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LAPACK 3.3.0
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LAPACK 3.3.0
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00001 SUBROUTINE DGSVJ0( JOBV, M, N, A, LDA, D, SVA, MV, V, LDV, EPS, 00002 + SFMIN, TOL, NSWEEP, WORK, LWORK, INFO ) 00003 * 00004 * -- LAPACK routine (version 3.3.0) -- 00005 * 00006 * -- Contributed by Zlatko Drmac of the University of Zagreb and -- 00007 * -- Kresimir Veselic of the Fernuniversitaet Hagen -- 00008 * November 2010 00009 * 00010 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00011 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00012 * 00013 * This routine is also part of SIGMA (version 1.23, October 23. 2008.) 00014 * SIGMA is a library of algorithms for highly accurate algorithms for 00015 * computation of SVD, PSVD, QSVD, (H,K)-SVD, and for solution of the 00016 * eigenvalue problems Hx = lambda M x, H M x = lambda x with H, M > 0. 00017 * 00018 IMPLICIT NONE 00019 * .. Scalar Arguments .. 00020 INTEGER INFO, LDA, LDV, LWORK, M, MV, N, NSWEEP 00021 DOUBLE PRECISION EPS, SFMIN, TOL 00022 CHARACTER*1 JOBV 00023 * .. 00024 * .. Array Arguments .. 00025 DOUBLE PRECISION A( LDA, * ), SVA( N ), D( N ), V( LDV, * ), 00026 + WORK( LWORK ) 00027 * .. 00028 * 00029 * Purpose 00030 * ======= 00031 * 00032 * DGSVJ0 is called from DGESVJ as a pre-processor and that is its main 00033 * purpose. It applies Jacobi rotations in the same way as DGESVJ does, but 00034 * it does not check convergence (stopping criterion). Few tuning 00035 * parameters (marked by [TP]) are available for the implementer. 00036 * 00037 * Further Details 00038 * ~~~~~~~~~~~~~~~ 00039 * DGSVJ0 is used just to enable SGESVJ to call a simplified version of 00040 * itself to work on a submatrix of the original matrix. 00041 * 00042 * Contributors 00043 * ~~~~~~~~~~~~ 00044 * Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany) 00045 * 00046 * Bugs, Examples and Comments 00047 * ~~~~~~~~~~~~~~~~~~~~~~~~~~~ 00048 * Please report all bugs and send interesting test examples and comments to 00049 * drmac@math.hr. Thank you. 00050 * 00051 * Arguments 00052 * ========= 00053 * 00054 * JOBV (input) CHARACTER*1 00055 * Specifies whether the output from this procedure is used 00056 * to compute the matrix V: 00057 * = 'V': the product of the Jacobi rotations is accumulated 00058 * by postmulyiplying the N-by-N array V. 00059 * (See the description of V.) 00060 * = 'A': the product of the Jacobi rotations is accumulated 00061 * by postmulyiplying the MV-by-N array V. 00062 * (See the descriptions of MV and V.) 00063 * = 'N': the Jacobi rotations are not accumulated. 00064 * 00065 * M (input) INTEGER 00066 * The number of rows of the input matrix A. M >= 0. 00067 * 00068 * N (input) INTEGER 00069 * The number of columns of the input matrix A. 00070 * M >= N >= 0. 00071 * 00072 * A (input/output) DOUBLE PRECISION array, dimension (LDA,N) 00073 * On entry, M-by-N matrix A, such that A*diag(D) represents 00074 * the input matrix. 00075 * On exit, 00076 * A_onexit * D_onexit represents the input matrix A*diag(D) 00077 * post-multiplied by a sequence of Jacobi rotations, where the 00078 * rotation threshold and the total number of sweeps are given in 00079 * TOL and NSWEEP, respectively. 00080 * (See the descriptions of D, TOL and NSWEEP.) 00081 * 00082 * LDA (input) INTEGER 00083 * The leading dimension of the array A. LDA >= max(1,M). 00084 * 00085 * D (input/workspace/output) DOUBLE PRECISION array, dimension (N) 00086 * The array D accumulates the scaling factors from the fast scaled 00087 * Jacobi rotations. 00088 * On entry, A*diag(D) represents the input matrix. 00089 * On exit, A_onexit*diag(D_onexit) represents the input matrix 00090 * post-multiplied by a sequence of Jacobi rotations, where the 00091 * rotation threshold and the total number of sweeps are given in 00092 * TOL and NSWEEP, respectively. 00093 * (See the descriptions of A, TOL and NSWEEP.) 00094 * 00095 * SVA (input/workspace/output) DOUBLE PRECISION array, dimension (N) 00096 * On entry, SVA contains the Euclidean norms of the columns of 00097 * the matrix A*diag(D). 00098 * On exit, SVA contains the Euclidean norms of the columns of 00099 * the matrix onexit*diag(D_onexit). 00100 * 00101 * MV (input) INTEGER 00102 * If JOBV .EQ. 'A', then MV rows of V are post-multipled by a 00103 * sequence of Jacobi rotations. 00104 * If JOBV = 'N', then MV is not referenced. 00105 * 00106 * V (input/output) DOUBLE PRECISION array, dimension (LDV,N) 00107 * If JOBV .EQ. 'V' then N rows of V are post-multipled by a 00108 * sequence of Jacobi rotations. 00109 * If JOBV .EQ. 'A' then MV rows of V are post-multipled by a 00110 * sequence of Jacobi rotations. 00111 * If JOBV = 'N', then V is not referenced. 00112 * 00113 * LDV (input) INTEGER 00114 * The leading dimension of the array V, LDV >= 1. 00115 * If JOBV = 'V', LDV .GE. N. 00116 * If JOBV = 'A', LDV .GE. MV. 00117 * 00118 * EPS (input) DOUBLE PRECISION 00119 * EPS = DLAMCH('Epsilon') 00120 * 00121 * SFMIN (input) DOUBLE PRECISION 00122 * SFMIN = DLAMCH('Safe Minimum') 00123 * 00124 * TOL (input) DOUBLE PRECISION 00125 * TOL is the threshold for Jacobi rotations. For a pair 00126 * A(:,p), A(:,q) of pivot columns, the Jacobi rotation is 00127 * applied only if DABS(COS(angle(A(:,p),A(:,q)))) .GT. TOL. 00128 * 00129 * NSWEEP (input) INTEGER 00130 * NSWEEP is the number of sweeps of Jacobi rotations to be 00131 * performed. 00132 * 00133 * WORK (workspace) DOUBLE PRECISION array, dimension (LWORK) 00134 * 00135 * LWORK (input) INTEGER 00136 * LWORK is the dimension of WORK. LWORK .GE. M. 00137 * 00138 * INFO (output) INTEGER 00139 * = 0 : successful exit. 00140 * < 0 : if INFO = -i, then the i-th argument had an illegal value 00141 * 00142 * ===================================================================== 00143 * 00144 * .. Local Parameters .. 00145 DOUBLE PRECISION ZERO, HALF, ONE, TWO 00146 PARAMETER ( ZERO = 0.0D0, HALF = 0.5D0, ONE = 1.0D0, 00147 + TWO = 2.0D0 ) 00148 * .. 00149 * .. Local Scalars .. 00150 DOUBLE PRECISION AAPP, AAPP0, AAPQ, AAQQ, APOAQ, AQOAP, BIG, 00151 + BIGTHETA, CS, MXAAPQ, MXSINJ, ROOTBIG, ROOTEPS, 00152 + ROOTSFMIN, ROOTTOL, SMALL, SN, T, TEMP1, THETA, 00153 + THSIGN 00154 INTEGER BLSKIP, EMPTSW, i, ibr, IERR, igl, IJBLSK, ir1, 00155 + ISWROT, jbc, jgl, KBL, LKAHEAD, MVL, NBL, 00156 + NOTROT, p, PSKIPPED, q, ROWSKIP, SWBAND 00157 LOGICAL APPLV, ROTOK, RSVEC 00158 * .. 00159 * .. Local Arrays .. 00160 DOUBLE PRECISION FASTR( 5 ) 00161 * .. 00162 * .. Intrinsic Functions .. 00163 INTRINSIC DABS, DMAX1, DBLE, MIN0, DSIGN, DSQRT 00164 * .. 00165 * .. External Functions .. 00166 DOUBLE PRECISION DDOT, DNRM2 00167 INTEGER IDAMAX 00168 LOGICAL LSAME 00169 EXTERNAL IDAMAX, LSAME, DDOT, DNRM2 00170 * .. 00171 * .. External Subroutines .. 00172 EXTERNAL DAXPY, DCOPY, DLASCL, DLASSQ, DROTM, DSWAP 00173 * .. 00174 * .. Executable Statements .. 00175 * 00176 APPLV = LSAME( JOBV, 'A' ) 00177 RSVEC = LSAME( JOBV, 'V' ) 00178 IF( .NOT.( RSVEC .OR. APPLV .OR. LSAME( JOBV, 'N' ) ) ) THEN 00179 INFO = -1 00180 ELSE IF( M.LT.0 ) THEN 00181 INFO = -2 00182 ELSE IF( ( N.LT.0 ) .OR. ( N.GT.M ) ) THEN 00183 INFO = -3 00184 ELSE IF( LDA.LT.M ) THEN 00185 INFO = -5 00186 ELSE IF( ( RSVEC.OR.APPLV ) .AND. ( MV.LT.0 ) ) THEN 00187 INFO = -8 00188 ELSE IF( ( RSVEC.AND.( LDV.LT.N ) ).OR. 00189 & ( APPLV.AND.( LDV.LT.MV ) ) ) THEN 00190 INFO = -10 00191 ELSE IF( TOL.LE.EPS ) THEN 00192 INFO = -13 00193 ELSE IF( NSWEEP.LT.0 ) THEN 00194 INFO = -14 00195 ELSE IF( LWORK.LT.M ) THEN 00196 INFO = -16 00197 ELSE 00198 INFO = 0 00199 END IF 00200 * 00201 * #:( 00202 IF( INFO.NE.0 ) THEN 00203 CALL XERBLA( 'DGSVJ0', -INFO ) 00204 RETURN 00205 END IF 00206 * 00207 IF( RSVEC ) THEN 00208 MVL = N 00209 ELSE IF( APPLV ) THEN 00210 MVL = MV 00211 END IF 00212 RSVEC = RSVEC .OR. APPLV 00213 00214 ROOTEPS = DSQRT( EPS ) 00215 ROOTSFMIN = DSQRT( SFMIN ) 00216 SMALL = SFMIN / EPS 00217 BIG = ONE / SFMIN 00218 ROOTBIG = ONE / ROOTSFMIN 00219 BIGTHETA = ONE / ROOTEPS 00220 ROOTTOL = DSQRT( TOL ) 00221 * 00222 * 00223 * -#- Row-cyclic Jacobi SVD algorithm with column pivoting -#- 00224 * 00225 EMPTSW = ( N*( N-1 ) ) / 2 00226 NOTROT = 0 00227 FASTR( 1 ) = ZERO 00228 * 00229 * -#- Row-cyclic pivot strategy with de Rijk's pivoting -#- 00230 * 00231 00232 SWBAND = 0 00233 *[TP] SWBAND is a tuning parameter. It is meaningful and effective 00234 * if SGESVJ is used as a computational routine in the preconditioned 00235 * Jacobi SVD algorithm SGESVJ. For sweeps i=1:SWBAND the procedure 00236 * ...... 00237 00238 KBL = MIN0( 8, N ) 00239 *[TP] KBL is a tuning parameter that defines the tile size in the 00240 * tiling of the p-q loops of pivot pairs. In general, an optimal 00241 * value of KBL depends on the matrix dimensions and on the 00242 * parameters of the computer's memory. 00243 * 00244 NBL = N / KBL 00245 IF( ( NBL*KBL ).NE.N )NBL = NBL + 1 00246 00247 BLSKIP = ( KBL**2 ) + 1 00248 *[TP] BLKSKIP is a tuning parameter that depends on SWBAND and KBL. 00249 00250 ROWSKIP = MIN0( 5, KBL ) 00251 *[TP] ROWSKIP is a tuning parameter. 00252 00253 LKAHEAD = 1 00254 *[TP] LKAHEAD is a tuning parameter. 00255 SWBAND = 0 00256 PSKIPPED = 0 00257 * 00258 DO 1993 i = 1, NSWEEP 00259 * .. go go go ... 00260 * 00261 MXAAPQ = ZERO 00262 MXSINJ = ZERO 00263 ISWROT = 0 00264 * 00265 NOTROT = 0 00266 PSKIPPED = 0 00267 * 00268 DO 2000 ibr = 1, NBL 00269 00270 igl = ( ibr-1 )*KBL + 1 00271 * 00272 DO 1002 ir1 = 0, MIN0( LKAHEAD, NBL-ibr ) 00273 * 00274 igl = igl + ir1*KBL 00275 * 00276 DO 2001 p = igl, MIN0( igl+KBL-1, N-1 ) 00277 00278 * .. de Rijk's pivoting 00279 q = IDAMAX( N-p+1, SVA( p ), 1 ) + p - 1 00280 IF( p.NE.q ) THEN 00281 CALL DSWAP( M, A( 1, p ), 1, A( 1, q ), 1 ) 00282 IF( RSVEC )CALL DSWAP( MVL, V( 1, p ), 1, 00283 + V( 1, q ), 1 ) 00284 TEMP1 = SVA( p ) 00285 SVA( p ) = SVA( q ) 00286 SVA( q ) = TEMP1 00287 TEMP1 = D( p ) 00288 D( p ) = D( q ) 00289 D( q ) = TEMP1 00290 END IF 00291 * 00292 IF( ir1.EQ.0 ) THEN 00293 * 00294 * Column norms are periodically updated by explicit 00295 * norm computation. 00296 * Caveat: 00297 * Some BLAS implementations compute DNRM2(M,A(1,p),1) 00298 * as DSQRT(DDOT(M,A(1,p),1,A(1,p),1)), which may result in 00299 * overflow for ||A(:,p)||_2 > DSQRT(overflow_threshold), and 00300 * undeflow for ||A(:,p)||_2 < DSQRT(underflow_threshold). 00301 * Hence, DNRM2 cannot be trusted, not even in the case when 00302 * the true norm is far from the under(over)flow boundaries. 00303 * If properly implemented DNRM2 is available, the IF-THEN-ELSE 00304 * below should read "AAPP = DNRM2( M, A(1,p), 1 ) * D(p)". 00305 * 00306 IF( ( SVA( p ).LT.ROOTBIG ) .AND. 00307 + ( SVA( p ).GT.ROOTSFMIN ) ) THEN 00308 SVA( p ) = DNRM2( M, A( 1, p ), 1 )*D( p ) 00309 ELSE 00310 TEMP1 = ZERO 00311 AAPP = ONE 00312 CALL DLASSQ( M, A( 1, p ), 1, TEMP1, AAPP ) 00313 SVA( p ) = TEMP1*DSQRT( AAPP )*D( p ) 00314 END IF 00315 AAPP = SVA( p ) 00316 ELSE 00317 AAPP = SVA( p ) 00318 END IF 00319 00320 * 00321 IF( AAPP.GT.ZERO ) THEN 00322 * 00323 PSKIPPED = 0 00324 * 00325 DO 2002 q = p + 1, MIN0( igl+KBL-1, N ) 00326 * 00327 AAQQ = SVA( q ) 00328 00329 IF( AAQQ.GT.ZERO ) THEN 00330 * 00331 AAPP0 = AAPP 00332 IF( AAQQ.GE.ONE ) THEN 00333 ROTOK = ( SMALL*AAPP ).LE.AAQQ 00334 IF( AAPP.LT.( BIG / AAQQ ) ) THEN 00335 AAPQ = ( DDOT( M, A( 1, p ), 1, A( 1, 00336 + q ), 1 )*D( p )*D( q ) / AAQQ ) 00337 + / AAPP 00338 ELSE 00339 CALL DCOPY( M, A( 1, p ), 1, WORK, 1 ) 00340 CALL DLASCL( 'G', 0, 0, AAPP, D( p ), 00341 + M, 1, WORK, LDA, IERR ) 00342 AAPQ = DDOT( M, WORK, 1, A( 1, q ), 00343 + 1 )*D( q ) / AAQQ 00344 END IF 00345 ELSE 00346 ROTOK = AAPP.LE.( AAQQ / SMALL ) 00347 IF( AAPP.GT.( SMALL / AAQQ ) ) THEN 00348 AAPQ = ( DDOT( M, A( 1, p ), 1, A( 1, 00349 + q ), 1 )*D( p )*D( q ) / AAQQ ) 00350 + / AAPP 00351 ELSE 00352 CALL DCOPY( M, A( 1, q ), 1, WORK, 1 ) 00353 CALL DLASCL( 'G', 0, 0, AAQQ, D( q ), 00354 + M, 1, WORK, LDA, IERR ) 00355 AAPQ = DDOT( M, WORK, 1, A( 1, p ), 00356 + 1 )*D( p ) / AAPP 00357 END IF 00358 END IF 00359 * 00360 MXAAPQ = DMAX1( MXAAPQ, DABS( AAPQ ) ) 00361 * 00362 * TO rotate or NOT to rotate, THAT is the question ... 00363 * 00364 IF( DABS( AAPQ ).GT.TOL ) THEN 00365 * 00366 * .. rotate 00367 * ROTATED = ROTATED + ONE 00368 * 00369 IF( ir1.EQ.0 ) THEN 00370 NOTROT = 0 00371 PSKIPPED = 0 00372 ISWROT = ISWROT + 1 00373 END IF 00374 * 00375 IF( ROTOK ) THEN 00376 * 00377 AQOAP = AAQQ / AAPP 00378 APOAQ = AAPP / AAQQ 00379 THETA = -HALF*DABS( AQOAP-APOAQ ) / 00380 + AAPQ 00381 * 00382 IF( DABS( THETA ).GT.BIGTHETA ) THEN 00383 * 00384 T = HALF / THETA 00385 FASTR( 3 ) = T*D( p ) / D( q ) 00386 FASTR( 4 ) = -T*D( q ) / D( p ) 00387 CALL DROTM( M, A( 1, p ), 1, 00388 + A( 1, q ), 1, FASTR ) 00389 IF( RSVEC )CALL DROTM( MVL, 00390 + V( 1, p ), 1, 00391 + V( 1, q ), 1, 00392 + FASTR ) 00393 SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO, 00394 + ONE+T*APOAQ*AAPQ ) ) 00395 AAPP = AAPP*DSQRT( DMAX1( ZERO, 00396 + ONE-T*AQOAP*AAPQ ) ) 00397 MXSINJ = DMAX1( MXSINJ, DABS( T ) ) 00398 * 00399 ELSE 00400 * 00401 * .. choose correct signum for THETA and rotate 00402 * 00403 THSIGN = -DSIGN( ONE, AAPQ ) 00404 T = ONE / ( THETA+THSIGN* 00405 + DSQRT( ONE+THETA*THETA ) ) 00406 CS = DSQRT( ONE / ( ONE+T*T ) ) 00407 SN = T*CS 00408 * 00409 MXSINJ = DMAX1( MXSINJ, DABS( SN ) ) 00410 SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO, 00411 + ONE+T*APOAQ*AAPQ ) ) 00412 AAPP = AAPP*DSQRT( DMAX1( ZERO, 00413 + ONE-T*AQOAP*AAPQ ) ) 00414 * 00415 APOAQ = D( p ) / D( q ) 00416 AQOAP = D( q ) / D( p ) 00417 IF( D( p ).GE.ONE ) THEN 00418 IF( D( q ).GE.ONE ) THEN 00419 FASTR( 3 ) = T*APOAQ 00420 FASTR( 4 ) = -T*AQOAP 00421 D( p ) = D( p )*CS 00422 D( q ) = D( q )*CS 00423 CALL DROTM( M, A( 1, p ), 1, 00424 + A( 1, q ), 1, 00425 + FASTR ) 00426 IF( RSVEC )CALL DROTM( MVL, 00427 + V( 1, p ), 1, V( 1, q ), 00428 + 1, FASTR ) 00429 ELSE 00430 CALL DAXPY( M, -T*AQOAP, 00431 + A( 1, q ), 1, 00432 + A( 1, p ), 1 ) 00433 CALL DAXPY( M, CS*SN*APOAQ, 00434 + A( 1, p ), 1, 00435 + A( 1, q ), 1 ) 00436 D( p ) = D( p )*CS 00437 D( q ) = D( q ) / CS 00438 IF( RSVEC ) THEN 00439 CALL DAXPY( MVL, -T*AQOAP, 00440 + V( 1, q ), 1, 00441 + V( 1, p ), 1 ) 00442 CALL DAXPY( MVL, 00443 + CS*SN*APOAQ, 00444 + V( 1, p ), 1, 00445 + V( 1, q ), 1 ) 00446 END IF 00447 END IF 00448 ELSE 00449 IF( D( q ).GE.ONE ) THEN 00450 CALL DAXPY( M, T*APOAQ, 00451 + A( 1, p ), 1, 00452 + A( 1, q ), 1 ) 00453 CALL DAXPY( M, -CS*SN*AQOAP, 00454 + A( 1, q ), 1, 00455 + A( 1, p ), 1 ) 00456 D( p ) = D( p ) / CS 00457 D( q ) = D( q )*CS 00458 IF( RSVEC ) THEN 00459 CALL DAXPY( MVL, T*APOAQ, 00460 + V( 1, p ), 1, 00461 + V( 1, q ), 1 ) 00462 CALL DAXPY( MVL, 00463 + -CS*SN*AQOAP, 00464 + V( 1, q ), 1, 00465 + V( 1, p ), 1 ) 00466 END IF 00467 ELSE 00468 IF( D( p ).GE.D( q ) ) THEN 00469 CALL DAXPY( M, -T*AQOAP, 00470 + A( 1, q ), 1, 00471 + A( 1, p ), 1 ) 00472 CALL DAXPY( M, CS*SN*APOAQ, 00473 + A( 1, p ), 1, 00474 + A( 1, q ), 1 ) 00475 D( p ) = D( p )*CS 00476 D( q ) = D( q ) / CS 00477 IF( RSVEC ) THEN 00478 CALL DAXPY( MVL, 00479 + -T*AQOAP, 00480 + V( 1, q ), 1, 00481 + V( 1, p ), 1 ) 00482 CALL DAXPY( MVL, 00483 + CS*SN*APOAQ, 00484 + V( 1, p ), 1, 00485 + V( 1, q ), 1 ) 00486 END IF 00487 ELSE 00488 CALL DAXPY( M, T*APOAQ, 00489 + A( 1, p ), 1, 00490 + A( 1, q ), 1 ) 00491 CALL DAXPY( M, 00492 + -CS*SN*AQOAP, 00493 + A( 1, q ), 1, 00494 + A( 1, p ), 1 ) 00495 D( p ) = D( p ) / CS 00496 D( q ) = D( q )*CS 00497 IF( RSVEC ) THEN 00498 CALL DAXPY( MVL, 00499 + T*APOAQ, V( 1, p ), 00500 + 1, V( 1, q ), 1 ) 00501 CALL DAXPY( MVL, 00502 + -CS*SN*AQOAP, 00503 + V( 1, q ), 1, 00504 + V( 1, p ), 1 ) 00505 END IF 00506 END IF 00507 END IF 00508 END IF 00509 END IF 00510 * 00511 ELSE 00512 * .. have to use modified Gram-Schmidt like transformation 00513 CALL DCOPY( M, A( 1, p ), 1, WORK, 1 ) 00514 CALL DLASCL( 'G', 0, 0, AAPP, ONE, M, 00515 + 1, WORK, LDA, IERR ) 00516 CALL DLASCL( 'G', 0, 0, AAQQ, ONE, M, 00517 + 1, A( 1, q ), LDA, IERR ) 00518 TEMP1 = -AAPQ*D( p ) / D( q ) 00519 CALL DAXPY( M, TEMP1, WORK, 1, 00520 + A( 1, q ), 1 ) 00521 CALL DLASCL( 'G', 0, 0, ONE, AAQQ, M, 00522 + 1, A( 1, q ), LDA, IERR ) 00523 SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO, 00524 + ONE-AAPQ*AAPQ ) ) 00525 MXSINJ = DMAX1( MXSINJ, SFMIN ) 00526 END IF 00527 * END IF ROTOK THEN ... ELSE 00528 * 00529 * In the case of cancellation in updating SVA(q), SVA(p) 00530 * recompute SVA(q), SVA(p). 00531 IF( ( SVA( q ) / AAQQ )**2.LE.ROOTEPS ) 00532 + THEN 00533 IF( ( AAQQ.LT.ROOTBIG ) .AND. 00534 + ( AAQQ.GT.ROOTSFMIN ) ) THEN 00535 SVA( q ) = DNRM2( M, A( 1, q ), 1 )* 00536 + D( q ) 00537 ELSE 00538 T = ZERO 00539 AAQQ = ONE 00540 CALL DLASSQ( M, A( 1, q ), 1, T, 00541 + AAQQ ) 00542 SVA( q ) = T*DSQRT( AAQQ )*D( q ) 00543 END IF 00544 END IF 00545 IF( ( AAPP / AAPP0 ).LE.ROOTEPS ) THEN 00546 IF( ( AAPP.LT.ROOTBIG ) .AND. 00547 + ( AAPP.GT.ROOTSFMIN ) ) THEN 00548 AAPP = DNRM2( M, A( 1, p ), 1 )* 00549 + D( p ) 00550 ELSE 00551 T = ZERO 00552 AAPP = ONE 00553 CALL DLASSQ( M, A( 1, p ), 1, T, 00554 + AAPP ) 00555 AAPP = T*DSQRT( AAPP )*D( p ) 00556 END IF 00557 SVA( p ) = AAPP 00558 END IF 00559 * 00560 ELSE 00561 * A(:,p) and A(:,q) already numerically orthogonal 00562 IF( ir1.EQ.0 )NOTROT = NOTROT + 1 00563 PSKIPPED = PSKIPPED + 1 00564 END IF 00565 ELSE 00566 * A(:,q) is zero column 00567 IF( ir1.EQ.0 )NOTROT = NOTROT + 1 00568 PSKIPPED = PSKIPPED + 1 00569 END IF 00570 * 00571 IF( ( i.LE.SWBAND ) .AND. 00572 + ( PSKIPPED.GT.ROWSKIP ) ) THEN 00573 IF( ir1.EQ.0 )AAPP = -AAPP 00574 NOTROT = 0 00575 GO TO 2103 00576 END IF 00577 * 00578 2002 CONTINUE 00579 * END q-LOOP 00580 * 00581 2103 CONTINUE 00582 * bailed out of q-loop 00583 00584 SVA( p ) = AAPP 00585 00586 ELSE 00587 SVA( p ) = AAPP 00588 IF( ( ir1.EQ.0 ) .AND. ( AAPP.EQ.ZERO ) ) 00589 + NOTROT = NOTROT + MIN0( igl+KBL-1, N ) - p 00590 END IF 00591 * 00592 2001 CONTINUE 00593 * end of the p-loop 00594 * end of doing the block ( ibr, ibr ) 00595 1002 CONTINUE 00596 * end of ir1-loop 00597 * 00598 *........................................................ 00599 * ... go to the off diagonal blocks 00600 * 00601 igl = ( ibr-1 )*KBL + 1 00602 * 00603 DO 2010 jbc = ibr + 1, NBL 00604 * 00605 jgl = ( jbc-1 )*KBL + 1 00606 * 00607 * doing the block at ( ibr, jbc ) 00608 * 00609 IJBLSK = 0 00610 DO 2100 p = igl, MIN0( igl+KBL-1, N ) 00611 * 00612 AAPP = SVA( p ) 00613 * 00614 IF( AAPP.GT.ZERO ) THEN 00615 * 00616 PSKIPPED = 0 00617 * 00618 DO 2200 q = jgl, MIN0( jgl+KBL-1, N ) 00619 * 00620 AAQQ = SVA( q ) 00621 * 00622 IF( AAQQ.GT.ZERO ) THEN 00623 AAPP0 = AAPP 00624 * 00625 * -#- M x 2 Jacobi SVD -#- 00626 * 00627 * -#- Safe Gram matrix computation -#- 00628 * 00629 IF( AAQQ.GE.ONE ) THEN 00630 IF( AAPP.GE.AAQQ ) THEN 00631 ROTOK = ( SMALL*AAPP ).LE.AAQQ 00632 ELSE 00633 ROTOK = ( SMALL*AAQQ ).LE.AAPP 00634 END IF 00635 IF( AAPP.LT.( BIG / AAQQ ) ) THEN 00636 AAPQ = ( DDOT( M, A( 1, p ), 1, A( 1, 00637 + q ), 1 )*D( p )*D( q ) / AAQQ ) 00638 + / AAPP 00639 ELSE 00640 CALL DCOPY( M, A( 1, p ), 1, WORK, 1 ) 00641 CALL DLASCL( 'G', 0, 0, AAPP, D( p ), 00642 + M, 1, WORK, LDA, IERR ) 00643 AAPQ = DDOT( M, WORK, 1, A( 1, q ), 00644 + 1 )*D( q ) / AAQQ 00645 END IF 00646 ELSE 00647 IF( AAPP.GE.AAQQ ) THEN 00648 ROTOK = AAPP.LE.( AAQQ / SMALL ) 00649 ELSE 00650 ROTOK = AAQQ.LE.( AAPP / SMALL ) 00651 END IF 00652 IF( AAPP.GT.( SMALL / AAQQ ) ) THEN 00653 AAPQ = ( DDOT( M, A( 1, p ), 1, A( 1, 00654 + q ), 1 )*D( p )*D( q ) / AAQQ ) 00655 + / AAPP 00656 ELSE 00657 CALL DCOPY( M, A( 1, q ), 1, WORK, 1 ) 00658 CALL DLASCL( 'G', 0, 0, AAQQ, D( q ), 00659 + M, 1, WORK, LDA, IERR ) 00660 AAPQ = DDOT( M, WORK, 1, A( 1, p ), 00661 + 1 )*D( p ) / AAPP 00662 END IF 00663 END IF 00664 * 00665 MXAAPQ = DMAX1( MXAAPQ, DABS( AAPQ ) ) 00666 * 00667 * TO rotate or NOT to rotate, THAT is the question ... 00668 * 00669 IF( DABS( AAPQ ).GT.TOL ) THEN 00670 NOTROT = 0 00671 * ROTATED = ROTATED + 1 00672 PSKIPPED = 0 00673 ISWROT = ISWROT + 1 00674 * 00675 IF( ROTOK ) THEN 00676 * 00677 AQOAP = AAQQ / AAPP 00678 APOAQ = AAPP / AAQQ 00679 THETA = -HALF*DABS( AQOAP-APOAQ ) / 00680 + AAPQ 00681 IF( AAQQ.GT.AAPP0 )THETA = -THETA 00682 * 00683 IF( DABS( THETA ).GT.BIGTHETA ) THEN 00684 T = HALF / THETA 00685 FASTR( 3 ) = T*D( p ) / D( q ) 00686 FASTR( 4 ) = -T*D( q ) / D( p ) 00687 CALL DROTM( M, A( 1, p ), 1, 00688 + A( 1, q ), 1, FASTR ) 00689 IF( RSVEC )CALL DROTM( MVL, 00690 + V( 1, p ), 1, 00691 + V( 1, q ), 1, 00692 + FASTR ) 00693 SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO, 00694 + ONE+T*APOAQ*AAPQ ) ) 00695 AAPP = AAPP*DSQRT( DMAX1( ZERO, 00696 + ONE-T*AQOAP*AAPQ ) ) 00697 MXSINJ = DMAX1( MXSINJ, DABS( T ) ) 00698 ELSE 00699 * 00700 * .. choose correct signum for THETA and rotate 00701 * 00702 THSIGN = -DSIGN( ONE, AAPQ ) 00703 IF( AAQQ.GT.AAPP0 )THSIGN = -THSIGN 00704 T = ONE / ( THETA+THSIGN* 00705 + DSQRT( ONE+THETA*THETA ) ) 00706 CS = DSQRT( ONE / ( ONE+T*T ) ) 00707 SN = T*CS 00708 MXSINJ = DMAX1( MXSINJ, DABS( SN ) ) 00709 SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO, 00710 + ONE+T*APOAQ*AAPQ ) ) 00711 AAPP = AAPP*DSQRT( DMAX1( ZERO, 00712 + ONE-T*AQOAP*AAPQ ) ) 00713 * 00714 APOAQ = D( p ) / D( q ) 00715 AQOAP = D( q ) / D( p ) 00716 IF( D( p ).GE.ONE ) THEN 00717 * 00718 IF( D( q ).GE.ONE ) THEN 00719 FASTR( 3 ) = T*APOAQ 00720 FASTR( 4 ) = -T*AQOAP 00721 D( p ) = D( p )*CS 00722 D( q ) = D( q )*CS 00723 CALL DROTM( M, A( 1, p ), 1, 00724 + A( 1, q ), 1, 00725 + FASTR ) 00726 IF( RSVEC )CALL DROTM( MVL, 00727 + V( 1, p ), 1, V( 1, q ), 00728 + 1, FASTR ) 00729 ELSE 00730 CALL DAXPY( M, -T*AQOAP, 00731 + A( 1, q ), 1, 00732 + A( 1, p ), 1 ) 00733 CALL DAXPY( M, CS*SN*APOAQ, 00734 + A( 1, p ), 1, 00735 + A( 1, q ), 1 ) 00736 IF( RSVEC ) THEN 00737 CALL DAXPY( MVL, -T*AQOAP, 00738 + V( 1, q ), 1, 00739 + V( 1, p ), 1 ) 00740 CALL DAXPY( MVL, 00741 + CS*SN*APOAQ, 00742 + V( 1, p ), 1, 00743 + V( 1, q ), 1 ) 00744 END IF 00745 D( p ) = D( p )*CS 00746 D( q ) = D( q ) / CS 00747 END IF 00748 ELSE 00749 IF( D( q ).GE.ONE ) THEN 00750 CALL DAXPY( M, T*APOAQ, 00751 + A( 1, p ), 1, 00752 + A( 1, q ), 1 ) 00753 CALL DAXPY( M, -CS*SN*AQOAP, 00754 + A( 1, q ), 1, 00755 + A( 1, p ), 1 ) 00756 IF( RSVEC ) THEN 00757 CALL DAXPY( MVL, T*APOAQ, 00758 + V( 1, p ), 1, 00759 + V( 1, q ), 1 ) 00760 CALL DAXPY( MVL, 00761 + -CS*SN*AQOAP, 00762 + V( 1, q ), 1, 00763 + V( 1, p ), 1 ) 00764 END IF 00765 D( p ) = D( p ) / CS 00766 D( q ) = D( q )*CS 00767 ELSE 00768 IF( D( p ).GE.D( q ) ) THEN 00769 CALL DAXPY( M, -T*AQOAP, 00770 + A( 1, q ), 1, 00771 + A( 1, p ), 1 ) 00772 CALL DAXPY( M, CS*SN*APOAQ, 00773 + A( 1, p ), 1, 00774 + A( 1, q ), 1 ) 00775 D( p ) = D( p )*CS 00776 D( q ) = D( q ) / CS 00777 IF( RSVEC ) THEN 00778 CALL DAXPY( MVL, 00779 + -T*AQOAP, 00780 + V( 1, q ), 1, 00781 + V( 1, p ), 1 ) 00782 CALL DAXPY( MVL, 00783 + CS*SN*APOAQ, 00784 + V( 1, p ), 1, 00785 + V( 1, q ), 1 ) 00786 END IF 00787 ELSE 00788 CALL DAXPY( M, T*APOAQ, 00789 + A( 1, p ), 1, 00790 + A( 1, q ), 1 ) 00791 CALL DAXPY( M, 00792 + -CS*SN*AQOAP, 00793 + A( 1, q ), 1, 00794 + A( 1, p ), 1 ) 00795 D( p ) = D( p ) / CS 00796 D( q ) = D( q )*CS 00797 IF( RSVEC ) THEN 00798 CALL DAXPY( MVL, 00799 + T*APOAQ, V( 1, p ), 00800 + 1, V( 1, q ), 1 ) 00801 CALL DAXPY( MVL, 00802 + -CS*SN*AQOAP, 00803 + V( 1, q ), 1, 00804 + V( 1, p ), 1 ) 00805 END IF 00806 END IF 00807 END IF 00808 END IF 00809 END IF 00810 * 00811 ELSE 00812 IF( AAPP.GT.AAQQ ) THEN 00813 CALL DCOPY( M, A( 1, p ), 1, WORK, 00814 + 1 ) 00815 CALL DLASCL( 'G', 0, 0, AAPP, ONE, 00816 + M, 1, WORK, LDA, IERR ) 00817 CALL DLASCL( 'G', 0, 0, AAQQ, ONE, 00818 + M, 1, A( 1, q ), LDA, 00819 + IERR ) 00820 TEMP1 = -AAPQ*D( p ) / D( q ) 00821 CALL DAXPY( M, TEMP1, WORK, 1, 00822 + A( 1, q ), 1 ) 00823 CALL DLASCL( 'G', 0, 0, ONE, AAQQ, 00824 + M, 1, A( 1, q ), LDA, 00825 + IERR ) 00826 SVA( q ) = AAQQ*DSQRT( DMAX1( ZERO, 00827 + ONE-AAPQ*AAPQ ) ) 00828 MXSINJ = DMAX1( MXSINJ, SFMIN ) 00829 ELSE 00830 CALL DCOPY( M, A( 1, q ), 1, WORK, 00831 + 1 ) 00832 CALL DLASCL( 'G', 0, 0, AAQQ, ONE, 00833 + M, 1, WORK, LDA, IERR ) 00834 CALL DLASCL( 'G', 0, 0, AAPP, ONE, 00835 + M, 1, A( 1, p ), LDA, 00836 + IERR ) 00837 TEMP1 = -AAPQ*D( q ) / D( p ) 00838 CALL DAXPY( M, TEMP1, WORK, 1, 00839 + A( 1, p ), 1 ) 00840 CALL DLASCL( 'G', 0, 0, ONE, AAPP, 00841 + M, 1, A( 1, p ), LDA, 00842 + IERR ) 00843 SVA( p ) = AAPP*DSQRT( DMAX1( ZERO, 00844 + ONE-AAPQ*AAPQ ) ) 00845 MXSINJ = DMAX1( MXSINJ, SFMIN ) 00846 END IF 00847 END IF 00848 * END IF ROTOK THEN ... ELSE 00849 * 00850 * In the case of cancellation in updating SVA(q) 00851 * .. recompute SVA(q) 00852 IF( ( SVA( q ) / AAQQ )**2.LE.ROOTEPS ) 00853 + THEN 00854 IF( ( AAQQ.LT.ROOTBIG ) .AND. 00855 + ( AAQQ.GT.ROOTSFMIN ) ) THEN 00856 SVA( q ) = DNRM2( M, A( 1, q ), 1 )* 00857 + D( q ) 00858 ELSE 00859 T = ZERO 00860 AAQQ = ONE 00861 CALL DLASSQ( M, A( 1, q ), 1, T, 00862 + AAQQ ) 00863 SVA( q ) = T*DSQRT( AAQQ )*D( q ) 00864 END IF 00865 END IF 00866 IF( ( AAPP / AAPP0 )**2.LE.ROOTEPS ) THEN 00867 IF( ( AAPP.LT.ROOTBIG ) .AND. 00868 + ( AAPP.GT.ROOTSFMIN ) ) THEN 00869 AAPP = DNRM2( M, A( 1, p ), 1 )* 00870 + D( p ) 00871 ELSE 00872 T = ZERO 00873 AAPP = ONE 00874 CALL DLASSQ( M, A( 1, p ), 1, T, 00875 + AAPP ) 00876 AAPP = T*DSQRT( AAPP )*D( p ) 00877 END IF 00878 SVA( p ) = AAPP 00879 END IF 00880 * end of OK rotation 00881 ELSE 00882 NOTROT = NOTROT + 1 00883 PSKIPPED = PSKIPPED + 1 00884 IJBLSK = IJBLSK + 1 00885 END IF 00886 ELSE 00887 NOTROT = NOTROT + 1 00888 PSKIPPED = PSKIPPED + 1 00889 IJBLSK = IJBLSK + 1 00890 END IF 00891 * 00892 IF( ( i.LE.SWBAND ) .AND. ( IJBLSK.GE.BLSKIP ) ) 00893 + THEN 00894 SVA( p ) = AAPP 00895 NOTROT = 0 00896 GO TO 2011 00897 END IF 00898 IF( ( i.LE.SWBAND ) .AND. 00899 + ( PSKIPPED.GT.ROWSKIP ) ) THEN 00900 AAPP = -AAPP 00901 NOTROT = 0 00902 GO TO 2203 00903 END IF 00904 * 00905 2200 CONTINUE 00906 * end of the q-loop 00907 2203 CONTINUE 00908 * 00909 SVA( p ) = AAPP 00910 * 00911 ELSE 00912 IF( AAPP.EQ.ZERO )NOTROT = NOTROT + 00913 + MIN0( jgl+KBL-1, N ) - jgl + 1 00914 IF( AAPP.LT.ZERO )NOTROT = 0 00915 END IF 00916 00917 2100 CONTINUE 00918 * end of the p-loop 00919 2010 CONTINUE 00920 * end of the jbc-loop 00921 2011 CONTINUE 00922 *2011 bailed out of the jbc-loop 00923 DO 2012 p = igl, MIN0( igl+KBL-1, N ) 00924 SVA( p ) = DABS( SVA( p ) ) 00925 2012 CONTINUE 00926 * 00927 2000 CONTINUE 00928 *2000 :: end of the ibr-loop 00929 * 00930 * .. update SVA(N) 00931 IF( ( SVA( N ).LT.ROOTBIG ) .AND. ( SVA( N ).GT.ROOTSFMIN ) ) 00932 + THEN 00933 SVA( N ) = DNRM2( M, A( 1, N ), 1 )*D( N ) 00934 ELSE 00935 T = ZERO 00936 AAPP = ONE 00937 CALL DLASSQ( M, A( 1, N ), 1, T, AAPP ) 00938 SVA( N ) = T*DSQRT( AAPP )*D( N ) 00939 END IF 00940 * 00941 * Additional steering devices 00942 * 00943 IF( ( i.LT.SWBAND ) .AND. ( ( MXAAPQ.LE.ROOTTOL ) .OR. 00944 + ( ISWROT.LE.N ) ) )SWBAND = i 00945 * 00946 IF( ( i.GT.SWBAND+1 ) .AND. ( MXAAPQ.LT.DBLE( N )*TOL ) .AND. 00947 + ( DBLE( N )*MXAAPQ*MXSINJ.LT.TOL ) ) THEN 00948 GO TO 1994 00949 END IF 00950 * 00951 IF( NOTROT.GE.EMPTSW )GO TO 1994 00952 00953 1993 CONTINUE 00954 * end i=1:NSWEEP loop 00955 * #:) Reaching this point means that the procedure has comleted the given 00956 * number of iterations. 00957 INFO = NSWEEP - 1 00958 GO TO 1995 00959 1994 CONTINUE 00960 * #:) Reaching this point means that during the i-th sweep all pivots were 00961 * below the given tolerance, causing early exit. 00962 * 00963 INFO = 0 00964 * #:) INFO = 0 confirms successful iterations. 00965 1995 CONTINUE 00966 * 00967 * Sort the vector D. 00968 DO 5991 p = 1, N - 1 00969 q = IDAMAX( N-p+1, SVA( p ), 1 ) + p - 1 00970 IF( p.NE.q ) THEN 00971 TEMP1 = SVA( p ) 00972 SVA( p ) = SVA( q ) 00973 SVA( q ) = TEMP1 00974 TEMP1 = D( p ) 00975 D( p ) = D( q ) 00976 D( q ) = TEMP1 00977 CALL DSWAP( M, A( 1, p ), 1, A( 1, q ), 1 ) 00978 IF( RSVEC )CALL DSWAP( MVL, V( 1, p ), 1, V( 1, q ), 1 ) 00979 END IF 00980 5991 CONTINUE 00981 * 00982 RETURN 00983 * .. 00984 * .. END OF DGSVJ0 00985 * .. 00986 END
1.7.3 
