(s(:,1).^2 + reg_param(j)^2*s(:,2).^2);
X    end
X  elseif (method(1:4)=='tsvd' | method(1:4)=='tgsv')
X    if (ps==1)
X      f(:,j) = [ones(reg_param(j),1);zeros(p-reg_param(j),1)];
X    else
X      f(:,j) = [zeros(p-reg_param(j),1);ones(reg_param(j),1)];
X    end
X  elseif (method(1:4)=='ttls')
X    if (ps==1)
X      coef = ((V1(p+1,:).^2)')/norm(V1(p+1,reg_param(j)+1:p+1))^2;
X      for i=1:p
X        k = reg_param(j);
X        f(i,j) = s(i)^2*...
X          sum( coef(k+1:p+1)./(s(i)+s1(k+1:p+1))./(s(i)-s1(k+1:p+1)) );
X        if (f(i,j) < 0), f(i,j) = eps; end
X        if (i > 1)
X          if (f(i-1,j) <= eps & f(i,j) > f(i-1,j)), f(i,j) = f(i-1,j); end
X        end
X      end
X    else
X      error('The SVD of [A,b] must be supplied')
X    end
X  elseif (method(1:4)=='mtsv')
X    error('Filter factors for MTSVD are not supported')
X  else
X    error('Illegal method')
X  end
Xend
END_OF_FILE
if test 2527 -ne `wc -c <'fil_fac.m'`; then
    echo shar: \"'fil_fac.m'\" unpacked with wrong size!
fi
# end of 'fil_fac.m'
fi
if test -f 'fnder.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'fnder.m'\"
else
echo shar: Extracting \"'fnder.m'\" \(79 characters\)
sed "s/^X//" >'fnder.m' <<'END_OF_FILE'
Xfunction fprime=fnder(f,dorder)
X%FNDER Dummy function for Regularization Tools
END_OF_FILE
if test 79 -ne `wc -c <'fnder.m'`; then
    echo shar: \"'fnder.m'\" unpacked with wrong size!
fi
# end of 'fnder.m'
fi
if test -f 'foxgood.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'foxgood.m'\"
else
echo shar: Extracting \"'foxgood.m'\" \(610 characters\)
sed "s/^X//" >'foxgood.m' <<'END_OF_FILE'
Xfunction [A,b,x] = foxgood(n)
X%FOXGOOD Severely ill-posed testproblem.
X%
X% [A,b,x] = foxgood(n)
X%
X% This is a model problem which does not satisfy the
X% discrete Picard condition for the small singular values.
X% The problem was first used by Fox & Goodwin.
X
X% Reference: C. T. H. Baker, "The Numerical Treatment of
X% Integral Equations", Clarendon Press, Oxford, 1977; p. 665.
X
X% Discretized by simple quadrature (midpoint rule).
X
X% Per Christian Hansen, UNI-C, 03/16/93.
X
X% Initialization.
Xh = 1/n; t = h*([1:n]' - 0.5);
X
XA = h*sqrt((t.^2)*ones(1,n) + ones(n,1)*(t.^2)');
Xx = t; b = ((1+t.^2).^1.5 - t.^3)/3;
END_OF_FILE
if test 610 -ne `wc -c <'foxgood.m'`; then
    echo shar: \"'foxgood.m'\" unpacked with wrong size!
fi
# end of 'foxgood.m'
fi
if test -f 'gcv.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'gcv.m'\"
else
echo shar: Extracting \"'gcv.m'\" \(3891 characters\)
sed "s/^X//" >'gcv.m' <<'END_OF_FILE'
Xfunction [reg_min,G,reg_param] = gcv(U,s,b,method)
X%GCV Plot the GCV function and find its minimum.
X%
X% [reg_min,G,reg_param] = gcv(U,s,b,method)
X% [reg_min,G,reg_param] = gcv(U,sm,b,method)  ,  sm = [sigma,mu]
X%
X% Plots the GCV-function
X%          || A*x - b ||^2
X%    G = -------------------
X%        (trace(I - A*A_I)^2
X% as a function of the regularization parameter reg_param.
X% Here, A_I is a matrix which produces the regularized solution.
X%
X% The following methods are allowed:
X%    method = 'Tikh' : Tikhonov regularization   (solid line )
X%    method = 'tsvd' : truncated SVD or GSVD     (o markers  )
X%    method = 'dsvd' : damped SVD or GSVD        (dotted line)
X% If method is not specified, 'Tikh' is default.
X%
X% If any output arguments are specified, then the minimum of G is
X% identified and the corresponding reg. parameter reg_min is returned.
X
X% Per Christian Hansen, UNI-C, 03/16/93.
X
X% Reference: G. Wahba, "Spline Models for Observational Data",
X% SIAM, 1990.
X
X% Set defaults.
Xif (nargin==3), method='Tikh'; end  % Default method.
Xnpoints = 100;                      % Number of points on the curve.
Xsmin_ratio = 16*eps;                % Smallest regularization parameter.
X
X% Initialization.
X[m,n] = size(U); [p,ps] = size(s);
Xbeta = U'*b; beta2 = b'*b - beta'*beta;
Xif (ps==2)
X  s = s(p:-1:1,1)./s(p:-1:1,2); beta = beta(p:-1:1);
Xend
Xif (nargout > 0), find_min = 1; else find_min = 0; end
X
Xif (method(1:4)=='Tikh' | method(1:4)=='tikh')
X
X  reg_param = zeros(npoints,1); G = reg_param; s2 = s.^2;
X  reg_param(npoints) = max([s(p),s(1)*smin_ratio]);
X  ratio = (s(1)/reg_param(npoints))^(1/(npoints-1));
X  ratio = 1.2*(s(1)/reg_param(npoints))^(1/(npoints-1));
X  for i=npoints-1:-1:1, reg_param(i) = ratio*reg_param(i+1); end
X  delta0 = 0;
X  if (m > n & beta2 > 0), delta0 = beta2; end
X  for i=1:npoints
X    f1 = (reg_param(i)^2)./(s2 + reg_param(i)^2);
X    fb = f1.*beta(1:p); rho2 = fb'*fb + delta0;
X    G(i) = rho2/(m - n + sum(f1))^2;
X  end 
X  loglog(reg_param,G,'-'), xlabel('lambda'), ylabel('G(lambda)')
X  title('GCV function')
X  if (find_min)
X    [minG,minGi] = min(G); reg_min = reg_param(minGi);
X    HoldState = ishold; hold on;
X    loglog(reg_min,minG,'*',[reg_min,reg_min],[minG/1000,minG],':')
X    title(['GCV function, minimum at ',num2str(reg_min)])
X    if (~HoldState), hold off; end
X  end
X
Xelseif (method(1:4)=='tsvd' | method(1:4)=='tgsv')
X
X  rho2(p-1) = beta(p)^2;
X  if (m > n & beta2 > 0), rho2(p-1) = rho2(p-1) + beta2; end
X  for k=p-2:-1:1, rho2(k) = rho2(k+1) + beta(k+1)^2; end
X  for k=1:p-1
X    G(k) = rho2(k)/(m - k + (n - p))^2;
X  end
X  reg_param = [1:p-1]';
X  semilogy(reg_param,G,'o'), xlabel('k'), ylabel('G(k)')
X  title('GCV function')
X  if (find_min)
X    [minG,reg_min] = min(G);
X    HoldState = ishold; hold on;
X    semilogy(reg_min,minG,'*',[reg_min,reg_min],[minG/1000,minG],'--')
X    title(['GCV function, minimum at ',num2str(reg_min)])
X    if (~HoldState), hold off; end
X  end
X
Xelseif (method(1:4)=='dsvd' | method(1:4)=='dgsv')
X
X  reg_param = zeros(npoints,1); G = reg_param;
X  reg_param(npoints) = max([s(p),s(1)*smin_ratio]);
X  ratio = (s(1)/reg_param(npoints))^(1/(npoints-1));
X  for i=npoints-1:-1:1, reg_param(i) = ratio*reg_param(i+1); end
X  delta0 = 0;
X  if (m > n & beta2 > 0), delta0 = beta2; end
X  for i=1:npoints
X    f1 = reg_param(i)./(s + reg_param(i));
X    fb = f1.*beta(1:p); rho2 = fb'*fb + delta0;
X    G(i) = rho2/(m - n + sum(f1))^2;
X  end 
X  loglog(reg_param,G,':'), xlabel('lambda'), ylabel('G(lambda)')
X  title('GCV function')
X  if (find_min)
X    [minG,minGi] = min(G); reg_min = reg_param(minGi);
X    HoldState = ishold; hold on;
X    loglog(reg_min,minG,'*',[reg_min,reg_min],[minG/1000,minG],'--')
X    tiel(['GCV function, minimum at ',num2str(reg_min)])
X    if (~HoldState), hold off; end
X  end
X
Xelseif (method(1:4)=='mtsv')
X
X  error('The MTSVD method is not supported')
X
Xelse, error('Illegal method'), end
END_OF_FILE
if test 3891 -ne `wc -c <'gcv.m'`; then
    echo shar: \"'gcv.m'\" unpacked with wrong size!
fi
# end of 'gcv.m'
fi
if test -f 'gen_form.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'gen_form.m'\"
else
echo shar: Extracting \"'gen_form.m'\" \(1350 characters\)
sed "s/^X//" >'gen_form.m' <<'END_OF_FILE'
Xfunction x = gen_form(L_p,x_s,A,b,K,M)
X%GEN_FORM Transform a standard-form problem back to the general-form setting.
X%
X% x = gen_form(L_p,x_s,A,b,K,M)    (method 1)
X% x = gen_form(L_p,x_s,x_0)        (method 2)
X%
X% Transforms the standard-form solution x_s back to the required
X% solution to the general-form problem:
X%    x = L_p*x_s + d ,
X% where L_p and d depend on the method as follows:
X%    method = 1: L_p = pseudoinverse of L, d  = K*(b - A*L_p*x_s)
X%    method = 2: L_p = A-weighted pseudoinverse of L, d = x_0.
X%
X% Usually, the standard-form problem is generated by means of
X% function std_form.
X%
X% Note that x_s may have more that one column.
X
X% References: L. Elden, "Algorithms for regularization of ill-
X% conditioned least-squares problems", BIT 17 (1977), 134-145.
X% L. Elden, "A weighted pseudoinverse, generalized singular values,
X% and constrained lest squares problems", BIT 22 (1982), 487-502.
X% M. Hanke, "Regularization with differential operators.  An itera-
X% tive approach", J. Numer. Funct. Anal. Optim. 13 (1992), 523-540.
X
X% Per Christian Hansen, UNI-C, 06/12/93.
X
X% Nargin determines which method.
Xif (nargin==6)
X  [p,q] = size(x_s); [Km,Kn] = size(K);
X  if (Km==0)
X    x = L_p*x_s;
X  else
X    x = L_p*x_s + K*(M*(b*ones(1,q) - A*(L_p*x_s)));
X  end
Xelse
X  x_0 = A; [p,q] = size(x_s);
X  x = L_p*x_s + x_0*ones(1,q);
Xend
END_OF_FILE
if test 1350 -ne `wc -c <'gen_form.m'`; then
    echo shar: \"'gen_form.m'\" unpacked with wrong size!
fi
# end of 'gen_form.m'
fi
if test -f 'gen_hh.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'gen_hh.m'\"
else
echo shar: Extracting \"'gen_hh.m'\" \(554 characters\)
sed "s/^X//" >'gen_hh.m' <<'END_OF_FILE'
Xfunction [x1,beta,v] = gen_hh(x)
X%GEN_HH Generate a Householder transformation.
X%
X% [x1,beta,v] = gen_hh(x)
X%
X% Given a vector x, gen_hh computes the scalar beta and the vector v
X% determining a Householder transformation
X%    H = (I - beta*v*v'),
X% such that H*x = +-norm(x)*e_1. x1 is the first element of H*x.
X
X% Per Christian Hansen, UNI-C, 11/11/1997.
X
Xv = x; alpha = norm(v);
Xif (alpha==0),
X  beta = 0;
Xelse
X  beta = 1/(alpha*(alpha + abs(v(1))));
Xend
Xif (v(1) >= 0)
X  v(1) = v(1) + alpha; x1 = -alpha;
Xelse
X  v(1) = v(1) - alpha; x1 = +alpha;
Xend
END_OF_FILE
if test 554 -ne `wc -c <'gen_hh.m'`; then
    echo shar: \"'gen_hh.m'\" unpacked with wrong size!
fi
# end of 'gen_hh.m'
fi
if test -f 'get_l.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'get_l.m'\"
else
echo shar: Extracting \"'get_l.m'\" \(793 characters\)
sed "s/^X//" >'get_l.m' <<'END_OF_FILE'
Xfunction [L,W] = get_l(n,d)
X%GET_L Compute discrete derivative operators.
X%
X% [L,W] = get_l(n,d)
X%
X% Computes the discrete approximation L to the derivative operator
X% of order d on a regular grid with n points, i.e. L is (n-d)-by-n.
X%
X% L is stored as a sparse matrix.
X%
X% Also computes W, an orthonormal basis for the null space of L.
X
X% Per Christian Hansen, UNI-C, 05/26/93.
X
X% Initialization.
Xif (d<1), error ('Order d must be positive'), end
Xnd = n-d;
X
X% Compute L.
Xc = [-1,1,zeros(1,d-1)];
Xfor i=2:d, c = [c(1:d),0] - [0,c(1:d)]; end
XL = sparse(nd,n);
Xfor i=1:d+1
X  L = L + sparse(1:nd,[1:nd]+i-1,c(i)*ones(1,nd),nd,n);
Xend
X
X% If required, compute the null vectors W.
Xif (nargout==2)
X  W = zeros(n,d);
X  W(:,1) = ones(n,1);
X  for i=2:d, W(:,i) = W(:,i-1).*[1:n]'; end
X  W = mgs(W);
Xend
END_OF_FILE
if test 793 -ne `wc -c <'get_l.m'`; then
    echo shar: \"'get_l.m'\" unpacked with wrong size!
fi
# end of 'get_l.m'
fi
if test -f 'gsvd.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'gsvd.m'\"
else
echo shar: Extracting \"'gsvd.m'\" \(1110 characters\)
sed "s/^X//" >'gsvd.m' <<'END_OF_FILE'
Xfunction [U,V,sm,X] = gsvd(A,L)
X%GSVD Generalized SVD of a matrix pair.
X%
X% sm = gsvd(A,L)
X% [U,V,sm,X] = gsvd(A,L) ,  sm = [sigma,mu]
X%
X% Computes the generalized SVD of the matrix pair (A,L):
X%    [ A ] = [ U  0 ]*[ diag(sigma)      0    ]*inv(X)
X%    [ L ]   [ 0  V ] [      0       eye(n-p) ]
X%                     [  diag(mu)        0    ]
X% where
X%    U  is  m-by-n ,    sigma  is  p-by-1
X%    V  is  p-by-p ,    mu     is  p-by-1
X%    X  is  n-by-n .
X%
X% It is assumed that m >= n >= p .
X
X% Reference: C. F. Van Loan, "Computing the CS and the generalized
X% singular value decomposition", Numer. Math. 46 (1985), 479-491.
X
X% Per Christian Hansen, UNI-C, 06/22/93.
X
X% Initialization.
X[m,n] = size(A); [p,n1] = size(L);
Xif (n1 ~= n | m < n | n < p)
X  error('Incorrect dimensions of A and L')
Xend
X
X% Compute the GSVD in compact form via the CS decomposition.
X[Q,d,X] = svd([full(A);full(L)]); Q = Q(:,1:n); d = diag(d);
Xif (nargout > 1)
X  [U,V,sm,Z] = csdecomp(Q(1:m,:),Q(m+1:m+p,:));
X  if (nargout==4)
X    for j=1:n, X(:,j) = X(:,j)/d(j); end
X    X = X*Z;
X  end
Xelse
X  U = csd(Q(1:m,:),Q(m+1:m+p,:));
Xend
END_OF_FILE
if test 1110 -ne `wc -c <'gsvd.m'`; then
    echo shar: \"'gsvd.m'\" unpacked with wrong size!
fi
# end of 'gsvd.m'
fi
if test -f 'heat.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'heat.m'\"
else
echo shar: Extracting \"'heat.m'\" \(1632 characters\)
sed "s/^X//" >'heat.m' <<'END_OF_FILE'
Xfunction [A,b,x] = heat(n,kappa)
X%HEAT Test problem: inverse heat equation.
X%
X% [A,b,x] = heat(n,kappa)
X%
X% A first kind Volterra integral equation with [0,1] as
X% integration interval.  The kernel is K(s,t) = k(s-t) with
X%    k(t) = t^(-3/2)/(2*kappa*sqrt(pi))*exp(-1/(4*kappa^2*t^2)) .
X% Here, kappa controls the ill-conditioning of the matrix:
X%    kappa = 5 gives a well-conditioned problem
X%    kappa = 1 gives an ill-conditioned problem.
X% The default is kappa = 1.
X%
X% An exact soltuion is constructed, and then the right-hand side
X% b is produced as b = A*x.
X
X% Reference: A. S. Carasso, "Determining surface temperatures
X% from interior observations", SIAM J. Appl. Math. 42 (1982),
X% 558-574.  See also L. Elden, "The numerical solution of a
X% non-characteristic Cauchy problem for a parabolic equation";
X% in P. Deuflhand & E. Hairer (Eds.), "Numerical Treatment of
X% Inverse Problems in Differential and Integral Equations",
X% Birkhauser, 1983.
X
X% Discretization by means of simple quadrature (midpoint rule).
X
X% Per Christian Hansen, UNI-C, 09/18/92.
X
X% Set default kappa.
Xif (nargin==1), kappa = 1; end
X
X% Initialization.
Xh = 1/n; t = h/2:h:1; e = ones(1,length(t));
Xc = h/(2*kappa*sqrt(pi)); d = 1/(4*kappa^2);
X
X% Compute the matrix A.
Xk = c*t.^(-1.5).*exp(-d*e./t);
Xr = zeros(1,length(t)); r(1) = k(1); A = toeplitz(k,r);
X
X% Compute the vectors x and b.
Xif (nargout>1)
X  x = zeros(n,1);
X  for i=1:n/2
X    ti = i*20/n;
X    if (ti < 2)
X      x(i) = 0.75*ti^2/4;
X    elseif (ti < 3)
X      x(i) = 0.75 + (ti-2)*(3-ti);
X    else
X      x(i) = 0.75*exp(-(ti-3)*2);
X    end
X  end
X  x(n/2+1:n) = zeros(1,n/2);
X  b = A*x;
Xend
END_OF_FILE
if test 1632 -ne `wc -c <'heat.m'`; then
    echo shar: \"'heat.m'\" unpacked with wrong size!
fi
# end of 'heat.m'
fi
if test -f 'heb_new.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'heb_new.m'\"
else
echo shar: Extracting \"'heb_new.m'\" \(1781 characters\)
sed "s/^X//" >'heb_new.m' <<'END_OF_FILE'
Xfunction lambda = heb_new(lambda_0,alpha,s,beta,omega)
X%HEB_NEW Newton iteration with Hebden model (utility routine for LSQI).
X%
X% lambda = heb_new(lambda_0,alpha,s,beta,omega)
X%
X% Uses Newton iteration with a Hebden (rational) model to find the
X% solution lambda to the secular equation
X%    || L (x_lambda - x_0) || = alpha ,
X% where x_lambda is the solution defined by Tikhonov regularization.
X%
X% The initial guess is lambda_0.
X%
X% The norm || L (x_lambda - x_0) || is computed via s, beta and omega.
X% Here, s holds either the singular values of A, if L = I, or the
X% c,s-pairs of the GSVD of (A,L), if L ~= I.  Moreover, beta = U'*b
X% and omega is either V'*x_0 or the first p elements of inv(X)*x_0.
X
X% Reference: T. F. Chan, J. Olkin & D. W. Cooley, "Solving quadratically
X% constrained least squares using block box unconstrained solvers",
X% BIT 32 (1992), 481-495.
X% Extension to the case x_0 ~= 0 by Per Chr. Hansen, UNI-C, 11/20/91.
X
X% Per Christian Hansen, UNI-C, 02/07/92.
X
X% Set defaults.
Xthr = eps;     % Relative stopping criterion.
Xit_max = 50;   % Max number of iterations.
X
X% Initialization.
Xif (lambda_0 < 0)
X  error('Initial guess lambda_0 must be nonnegative')
Xend
X[p,ps] = size(s);
Xif (ps==2), mu = s(:,2); s = s(:,1)./s(:,2); end
Xs2 = s.^2;
X
X% Iterate.
Xlambda = lambda_0^2; step = 1; it = 0;
Xwhile (abs(step) > thr*lambda & it < it_max), it = it+1;
X  e = s./(s2 + lambda); f = 2.*e;
X  if (ps==1)
X    Lx = e.*beta - f.*omega;
X  else
X    Lx = e.*beta - f.*mu.*omega;
X  end
X  norm_Lx = norm(Lx);
X  Lv = Lx./(s2 + lambda);
X  step = (norm_Lx - alpha)*norm_Lx^2/((Lv'*Lx)*alpha);
X  lambda = lambda + step;
Xend
X
X% Terminate with an error if too many iterations.
Xif (abs(step) > thr*lambda), error('Max. number of iterations reached'), end
X
Xlambda = sqrt(lambda);
END_OF_FILE
if test 1781 -ne `wc -c <'heb_new.m'`; then
    echo shar: \"'heb_new.m'\" unpacked with wrong size!
fi
# end of 'heb_new.m'
fi
if test -f 'ilaplace.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'ilaplace.m'\"
else
echo shar: Extracting \"'ilaplace.m'\" \(1976 characters\)
sed "s/^X//" >'ilaplace.m' <<'END_OF_FILE'
Xfunction [A,b,x] = ilaplace(n,example)
X%ILAPLACE Test problem: inverse Laplace transformation.
X%
X% [A,b,x] = ilaplace(n,example)
X%
X% Discretization of the inverse Laplace transformation by means of
X% Gauss-Laguerre quadrature.  The kernel K is given by
X%    K(s,t) = exp(-s*t) ,
X% and both integration intervals are [0,inf).
X% The following examples are implemented, where f denotes
X% the solution, and g denotes the right-hand side:
X%    1: f(t) = exp(-t/2),        g(s) = 1/(s + 0.5)
X%    2: f(t) = 1 - exp(-t/2),    g(s) = 1/s - 1/(s + 0.5)
X%    3: f(t) = t^2*exp(-t/2),    g(s) = 2/(s + 0.5)^3
X%    4: f(t) = | 0 , t <= 2,     g(s) = exp(-2*s)/s.
X%              | 1 , t >  2
X
X% Reference: J. M. Varah, "Pitfalls in the numerical solution of linear
X% ill-posed problems", SIAM J. Sci. Stat. Comput. 4 (1983), 164-176.
X
X% Per Christian Hansen, UNI-C, 09/18/92.
X
X% Initialization.
Xif (n <= 0), error('The order n must be positive'); end
Xif (nargin == 1), example = 1; end
X
X% Compute equidistand collocation points s.
Xs = (10/n)*[1:n]';
X
X% Compute abscissas t and weights w from the eigensystem of the
X% symmetric tridiagonal system derived from the recurrence
X% relation for the Laguerre polynomials.  Sorting of the
X% eigenvalues and -vectors is necessary.
Xt = diag(2*[1:n]-1) - diag([1:n-1],1) - diag([1:n-1],-1);
X[Q,t] = eig(t); t = diag(t); [t,indx] = sort(t);
Xw = Q(1,indx).^2; Q = [];
X
X% Set up the coefficient matrix A.
XA = zeros(n,n);
Xfor i=1:n
X  for j=1:n
X    A(i,j) = (1-s(i))*t(j);
X  end
Xend
XA = exp(A)*diag(w);
X
X% Compute the right-hand side b and the solution x by means of
X% simple collocation.
Xif (example==1)
X  b = ones(n,1)./(s + .5);
X  x = exp(-t/2);
Xelseif (example==2)
X  b = ones(n,1)./s - ones(n,1)./(s + .5);
X  x = ones(n,1) - exp(-t/2);
Xelseif (example==3)
X  b = 2*ones(n,1)./((s + .5).^3);
X  x = (t.^2).*exp(-t/2);
Xelseif (example==4)
X  b = exp(-2*s)./s;
X  x = ones(n,1); f = find(t<=2); x(f) = zeros(length(f),1);
Xelse
X  error('Illegal example')
Xend
END_OF_FILE
if test 1976 -ne `wc -c <'ilaplace.m'`; then
    echo shar: \"'ilaplace.m'\" unpacked with wrong size!
fi
# end of 'ilaplace.m'
fi
if test -f 'l_corner.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'l_corner.m'\"
else
echo shar: Extracting \"'l_corner.m'\" \(8126 characters\)
sed "s/^X//" >'l_corner.m' <<'END_OF_FILE'
Xfunction [reg_c,rho_c,eta_c] = l_corner(rho,eta,reg_param,U,s,b,method,M)
X%L_CORNER Locate the "corner" of the L-curve.
X%
X% [reg_c,rho_c,eta_c] =
X%        l_corner(rho,eta,reg_param)
X%        l_corner(rho,eta,reg_param,U,s,b,method,M)
X%        l_corner(rho,eta,reg_param,U,sm,b,method,M) ,  sm = [sigma,mu]
X%
X% Locates the "corner" of the L-curve in log-log scale.
X%
X% It is assumed that corresponding values of || A x - b ||, || L x ||,
X% and the regularization parameter are stored in the arrays rho, eta,
X% and reg_param, respectively (such as the output from routine l_curve).
X%
X% If nargin = 3, then no particular method is assumed, and if
X% nargin = 2 then it is issumed that reg_param = 1:length(rho).
X%
X% If nargin >= 6, then the following methods are allowed:
X%    method = 'Tikh'  : Tikhonov regularization
X%    method = 'tsvd'  : truncated SVD or GSVD
X%    method = 'dsvd'  : damped SVD or GSVD
X%    method = 'mtsvd' : modified TSVD,
X% and if no method is specified, 'Tikh' is default.  If the Spline Toolbox
X% is not available, then only 'Tikh' and 'dsvd' can be used.
X%
X% An eighth argument M specifies an upper bound for eta, below which
X% the corner should be found.
X
X% The following routines from the Spline Toolbox are needed if
X% method differs from 'Tikh' or 'dsvd':
X% fnder, ppbrk, ppcut, ppmak, sp2pp, spbrk, spmak.
X
X% Per Christian Hansen, UNI-C, 03/17/93.
X
X% Set default regularization method.
Xif (nargin <= 3)
X  method = 'none';
X  if (nargin==2), reg_param = [1:length(rho)]'; end
Xelse
X  if (nargin==6), method = 'Tikh'; end
Xend
X
X% Set threshold for skipping very small singular values in the
X% L-curve analysis.
Xs_thr = eps;  % Neglect singular values less than s_thr.
X
X% Set default parameters for treatment of discrete L-curve.
Xdeg   = 2;  % Degree of local smooting polynomial.
Xq     = 2;  % Half-width of local smoothing interval.
Xorder = 4;  % Order of fitting 2-D spline curve.
X
X% Initialization.
Xif (length(rho) < order)
X  error('Too few data points for L-curve analysis')
Xend
Xif (nargin > 3)
X  [p,ps] = size(s); [m,n] = size(U);
X  if (ps==2), s = s(p:-1:1,1)./s(p:-1:1,2); U = U(:,p:-1:1); end
X  beta = U'*b; xi = beta./s;
Xend
X
X% Restrict the analysis of the L-curve according to M (if specified)
X% and s_thr.
Xif (nargin==8)
X  index = find(eta < M);
X  rho = rho(index); eta = eta(index); reg_param = reg_param(index);
X  s = s(index); beta = beta(index); xi = xi(index);
Xend
X
Xif (method(1:4)=='Tikh' | method(1:4)=='tikh')
X
X  % The L-curve is differentiable; computation of curvature in
X  % log-log scale is easy.
X
X  % Initialization.
X  [reg_m,reg_n] = size(reg_param);
X  phi = zeros(reg_m,reg_n); dphi = phi; psi = phi; dpsi = phi;
X  s2 = s.^2; beta2 = beta.^2; xi2 = xi.^2;
X
X  % Compute some intermediate quantities.
X  for i = 1:length(reg_param)
X    f  = s2./(s2 + reg_param(i)^2); cf = 1 - f;
X    f1 = -2*f.*cf/reg_param(i);
X    f2 = -f1.*(3-4*f)/reg_param(i);
X    phi(i)  = sum(f.*f1.*xi2);
X    psi(i)  = sum(cf.*f1.*beta2);
X    dphi(i) = sum((f1.^2 + f.*f2).*xi2);
X    dpsi(i) = sum((-f1.^2 + cf.*f2).*beta2);
X  end
X
X  % Now compute the first and second derivatives of eta and rho
X  % with respect to lambda;
X  deta  =  phi./eta;
X  drho  = -psi./rho;
X  ddeta =  dphi./eta - deta.*(deta./eta);
X  ddrho = -dpsi./rho - drho.*(drho./rho);
X
X  % Convert to derivatives of log(eta) and log(rho).
X  dlogeta  = deta./eta;
X  dlogrho  = drho./rho;
X  ddlogeta = ddeta./eta - (dlogeta).^2;
X  ddlogrho = ddrho./rho - (dlogrho).^2;
X
X  % Let g = curvature.
X  g = (dlogrho.*ddlogeta - ddlogrho.*dlogeta)./...
X      (dlogrho.^2 + dlogeta.^2).^(1.5);
X
X  % Locate the corner.  If the curvature is negative everywhere,
X  % then define the leftmost point of the L-curve as the corner.
X  [gmax,gi] = max(g);
X  if (gmax < 0)
X    lr = length(rho);
X    reg_c = reg_param(lr); rho_c = rho(lr); eta_c = eta(lr);
X  else
X    rho_c = rho(gi); eta_c = eta(gi); reg_c = reg_param(gi);
X  end
X
Xelseif (method(1:4)=='tsvd' | method(1:4)=='tgsv' | ...
X        method(1:4)=='mtsv' | method(1:4)=='none')
X
X  % The L-curve is discrete and may include unwanted fine-grained
X  % corners.  Use local smoothing, followed by fitting a 2-D spline
X  % curve to the smoothed discrete L-curve.
X
X  % Check if the Spline Toolbox exists, otherwise return.
X  if (exist('spdemos')~=2)
X    error('The Spline Toolbox in not available so l_corner cannot be used')
X  end
X
X  % For TSVD, TGSVD, and MTSVD, restrict the analysis of the L-curve
X  % according to s_thr.
X  if (nargin > 3)
X    index = find(s > s_thr);
X    rho = rho(index); eta = eta(index); reg_param = reg_param(index);
X    s = s(index); beta = beta(index); xi = xi(index);
X  end
X
X  % Convert to logarithms.
X  lr = length(rho);
X  lrho = log(rho); leta = log(eta); slrho = lrho; sleta = leta;
X
X  % For all interior points k = q+1:length(rho)-q-1 on the discrete
X  % L-curve, perform local smoothing with a polynomial of degree deg
X  % to the points k-q:k+q.
X  v = [-q:q]'; A = zeros(2*q+1,deg+1); A(:,1) = ones(length(v),1);
X  for j = 2:deg+1, A(:,j) = A(:,j-1).*v; end
X  for k = q+1:lr-q-1
X    cr = A\lrho(k+v); slrho(k) = cr(1);
X    ce = A\leta(k+v); sleta(k) = ce(1);
X  end
X
X  % Fit a 2-D spline curve to the smoothed discrete L-curve.
X  sp = spmak([1:lr+order],[slrho';sleta']);
X  pp = ppcut(sp2pp(sp),[4,lr+1]);
X
X  % Extract abscissa and ordinate splines and differentiate them.
X  P     = spleval(pp);  dpp   = fnder(pp);
X  D     = spleval(dpp); ddpp  = fnder(pp,2);
X  DD    = spleval(ddpp);
X  ppx   = P(1,:);       ppy   = P(2,:);
X  dppx  = D(1,:);       dppy  = D(2,:);
X  ddppx = DD(1,:);      ddppy = DD(2,:);
X
X  % Compute the corner of the spline curve via max. curvature.
X  % Define curvature = 0 where both dppx and dppy are zero.
X  k1    = dppx.*ddppy - ddppx.*dppy;
X  k2    = (dppx.^2 + dppy.^2).^(1.5);
X  I_nz  = find(k2 ~= 0);
X  kappa = zeros(1,length(dppx));
X  kappa(I_nz) = -k1(I_nz)./k2(I_nz);
X  [kmax,ikmax] = max(kappa);
X  x_corner = ppx(ikmax); y_corner = ppy(ikmax);
X
X  % Locate the point on the discrete L-curve which is closest to the
X  % corner of the spline curve.  Prefer a point below and to the
X  % left of the corner.  If the curvature is negative everywhere,
X  % then define the leftmost point of the L-curve as the corner.
X  if (kmax < 0)
X    reg_c = reg_param(lr); rho_c = rho(lr); eta_c = eta(lr);
X  else
X    index = find(lrho < x_corner & leta < y_corner);
X    if (length(index) > 0)
X      [dummy,rpi] = min((lrho(index)-x_corner).^2 + (leta(index)-y_corner).^2);
X      rpi = index(rpi);
X    else
X      [dummy,rpi] = min((lrho-x_corner).^2 + (leta-y_corner).^2);
X    end
X    reg_c = reg_param(rpi); rho_c = rho(rpi); eta_c = eta(rpi);
X  end
X
Xelseif (method(1:4)=='dsvd' | method(1:4)=='dgsv')
X
X  % The L-curve is differentiable; computation of curvature in
X  % log-log scale is easy.
X
X  % Initialization.
X  [reg_m,reg_n] = size(reg_param);
X  phi = zeros(reg_m,reg_n); dphi = phi; psi = phi; dpsi = phi;
X  beta2 = beta.^2; xi2 = xi.^2;
X
X  % Compute some intermediate quantities.
X  for i = 1:length(reg_param)
X    f  = s./(s + reg_param(i)); cf = 1 - f;
X    f1 = -f.*cf/reg_param(i);
X    f2 = -2*f1.*cf/reg_param(i);
X    phi(i)  = sum(f.*f1.*xi2);
X    psi(i)  = sum(cf.*f1.*beta2);
X    dphi(i) = sum((f1.^2 + f.*f2).*xi2);
X    dpsi(i) = sum((-f1.^2 + cf.*f2).*beta2);
X  end
X
X  % Now compute the first and second derivatives of eta and rho
X  % with respect to lambda;
X  deta  =  phi./eta;
X  drho  = -psi./rho;
X  ddeta =  dphi./eta - deta.*(deta./eta);
X  ddrho = -dpsi./rho - drho.*(drho./rho);
X
X  % Convert to derivatives of log(eta) and log(rho).
X  dlogeta  = deta./eta;
X  dlogrho  = drho./rho;
X  ddlogeta = ddeta./eta - (dlogeta).^2;
X  ddlogrho = ddrho./rho - (dlogrho).^2;
X
X  % Let g = curvature.
X  g = (dlogrho.*ddlogeta - ddlogrho.*dlogeta)./...
X      (dlogrho.^2 + dlogeta.^2).^(1.5);
X
X  % Locate the corner.  If the curvature is negative everywhere,
X  % then define the leftmost point of the L-curve as the corner.
X  [gmax,gi] = max(g);
X  if (gmax < 0)
X    lr = length(rho);
X    reg_c = reg_param(lr); rho_c = rho(lr); eta_c = eta(lr);
X  else
X    rho_c = rho(gi); eta_c = eta(gi); reg_c = reg_param(gi);
X  end
X
Xelse, error('Illegal method'), end
END_OF_FILE
if test 8126 -ne `wc -c <'l_corner.m'`; then
    echo shar: \"'l_corner.m'\" unpacked with wrong size!
fi
# end of 'l_corner.m'
fi
if test -f 'l_curve.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'l_curve.m'\"
else
echo shar: Extracting \"'l_curve.m'\" \(4936 characters\)
sed "s/^X//" >'l_curve.m' <<'END_OF_FILE'
Xfunction [reg_corner,rho,eta,reg_param] = l_curve(U,sm,b,method,L,V)
X%L_CURVE Plot the L-curve and find its "corner".
X%
X% [reg_corner,rho,eta,reg_param] =
X%                  l_curve(U,s,b,method)
X%                  l_curve(U,sm,b,method)  ,  sm = [sigma,mu]
X%                  l_curve(U,s,b,method,L,V)
X%
X% Plots the L-shaped curve of eta, the solution norm || x || or
X% semi-norm || L x ||, as a function of rho, the residual norm
X% || A x - b ||, for the following methods:
X%    method = 'Tikh'  : Tikhonov regularization   (solid line )
X%    method = 'tsvd'  : truncated SVD or GSVD     (o markers  )
X%    method = 'dsvd'  : damped SVD or GSVD        (dotted line)
X%    method = 'mtsvd' : modified TSVD             (x markers  )
X% The corresponding reg. parameters are returned in reg_param.
X% If no method is specified, 'Tikh' is default.
X%
X% If any output arguments are specified, then the corner of the L-curve
X% is identified and the corresponding reg. parameter reg_corner is
X% returned.  Use routine l_corner if an upper bound on eta is required.
X%
X% If the Spline Toolbox is not available and reg_corner is requested,
X% then the routine returns reg_corner = NaN for 'tsvd' and 'mtsvd'.
X
X% Reference: P. C. Hansen & D. P. O'Leary, "The use of the L-curve in
X% the regularization of discrete ill-posed problems", Report UMIACS-
X% TR-91-142, Dept. of Computer Science, Univ. of Maryland, 1991;
X% to appear in SIAM J. Sci. Comp.
X
X% Per Christian Hansen, UNI-C, 03/17/93.
X
X% Set defaults.
Xif (nargin==3), method='Tikh'; end  % Tikhonov reg. is default.
Xnpoints = 100;  % Number of points on the L-curve for Tikh and dsvd.
Xsmin_ratio = 16*eps;  % Smallest regularization parameter.
X
X% Initialization.
X[m,n] = size(U); [p,ps] = size(sm);
Xif (nargout > 0), locate = 1; else locate = 0; end
Xbeta = U'*b; beta2 = b'*b - beta'*beta;
Xif (ps==1)
X  s = sm; beta = beta(1:p);
Xelse
X  s = sm(p:-1:1,1)./sm(p:-1:1,2); beta = beta(p:-1:1);
Xend
Xxi = beta(1:p)./s;
X
Xif (method(1:4)=='Tikh' | method(1:4)=='tikh')
X
X  eta = zeros(npoints,1); rho = eta; reg_param = eta; s2 = s.^2;
X  reg_param(npoints) = max([s(p),s(1)*smin_ratio]);
X  ratio = (s(1)/reg_param(npoints))^(1/(npoints-1));
X  ratio = (s(1)/reg_param(npoints))^(1/(npoints-1));
X  for i=npoints-1:-1:1, reg_param(i) = ratio*reg_param(i+1); end
X  for i=1:npoints
X    f = s2./(s2 + reg_param(i)^2);
X    eta(i) = norm(f.*xi);
X    rho(i) = norm((1-f).*beta(1:p));
X  end
X  if (m > n & beta2 > 0), rho = sqrt(rho.^2 + beta2); end
X  marker = '-'; pos = .8; txt = 'Tikh.';
X
Xelseif (method(1:4)=='tsvd' | method(1:4)=='tgsv')
X
X  eta = zeros(p,1); rho = eta;
X  eta(1) = xi(1)^2;
X  for k=2:p, eta(k) = eta(k-1) + xi(k)^2; end
X  eta = sqrt(eta);
X  if (m > n)
X    if (beta2 > 0), rho(p) = beta2; else rho(p) = eps^2; end
X  else
X    rho(p) = eps^2;
X  end
X  for k=p-1:-1:1, rho(k) = rho(k+1) + beta(k+1)^2; end
X  rho = sqrt(rho);
X  reg_param = [1:p]'; marker = 'o'; pos = .75;
X  if (ps==1)
X    U = U(:,1:p); txt = 'TSVD';
X  else
X    U = U(:,1:p); txt = 'TGSVD';
X  end
X
Xelseif (method(1:4)=='dsvd' | method(1:4)=='dgsv')
X
X  eta = zeros(npoints,1); rho = eta; reg_param = eta;
X  reg_param(npoints) = max([s(p),s(1)*smin_ratio]);
X  ratio = (s(1)/reg_param(npoints))^(1/(npoints-1));
X  for i=npoints-1:-1:1, reg_param(i) = ratio*reg_param(i+1); end
X  for i=1:npoints
X    f = s./(s + reg_param(i));
X    eta(i) = norm(f.*xi);
X    rho(i) = norm((1-f).*beta(1:p));
X  end
X  if (m > n & beta2 > 0), rho = sqrt(rho.^2 + beta2); end
X  marker = ':'; pos = .85;
X  if (ps==1), txt = 'DSVD'; else txt = 'DGSVD'; end
X
Xelseif (method(1:4)=='mtsv')
X
X  if (nargin~=6)
X    error('The matrices L and V must also be specified')
X  end
X  [p,n] = size(L); rho = zeros(p,1); eta = rho;
X  [Q,R] = qr(L*V(:,n:-1:n-p));
X  for i=1:p
X    k = n-p+i;
X    Lxk = L*V(:,1:k)*xi(1:k);
X    zk = R(1:n-k,1:n-k)\(Q(:,1:n-k)'*Lxk); zk = zk(n-k:-1:1);
X    eta(i) = norm(Q(:,n-k+1:p)'*Lxk);
X    if (i < p)
X      rho(i) = norm(beta(k+1:n) + s(k+1:n).*zk);
X    else
X      rho(i) = eps;
X    end
X  end
X  if (m > n & beta2 > 0), rho = sqrt(rho.^2 + beta2); end
X  reg_param = [n-p+1:n]'; txt = 'MTSVD';
X  U = U(:,reg_param); sm = sm(reg_param);
X  marker = 'x'; pos = .7; ps = 2;  % General form regularization.
X 
Xelse, error('Illegal method'), end
X
X% Locate the "corner" of the L-curve, if required.  If the Spline
X% Toolbox is not available, return NaN for reg_corner.
Xif (locate)
X  SkipCorner = ( (method(1:4)=='tsvd' | method(1:4)=='tgsv' | ...
X                  method(1:4)=='mtsv') & exist('spdemos')~=2 );
X  if (SkipCorner)
X    reg_corner = NaN;
X  else
X    [reg_corner,rho_c,eta_c] = l_corner(rho,eta,reg_param,U,sm,b,method);
X  end
Xend
X
X% Make plot.
Xplot_lc(rho,eta,marker,ps,reg_param);
Xif (locate & ~SkipCorner)
X  HoldState = ishold; hold on;
X  loglog([min(rho)/100,rho_c],[eta_c,eta_c],'--',...
X         [rho_c,rho_c],[min(eta)/100,eta_c],'--')
X  title(['L-curve, ',txt,' corner at ',num2str(reg_corner)]);
X  if (~HoldState), hold off; end
Xend
END_OF_FILE
if test 4936 -ne `wc -c <'l_curve.m'`; then
    echo shar: \"'l_curve.m'\" unpacked with wrong size!
fi
# end of 'l_curve.m'
fi
if test -f 'lagrange.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'lagrange.m'\"
else
echo shar: Extracting \"'lagrange.m'\" \(1654 characters\)
sed "s/^X//" >'lagrange.m' <<'END_OF_FILE'
Xfunction [La,dLa,lambda0] = lagrange(U,s,b,more)
X%LAGRANGE Plot the Lagrange function for Tikhonov regularization.
X%
X% [La,dLa,lambda0] = lagrange(U,s,b,more)
X% [La,dLa,lambda0] = lagrange(U,sm,b,more)  ,  sm = [sigma,mu]
X%
X% Plots the Lagrange function
X%    La(lambda) = || A x - b ||^2 + lambda^2*|| L x ||^2
X% and its first derivative dLa = dLa/dlambda versus lambda.
X% Here, x is the Tikhonov regularized solution.
X%
X% If nargin = 4, || A x - b || and || L x || are also plotted.
X%
X% Returns La, dLa, and the value lambda0 of lambda for which
X% dLa has its minimum.
X
X% Per Christian Hansen, UNI-C, 07/14/92.
X
X% Initialization.
X[m,n] = size(U); [p,ps] = size(s); npoints = 40;
Xbeta = U'*b; beta2 = b'*b - beta'*beta;
Xif (ps==2)
X  s = s(p:-1:1,1)./s(p:-1:1,2); beta = beta(p:-1:1);
Xend
Xxi = beta(1:p)./s;
X
X% Compute the L-curve.
Xeta = zeros(npoints,1); rho = eta;
Xlambda(npoints,1) = s(p);
Xratio = (s(1)/s(p))^(1/(npoints-1));
Xfor i=npoints-1:-1:1, lambda(i) = ratio*lambda(i+1); end
Xfor i=1:npoints
X  f = fil_fac(s,lambda(i));
X  eta(i) = norm(f.*xi);
X  rho(i) = norm((1-f).*beta(1:p));
Xend
Xif (m > n & beta2 > 0), rho = sqrt(rho.^2 + beta2); end
X
X% Compute the Lagrange function and its derivative.
XLa = rho.^2 + (lambda.^2).*(eta.^2);
XdLa = 2*lambda.*(eta.^2);
X[mindLa,mindLi] = min(dLa); lambda0 = lambda(mindLi);
X
X% Plot the functions.
Xif (nargin==3)
X  loglog(lambda,La,'-',lambda,dLa,'--',lambda0,mindLa,'o')
X  title('--- La   - - - dLa/dlambda')
Xelse
X  loglog(lambda,La,'-',lambda,dLa,'--',lambda,eta,':',lambda,rho,'-.',...
X         lambda0,mindLa,'o')
X  title('--- La   - - - dLa/dlambda   ... || Lx ||   -.-. || Ax-b ||')
Xend
Xxlabel('lambda')
END_OF_FILE
if test 1654 -ne `wc -c <'lagrange.m'`; then
    echo shar: \"'lagrange.m'\" unpacked with wrong size!
fi
# end of 'lagrange.m'
fi
if test -f 'lanc_b.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'lanc_b.m'\"
else
echo shar: Extracting \"'lanc_b.m'\" \(2826 characters\)
sed "s/^X//" >'lanc_b.m' <<'END_OF_FILE'
Xfunction [U,B_k,V] = lanc_b(A,p,k,reorth)
X%LANC_B Lanczos bidiagonalization.
X%
X% B_k = lanc_b(A,p,k,reorth)
X% [U,B_k,V] = lanc_b(A,p,k,reorth)
X%
X% Performs k steps of the Lanczos bidiagonalization process
X% with starting vector p, producing a lower bidiagonal matrix
X%         [b_11               ]                   [b_21    b_11]
X%         [b_21 b_22          ]                   [b_32    b_22]
X%     B = [     b_32 .        ]  stored as  B_k = [ .       .  ] .
X%         [          . b_kk   ]                   [b_k+1,k b_kk]
X%         [            b_k+1,k]
X% U and V consist of the left and right Lanczos vectors.
X%
X% Reorthogonalization is controlled by means of reorth:
X%    reorth = 0 : no reorthogonalization,
X%    reorth = 1 : reorthogonalization by means of MGS,
X%    reorth = 2 : Householder-reorthogonalization.
X% No reorthogonalization is assumed if reorth is not specified.
X
X% Reference: G. H. Golub & C. F. Van Loan, "Matrix Computations",
X% 2. Ed., Johns Hopkins, 1989.  Section 9.3.4.
X
X% Per Christian Hansen, UNI-C, 05/25/93.
X
X% Initialization.
Xif (k<1), error('Number of steps k must be positive'), end
Xif (nargin < 4), reorth = 0; end
Xif (reorth < 0 | reorth > 2), error('Illegal reorth'), end
Xif (nargout==2), error('Not enough output arguments'), end
X[m,n] = size(A);
Xif (nargout>1 | reorth==1)
X  U = zeros(m,k); V = zeros(n,k); UV = 1;
Xelse
X  UV = 0;
Xend
Xif (reorth==2)
X  if (k>=n), error('No. of iterations must satisfy k < n'), end
X  HHU = zeros(m,k); HHV = zeros(n,k);
X  HHalpha = zeros(1,k); HHbeta = HHalpha;
Xend
X
X% Prepare for Lanczos iteration.
Xv = zeros(n,1);
Xbeta = norm(p);
Xif (beta==0), error('Starting vector must be nonzero'), end
Xif (reorth==2)
X  [beta,HHbeta(1),HHU(:,1)] = gen_hh(p);
Xend
Xu = p/beta;
Xif (UV), U(:,1) = u; end
X
X% Perform Lanczos bidiagonalization with/without reorthogonalization.
Xfor i=1:k
X
X  r = A'*u - beta*v;
X  if (reorth==0)
X    alpha = norm(r); v = r/alpha;
X  elseif (reorth==1)
X    for j=1:i-1, r = r - (V(:,j)'*r)*V(:,j); end
X    alpha = norm(r); v = r/alpha;
X  else
X    for j=1:i-1
X      r(j:n) = app_hh(r(j:n),HHalpha(j),HHV(j:n,j));
X    end
X    [alpha,HHalpha(i),HHV(i:n,i)] = gen_hh(r(i:n));
X    v = zeros(n,1); v(i) = 1;
X    for j=i:-1:1
X      v(j:n) = app_hh(v(j:n),HHalpha(j),HHV(j:n,j));
X    end
X  end
X  B_k(i,2) = alpha; if (UV), V(:,i) = v; end
X
X  p = A*v - alpha*u;
X  if (reorth==0)
X    beta = norm(p); u = p/beta;
X  elseif (reorth==1)
X    for j=1:i, p = p - (U(:,j)'*p)*U(:,j); end
X    beta = norm(p); u = p/beta;
X  else
X    for j=1:i
X      p(j:m) = app_hh(p(j:m),HHbeta(j),HHU(j:m,j));
X    end
X    [beta,HHbeta(i+1),HHU(i+1:m,i+1)] = gen_hh(p(i+1:m));
X    u = zeros(m,1); u(i+1) = 1;
X    for j=i+1:-1:1
X      u(j:m) = app_hh(u(j:m),HHbeta(j),HHU(j:m,j));
X    end
X  end
X  B_k(i,1) = beta; if (UV), U(:,i+1) = u; end
X
Xend
X
Xif (nargout==1), U = B_k; end
END_OF_FILE
if test 2826 -ne `wc -c <'lanc_b.m'`; then
    echo shar: \"'lanc_b.m'\" unpacked with wrong size!
fi
# end of 'lanc_b.m'
fi
if test -f 'lsolve.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'lsolve.m'\"
else
echo shar: Extracting \"'lsolve.m'\" \(1089 characters\)
sed "s/^X//" >'lsolve.m' <<'END_OF_FILE'
Xfunction x = lsolve(L,y,W,NAA)
X%LSOLVE Utility routine for "preconditioned" iterative methods.
X%
X% x = lsolve(L,y,W,NAA)
X%
X% Computes the vector
X%    x = L_p*y
X% where L_p is the A-weighted generalized inverse of L.
X%
X% Typically, L is a p-by-n band matrix with bandwidth n-p+1, W holds
X% a basis for the null space of L, and NAA is a utility matrix which
X% should be computed by routine pinit.
X%
X% Alternatively, L is square and dense, and W and NAA are not needed.
X%
X% Notice that x and y may be matrices, in which case
X%    x(:,i) = L_p*y(:,i) .
X
X% Reference: M. Hanke, "Regularization with differential operators.
X% An iterative approach", J. Numer. Funct. Anal. Optim. 13 (1992),
X% 523-540.
X
X% Per Christian Hansen, UNI-C, and Martin Hanke, Institut fuer
X% Praktische Mathematik, Universitaet Karlsruhe, 05/26/93.
X
X% Initialization.
X[p,n] = size(L); nu = n-p; [py,ly] = size(y);
X
X% Special treatment of square L.
Xif (nu==0), x = L\y; return; end
X 
X% Compute a particular solution
Xx = [[eye(nu),zeros(nu,p)];L]\[zeros(nu,ly);y];
X
X% Perform the necessary projection.
Xx = x - W*(NAA*x);
END_OF_FILE
if test 1089 -ne `wc -c <'lsolve.m'`; then
    echo shar: \"'lsolve.m'\" unpacked with wrong size!
fi
# end of 'lsolve.m'
fi
if test -f 'lsqi.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'lsqi.m'\"
else
echo shar: Extracting \"'lsqi.m'\" \(2636 characters\)
sed "s/^X//" >'lsqi.m' <<'END_OF_FILE'
Xfunction [x_alpha,lambda] = lsqi(U,s,V,b,alpha,x_0)
X%LSQI Least squares minimizaiton with a quadratic inequality constraint.
X%
X% [x_alpha,lambda] = lsqi(U,s,V,b,alpha,x_0)
X% [x_alpha,lambda] = lsqi(U,sm,X,b,alpha,x_0)  ,  sm = [sigma,mu]
X%
X% Least squares minimization with a quadratic inequality constraint:
X%    min || A x - b ||   subject to   || x - x_0 ||     <= alpha
X%    min || A x - b ||   subject to   || L (x - x_0) || <= alpha
X% where x_0 is an initial guess of the solution, and alpha is a
X% positive constant.  Requires either the compact SVD of A saved as
X% U, s, and V, or part of the GSVD of (A,L) saved as U, sm, and X.
X% The regularization parameter lambda is also returned.
X%
X% If alpha is a vector, then x_alpha is a matrix such that
X%    x_alpha = [ x_alpha(1), x_alpha(2), ... ] .
X%
X% If x_0 is not specified, x_0 = 0 is used.
X
X% Reference: T. F. Chan, J. Olkin & D. W. Cooley, "Solving quadratically
X% constrained least squares using block box unconstrained solvers",
X% BIT 32 (1992), 481-495.
X% Extension to the case x_0 ~= 0 by Per Chr. Hansen, UNI-C, 11/20/91.
X% Key point: the initial lambda is almost unaffected by x_0 because
X% || x_unreg || >> || x_0 ||.
X
X% Per Christian Hansen, UNI-C, 11/22/91.
X
X% Initialization.
X[n,p] = size(V); [p,ps] = size(s);
Xif (min(alpha)<0)
X  error('Illegal inequality constraint alpha')
Xend
Xif (nargin==5), x_0 = zeros(n,1); end
Xla  = length(alpha);
Xx_k = zeros(n,la);            lambda = zeros(la,1);
Xsnz = length(find(s(:,1)>0)); beta   = U'*b;
X
Xif (ps == 1)
X  xi = beta(1:snz)./s(1:snz); omega = V'*x_0; s2 = s.^2;
X  x_unreg = V(:,1:snz)*xi; norm_x_unreg = norm(x_unreg - x_0);
X  for k=1:la
X    if (norm_x_unreg <= alpha(k))
X      x_alpha(:,k) = x_unreg; lambda(k) = 0;
X    else
X      lambda_0 = s(snz)*(norm_x_unreg/alpha(k) - 1);
X      lambda(k) = heb_new(lambda_0,alpha(k),s,beta,omega);
X      e = s./(s2 + lambda(k)^2); f = s.*e;
X      x_alpha(:,k) = V(:,1:p)*(e.*beta + (1-f).*omega);
X    end
X  end
Xelse
X  x_u   = V(:,p+1:n)*beta(p+1:n);  ps1   = p-snz+1;
X  xi    = beta(ps1:p)./s(ps1:p,1); gamma = s(:,1)./s(:,2);
X  omega = V\x_0; omega = omega(1:p);
X  x_unreg = V(:,ps1:p)*xi + x_u;
X  norm_Lx_unreg = norm(s(ps1:p,2).*(xi - omega(ps1:p)));
X  for k=1:la
X    if (norm_Lx_unreg <= alpha(k))
X      x_alpha(:,k) = x_unreg; lambda(k) = 0;
X    else
X      lambda_0 = (s(ps1,1)/s(ps1,2))*(norm_Lx_unreg/alpha(k) - 1);
X      lambda(k) = heb_new(lambda_0,alpha(k),s,beta(1:p),omega);
X      e = gamma./(gamma.^2 + lambda(k)^2); f = gamma.*e;
X      x_alpha(:,k) = V(:,1:p)*(e.*beta(1:p)./s(:,2) + ...
X                               (1-f).*s(:,2).*omega) + x_u;
X    end
X  end
Xend
END_OF_FILE
if test 2636 -ne `wc -c <'lsqi.m'`; then
    echo shar: \"'lsqi.m'\" unpacked with wrong size!
fi
# end of 'lsqi.m'
fi
if test -f 'lsqr.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'lsqr.m'\"
else
echo shar: Extracting \"'lsqr.m'\" \(4571 characters\)
sed "s/^X//" >'lsqr.m' <<'END_OF_FILE'
Xfunction [X,rho,eta,F] = lsqr(A,b,k,reorth,s)
X%LSQR Solution of least squares problems by Lanczos bidiagonalization.
X%
X% [X,rho,eta,F] = lsqr(A,b,k,reorth,s)
X%
X% Performs k steps of the LSQR Lanczos bidiagonalization algorithm
X% applied to the system
X%    min || A x - b || .
X% The routine returns all k solutions, stored as columns of
X% the matrix X.  The solution norm and residual norm are returned
X% in eta and rho, respectively.
X%
X% If the singular values s are also provided, lsqr computes the
X% filter factors associated with each step and stores them columnwise
X% in the matrix F.
X%
X% Reorthogonalization is controlled by means of reorth:
X%    reorth = 0 : no reorthogonalization (default),
X%    reorth = 1 : reorthogonalization by means of MGS,
X%    reorth = 2 : Householder-reorthogonalization.
X
X% Reference: C. C. Paige & M. A. Saunders, "LSQR: an algorithm for
X% sparse linear equations and sparse least squares", ACM Trans.
X% Math. Software 8 (1982), 43-71.
X
X% Per Christian Hansen, UNI-C, 05/25/93.
X
X% The fudge threshold is used to prevent filter factors from exploding.
Xfudge_thr = 1e-4;
X
X% Initialization.
Xif (k < 1), error('Number of steps k must be positive'), end
Xif (nargin==3), reorth = 0; end
Xif (nargout==4 & nargin<5), error('Too few input arguments'), end
X[m,n] = size(A); X = zeros(n,k);
Xif (reorth==0)
X  UV = 0;
Xelseif (reorth==1)
X  U = zeros(m,k); V = zeros(n,k); UV = 1;
Xelseif (reorth==2)
X  if (k>=n), error('No. of iterations must satisfy k < n'), end
X  UV = 0; HHU = zeros(m,k); HHV = zeros(n,k);
X  HHalpha = zeros(1,k); HHbeta = HHalpha;
Xelse
X  error('Illegal reorth')
Xend
Xif (nargout > 1)
X  eta = zeros(k,1); rho = eta;
X  c2 = -1; s2 = 0; xnorm = 0; z = 0;
Xend
Xif (nargin==5)
X  ls = length(s);
X  F = zeros(ls,k); Fv = zeros(ls,1); Fw = Fv;
X  s = s.^2;
Xend
X
X% Prepare for LSQR iteration.
Xv = zeros(n,1); x = v; beta = norm(b); 
Xif (beta==0), error('Right-hand side must be nonzero'), end
Xif (reorth==2)
X  [beta,HHbeta(1),HHU(:,1)] = gen_hh(b);
Xend
Xu = b/beta; if (UV), U(:,1) = u; end
Xr = A'*u; alpha = norm(r);
Xif (reorth==2)
X  [alpha,HHalpha(1),HHV(:,1)] = gen_hh(r);
Xend
Xv = r/alpha; if (UV), V(:,1) = v; end
Xphi_bar = beta; rho_bar = alpha; w = v;
Xif (nargin==5), Fv = s/(alpha*beta); Fw = Fv; end
X
X% Perform Lanczos bidiagonalization with/without reorthogonalization.
Xfor i=2:k+1
X
X  alpha_old = alpha; beta_old = beta;
X
X  % Compute A*v - alpha*u.
X  p = A*v - alpha*u;
X  if (reorth==0)
X    beta = norm(p); u = p/beta;
X  elseif (reorth==1)
X    for j=1:i-1, p = p - (U(:,j)'*p)*U(:,j); end
X    beta = norm(p); u = p/beta;
X  else
X    for j=1:i-1
X      p(j:m) = app_hh(p(j:m),HHbeta(j),HHU(j:m,j));
X    end
X    [beta,HHbeta(i),HHU(i:m,i)] = gen_hh(p(i:m));
X    u = zeros(m,1); u(i) = 1;
X    for j=i:-1:1
X      u(j:m) = app_hh(u(j:m),HHbeta(j),HHU(j:m,j));
X    end
X  end
X
X  % Compute A'*u - beta*v.
X  r = A'*u - beta*v;
X  if (reorth==0)
X    alpha = norm(r); v = r/alpha;
X  elseif (reorth==1)
X    for j=1:i-1, r = r - (V(:,j)'*r)*V(:,j); end
X    alpha = norm(r); v = r/alpha;
X  else
X    for j=1:i-1
X      r(j:n) = app_hh(r(j:n),HHalpha(j),HHV(j:n,j));
X    end
X    [alpha,HHalpha(i),HHV(i:n,i)] = gen_hh(r(i:n));
X    v = zeros(n,1); v(i) = 1;
X    for j=i:-1:1
X      v(j:n) = app_hh(v(j:n),HHalpha(j),HHV(j:n,j));
X    end
X  end
X
X  % Store U and V if necessary.
X  if (UV), U(:,i) = u; V(:,i) = v; end
X
X  % Construct and apply orthogonal transformation.
X  rrho = pythag(rho_bar,beta); c1 = rho_bar/rrho;
X  s1 = beta/rrho; theta = s1*alpha; rho_bar = -c1*alpha;
X  phi = c1*phi_bar; phi_bar = s1*phi_bar;
X
X  % Compute solution norm and residual norm if necessary;
X  if (nargout > 1)
X    delta = s2*rrho; gamma_bar = -c2*rrho; rhs = phi - delta*z;
X    z_bar = rhs/gamma_bar; eta(i-1) = pythag(xnorm,z_bar);
X    gamma = pythag(gamma_bar,theta);
X    c2 = gamma_bar/gamma; s2 = theta/gamma;
X    z = rhs/gamma; xnorm = pythag(xnorm,z);
X    rho(i-1) = abs(phi_bar);
X  end
X
X  % If required, compute the filter factors.
X  if (nargin==5)
X
X    if (i==2)
X      Fv_old = Fv;
X      Fv = Fv.*(s - beta^2 - alpha_old^2)/(alpha*beta);
X      F(:,i-1) = (phi/rrho)*Fw;
X    else
X      tmp = Fv;
X      Fv = (Fv.*(s - beta^2 - alpha_old^2) - ...
X                 Fv_old*alpha_old*beta_old)/(alpha*beta);
X      Fv_old = tmp;
X      F(:,i-1) = F(:,i-2) + (phi/rrho)*Fw;
X    end
X    if (i > 3)
X      f = find(abs(F(:,i-2)-1) < fudge_thr & abs(F(:,i-3)-1) < fudge_thr);
X      if (length(f) > 0), F(f,i-1) = ones(length(f),1); end
X    end
X    Fw = Fv - (theta/rrho)*Fw;
X
X  end
X
X  % Update the solution.
X  x = x + (phi/rrho)*w; w = v - (theta/rrho)*w;
X  X(:,i-1) = x;
X
Xend
END_OF_FILE
if test 4571 -ne `wc -c <'lsqr.m'`; then
    echo shar: \"'lsqr.m'\" unpacked with wrong size!
fi
# end of 'lsqr.m'
fi
if test -f 'ltsolve.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'ltsolve.m'\"
else
echo shar: Extracting \"'ltsolve.m'\" \(1083 characters\)
sed "s/^X//" >'ltsolve.m' <<'END_OF_FILE'
Xfunction x = ltsolve(L,y,W,NAA)
X%LTSOLVE Utility routine for "preconditioned" iterative methods.
X%
X% x = ltsolve(L,y,W,NAA)
X%
X% Computes the vector x from the relation
X%    [ x ] = inv([  L  ]')*y .
X%    [ z ]      ([ I 0 ] )
X% Typically, L is a p-by-n band matrix with bandwidth n-p+1.
X% Alternatively, L is square and dense.
X%
X% If W and NAA are also specified, then x = L_p*y instead, where
X% L_p is the A-weighted generalized inverse of L.
X%
X% Notice that x and y may be matrices, in which case x(:,i)
X% corresponds to y(:,i).
X
X% Reference: M. Hanke, "Regularization with differential operators.
X% An iterative approach", J. Numer. Funct. Anal. Optim. 13 (1992),
X% 523-540.
X
X% Per Christian Hansen, UNI-C, and Martin Hanke, Institut fuer
X% Praktische Mathematik, Universitaet Karlsruhe, 05/26/93.
X
X% Initialization.
X[p,n] = size(L); nu = n-p; [ny,ly] = size(y);
X
X% Special treatment of square L.
Xif (nu==0), x = (L')\y; return; end
X 
X% Perform the projection, if necessary.
Xif (nargin > 2), y = y - NAA'*(W'*y); end
X
X% Compute x.
Xx = y'/[L;[zeros(nu,p),eye(nu)]];
Xx = x(:,1:p)';
END_OF_FILE
if test 1083 -ne `wc -c <'ltsolve.m'`; then
    echo shar: \"'ltsolve.m'\" unpacked with wrong size!
fi
# end of 'ltsolve.m'
fi
if test -f 'maxent.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'maxent.m'\"
else
echo shar: Extracting \"'maxent.m'\" \(4639 characters\)
sed "s/^X//" >'maxent.m' <<'END_OF_FILE'
Xfunction [x_lambda,rho,eta,data,X] = maxent(A,b,lambda,w,x0)
X%MAXENT Maximum entropy regularization.
X%
X% [x_lambda,rho,eta] = maxent(A,b,lambda)
X%
X% Maximum entropy regularization:
X%    min { || A x - b ||^2 + lambda^2*x'*log(diag(w)*x) } ,
X% where -x'*log(diag(w)*x) is the entropy of the solution x.
X% If no weights w are specified, unit weights are used.
X%
X% If lambda is a vector, then x_lambda is a matrix such that
X%    x_lambda = [x_lambda(1), x_lambda(2), ... ] .
X%
X% This routine uses a nonlinear conjugate gradient algorithm with "soft"
X% line search and a step-length control that ensures a positive solution.
X% If the starting vector x0 is not specified, then the default is
X%    x0 = norm(b)/norm(A,1)*ones(n,1) .
X
X% Per Christian Hansen, UNI-C and Tommy Elfving, Dept. of Mathematics,
X% Linkoping University, 06/10/92.
X
X% Reference: R. Fletcher, "Practical Methods for Optimization",
X% Second Edition, Wiley, Chichester, 1987.
X
X% Set defaults.
Xflat = 1e-3;     % Measures a flat minimum.
Xflatrange = 10;  % How many iterations before a minimum is considered flat.
Xmaxit = 150;     % Maximum number of CG iterations;
Xminstep = 1e-12; % Determines the accuracy of x_lambda.
Xsigma = 0.5;     % Threshold used in descent test.
Xtau0 = 1e-3;     % Initial threshold used in secant root finder.
X
X% Initialization.
X[m,n] = size(A); x_lambda = zeros(n,length(lambda)); F = zeros(maxit,1);
Xif (min(lambda) <= 0)
X  error('Regularization parameter lambda must be positive')
Xend
Xif (nargin ==3), w  = ones(n,1); end
Xif (nargin < 5), x0 = ones(n,1); end
X
X% Treat each lambda separately.
Xfor j=1:length(lambda);
X
X  % Prepare for nonlinear CG iteration.
X  l2 = lambda(j)^2;
X  x  = x0; Ax = A*x;
X  g  = 2*A'*(Ax - b) + l2*(1 + log(w.*x));
X  p  = -g;
X  r  = Ax - b;
X
X  % Start the nonlinear CG iteration here.
X  delta_x = x; dF = 1; it = 0; phi0 = p'*g;
X  while (norm(delta_x) > minstep*norm(x) & dF > flat & it < maxit & phi0 < 0)
X    it = it + 1;
X
X    % Compute some CG quantities.
X    Ap = A*p; gamma = Ap'*Ap; v = A'*Ap;
X
X    % Determine the steplength alpha by "soft" line search in which
X    % the minimum of phi(alpha) = p'*g(x + alpha*p) is determined to
X    % a certain "soft" tolerance.
X    % First compute initial parameters for the root finder.
X    alpha_left = 0; phi_left = phi0;
X    if (min(p) >= 0)
X      alpha_right = -phi0/(2*gamma);
X      h = 1 + alpha_right*p./x;
X    else
X      % Step-length control to ensure a positive x + alpha*p.
X      I = find(p < 0);
X      alpha_right = min(-x(I)./p(I));
X      h = 1 + alpha_right*p./x; delta = eps;
X      while (min(h) <= 0)
X        alpha_right = alpha_right*(1 - delta);
X        h = 1 + alpha_right*p./x;
X        delta = delta*2;
X      end
X    end
X    z = log(h);
X    phi_right = phi0 + 2*alpha_right*gamma + l2*p'*z;
X    alpha = alpha_right; phi = phi_right;
X
X    if (phi_right <= 0)
X
X      % Special treatment of the case when phi(alpha_right) = 0.
X      z = log(1 + alpha*p./x);
X      g_new = g + l2*z + 2*alpha*v; t = g_new'*g_new;
X      beta = (t - g'*g_new)/(phi - phi0);
X
X    else
X
X      % The regular case: improve the steplength alpha iteratively
X      % until the new step is a descent step.
X      t = 1; u = 1; tau = tau0;
X      while (u > -sigma*t)
X
X        % Use the secant method to improve the root of phi(alpha) = 0
X        % to within an accuracy determined by tau.
X        while (abs(phi/phi0) > tau)
X          alpha = (alpha_left*phi_right - alpha_right*phi_left)/...
X                  (phi_right - phi_left);
X          z = log(1 + alpha*p./x);
X          phi = phi0 + 2*alpha*gamma + l2*p'*z;
X          if (phi > 0)
X            alpha_right = alpha; phi_right = phi;
X          else
X            alpha_left  = alpha; phi_left  = phi;
X          end
X        end
X
X        % To check the descent step, compute u = p'*g_new and
X        % t = norm(g_new)^2, where g_new is the gradient at x + alpha*p.
X        g_new = g + l2*z + 2*alpha*v; t = g_new'*g_new;
X        beta = (t - g'*g_new)/(phi - phi0);
X        u = -t + beta*phi;
X        tau = tau/10;
X
X      end  % End of improvement iteration.
X
X    end  % End of regular case.
X    
X    % Update the iteration vectors.
X    g = g_new; delta_x = alpha*p;
X    x = x + delta_x;
X    p = -g + beta*p;
X    r = r + alpha*Ap;
X    phi0 = p'*g;
X
X    % Compute some norms and check for flat minimum.
X    rho(j,1) = norm(r); eta(j,1) = x'*log(w.*x);
X    F(it) = rho(j,1)^2 + l2*eta(j,1);
X    if (it <= flatrange)
X      dF = 1;
X    else
X      dF = abs(F(it) - F(it-flatrange))/abs(F(it));
X    end
X
X    data(it,:) = [F(it),norm(delta_x),norm(g)];
X    X(:,it) = x;
X
X  end  % End of iteration for x_lambda(j).
X
X  x_lambda(:,j) = x;
X
Xend
END_OF_FILE
if test 4639 -ne `wc -c <'maxent.m'`; then
    echo shar: \"'maxent.m'\" unpacked with wrong size!
fi
# end of 'maxent.m'
fi
if test -f 'mgs.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'mgs.m'\"
else
echo shar: Extracting \"'mgs.m'\" \(432 characters\)
sed "s/^X//" >'mgs.m' <<'END_OF_FILE'
Xfunction Q = mgs(A)
X%MGS Modified Gram-Schmidt orthonormalization.
X%
X% Q = mgs(A)
X%
X% Applies the modified Gram-Schmidt process to the matrix A
X% (assuming that A has full rank), and returns a matrix Q whose
X% columns are orthonormal and span the range of A.
X
X% Per Christian Hansen, UNI-C, 04/11/90.
X
XQ = A; [m,n] = size(A);
X
Xfor k=1:n
X   Q(:,k) = Q(:,k)/norm(Q(:,k));
X   Q(:,k+1:n) = Q(:,k+1:n) - Q(:,k)*(Q(:,k)'*Q(:,k+1:n));
Xend
END_OF_FILE
if test 432 -ne `wc -c <'mgs.m'`; then
    echo shar: \"'mgs.m'\" unpacked with wrong size!
fi
# end of 'mgs.m'
fi
if test -f 'mtsvd.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'mtsvd.m'\"
else
echo shar: Extracting \"'mtsvd.m'\" \(1441 characters\)
sed "s/^X//" >'mtsvd.m' <<'END_OF_FILE'
Xfunction x_k = mtsvd(U,s,V,b,k,L)
X%MTSVD Modified truncated SVD regularization.
X%
X% x_k = mtsvd(U,s,V,b,k,L)
X%
X% Computes the modified TSVD solution:
X%    x_k = V*[ xi_k ] .
X%            [ xi_0 ]
X% Here, xi_k defines the usual TSVD solution
X%    xi_k = inv(diag(s(1:k)))*U(:,1:k)'*b ,
X% and xi_0 is chosen so as to minimize the seminorm || L x_k ||.
X% This leads to choosing xi_0 as follows:
X%    xi_0 = -pinv(L*V(:,k+1:n))*L*V(:,1:k)*xi_k .
X%
X% The truncation parameter must satisfy k > n-p.
X%
X% If k is a vector, then x_k is a matrix such that
X%     x_k = [ x_k(1), x_k(2), ... ] .
X
X% Reference: P. C. Hansen, T. Sekii & H. Shibahashi, "The modified
X% truncated-SVD method for regularization in general form", SIAM J.
X% Sci. Stat. Comput. 13 (1992), 1142-1150.
X
X% Per Christian Hansen, UNI-C, 05/26/93.
X
X% Initialization.
X[m,n1] = size(U); [p,n] = size(L);
Xlk = length(k); kmin = min(k);
Xif (kmin<n-p+1 | max(k)>n)
X  error('Illegal truncation parameter k')
Xend
Xx_k = zeros(n,lk); xi = (U(:,1:n)'*b)./s;
X
X% Compute large enough QR factorization.
X[Q,R] = qr(L*V(:,n:-1:kmin+1));
X
X% Treat each k separately.
Xfor j=1:lk
X  kj = k(j); tmp = V(:,1:kj)*xi(1:kj);
X  if (kj==n)
X    x_k(:,j) = tmp;
X  else
X    z = R(1:n-kj,1:n-kj)\(Q(:,1:n-kj)'*(L*tmp));
X    z = z(n-kj:-1:1);
X    x_k(:,j) = tmp - V(:,kj+1:n)*z;
X  end
X  if (m > n)
X    beta  = U(:,1:n)'*b;
X    beta2 = b'*b - beta'*beta;
X    if (beta2 > 0), rho = sqrt(rho.^2 + beta2); end
X  end
Xend
END_OF_FILE
if test 1441 -ne `wc -c <'mtsvd.m'`; then
    echo shar: \"'mtsvd.m'\" unpacked with wrong size!
fi
# end of 'mtsvd.m'
fi
if test -f 'newton.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'newton.m'\"
else
echo shar: Extracting \"'newton.m'\" \(1866 characters\)
sed "s/^X//" >'newton.m' <<'END_OF_FILE'
Xfunction lambda = newton(lambda_0,delta,s,beta,omega,delta_0)
X%NEWTON Newton iteration (utility routine for DISCREP).
X%
X% lambda = newton(lambda_0,delta,s,beta,omega,delta_0)
X%
X% Uses Newton iteration to find the solution lambda to the equation
X%    || A x_lambda - b || = delta ,
X% where x_lambda is the solution defined by Tikhonov regularization.
X%
X% The initial guess is lambda_0.
X%
X% The norm || A x_lambda - b || is computed via s, beta, omega and
X% delta_0.  Here, s holds either the singular values of A, if L = I,
X% or the c,s-pairs of the GSVD of (A,L), if L ~= I.  Moreover,
X% beta = U'*b and omega is either V'*x_0 or the first p elements of
X% inv(X)*x_0.  Finally, delta_0 is the incompatibility measure.
X
X% Reference: V. A. Morozov, "Methods for Solving Incorrectly Posed
X% Problems", Springer, 1984; Chapter 26.
X
X% Per Christian Hansen, UNI-C, 02/21/92.
X
X% Set defaults.
Xthr = 100*sqrt(eps);  % Relative stopping criterion.
Xit_max = 50;      % Max number of iterations.
X% lambda_0 = eps;
X
X% Initialization.
Xif (lambda_0 < 0)
X  error('Initial guess lambda_0 must be nonnegative')
Xend
X[p,ps] = size(s);
Xif (ps==2), sigma = s(:,1); mu = s(:,2); s = s(:,1)./s(:,2); end
Xs2 = s.^2;
X
X% Iterate; avoid negative values of lambda.
Xlambda = lambda_0^2; step = 1; it = 0;
Xwhile (abs(step) > thr*lambda & abs(step) > thr & it < it_max), it = it+1;
X  f = s2./(s2 + lambda);
X  if (ps==1)
X    r  = (1-f).*(beta - s.*omega);
X    dr = f.*(beta - omega)./(s2 + lambda);
X  else
X    r  = (1-f).*(beta - sigma.*omega);
X    dr = f.*(beta - sigma.*omega)./(s2 + lambda);
X  end
X  res = sqrt(r'*r + delta_0^2);
X  step = -(res - delta)*res/(dr'*r);
X  lambda = lambda + step;
X  if (lambda <= 0), lambda = eps*max(s2); end
Xend
X
X% Terminate with an error if too many iterations.
Xif (abs(step) > thr*lambda)
X  error('Max. number of iterations reached')
Xend
X
Xlambda = sqrt(lambda);
END_OF_FILE
if test 1866 -ne `wc -c <'newton.m'`; then
    echo shar: \"'newton.m'\" unpacked with wrong size!
fi
# end of 'newton.m'
fi
if test -f 'nu.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'nu.m'\"
else
echo shar: Extracting \"'nu.m'\" \(2725 characters\)
sed "s/^X//" >'nu.m' <<'END_OF_FILE'
Xfunction [X,rho,eta,F] = nu(A,b,k,nu,s)
X%NU Brakhage's nu-method.
X%
X% [X,rho,eta,F] = nu(A,b,k,nu,s)
X%
X% Performs k steps of Brakhage's nu-method for the problem
X%    min || A x - b || .
X% The routine returns all k solutions, stored as columns of
X% the matrix X.  The solution norm and residual norm are returned
X% in eta and rho, respectively.
X%
X% If nu is not specified, nu = .5 is the default value, which gives
X% the Chebychev method of Nemirovskii and Polyak.
X%
X% If the singular values s are also provided, nu computes the
X% filter factors associated with each step and stores them
X% columnwise in the matrix F.
X
X% Reference: H. Brakhage, "On ill-posed problems and the method of
X% conjugate gradients"; in H. W. Engl & G. W. Groetsch, "Inverse and
X% Ill-Posed Problems", Academic Press, 1987.
X
X% Martin Hanke, Institut fuer Praktische Mathematik, Universitaet
X% Karlsruhe and Per Christian Hansen, UNI-C, 03/21/92.
X
X% Set parameter.
Xl_steps = 3;      % Number of Lanczos steps for est. of || A ||.
Xfudge   = 0.99;   % Scale A and b by fudge/|| A*L_p ||.
Xfudge_thr = 1e-4; % Used to prevent filter factors from exploding.
X
X% Initialization.
Xif (k < 1), error('Number of steps k must be positive'), end
Xif (nargin==3), nu = .5; end
X[m,n] = size(A); X = zeros(n,k);
Xif (nargout > 1)
X  rho = zeros(k,1); eta = rho;
Xend;
Xif (nargin==5)
X  F = zeros(n,k); Fd = zeros(n,1); s = s.^2;
Xend
XV = zeros(n,l_steps); B = zeros(l_steps+1,l_steps);
Xv = zeros(n,1); eta = zeros(l_steps+1,1);
X
X% Compute a rough estimate of the norm of A by means of a few
X% steps of Lanczos bidiagonalization, and scale A and b such
X% that || A || is slightly less than one.
Xbeta = norm(b); u = b/beta;
Xfor i=1:l_steps
X  r = A'*u - beta*v;
X  alpha = norm(r); v = r/alpha;
X  B(i,i) = alpha; V(:,i) = v;
X  p = A*v - alpha*u;
X  beta = norm(p); u = p/beta;
X  B(i+1,i) = beta;
Xend
Xscale = fudge/norm(B); A = scale*A; b = scale*b;
Xif (nargin==5), s = scale^2*s; end
X
X% Prepare for iteration.
Xx = zeros(n,1);
Xd = A'*b;
Xr = d;
Xif (nargout>1), z = b; end
X
X% Iterate.
Xfor j=0:k-1
X
X  alpha = 4*(j+nu)*(j+nu+0.5)/(j+2*nu)/(j+2*nu+0.5);
X  beta  = (j+nu)*(j+1)*(j+0.5)/(j+2*nu)/(j+2*nu+0.5)/(j+nu+1);
X  Ad = A*d; AAd = A'*Ad;
X  x  = x + alpha*d;
X  r  = r - alpha*AAd;
X  d  = r + beta*d;
X  X(:,j+1) = x;
X  if (nargout>1)
X    z = z - alpha*Ad; rho(j+1) = norm(z)/scale;
X  end;
X  if (nargout>2), eta(j+1) = norm(x); end;
X
X  if (nargin==5)
X    if (j==0)
X      F(:,1) = alpha*s;
X      Fd = s - s.*F(:,1) + beta*s;
X    else
X      F(:,j+1) = F(:,j) + alpha*Fd;
X      Fd = s - s.*F(:,j+1) + beta*Fd;
X    end
X    if (j > 1)
X      f = find(abs(F(:,j)-1) < fudge_thr & abs(F(:,j-1)-1) < fudge_thr);
X      if (length(f) > 0), F(f,j+1) = ones(length(f),1); end
X    end
X  end
X
Xend
END_OF_FILE
if test 2725 -ne `wc -c <'nu.m'`; then
    echo shar: \"'nu.m'\" unpacked with wrong size!
fi
# end of 'nu.m'
fi
if test -f 'parallax.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'parallax.m'\"
else
echo shar: Extracting \"'parallax.m'\" \(1856 characters\)
sed "s/^X//" >'parallax.m' <<'END_OF_FILE'
Xfunction [A,b] = parallax(n)
X%PARALLAX Stellar parallax problem with 28 fixed, real observations.
X%
X% [A,b] = parallax(n)
X%
X% Stellar parallax problem with 28 fixed, real observations.
X%
X% The underlying problem is a Fredholm integral equation of the
X% first kind with kernel
X%    K(s,t) = (1/sigma*sqrt(2*pi))*esp(-0.5*((s-t)/sigma)^2) ,
X% and it is discretized by means of a Galerkin method with n
X% orthonormal basis functions.  The right-hand side consists of
X% a measured distribution function of stellar parallaxes, and its
X% length is fixed, m = 26.  The exact solution, which represents
X% the true distribution of stellar parallaxes, in not known.
X
X% Reference: W. M. Smart, "Stellar Dynamics", Cambridge
X% University Press, 1938; p. 30.
X
X% Discretized by Galerkin method with orthonormal box functions;
X% 2-D integration is done by means of the computational molecule:
X%       1   4   1
X%       4  16   1
X%       1   4   1
X
X% Per Christian Hansen, UNI-C, 09/16/92.
X
X% Initialization.
Xa = 0; b = 0.1; m = 26; sigma = 0.014234;
Xhs = 0.130/m; hx = (b-a)/n; hsh = hs/2; hxh = hx/2;
Xss = (-0.03 + [0:m-1]'*hs)*ones(1,n);
Xxx = ones(m,1)*(a + [0:n-1]*hx);
X
X% Set up the matrix.
XA =     16*exp(-0.5*((ss+hsh - xx-hxh)/sigma).^2);
XA = A + 4*(exp(-0.5*((ss+hsh - xx    )/sigma).^2) + ...
X           exp(-0.5*((ss+hsh - xx-hx )/sigma).^2) + ...
X           exp(-0.5*((ss     - xx-hxh)/sigma).^2) + ...
X           exp(-0.5*((ss+hs  - xx-hxh)/sigma).^2));
XA = A +   (exp(-0.5*((ss     - xx    )/sigma).^2) + ...
X           exp(-0.5*((ss+hs  - xx    )/sigma).^2) + ...
X           exp(-0.5*((ss     - xx-hx )/sigma).^2) + ...
X           exp(-0.5*((ss+hs  - xx-hx )/sigma).^2));
XA = sqrt(hs*hx)/(36*sigma*sqrt(2*pi))*A;
X
X% Set up the normalized right-hand side.
Xb = [3;7;7;17;27;39;46;51;56;50;43;45;43;32;33;29;...
X     21;12;17;13;15;12;6;6;5;5]/(sqrt(hs)*640);
END_OF_FILE
if test 1856 -ne `wc -c <'parallax.m'`; then
    echo shar: \"'parallax.m'\" unpacked with wrong size!
fi
# end of 'parallax.m'
fi
if test -f 'pcgls.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'pcgls.m'\"
else
echo shar: Extracting \"'pcgls.m'\" \(2810 characters\)
sed "s/^X//" >'pcgls.m' <<'END_OF_FILE'
Xfunction [X,rho,eta,F] = pcgls(A,L,W,b,k,sm)
X%PCGLS "Preconditioned" conjugate gradients appl. implicitly to normal equations.
X% [X,rho,eta,F] = pcgls(A,L,W,b,k,sm)
X%
X% Performs k steps of the `preconditioned' conjugate gradient
X% algorithm applied implicitly to the normal equations
X%    (A*L_p)'*(A*L_p)*x = (A*L_p)'*b ,
X% where L_p is the A-weighted generalized inverse of L.  Notice
X% that the matrix W holding a basis for the null space of L must
X% also be specified.
X%
X% The routine returns all k solutions, stored as columns of the
X% matrix X.  The solution seminorm and residual norm are returned in eta
X% and rho, respectively.
X%
X% If the generalized singular values sm of (A,L) are also provided,
X% pcgls computes the filter factors associated with each step and
X% stores them columnwise in the matrix F.
X
X% References: A. Bjorck, "Least Squares Methods", in P. G.
X% Ciarlet & J. L Lions (Eds.), "Handbook of  Numerical Analysis,
X% Vol. I", Elsevier, Amsterdam, 1990; p. 560.
X% M. Hanke, "Regularization with differential operators.  An itera-
X% tive approach", J. Numer. Funct. Anal. Optim. 13 (1992), 523-540.
X% C. R. Vogel, "Solving ill-conditioned linear systems using the 
X% conjugate gradient method", Report, Dept. of Mathematical
X% Sciences, Montana State University, 1987.
X 
X% Per Christian Hansen, UNI-C and Martin Hanke, Institut fuer
X% Praktische Mathematik, Universitaet Karlsruhe, 11/05/92.
X
X% The fudge threshold is used to prevent filter factors from exploding.
Xfudge_thr = 1e-4;
X 
X% Initialization
Xif (k < 1), error('Number of steps k must be positive'), end
X[m,n] = size(A); [p,n1] = size(L); X = zeros(n,k);
Xif (nargout > 1)
X  eta = zeros(k,1); rho = eta;
Xend
Xif (nargout>3 & nargin==5), error('Too few imput arguments'), end
Xif (nargin==6)
X  F = zeros(p,k); Fd = zeros(p,1); gamma = (sm(:,1)./sm(:,2)).^2;
Xend
X
X% Prepare for computations with L_p.
X[NAA,x_0] = pinit(W,A,b);
X
X% Prepare for CG iteartion.
Xx  = x_0;
Xr  = b - A*x_0; s = A'*r;
Xq1 = ltsolve(L,s);
Xq  = lsolve(L,q1,W,NAA);
Xz  = q;
Xdq = s'*q;
Xif (nargout>2), z1 = q1; x1 = zeros(p,1); end
X
X% Iterate.
Xfor j=1:k
X
X  Az  = A*z; alpha = dq/(Az'*Az);
X  x   = x + alpha*z;
X  r   = r - alpha*Az; s = A'*r;
X  q1  = ltsolve(L,s);
X  q   = lsolve(L,q1,W,NAA);
X  dq2 = s'*q; beta = dq2/dq;
X  dq  = dq2;
X  z   = q + beta*z;
X  X(:,j) = x;
X  if (nargout>1), rho(j) = norm(r); end
X  if (nargout>2)
X    x1 = x1 + alpha*z1; z1 = q1 + beta*z1; eta(j) = norm(x1);
X  end
X
X  if (nargin==6)
X    if (j==1)
X      F(:,1) = alpha*gamma;
X      Fd = gamma - gamma.*F(:,1) + beta*gamma;
X    else
X      F(:,j) = F(:,j-1) + alpha*Fd;
X      Fd = gamma - gamma.*F(:,j) + beta*Fd;
X    end
X    if (j > 2)
X      f = find(abs(F(:,j-1)-1) < fudge_thr & abs(F(:,j-2)-1) < fudge_thr);
X      if (length(f) > 0), F(f,j) = ones(length(f),1); end
X    end
X  end
X
Xend
END_OF_FILE
if test 2810 -ne `wc -c <'pcgls.m'`; then
    echo shar: \"'pcgls.m'\" unpacked with wrong size!
fi
# end of 'pcgls.m'
fi
if test -f 'phillips.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'phillips.m'\"
else
echo shar: Extracting \"'phillips.m'\" \(1685 characters\)
sed "s/^X//" >'phillips.m' <<'END_OF_FILE'
Xfunction [A,b,x] = phillips(n)
X%PHILLIPS Phillips' "famous" test problem.
X%
X% [A,b,x] = phillips(n)
X%
X% Discretization of the `famous' first-kind Fredholm integral
X% equation deviced by D. L. Phillips.  Define the function
X%    phi(x) = | 1 + cos(x*pi/3) ,  |x| <  3 .
X%             | 0               ,  |x| >= 3
X% Then the kernel K, the solution f, and the right-hand side
X% g are given by:
X%    K(s,t) = phi(s-t) ,
X%    f(t)   = phi(t) ,
X%    g(s)   = (6-|s|)*(1+.5*cos(s*pi/3)) + 9/(2*pi)*sin(|s|*pi/3) .
X% Both integration intervals are [-6,6].
X%
X% The order n must be a multiple of 4.
X
X% Reference: D. L. Phillips, "A technique for the numerical solution
X% of certain integral equations of the first kind", J. ACM 9
X% (1962), 84-97.
X
X% Discretized by Galerkin method with orthonormal box functions.
X
X% Per Christian Hansen, UNI-C, 09/17/92.
X
X% Check input.
Xif (rem(n,4)~=0), error('The order n must be a multiple of 4'), end
X
X% Compute the matrix A.
Xh = 12/n; n4 = n/4; r1 = zeros(1,n);
Xc = cos([-1:n4]*4*pi/n);
Xr1(1:n4) = h + 9/(h*pi^2)*(2*c(2:n4+1) - c(1:n4) - c(3:n4+2));
Xr1(n4+1) = h/2 + 9/(h*pi^2)*(cos(4*pi/n)-1);
XA = toeplitz(r1);
X
X% Compute the right-hand side b.
Xif (nargout>1),
X  b = zeros(n,1); c = pi/3;
X  for i=n/2+1:n
X    t1 = -6 + i*h; t2 = t1 - h;
X    b(i) =   t1*(6-abs(t1)/2) ...
X           + ((3-abs(t1)/2)*sin(c*t1) - 2/c*(cos(c*t1) - 1))/c ...
X           - t2*(6-abs(t2)/2) ...
X           - ((3-abs(t2)/2)*sin(c*t2) - 2/c*(cos(c*t2) - 1))/c;
X    b(n-i+1) = b(i);
X  end
X  b = b/sqrt(h);
Xend
X
X% Compute the solution x.
Xif (nargout==3),
X  x = zeros(n,1);
X  x(2*n4+1:3*n4) = (h + diff(sin([0:h:(3+10*eps)]'*c))/c)/sqrt(h);
X  x(n4+1:2*n4) = x(3*n4:-1:2*n4+1);
Xend
END_OF_FILE
if test 1685 -ne `wc -c <'phillips.m'`; then
    echo shar: \"'phillips.m'\" unpacked with wrong size!
fi
# end of 'phillips.m'
fi
if test -f 'picard.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'picard.m'\"
else
echo shar: Extracting \"'picard.m'\" \(1286 characters\)
sed "s/^X//" >'picard.m' <<'END_OF_FILE'
Xfunction xi = picard(U,s,b,d)
X%PICARD Visual inspection of the Picard condition.
X%
X% xi = picard(U,s,b,d)
X% xi = picard(U,sm,b,d)  ,  sm = [sigma,mu]
X%
X% Plots the singular values, s(i), the abs. value of the Fourier
X% coefficients, |U(:,i)'*b|, and a (possibly smoothed) curve of
X% the solution coefficients xi(i) = |U(:,i)'*b|/s(i).
X%
X% If s = [sigma,mu], where gamma = sigma./mu are the generalized
X% singular values, then this routine plots gamma(i), |U(:,i)'*b|,
X% and (smoothed) xi(i) = |U(:,i)'*b|/gamma(i).
X%
X% The smoothing is a geometric mean over 2*d+1 points, centered
X% at point # i. If nargin = 3, then d = 0 (i.e, no smothing).
X
X% Reference: P. C. Hansen, "The discrete Picard condition for
X% discrete ill-posed problems", BIT 30 (1990), 658-672.
X
X% Per Christian Hansen, UNI-C, 04/11/90.
X
X% Initialization.
X[n,ps] = size(s); beta = abs(U(:,1:n)'*b); eta = zeros(n,1);
Xif (nargin==3), d = 0; end;
Xif (ps==2), s = s(:,1)./s(:,2); end
Xd21 = 2*d+1; keta = 1+d:n-d;
Xfor i=keta
X  eta(i) = (prod(beta(i-d:i+d))^(1/d21))/s(i);
Xend
X
X% Plot the data.
Xsemilogy(1:n,s,'-',1:n,beta,'x',keta,eta(keta),'o')
Xxlabel('i')
Xif (ps==1)
X  title('--- = sing. values     x = rhs. coef.     o = solution coef.')
Xelse
X  title('--- = gen. sing. values     x = rhs. coef.     o = sol. coef.')
Xend
END_OF_FILE
if test 1286 -ne `wc -c <'picard.m'`; then
    echo shar: \"'picard.m'\" unpacked with wrong size!
fi
# end of 'picard.m'
fi
if test -f 'pinit.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'pinit.m'\"
else
echo shar: Extracting \"'pinit.m'\" \(1004 characters\)
sed "s/^X//" >'pinit.m' <<'END_OF_FILE'
Xfunction [NAA,x_0] = pinit(W,A,b)
X%PINIT Utility init.-procedure for "preconditioned" iterative methods.
X%
X% NAA = pinit(W,A)
X% [NAA,x_0] = pinit(W,A,b)
X%
X% Initialization for `preconditioning' of general-form problems.
X% Here, W holds a basis for the null space of L.
X%
X% Determines the matrix NAA needed in the iterative routines for
X% treating regularization problems in general form.
X%
X% If b is also specified then x_0, the component of the solution in
X% the null space of L, is also computed.
X
X% Reference: M. Hanke, "Regularization with differential operators.
X% An iterative approach", J. Numer. Funct. Anal. Optim. 13 (1992),
X% 523-540.
X
X% Per Christian Hansen, UNI-C, and Martin Hanke, Institut fuer
X% Praktische Mathematik, Universitaet Karlsruhe, 05/26/93.
X 
X% Initialization.
X[n,nu] = size(W);
X
X% Special treatment of square L.
Xif (nu==0), NAA = []; x_0 = zeros(n,1); return, end
X
X% Compute NAA.
XT = pinv(A*W);
XNAA = T*A;
X
X% If required, also compute x_0.
Xif (nargin==3), x_0 = W*(T*b); end
END_OF_FILE
if test 1004 -ne `wc -c <'pinit.m'`; then
    echo shar: \"'pinit.m'\" unpacked with wrong size!
fi
# end of 'pinit.m'
fi
if test -f 'plot_lc.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'plot_lc.m'\"
else
echo shar: Extracting \"'plot_lc.m'\" \(1739 characters\)
sed "s/^X//" >'plot_lc.m' <<'END_OF_FILE'
Xfunction plot_lc(rho,eta,marker,ps,reg_param)
X%PLOT_LC Plot the L-curve.
X%
X% plot_lc(rho,eta,marker,ps,reg_param)
X%
X% Plots the L-shaped curve of the solution norm
X%    eta = || x ||      if   ps = 1
X%    eta = || L x ||    if   ps = 2
X% as a function of the residual norm rho = || A x - b ||.  If ps is
X% not specified, the value ps = 1 is assumed.
X%
X% The text string marker is used as marker.  If marker is not
X% specified, the marker '-' is used.
X%
X% If a fifth argument reg_param is present, holding the regularization
X% parameters corresponding to rho and eta, then some points on the
X% L-curve are identified by their corresponding parameter.
X
X% Per Christian Hansen, UNI-C, 03/17/93.
X
X% Set defaults.
Xif (nargin==2), marker = '-'; end  % Default marker.
Xif (nargin < 4), ps = 1; end       % Std. form is default.
Xnp = 10;                           % Number of identified points.
X
X% Initialization.
Xif (ps < 1 | ps > 2), error('Illegal value of ps'), end
Xn = length(rho); ni = round(n/np);
X
X% Make plot.
Xif (max(eta)/min(eta) > 10 | max(rho)/min(rho) > 10)
X  if (nargin < 5)
X    loglog(rho,eta,marker)
X  else
X    loglog(rho,eta,marker,rho(ni:ni:n),eta(ni:ni:n),'x')
X    HoldState = ishold; hold on;
X    for k = ni:ni:n
X      text(rho(k),eta(k),num2str(reg_param(k)));
X    end
X    if (~HoldState), hold off; end
X  end
Xelse
X  if (nargin < 5)
X    plot(rho,eta,marker)
X  else
X    plot(rho,eta,marker,rho(ni:ni:n),eta(ni:ni:n),'x')
X    HoldState = ishold; hold on;
X    for k = ni:ni:n
X      text(rho(k),eta(k),num2str(reg_param(k)));
X    end
X    if (~HoldState), hold off; end
X  end
Xend
Xxlabel('residual norm || A x - b ||')
Xif (ps==1)
X  ylabel('solution norm || x ||')
Xelse
X  ylabel('solution semi-norm || L x ||')
Xend
Xtitle('L-curve')
END_OF_FILE
if test 1739 -ne `wc -c <'plot_lc.m'`; then
    echo shar: \"'plot_lc.m'\" unpacked with wrong size!
fi
# end of 'plot_lc.m'
fi
if test -f 'plsqr.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'plsqr.m'\"
else
echo shar: Extracting \"'plsqr.m'\" \(5334 characters\)
sed "s/^X//" >'plsqr.m' <<'END_OF_FILE'
Xfunction [X,rho,eta,F] = plsqr(A,L,W,b,k,reorth,sm)
X%PLSQR "Preconditioned" version of the LSQR Lanczos bidiagonalization algorithm.
X%
X% [X,rho,eta,F] = plsqr(A,L,W,b,k,reorth,sm)
X%
X% Performs k steps of the `preconditioned' LSQR Lanczos
X% bidiagonalization algorithm applied to the system
X%    min || (A*L_p) x - b || ,
X% where L_p is the A-weighted generalized inverse of L.  Notice
X% that the matrix W holding a basis for the null space of L must
X% also be specified.
X%
X% The routine returns all k solutions, stored as columns of
X% the matrix X.  The solution seminorm and the residual norm are
X% returned in eta and rho, respectively.
X%
X% If the generalized singular values sm of (A,L) are also provided,
X% then glsqr computes the filter factors associated with each step
X% and stores them columnwise in the matrix F.
X%
X% Reorthogonalization is controlled by means of reorth:
X%    reorth = 0 : no reorthogonalization (default),
X%    reorth = 1 : reorthogonalization by means of MGS,
X%    reorth = 2 : Householder-reorthogonalization.
X
X% References: C. C. Paige & M. A. Saunders, "LSQR: an algorithm for
X% sparse linear equations and sparse least squares", ACM Trans.
X% Math. Software 8 (1982), 43-71.
X% M. Hanke, "Regularization with differential operators.  An itera-
X% tive approach", J. Numer. Funct. Anal. Optim. 13 (1992), 523-540.
X
X% Per Christian Hansen, UNI-C, 05/26/93.
X
X% The fudge threshold is used to prevent filter factors from exploding.
Xfudge_thr = 1e-4;
X
X% Initialization
Xif (k < 1), error('Number of steps k must be positive'), end
Xif (nargin==5), reorth = 0; end
Xif (nargout==4 & nargin<7), error('Too few input arguments'), end
X[m,n] = size(A); X = zeros(n,k); [pp,n1] = size(L);
Xif (n1 ~= n | m < n | n < pp)
X  error('Incorrect dimensions of A and L')
Xend
Xif (reorth==0)
X  UV = 0;
Xelseif (reorth==1)
X  U = zeros(m,k); V = zeros(pp,k); UV = 1;
Xelseif (reorth==2)
X  if (k>=n), error('No. of iterations must satisfy k < n'), end
X  UV = 0; HHU = zeros(m,k); HHV = zeros(pp,k);
X  HHalpha = zeros(1,k); HHbeta = HHalpha;
Xelse
X  error('Illegal reorth')
Xend
Xif (nargout > 1)
X  eta = zeros(k,1); rho = eta;
X  c2 = -1; s2 = 0; xnorm = 0; z = 0;
Xend
Xif (nargin==7)
X  [ls,ms] = size(sm);
X  F = zeros(ls,k); Fv = zeros(ls,1); Fw = Fv;
X  s = (sm(:,1)./sm(:,2)).^2;
Xend
X
X% Prepare for computations with L_p.
X[NAA,x_0] = pinit(W,A,b);
X
X% By subtracting the component A*x_0 from b we ensure that
X% the corrent residual norms are computed.
Xb = b - A*x_0;
X
X% Prepare for LSQR iteration.
Xv = zeros(pp,1); x = v; beta = norm(b);
Xif (beta==0), error('Right-hand side must be nonzero'), end
Xif (reorth==2)
X  [beta,HHbeta(1),HHU(:,1)] = gen_hh(b);
Xend
Xu = b/beta; if (UV), U(:,1) = u; end
Xr = ltsolve(L,A'*u,W,NAA); alpha = norm(r);
Xif (reorth==2)
X  [alpha,HHalpha(1),HHV(:,1)] = gen_hh(r);
Xend
Xv = r/alpha; if (UV), V(:,1) = v; end
Xphi_bar = beta; rho_bar = alpha; w = v;
Xif (nargin==7), Fv = s/(alpha*beta); Fw = Fv; end
X
X% Perform Lanczos bidiagonalization with/without reorthogonalization.
Xfor i=2:k+1
X
X  alpha_old = alpha; beta_old = beta;
X
X  % Compute (A*L_p)*v - alpha*u.
X  p = A*lsolve(L,v,W,NAA) - alpha*u;
X  if (reorth==0)
X    beta = norm(p); u = p/beta;
X  elseif (reorth==1)
X    for j=1:i-1, p = p - (U(:,j)'*p)*U(:,j); end
X    beta = norm(p); u = p/beta;
X  else
X    for j=1:i-1
X      p(j:m) = app_hh(p(j:m),HHbeta(j),HHU(j:m,j));
X    end
X    [beta,HHbeta(i),HHU(i:m,i)] = gen_hh(p(i:m));
X    u = zeros(m,1); u(i) = 1;
X    for j=i:-1:1
X      u(j:m) = app_hh(u(j:m),HHbeta(j),HHU(j:m,j));
X    end
X  end
X
X  % Compute L_p'*A'*u - beta*v.
X  r = ltsolve(L,A'*u,W,NAA) - beta*v;
X  if (reorth==0)
X    alpha = norm(r); v = r/alpha;
X  elseif (reorth==1)
X    for j=1:i-1, r = r - (V(:,j)'*r)*V(:,j); end
X    alpha = norm(r); v = r/alpha;
X  else
X    for j=1:i-1
X      r(j:pp) = app_hh(r(j:pp),HHalpha(j),HHV(j:pp,j));
X    end
X    [alpha,HHalpha(i),HHV(i:pp,i)] = gen_hh(r(i:pp));
X    v = zeros(pp,1); v(i) = 1;
X    for j=i:-1:1
X      v(j:pp) = app_hh(v(j:pp),HHalpha(j),HHV(j:pp,j));
X    end
X  end
X
X  % Store U and V if necessary.
X  if (UV), U(:,i) = u; V(:,i) = v; end
X
X  % Construct and apply orthogonal transformation.
X  rrho = pythag(rho_bar,beta); c1 = rho_bar/rrho;
X  s1 = beta/rrho; theta = s1*alpha; rho_bar = -c1*alpha;
X  phi = c1*phi_bar; phi_bar = s1*phi_bar;
X
X  % Compute solution norm and residual norm if necessary;
X  if (nargout > 1)
X    delta = s2*rrho; gamma_bar = -c2*rrho; rhs = phi - delta*z;
X    z_bar = rhs/gamma_bar; eta(i-1) = pythag(xnorm,z_bar);
X    gamma = pythag(gamma_bar,theta);
X    c2 = gamma_bar/gamma; s2 = theta/gamma;
X    z = rhs/gamma; xnorm = pythag(xnorm,z);
X    rho(i-1) = abs(phi_bar);
X  end
X
X  % If required, compute the filter factors.
X  if (nargin==7)
X
X    if (i==2)
X      Fv_old = Fv;
X      Fv  = Fv.*(s - beta^2 - alpha_old^2)/(alpha*beta);
X      F(:,i-1) = (phi/rrho)*Fw;
X    else
X      tmp = Fv;
X      Fv = (Fv.*(s - beta^2 - alpha_old^2) - ...
X                 Fv_old*alpha_old*beta_old)/(alpha*beta);
X      Fv_old = tmp;
X      F(:,i-1) = F(:,i-2) + (phi/rrho)*Fw;
X    end
X    if (i > 3)
X      f = find(abs(F(:,i-2)-1) < fudge_thr & abs(F(:,i-3)-1) < fudge_thr);
X      if (length(f) > 0), F(f,i-1) = ones(length(f),1); end
X    end
X    Fw = Fv - (theta/rrho)*Fw;
X
X  end
X
X  % Update the solution.
X  x = x + (phi/rrho)*w; w = v - (theta/rrho)*w;
X  X(:,i-1) = lsolve(L,x,W,NAA) + x_0;
X
Xend
END_OF_FILE
if test 5334 -ne `wc -c <'plsqr.m'`; then
    echo shar: \"'plsqr.m'\" unpacked with wrong size!
fi
# end of 'plsqr.m'
fi
if test -f 'pnu.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'pnu.m'\"
else
echo shar: Extracting \"'pnu.m'\" \(3323 characters\)
sed "s/^X//" >'pnu.m' <<'END_OF_FILE'
Xfunction [X,rho,eta,F] = pnu(A,L,W,b,k,nu,sm)
X%PNU "Preconditioned" version of Brakhage's nu-method.
X%
X% [X,rho,eta,F] = pnu(A,L,W,b,k,nu,sm)
X%
X% Performs k steps of a `preconditioned' version of Brakhage's
X% nu-method for the problem
X%    min || (A*L_p) x - b || ,
X% where L_p is the A-weighted generalized inverse of L.  Notice
X% that the matrix W holding a basis for the null space of L must
X% also be specified.
X%
X% The routine returns all k solutions, stored as columns of
X% the matrix X.  The solution seminorm and residual norm are returned
X% in eta and rho, respectively.
X%
X% If nu is not specified, nu = .5 is the default value, which gives
X% the Chebychev method of Nemirovskii and Polyak.
X%
X% If the generalized singular values sm of (A,L) are also provided,
X% then pnu computes the filter factors associated with each step and
X% stores them columnwise in the matrix F.
X
X% Reference: H. Brakhage, "On ill-posed problems and the method of
X% conjugate gradients"; in H. W. Engl & G. W. Groetsch, "Inverse and
X% Ill-Posed Problems", Academic Press, 1987.
X 
X% Martin Hanke, Institut fuer Praktische Mathematik, Universitaet
X% Karlsruhe and Per Christian Hansen, UNI-C, 06/25/92.
X
X% Set parameters.
Xl_steps = 3;      % Number of Lanczos steps for est. of || A*L_p ||.
Xfudge   = 0.99;   % Scale A and b by fudge/|| A*L_p ||.
Xfudge_thr = 1e-4; % Used to prevent filter factors from exploding.
X 
X% Initialization.
Xif (k < 1), error('Number of steps k must be positive'), end
Xif (nargin==5), nu = .5; end
X[m,n] = size(A); [p,n1] = size(L); X = zeros(n,k);
Xif (nargout > 1)
X  rho = zeros(k,1); eta = rho;
Xend;
Xif (nargin==7)
X  F = zeros(n,k); Fd = zeros(n,1); s = (sm(:,1)./sm(:,2)).^2;
Xend
XV = zeros(p,l_steps); B = zeros(l_steps+1,l_steps);
Xv = zeros(p,1); eta = zeros(l_steps+1,1);
X 
X% Prepare for computations with L_p.
X[NAA,x_0] = pinit(W,A,b); x1 = x_0;
X
X% Compute a rough estimate of || A*L_p || by means of a few
X% steps of Lanczos bidiagonalization, and scale A and b such
X% that || A*L_p || is slightly less than one.
Xb_0 = b - A*x_0; beta = norm(b_0); u = b_0/beta;
Xfor i=1:l_steps
X  r = ltsolve(L,A'*u,W,NAA) - beta*v;
X  alpha = norm(r); v = r/alpha;
X  B(i,i) = alpha; V(:,i) = v;
X  p = A*lsolve(L,v,W,NAA) - alpha*u;
X  beta = norm(p); u = p/beta;
X  B(i+1,i) = beta;
Xend
Xscale = fudge/norm(B); A = scale*A; b = scale*b;
Xif (nargin==7), s = scale^2*s; end
X
X% Prepare for iteration.
Xx  = x_0;
Xz  = -scale*b_0;
Xr  = A'*z;
Xd1 = ltsolve(L,r);
Xd  = lsolve(L,d1,W,NAA);
Xif (nargout>2), x1 = L*x_0; end
X
X% Iterate.
Xfor j=0:k-1
X
X  alpha = 4*(j+nu)*(j+nu+0.5)/(j+2*nu)/(j+2*nu+0.5);
X  beta  = -(j+nu)*(j+1)*(j+0.5)/(j+2*nu)/(j+2*nu+0.5)/(j+nu+1);
X  Ad  = A*d; AAd = A'*Ad;
X  x   = x - alpha*d;
X  r   = r - alpha*AAd;
X  rr1 = ltsolve(L,r);
X  rr  = lsolve(L,rr1,W,NAA);
X  d   = rr - beta*d;
X  X(:,j+1) = x;
X  if (nargout>1 )
X    z = z - alpha*Ad; rho(j+1) = norm(z)/scale;
X  end;
X  if (nargout>2)
X    x1 = x1 - alpha*d1; d1 = rr1 - beta*d1;
X    eta(j+1) = norm(x1);
X  end;
X
X  if (nargin==7)
X    if (j==0)
X      F(:,1) = alpha*s;
X      Fd = s - s.*F(:,1) + beta*s;
X    else
X      F(:,j+1) = F(:,j) + alpha*Fd;
X      Fd = s - s.*F(:,j+1) + beta*Fd;
X    end
X    if (j > 1)
X      f = find(abs(F(:,j)-1) < fudge_thr & abs(F(:,j-1)-1) < fudge_thr);
X      if (length(f) > 0), F(f,j+1) = ones(length(f),1); end
X    end
X  end
X
Xend
END_OF_FILE
if test 3323 -ne `wc -c <'pnu.m'`; then
    echo shar: \"'pnu.m'\" unpacked with wrong size!
fi
# end of 'pnu.m'
fi
if test -f 'ppbrk.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'ppbrk.m'\"
else
echo shar: Extracting \"'ppbrk.m'\" \(93 characters\)
sed "s/^X//" >'ppbrk.m' <<'END_OF_FILE'
Xfunction [breaks,coefs,l,k,d]=ppbrk(pp,print)
X%PPBRK Dummy function for Regularization Tools
END_OF_FILE
if test 93 -ne `wc -c <'ppbrk.m'`; then
    echo shar: \"'ppbrk.m'\" unpacked with wrong size!
fi
# end of 'ppbrk.m'
fi
if test -f 'ppcut.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'ppcut.m'\"
else
echo shar: Extracting \"'ppcut.m'\" \(76 characters\)
sed "s/^X//" >'ppcut.m' <<'END_OF_FILE'
Xfunction pc=ppcut(pp,bounds)
X%PPCUT Dummy function for Regularization Tools
END_OF_FILE
if test 76 -ne `wc -c <'ppcut.m'`; then
    echo shar: \"'ppcut.m'\" unpacked with wrong size!
fi
# end of 'ppcut.m'
fi
if test -f 'ppmak.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'ppmak.m'\"
else
echo shar: Extracting \"'ppmak.m'\" \(81 characters\)
sed "s/^X//" >'ppmak.m' <<'END_OF_FILE'
Xfunction pp=ppmak(breaks,coefs,d)
X%PPMAK Dummy function for Regularization Tools
END_OF_FILE
if test 81 -ne `wc -c <'ppmak.m'`; then
    echo shar: \"'ppmak.m'\" unpacked with wrong size!
fi
# end of 'ppmak.m'
fi
if test -f 'ppual.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'ppual.m'\"
else
echo shar: Extracting \"'ppual.m'\" \(71 characters\)
sed "s/^X//" >'ppual.m' <<'END_OF_FILE'
Xfunction v=ppual(pp,xx)
X%PPUAL Dummy function for Regularization Tools
END_OF_FILE
if test 71 -ne `wc -c <'ppual.m'`; then
    echo shar: \"'ppual.m'\" unpacked with wrong size!
fi
# end of 'ppual.m'
fi
if test -f 'pythag.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'pythag.m'\"
else
echo shar: Extracting \"'pythag.m'\" \(314 characters\)
sed "s/^X//" >'pythag.m' <<'END_OF_FILE'
Xfunction x = pythag(y,z)
X%PYTHAG Computes sqrt( y^2 + z^2 ).
X%
X% x = pythag(y,z)
X%
X% Returns sqrt(y^2 + z^2) but is careful to scale to avoid overflow.
X
X% Christian H. Bischof, Argonne National Laboratory, 03/31/89.
X
Xrmax = max(abs([y;z]));
Xif (rmax==0)
X  x = 0;
Xelse
X  x = rmax*sqrt((y/rmax)^2 + (z/rmax)^2);
Xend
END_OF_FILE
if test 314 -ne `wc -c <'pythag.m'`; then
    echo shar: \"'pythag.m'\" unpacked with wrong size!
fi
# end of 'pythag.m'
fi
if test -f 'quasiopt.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'quasiopt.m'\"
else
echo shar: Extracting \"'quasiopt.m'\" \(2949 characters\)
sed "s/^X//" >'quasiopt.m' <<'END_OF_FILE'
Xfunction [reg_min,Q,reg_param] = quasiopt(U,s,b,method)
X%QUASIOPT Quasi-optimality criterion for choosing the regularization parameter.
X%
X% [reg_min,Q,reg_param] = quasiopt(U,s,b,method)
X% [reg_min,Q,reg_param] = quasiopt(U,sm,b,method)  ,  sm = [sigma,mu]
X%
X% Plots the quasi-optimality function Q for the following methods:
X%    method = 'Tikh' : Tikhonov regularization   (solid line )
X%    method = 'tsvd' : truncated SVD or GSVD     (o markers  )
X%    method = 'dsvd' : damped SVD or GSVD        (dotted line)
X% If no method is specified, 'Tikh' is default.
X%
X% If any output arguments are specified, then the minimum of Q is
X% identified and the corresponding reg. parameter reg_min is returned.
X
X% Reference: T. Kitagawa, "A deterministic approach to optimal
X% regularization - the finite dimensional case", Japan J. Appl.
X% Math. 4 (1987), 371-391.
X
X% Per Christian Hansen, UNI-C, 03/17/93.
X
X% Set defaults.
Xnpoints = 80;  % Number of points for 'Tikh' and 'dsvd'.
Xif (nargin==3), method = 'Tikh'; end   % Default method.
X
X% Initialization.
X[m,n] = size(U); [p,ps] = size(s);
Xif (ps==2), s = s(p:-1:1,1)./s(p:-1:1,2); U = U(:,p:-1:1); end
Xxi = (U'*b)./s;
Xif (nargout > 0), find_min = 1; else find_min = 0; end
X
X% Compute the quasioptimality function Q.
Xif (method(1:4)=='Tikh' | method(1:4)=='tikh')
X
X  Q = zeros(npoints,1); reg_param = Q;
X  reg_param(npoints) = s(p);
X  ratio = (s(1)/s(p))^(1/(npoints-1));
X  for i=npoints-1:-1:1, reg_param(i) = ratio*reg_param(i+1); end
X  for i=1:npoints
X    f = (s.^2)./(s.^2 + reg_param(i)^2);
X    Q(i) = norm((1 - f).*f.*xi);
X  end
X
Xelseif (method(1:4)=='dsvd' | method(1:4)=='dgsv')
X
X  Q = zeros(npoints,1); reg_param = Q;
X  reg_param(npoints,1) = s(p);
X  ratio = (s(1)/s(p))^(1/(npoints-1));
X  for i=npoints-1:-1:1, reg_param(i) = ratio*reg_param(i+1); end
X  for i=1:npoints
X    f = s./(s + reg_param(i));
X    Q(i) = norm((1 - f).*f.*xi);
X  end
X
Xelseif (method(1:4)=='tsvd' | method(1:4)=='tgsv')
X
X  Q = abs(xi); reg_param = [1:p]';
X
Xelse, error('Illegal method'), end
X
X% Plot the function.
Xif (method=='tsvd' | method=='tgsv')
X  semilogy(reg_param,Q,'o'), xlabel('k')
X  title('Quasi-optimality function')
X  if (find_min)
X    [minQ,minQi] = min(Q); reg_min = reg_param(minQi);
X    HoldState = ishold; hold on;
X    semilogy([reg_min,reg_min],[minQ,minQ/1000],'--')
X    if (~HoldState), hold off; end
X    title(['Quasi-optimality function, minimum at ',num2str(reg_min)])
X  end
Xelse
X  if (method(1:4)=='Tikh' | method(1:4)=='tikh' | ...
X      method(1:4)=='dsvd' | method(1:4)=='dgsv'      )
X    loglog(reg_param,Q), xlabel('lambda')
X  else
X    loglog(reg_param,Q,':'), xlabel('lambda')
X  end
X  title('Quasi-optimality function')
X  if (find_min)
X    [minQ,minQi] = min(Q); reg_min = reg_param(minQi);
X    HoldState = ishold; hold on;
X    loglog([reg_min,reg_min],[minQ,minQ/1000],'--')
X    if (~HoldState), hold off; end
X    title(['Quasi-optimality function, minimum at ',num2str(reg_min)])
X  end
Xend
END_OF_FILE
if test 2949 -ne `wc -c <'quasiopt.m'`; then
    echo shar: \"'quasiopt.m'\" unpacked with wrong size!
fi
# end of 'quasiopt.m'
fi
if test -f 'regudemo.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'regudemo.m'\"
else
echo shar: Extracting \"'regudemo.m'\" \(5039 characters\)
sed "s/^X//" >'regudemo.m' <<'END_OF_FILE'
X%REGUDEMO Tutorial script for Regularization Tools.
X
X% Per Christian Hansen, UNI-C, 03/17/93.
X
Xecho on, clf
X
X% Part 1.  The discrete Picard condition
X% --------------------------------------
X%
X% First generate a "pure" test problem where only rounding
X% errors are present.  Then generate another "noisy" test
X% problem by adding white noise to the right-hand side.
X%
X% Next compute the SVD of the coefficient matrix A.
X%
X% Finally, check the Picard condition for both test problems
X% graphically.  Notice that for both problems the condition is
X% indeed satisfied for the coefficients corresponding to the
X% larger singular values, while the noise eventually starts to
X% dominate.
X
X[A,b_bar,x] = shaw(32);
Xrandn('seed',70957);
Xe = 1e-3*randn(size(b_bar)); b = b_bar + e;
X[U,s,V] = csvd(A);
Xpause % Strike any key to continue
Xsubplot(2,1,1); picard(U,s,b_bar);
Xsubplot(2,1,2); picard(U,s,b); pause
Xclf
X
X% Part 2.  Filter factors
X% -----------------------
X%
X% Compute regularized solutions to the "noisy" problem from Part 1 
X% by means of Tikhonov's method and LSQR without reorthogonalization.
X% Also, compute the corresponding filter factors.
X%
X% A surface (or mesh) plot of the solutions clearly shows their dependence
X% on the regularization parameter (lambda or the iteration number).
X
Xlambda = [1,3e-1,1e-1,3e-2,1e-2,3e-3,1e-3,3e-4,1e-4,3e-5];
XX_tikh = tikhonov(U,s,V,b,lambda);
XF_tikh = fil_fac(s,lambda);
Xiter = 30; reorth = 0;
X[X_lsqr,rho,eta,F_lsqr] = lsqr(A,b,iter,reorth,s);
Xpause % Strike any key to continue
Xsubplot(2,2,1); surf(X_tikh), title('Tikhonov solutions'), axis('ij')
Xsubplot(2,2,2); surf(log10(F_tikh)), axis('ij')
X                title('Tikh filter factors, log scale')
Xsubplot(2,2,3); surf(X_lsqr(:,1:17)), title('LSQR solutions'), axis('ij')
Xsubplot(2,2,4); surf(log10(F_lsqr(:,1:17))), axis('ij')
X                title('LSQR filter factors, log scale'), pause
Xclf
X
X% Part 3.  The L-curve
X% --------------------
X%
X% Plot the L-curves for Tikhonov regularization and for
X% LSQR for the "noisy" test problem from Part 1.
X%
X% Notice the similarity between the two L-curves and thus,
X% in turn, by the two methods.
X
Xpause % Strike any key to continue
Xsubplot(1,2,1); l_curve(U,s,b); axis([1e-3,1,1,1e3])
Xsubplot(1,2,2); plot_lc(rho,eta,'o'); axis([1e-3,1,1,1e3]), pause
Xclf
X
X% Part 4.  Regularization parameters
X% ----------------------------------
X%
X% Use the L-curve criterion and GCV to determine the regularization
X% parameters for Tikhonov regularization and truncated SVD.
X%
X% Then compute the relative errors for the four solutions.
X
Xpause % Strike any key to continue
Xlambda_l = l_curve(U,s,b);   axis([1e-3,1,1,1e3]),      pause
Xk_l = l_curve(U,s,b,'tsvd'); axis([1e-3,1,1,1e3]),      pause
Xlambda_gcv = gcv(U,s,b);     axis([1e-6,1,1e-9,1e-1]),  pause
Xk_gcv = gcv(U,s,b,'tsvd');   axis([0,20,1e-9,1e-1]),    pause
X
Xx_tikh_l   = tikhonov(U,s,V,b,lambda_l);
Xx_tikh_gcv = tikhonov(U,s,V,b,lambda_gcv);
Xif isnan(k_l)
X  x_tsvd_l = zeros(32,1); % Spline Toolbox not available.
Xelse
X  x_tsvd_l = tsvd(U,s,V,b,k_l);
Xend
Xx_tsvd_gcv = tsvd(U,s,V,b,k_gcv);
X[norm(x-x_tikh_l),norm(x-x_tikh_gcv),...
X norm(x-x_tsvd_l),norm(x-x_tsvd_gcv)]/norm(x)
Xpause % Strike any key to continue
X
X% Part 5.  Standard form versus general form
X% ------------------------------------------
X%
X% Generate a new test problem: inverse Laplace transformation
X% with white noise in the right-hand side.
X%
X% For the general-form regularization, choose minimization of
X% the first derivative.
X%
X% First display some left singular vectors of SVD and GSVD; then
X% compare truncated SVD solutions with truncated GSVD solutions.
X% Notice that TSVD cannot reproduce the asymptotic part of the
X% solution in the right part of the figure.
X
Xn = 16; [A,b,x] = ilaplace(n,2);
Xb = b + 1e-4*randn(size(b));
XL = get_l(n,1);
X[U,s,V] = csvd(A); [UU,VV,sm,XX] = gsvd(A,L);
Xpause % Strike any key to continue
XI = 1;
Xfor i=[3,6,9,12]
X  subplot(2,2,I); plot(1:n,V(:,i)); axis([1,n,-1,1])
X  xlabel(['i = ',num2str(i)]), I = I + 1;
Xend
Xsubplot(2,2,1), text(12,1.2,'Right singular vectors V(:,i)'), pause
Xclf
XI = 1;
Xfor i=[n-2,n-5,n-8,n-11]
X  subplot(2,2,I); plot(1:n,XX(:,i)), axis([1,n,-1,1]);
X  xlabel(['i = ',num2str(i)]), I = I + 1;
Xend
Xsubplot(2,2,1)
Xtext(10,1.2,'Right generalized singular vectors XX(:,i)'), pause
Xclf
X
Xk_tsvd = 7; k_tgsvd = 6;
XX_I = tsvd(U,s,V,b,1:k_tsvd);
XX_L = tgsvd(UU,sm,XX,b,1:k_tgsvd);
Xpause % Strike any key to continue
Xsubplot(2,1,1);
X  plot(1:n,X_I,1:n,x,'x'), axis([1,n,0,1.2]), xlabel('L = I')
Xsubplot(2,1,2);
X  plot(1:n,X_L,1:n,x,'x'), axis([1,n,0,1.2]), xlabel('L ~= I'), pause
Xclf
X
X% Part 6.  No square integrable solution
X% --------------------------------------
X%
X% In the last example there is no square integrable solution to
X% the underlying integral equation (NB: no noise is added).
X%
X% Notice that the discrete Picard condition does not seem to
X% be satisfied, which indicates trouble!
X
X[A,b] = ursell(32); [U,s,V] = csvd(A);
Xpause % Strike any key to continue
Xpicard(U,s,b); pause
X
X% This concludes the demo.
Xecho off
END_OF_FILE
if test 5039 -ne `wc -c <'regudemo.m'`; then
    echo shar: \"'regudemo.m'\" unpacked with wrong size!
fi
# end of 'regudemo.m'
fi
if test -f 'shaw.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'shaw.m'\"
else
echo shar: Extracting \"'shaw.m'\" \(1404 characters\)
sed "s/^X//" >'shaw.m' <<'END_OF_FILE'
Xfunction [A,b,x] = shaw(n)
X%SHAW Test problem: one-dimensional iimage restoration model.
X%
X% [A,b,x] = shaw(n)
X%
X% Discretization of a first kind Fredholm integral equation with
X% [-pi/2,pi/2] as both integration intervals.  The kernel K and
X% the solution f, which are given by
X%    K(s,t) = (cos(s) + cos(t))*(sin(u)/u)^2
X%    u = pi*(sin(s) + sin(t))
X%    f(t) = a1*exp(-c1*(t - t1)^2) + a2*exp(-c2*(t - t2)^2) ,
X% are discretized by simple quadrature to produce A and x.
X% Then the discrete right-hand b side is produced as b = A*x.
X%
X% The order n must be even.
X
X% Reference: C. B. Shaw, Jr., "Improvements of the resolution of
X% an instrument by numerical solution of an integral equation",
X% J. Math. Anal. Appl. 37 (1972), 83-112.
X
X% Per Christian Hansen, UNI-C, 08/20/91.
X
X% Check input.
Xif (rem(n,2)~=0), error('The order n must be even'), end
X
X% Initialization.
Xh = pi/n; A = zeros(n,n);
X
X% Compute the matrix A.
Xco = cos(-pi/2 + [.5:n-.5]*h);
Xpsi = pi*sin(-pi/2 + [.5:n-.5]*h);
Xfor i=1:n/2
X  for j=i:n-i
X    ss = psi(i) + psi(j);
X    A(i,j) = ((co(i) + co(j))*sin(ss)/ss)^2;
X    A(n-j+1,n-i+1) = A(i,j);
X  end
X  A(i,n-i+1) = (2*co(i))^2;
Xend
XA = A + triu(A,1)'; A = A*h;
X
X% Compute the vectors x and b.
Xa1 = 2; c1 = 6; t1 =  .8;
Xa2 = 1; c2 = 2; t2 = -.5;
Xif (nargout>1)
X  x =   a1*exp(-c1*(-pi/2 + [.5:n-.5]'*h - t1).^2) ...
X      + a2*exp(-c2*(-pi/2 + [.5:n-.5]'*h - t2).^2);
X  b = A*x;
Xend
END_OF_FILE
if test 1404 -ne `wc -c <'shaw.m'`; then
    echo shar: \"'shaw.m'\" unpacked with wrong size!
fi
# end of 'shaw.m'
fi
if test -f 'sorted.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'sorted.m'\"
else
echo shar: Extracting \"'sorted.m'\" \(87 characters\)
sed "s/^X//" >'sorted.m' <<'END_OF_FILE'
Xfunction pointer=sorted(knots, points)
X%SORTED Dummy function for Regularization Tools
END_OF_FILE
if test 87 -ne `wc -c <'sorted.m'`; then
    echo shar: \"'sorted.m'\" unpacked with wrong size!
fi
# end of 'sorted.m'
fi
if test -f 'sp2pp.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'sp2pp.m'\"
else
echo shar: Extracting \"'sp2pp.m'\" \(73 characters\)
sed "s/^X//" >'sp2pp.m' <<'END_OF_FILE'
Xfunction pp=sp2pp(spline)
X%SP2PP Dummy function for Regularization Tools
END_OF_FILE
if test 73 -ne `wc -c <'sp2pp.m'`; then
    echo shar: \"'sp2pp.m'\" unpacked with wrong size!
fi
# end of 'sp2pp.m'
fi
if test -f 'spbrk.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'spbrk.m'\"
else
echo shar: Extracting \"'spbrk.m'\" \(96 characters\)
sed "s/^X//" >'spbrk.m' <<'END_OF_FILE'
Xfunction [knots,coefs,n,k,d]=spbrk(spline,print)
X%SPBRK Dummy function for Regularization Tools
END_OF_FILE
if test 96 -ne `wc -c <'spbrk.m'`; then
    echo shar: \"'spbrk.m'\" unpacked with wrong size!
fi
# end of 'spbrk.m'
fi
if test -f 'spikes.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'spikes.m'\"
else
echo shar: Extracting \"'spikes.m'\" \(1214 characters\)
sed "s/^X//" >'spikes.m' <<'END_OF_FILE'
Xfunction [A,b,x] = spikes(n,t_max)
X%SPIKES Test problem with a "spiky" solution.
X%
X% [A,b,x] = spikes(n,t_max)
X%
X% Artificially generated discrete ill-posed problem.
X%
X% The solution x consists of a unit step at t = .5, and a pulse train
X% of spikes of decrasing magnitude at t = .5, 1.5, 2.5, ...
X%
X% The parameter t_max is optional; its default value is 5.
X% It controls the length of the pulse train.
X
X% Per Christian Hansen, UNI-C, 06/24/91.
X
X% Initialization.
Xif (nargin == 1), t_max = 5; end
Xt = t_max*[1:n]/n; del = t_max/n;
X
X% Compute the matrix A.
X[t,sigma] = meshdom(del:del:t_max,del:del:t_max); sigma = flipud(sigma);
XA = sigma./(2*sqrt(pi*t.^3)).*exp(-(sigma.^2)./(4*t));
X
X% Compute the right-hand side b and the solution x.
Xif (nargout > 1)
X  heights = 2*ones(t_max,1); heights(1) = 25;
X  heights(2) = 9; heights(3) = 5; heights(4) = 4; heights(5) = 3;
X  x = zeros(n,1); n_h = 1;
X  peak = 0.5/t_max; peak_dist = 1/t_max;
X  if (peak < 1)
X    n_peak = round(peak*n); x(n_peak) = heights(n_h);
X    x(n_peak+1:n) = ones(n-n_peak,1);
X    peak = peak + peak_dist; n_h = n_h + 1;
X  end
X  while (peak < 1)
X    x(round(peak*n)) = heights(n_h);
X    peak = peak + peak_dist; n_h = n_h + 1;
X  end
X  b = A*x;
Xend
END_OF_FILE
if test 1214 -ne `wc -c <'spikes.m'`; then
    echo shar: \"'spikes.m'\" unpacked with wrong size!
fi
# end of 'spikes.m'
fi
if test -f 'spleval.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'spleval.m'\"
else
echo shar: Extracting \"'spleval.m'\" \(521 characters\)
sed "s/^X//" >'spleval.m' <<'END_OF_FILE'
Xfunction points = spleval(f)
X%SPLEVAL Evaluation of a spline or spline curve.
X%
X% points = spleval(f)
X%
X% Computes points on the given spline or spline curve f between
X% its extreme breaks.
X
X% Original routine fnplt by C. de Boor / latest change: Feb.25, 1989
X% Simplified by Per Christian Hansen, UNI-C, 11/19/91.
X
Xif (f(1)==11), f = sp2pp(f); end
X
X[breaks,coefs,l,k,d] = ppbrk(f);
Xnpoints=100;
Xx = breaks(1) + [0:npoints]*((breaks(l+1)-breaks(1))/npoints);
Xv=ppual(f,x);
X
Xif (d==1), points=[x;v]; else, points = v; end
END_OF_FILE
if test 521 -ne `wc -c <'spleval.m'`; then
    echo shar: \"'spleval.m'\" unpacked with wrong size!
fi
# end of 'spleval.m'
fi
if test -f 'spmak.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'spmak.m'\"
else
echo shar: Extracting \"'spmak.m'\" \(82 characters\)
sed "s/^X//" >'spmak.m' <<'END_OF_FILE'
Xfunction spline=spmak(knots,coefs)
X%SPMAK Dummy function for Regularization Tools
END_OF_FILE
if test 82 -ne `wc -c <'spmak.m'`; then
    echo shar: \"'spmak.m'\" unpacked with wrong size!
fi
# end of 'spmak.m'
fi
if test -f 'sprpp.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'sprpp.m'\"
else
echo shar: Extracting \"'sprpp.m'\" \(76 characters\)
sed "s/^X//" >'sprpp.m' <<'END_OF_FILE'
Xfunction [v,b] = sprpp(tx,a)
X%SPRPP Dummy function for Regularization Tools
END_OF_FILE
if test 76 -ne `wc -c <'sprpp.m'`; then
    echo shar: \"'sprpp.m'\" unpacked with wrong size!
fi
# end of 'sprpp.m'
fi
if test -f 'std_form.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'std_form.m'\"
else
echo shar: Extracting \"'std_form.m'\" \(2544 characters\)
sed "s/^X//" >'std_form.m' <<'END_OF_FILE'
Xfunction [A_s,b_s,L_p,K,M] = std_form(A,L,b,W)
X%STD_FORM Transform a general-form reg. problem into one in standard form.
X%
X% [A_s,b_s,L_p,K,M] = std_form(A,L,b)      (method 1)
X% [A_s,b_s,L_p,x_0] = std_form(A,L,b,W)    (method 2)
X%
X% Transforms a regularization problem in general form
X%    min { || A x - b ||^2 + lambda^2 || L x ||^2 }
X% into one in standard form
X%    min { || A_s x_s - b_s ||^2 + lambda^2 || x_s ||^2 } .
X%
X% Two methods are available.  In both methods, the regularized
X% solution to the original problem can be written as
X%    x = L_p*x_s + d
X% where L_p and d depend on the method as follows:
X%    method = 1: L_p = pseudoinverse of L, d  = K*M*(b - A*L_p*x_s)
X%    method = 2: L_p = A-weighted pseudoinverse of L, d = x_0.
X%
X% The transformation from x_s back to x can be carried out by means
X% of the subroutine gen_form.
X
X% References: L. Elden, "Algorithms for regularization of ill-
X% conditioned least-squares problems", BIT 17 (1977), 134-145.
X% L. Elden, "A weighted pseudoinverse, generalized singular values,
X% and constrained lest squares problems", BIT 22 (1982), 487-502.
X% M. Hanke, "Regularization with differential operators.  An itera-
X% tive approach", J. Numer. Funct. Anal. Optim. 13 (1992), 523-540.
X
X% Per Christian Hansen, UNI-C, 05/26/93.
X
X% Nargin determines which method.
Xif (nargin==3)
X
X  % Initialization for method 1.
X  [m,n] = size(A); [p,np] = size(L);
X  if (np~=n), error('A and L must have the same number of columns'), end
X
X  % Special treatment of the case where L is square.
X  if (p==n)
X    L_p = inv(L); K = []; M = []; A_s = A/L; b_s = b;
X    return
X  end
X
X  % Compute a QR factorization of L'.
X  [K,R] = qr(full(L')); R = R(1:p,:);
X
X  % Compute a QR factorization of A*K(:,p+1:n)).
X  [H,T] = qr(A*K(:,p+1:n)); T = T(1:n-p,:);
X
X  % Compute the transformed quantities.
X  L_p = (R\(K(:,1:p)'))';
X  K   = K(:,p+1:n);
X  M   = T\(H(:,1:n-p)');
X  A_s = H(:,n-p+1:m)'*A*L_p;
X  b_s = H(:,n-p+1:m)'*b;
X
Xelse
X
X  % Initialization for method 2.
X  [m,n] = size(A); [p,nl] = size(L); nu = n-p;
X  if (nl~=n), error('A and L must have the same number of columns'), end
X
X  % Special treatment of the case where L is square.
X  if (p==n)
X    L_p = inv(L); A_s = A/L; b_s = b;
X    x_0 = zeros(n,1); K = x_0; % Fix output name.
X    return
X  end
X
X  % Compute NAA and x_0;
X  [NAA,x_0] = pinit(W,A,b);
X  b_s = b - A*x_0;
X
X  % Compute the transformed quantities.
X  L1  = inv([[eye(nu),zeros(nu,p)];L]); L1 = full(L1(:,nu+1:n));
X  L_p = L1 - W*(NAA*L1);
X  A_s = A*L_p;
X
X  % Fix output name.
X  K = x_0;
X
Xend
END_OF_FILE
if test 2544 -ne `wc -c <'std_form.m'`; then
    echo shar: \"'std_form.m'\" unpacked with wrong size!
fi
# end of 'std_form.m'
fi
if test -f 'tgsvd.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'tgsvd.m'\"
else
echo shar: Extracting \"'tgsvd.m'\" \(891 characters\)
sed "s/^X//" >'tgsvd.m' <<'END_OF_FILE'
Xfunction x_k = tgsvd(U,sm,X,b,k)
X%TGSVD Truncated GSVD regularization.
X%
X% x_k = tgsvd(U,sm,X,b,k) ,  sm = [sigma,mu]
X%
X% Computes the truncated GSVD solution
X%            [ 0              0                 0    ]
X%    x_k = X*[ 0  inv(diag(sigma(p-k+1:p)))     0    ]*U'*b .
X%            [ 0              0             eye(n-p) ]
X% If k is a vector, then x_k is a matrix such that
X%    x_k = [ x_k(1), x_k(2), ... ] .
X
X% Reference: P. C. Hansen, "Regularization, GSVD and truncated GSVD",
X% BIT 29 (1989), 491-504.
X
X% Per Christian Hansen, UNI-C, 11/18/91.
X
X% Initialization.
X[n,n] = size(X); p = length(sm(:,1)); lk = length(k);
Xif (min(k)<1 | max(k)>p)
X  error('Illegal truncation parameter k')
Xend
X
X% Treat each k separately.
Xx_k = zeros(n,lk); xi = (U(:,1:p)'*b)./sm(:,1);
Xx_0 = X(:,p+1:n)*U(:,p+1:n)'*b;
Xfor j=1:lk
X  i = k(j); pi1 = p-i+1;
X  x_k(:,j) = X(:,pi1:p)*xi(pi1:p) + x_0;
Xend
END_OF_FILE
if test 891 -ne `wc -c <'tgsvd.m'`; then
    echo shar: \"'tgsvd.m'\" unpacked with wrong size!
fi
# end of 'tgsvd.m'
fi
if test -f 'tikhonov.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'tikhonov.m'\"
else
echo shar: Extracting \"'tikhonov.m'\" \(1291 characters\)
sed "s/^X//" >'tikhonov.m' <<'END_OF_FILE'
Xfunction x_lambda = tikhonov(U,s,V,b,lambda)
X%TIKHONOV Tikhonov regularization.
X%
X% x_lambda = tikhonov(U,s,V,b,lambda)
X% x_lambda = tikhonov(U,sm,X,b,lambda) ,  sm = [sigma,mu]
X%
X% Computes the Tikhonov regularized solution x_lambda.  If the
X% SVD is used, i.e. if U, s, and V are specified, then standard-
X% form regularization is applied:
X%    min { || A x - b ||^2 + lambda^2 || x ||^2 } .
X% If, on the other hand, the GSVD is used, i.e. if U, sm, and X
X% are specified, then general-form regularization is applied:
X%    min { || A x - b ||^2 + lambda^2 || L x ||^2 } .
X%
X% If lambda is a vector, then x_lambda is a matrix such that
X%    x_lambda = [ x_lambda(1), x_lambda(2), ... ] .
X
X% Per Christian Hansen, UNI-C, 03/10/90.
X
X% Reference: A. N. Tikhonov & V. Y. Arsenin, "Solutions of
X% Ill-Posed Problems", Wiley, 1977.
X
X% Initialization.
Xif (min(lambda)<0)
X  error('Illegal regularization parameter lambda')
Xend
X[n,pv] = size(V); [p,ps] = size(s);
Xeta = s(:,1).*(U(:,1:p)'*b);
Xll = length(lambda); x_lambda = zeros(n,ll);
X
X% Treat each lambda separately.
Xif (ps==1)
X  for i=1:ll
X    x_lambda(:,i) = V(:,1:p)*(eta./(s.^2 + lambda(i)^2));
X  end
Xelse
X  x0 = V(:,p+1:n)*U(:,p+1:n)'*b; 
X  for i=1:ll
X    x_lambda(:,i) = V(:,1:p)*(eta./(s(:,1).^2 + lambda(i)^2*s(:,2).^2)) + x0;
X  end
Xend
END_OF_FILE
if test 1291 -ne `wc -c <'tikhonov.m'`; then
    echo shar: \"'tikhonov.m'\" unpacked with wrong size!
fi
# end of 'tikhonov.m'
fi
if test -f 'tsvd.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'tsvd.m'\"
else
echo shar: Extracting \"'tsvd.m'\" \(566 characters\)
sed "s/^X//" >'tsvd.m' <<'END_OF_FILE'
Xfunction x_k = tsvd(U,s,V,b,k)
X%TSVD Truncated SVD regularization.
X%
X% x_k = tsvd(U,s,V,b,k)
X%
X% Computes the truncated SVD solution
X%    x_k = V(:,1:k)*inv(diag(s(1:k)))*U(:,1:k)'*b .
X% If k is a vector, then x_k is a matrix such that
X%    x_k = [ x_k(1), x_k(2), ... ] .
X
X% Per Christian Hansen, UNI-C, 11/18/91.
X
X% Initialization.
X[n,p] = size(V); lk = length(k);
Xif (min(k)<1 | max(k)>p)
X  error('Illegal truncation parameter k')
Xend
Xx_k = zeros(n,lk); xi = (U(:,1:p)'*b)./s;
X
X% Treat each k separately.
Xfor j=1:lk
X  i = k(j);
X  x_k(:,j) = V(:,1:i)*xi(1:i);
Xend
END_OF_FILE
if test 566 -ne `wc -c <'tsvd.m'`; then
    echo shar: \"'tsvd.m'\" unpacked with wrong size!
fi
# end of 'tsvd.m'
fi
if test -f 'ttls.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'ttls.m'\"
else
echo shar: Extracting \"'ttls.m'\" \(1308 characters\)
sed "s/^X//" >'ttls.m' <<'END_OF_FILE'
Xfunction [x_k,rho,eta] = ttls(V1,k,s1)
X%TTLS Truncated TLS regularization.
X%
X% [x_k,rho,eta] = ttls(V1,k,s1)
X%
X% Computes the truncated TLS solution
X%    x_k = - V1(1:n,k+1:n+1)*pinv(V1(n+1,k+1:n+1))
X% where V1 is the right singular matrix in the SVD of the matrix
X%    [A,b] = U1*diag(s1)*V1' .
X%
X% If k is a vector, then x_k is a matrix such that
X%    x_k = [ x_k(1), x_k(2), ... ] .
X% If k is not specified, k = n is used.
X%
X% The solution norms and TLS residual norms corresponding to x_k are
X% returned in eta and rho, respectively.  Notice that the singular
X% values s1 are required to compute rho.
X
X% Per Christian Hansen, UNI-C, 03/18/93.
X
X% Initialization.
X[n1,m1] = size(V1); n = n1-1;
Xif (m1 ~= n1), error('The matrix V1 must be square'), end
Xif (nargin == 1), k = n; end
Xlk = length(k);
Xif (min(k) < 1 | max(k) > n)
X  error('Illegal truncation parameter k')
Xend
Xx_k = zeros(n,lk);
Xif (nargout > 1)
X  if (nargin < 3)
X    error('The singular values must also be specified')
X  end
X  ns = length(s1); rho = zeros(lk,1);
Xend
Xif (nargout==3), eta = zeros(lk,1); end
X
X% Treat each k separately.
Xfor j=1:lk
X  i = k(j);
X  v = V1(n1,i+1:n1); gamma = 1/(v*v');
X  x_k(:,j) = - V1(1:n,i+1:n1)*v'*gamma;
X  if (nargout > 1), rho(j) = norm(s1(i+1:ns)); end
X  if (nargout == 3), eta(j) = sqrt(gamma - 1); end
Xend
END_OF_FILE
if test 1308 -ne `wc -c <'ttls.m'`; then
    echo shar: \"'ttls.m'\" unpacked with wrong size!
fi
# end of 'ttls.m'
fi
if test -f 'ursell.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'ursell.m'\"
else
echo shar: Extracting \"'ursell.m'\" \(1026 characters\)
sed "s/^X//" >'ursell.m' <<'END_OF_FILE'
Xfunction [A,b] = ursell(n)
X%URSELL Test problem: integral equation wiht no square integrable solution.
X%
X% [A,b] = ursell(n)
X%
X% Discretization of a first kind Fredholm integral equation with
X% kernel K and right-hand side g given by
X%    K(s,t) = 1/(s+t+1) ,  g(s) = 1 ,
X% where both integration itervals are [0,1].
X%
X% Note: this integral equation has NO square integrable solution.
X
X% Reference: F. Ursell, "Introduction to the theory of linear
X% integral equations", Chapter 1 in L. M. Delves & J. Walsh (Eds.),
X% "Numerical Solution of Integral Equations", Clarendon Press, 1974.
X
X% Discretized by Galerkin method with orthonormal box functions.
X
X% Per Christian Hansen, UNI-C, 09/16/92.
X
X% Compute the matrix A.
Xfor k = 1:n
X  d1 = 1 + (1+k)/n; d2 = 1 + k/n; d3 = 1 + (k-1)/n;
X  c(k) = n*(d1*log(d1) + d3*log(d3) - 2*d2*log(d2));
X  e1 = 1 + (n+k)/n; e2 = 1 + (n+k-1)/n; e3 = 1 + (n+k-2)/n;
X  r(k) = n*(e1*log(e1) + e3*log(e3) - 2*e2*log(e2));
Xend
XA = hankel(c,r);
X
X% Compute the right-hand side b.
Xb = ones(n,1)/sqrt(n);
END_OF_FILE
if test 1026 -ne `wc -c <'ursell.m'`; then
    echo shar: \"'ursell.m'\" unpacked with wrong size!
fi
# end of 'ursell.m'
fi
if test -f 'wing.m' -a "${1}" != "-c" ; then 
  echo shar: Will not clobber existing file \"'wing.m'\"
else
echo shar: Extracting \"'wing.m'\" \(1306 characters\)
sed "s/^X//" >'wing.m' <<'END_OF_FILE'
Xfunction [A,b,x] = wing(n,t1,t2)
X%WING Test problem with a discontinuous solution.
X%
X% [A,b,x] = wing(n,t1,t2)
X%
X% Discretization of a first kind Fredholm integral eqaution with
X% kernel K and right-hand side g given by
X%    K(s,t) = t*exp(-s*t^2)                       0 < s,t < 1
X%    g(s)   = (exp(-s*t1^2) - exp(-s*t2^2)/(2*s)  0 < s   < 1
X% and with the solution f given by
X%    f(t) = | 1  for  t1 < t < t2
X%           | 0  elsewhere.
X%
X% Here, t1 and t2 are constants satisfying t1 < t2.  If they are
X% not speficied, the values t1 = 1/3 and t2 = 2/3 are used.
X
X% Reference: G. M. Wing, "A Primer on Integral Equations of the
X% First Kind", SIAM, 1991.
X
X% Discretized by Galerkin method with orthonormal box functions;
X% both integrations are done by the midpoint rule.
X
X% Per Christian Hansen, UNI-C, 09/17/92.
X
X% Initialization.
Xif (nargin==1)
X  t1 = 1/3; t2 = 2/3;
Xelse
X  if (t1 > t2), error('t1 must be smaller than t2'), end
Xend
XA = zeros(n,n); h = 1/n; sh = sqrt(h);
X
X% Set up matrix.
Xsti = ([1:n]-0.5)*h;
Xfor i=1:n
X  A(i,:) = h*sti.*exp(-sti(i)*sti.^2);
Xend
X
X% Set up right-hand side.
Xif (nargout > 1)
X  b = sqrt(h)*0.5*(exp(-sti*t1^2)' - exp(-sti*t2^2)')./sti';
Xend
X
X% Set up solution.
Xif (nargout==3)
X  I = find(t1 < sti & sti < t2);
X  x = zeros(n,1); x(I) = sqrt(h)*ones(length(I),1);
Xend
END_OF_FILE
if test 1306 -ne `wc -c <'wing.m'`; then
    echo shar: \"'wing.m'\" unpacked with wrong size!
fi
# end of 'wing.m'
fi
echo shar: End of shell archive.
exit 0

 Michela Redivo-Zaglia
 Universita` di Padova - Dipartimento di Elettronica e Informatica
 Via G. Gradenigo 6/A  - 35131 Padova - Italy
  
 Phone ++39-49-8277625                    e-mail: michela@dei.unipd.it
 Fax   ++39-49-8277699