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LAPACK 3.3.1
Linear Algebra PACKage
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LAPACK 3.3.1
Linear Algebra PACKage
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00001 SUBROUTINE SGSVJ0( JOBV, M, N, A, LDA, D, SVA, MV, V, LDV, EPS, 00002 $ SFMIN, TOL, NSWEEP, WORK, LWORK, INFO ) 00003 * 00004 * -- LAPACK routine (version 3.3.1) -- 00005 * 00006 * -- Contributed by Zlatko Drmac of the University of Zagreb and -- 00007 * -- Kresimir Veselic of the Fernuniversitaet Hagen -- 00008 * -- April 2011 -- 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 * .. 00020 * .. Scalar Arguments .. 00021 INTEGER INFO, LDA, LDV, LWORK, M, MV, N, NSWEEP 00022 REAL EPS, SFMIN, TOL 00023 CHARACTER*1 JOBV 00024 * .. 00025 * .. Array Arguments .. 00026 REAL A( LDA, * ), SVA( N ), D( N ), V( LDV, * ), 00027 $ WORK( LWORK ) 00028 * .. 00029 * 00030 * Purpose 00031 * ======= 00032 * 00033 * SGSVJ0 is called from SGESVJ as a pre-processor and that is its main 00034 * purpose. It applies Jacobi rotations in the same way as SGESVJ does, but 00035 * it does not check convergence (stopping criterion). Few tuning 00036 * parameters (marked by [TP]) are available for the implementer. 00037 * 00038 * Further Details 00039 * ~~~~~~~~~~~~~~~ 00040 * SGSVJ0 is used just to enable SGESVJ to call a simplified version of 00041 * itself to work on a submatrix of the original matrix. 00042 * 00043 * Contributors 00044 * ~~~~~~~~~~~~ 00045 * Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany) 00046 * 00047 * Bugs, Examples and Comments 00048 * ~~~~~~~~~~~~~~~~~~~~~~~~~~~ 00049 * Please report all bugs and send interesting test examples and comments to 00050 * drmac@math.hr. Thank you. 00051 * 00052 * Arguments 00053 * ========= 00054 * 00055 * JOBV (input) CHARACTER*1 00056 * Specifies whether the output from this procedure is used 00057 * to compute the matrix V: 00058 * = 'V': the product of the Jacobi rotations is accumulated 00059 * by postmulyiplying the N-by-N array V. 00060 * (See the description of V.) 00061 * = 'A': the product of the Jacobi rotations is accumulated 00062 * by postmulyiplying the MV-by-N array V. 00063 * (See the descriptions of MV and V.) 00064 * = 'N': the Jacobi rotations are not accumulated. 00065 * 00066 * M (input) INTEGER 00067 * The number of rows of the input matrix A. M >= 0. 00068 * 00069 * N (input) INTEGER 00070 * The number of columns of the input matrix A. 00071 * M >= N >= 0. 00072 * 00073 * A (input/output) REAL array, dimension (LDA,N) 00074 * On entry, M-by-N matrix A, such that A*diag(D) represents 00075 * the input matrix. 00076 * On exit, 00077 * A_onexit * D_onexit represents the input matrix A*diag(D) 00078 * post-multiplied by a sequence of Jacobi rotations, where the 00079 * rotation threshold and the total number of sweeps are given in 00080 * TOL and NSWEEP, respectively. 00081 * (See the descriptions of D, TOL and NSWEEP.) 00082 * 00083 * LDA (input) INTEGER 00084 * The leading dimension of the array A. LDA >= max(1,M). 00085 * 00086 * D (input/workspace/output) REAL array, dimension (N) 00087 * The array D accumulates the scaling factors from the fast scaled 00088 * Jacobi rotations. 00089 * On entry, A*diag(D) represents the input matrix. 00090 * On exit, A_onexit*diag(D_onexit) represents the input matrix 00091 * post-multiplied by a sequence of Jacobi rotations, where the 00092 * rotation threshold and the total number of sweeps are given in 00093 * TOL and NSWEEP, respectively. 00094 * (See the descriptions of A, TOL and NSWEEP.) 00095 * 00096 * SVA (input/workspace/output) REAL array, dimension (N) 00097 * On entry, SVA contains the Euclidean norms of the columns of 00098 * the matrix A*diag(D). 00099 * On exit, SVA contains the Euclidean norms of the columns of 00100 * the matrix onexit*diag(D_onexit). 00101 * 00102 * MV (input) INTEGER 00103 * If JOBV .EQ. 'A', then MV rows of V are post-multipled by a 00104 * sequence of Jacobi rotations. 00105 * If JOBV = 'N', then MV is not referenced. 00106 * 00107 * V (input/output) REAL array, dimension (LDV,N) 00108 * If JOBV .EQ. 'V' then N rows of V are post-multipled by a 00109 * sequence of Jacobi rotations. 00110 * If JOBV .EQ. 'A' then MV rows of V are post-multipled by a 00111 * sequence of Jacobi rotations. 00112 * If JOBV = 'N', then V is not referenced. 00113 * 00114 * LDV (input) INTEGER 00115 * The leading dimension of the array V, LDV >= 1. 00116 * If JOBV = 'V', LDV .GE. N. 00117 * If JOBV = 'A', LDV .GE. MV. 00118 * 00119 * EPS (input) INTEGER 00120 * EPS = SLAMCH('Epsilon') 00121 * 00122 * SFMIN (input) INTEGER 00123 * SFMIN = SLAMCH('Safe Minimum') 00124 * 00125 * TOL (input) REAL 00126 * TOL is the threshold for Jacobi rotations. For a pair 00127 * A(:,p), A(:,q) of pivot columns, the Jacobi rotation is 00128 * applied only if ABS(COS(angle(A(:,p),A(:,q)))) .GT. TOL. 00129 * 00130 * NSWEEP (input) INTEGER 00131 * NSWEEP is the number of sweeps of Jacobi rotations to be 00132 * performed. 00133 * 00134 * WORK (workspace) REAL array, dimension LWORK. 00135 * 00136 * LWORK (input) INTEGER 00137 * LWORK is the dimension of WORK. LWORK .GE. M. 00138 * 00139 * INFO (output) INTEGER 00140 * = 0 : successful exit. 00141 * < 0 : if INFO = -i, then the i-th argument had an illegal value 00142 * 00143 * ===================================================================== 00144 * 00145 * .. Local Parameters .. 00146 REAL ZERO, HALF, ONE, TWO 00147 PARAMETER ( ZERO = 0.0E0, HALF = 0.5E0, ONE = 1.0E0, 00148 $ TWO = 2.0E0 ) 00149 * .. 00150 * .. Local Scalars .. 00151 REAL AAPP, AAPP0, AAPQ, AAQQ, APOAQ, AQOAP, BIG, 00152 $ BIGTHETA, CS, MXAAPQ, MXSINJ, ROOTBIG, ROOTEPS, 00153 $ ROOTSFMIN, ROOTTOL, SMALL, SN, T, TEMP1, THETA, 00154 $ THSIGN 00155 INTEGER BLSKIP, EMPTSW, i, ibr, IERR, igl, IJBLSK, ir1, 00156 $ ISWROT, jbc, jgl, KBL, LKAHEAD, MVL, NBL, 00157 $ NOTROT, p, PSKIPPED, q, ROWSKIP, SWBAND 00158 LOGICAL APPLV, ROTOK, RSVEC 00159 * .. 00160 * .. Local Arrays .. 00161 REAL FASTR( 5 ) 00162 * .. 00163 * .. Intrinsic Functions .. 00164 INTRINSIC ABS, AMAX1, AMIN1, FLOAT, MIN0, SIGN, SQRT 00165 * .. 00166 * .. External Functions .. 00167 REAL SDOT, SNRM2 00168 INTEGER ISAMAX 00169 LOGICAL LSAME 00170 EXTERNAL ISAMAX, LSAME, SDOT, SNRM2 00171 * .. 00172 * .. External Subroutines .. 00173 EXTERNAL SAXPY, SCOPY, SLASCL, SLASSQ, SROTM, SSWAP 00174 * .. 00175 * .. Executable Statements .. 00176 * 00177 * Test the input parameters. 00178 * 00179 APPLV = LSAME( JOBV, 'A' ) 00180 RSVEC = LSAME( JOBV, 'V' ) 00181 IF( .NOT.( RSVEC .OR. APPLV .OR. LSAME( JOBV, 'N' ) ) ) THEN 00182 INFO = -1 00183 ELSE IF( M.LT.0 ) THEN 00184 INFO = -2 00185 ELSE IF( ( N.LT.0 ) .OR. ( N.GT.M ) ) THEN 00186 INFO = -3 00187 ELSE IF( LDA.LT.M ) THEN 00188 INFO = -5 00189 ELSE IF( ( RSVEC.OR.APPLV ) .AND. ( MV.LT.0 ) ) THEN 00190 INFO = -8 00191 ELSE IF( ( RSVEC.AND.( LDV.LT.N ) ).OR. 00192 $ ( APPLV.AND.( LDV.LT.MV ) ) ) THEN 00193 INFO = -10 00194 ELSE IF( TOL.LE.EPS ) THEN 00195 INFO = -13 00196 ELSE IF( NSWEEP.LT.0 ) THEN 00197 INFO = -14 00198 ELSE IF( LWORK.LT.M ) THEN 00199 INFO = -16 00200 ELSE 00201 INFO = 0 00202 END IF 00203 * 00204 * #:( 00205 IF( INFO.NE.0 ) THEN 00206 CALL XERBLA( 'SGSVJ0', -INFO ) 00207 RETURN 00208 END IF 00209 * 00210 IF( RSVEC ) THEN 00211 MVL = N 00212 ELSE IF( APPLV ) THEN 00213 MVL = MV 00214 END IF 00215 RSVEC = RSVEC .OR. APPLV 00216 00217 ROOTEPS = SQRT( EPS ) 00218 ROOTSFMIN = SQRT( SFMIN ) 00219 SMALL = SFMIN / EPS 00220 BIG = ONE / SFMIN 00221 ROOTBIG = ONE / ROOTSFMIN 00222 BIGTHETA = ONE / ROOTEPS 00223 ROOTTOL = SQRT( TOL ) 00224 * 00225 * .. Row-cyclic Jacobi SVD algorithm with column pivoting .. 00226 * 00227 EMPTSW = ( N*( N-1 ) ) / 2 00228 NOTROT = 0 00229 FASTR( 1 ) = ZERO 00230 * 00231 * .. Row-cyclic pivot strategy with de Rijk's pivoting .. 00232 * 00233 00234 SWBAND = 0 00235 *[TP] SWBAND is a tuning parameter. It is meaningful and effective 00236 * if SGESVJ is used as a computational routine in the preconditioned 00237 * Jacobi SVD algorithm SGESVJ. For sweeps i=1:SWBAND the procedure 00238 * ...... 00239 00240 KBL = MIN0( 8, N ) 00241 *[TP] KBL is a tuning parameter that defines the tile size in the 00242 * tiling of the p-q loops of pivot pairs. In general, an optimal 00243 * value of KBL depends on the matrix dimensions and on the 00244 * parameters of the computer's memory. 00245 * 00246 NBL = N / KBL 00247 IF( ( NBL*KBL ).NE.N )NBL = NBL + 1 00248 00249 BLSKIP = ( KBL**2 ) + 1 00250 *[TP] BLKSKIP is a tuning parameter that depends on SWBAND and KBL. 00251 00252 ROWSKIP = MIN0( 5, KBL ) 00253 *[TP] ROWSKIP is a tuning parameter. 00254 00255 LKAHEAD = 1 00256 *[TP] LKAHEAD is a tuning parameter. 00257 SWBAND = 0 00258 PSKIPPED = 0 00259 * 00260 DO 1993 i = 1, NSWEEP 00261 * .. go go go ... 00262 * 00263 MXAAPQ = ZERO 00264 MXSINJ = ZERO 00265 ISWROT = 0 00266 * 00267 NOTROT = 0 00268 PSKIPPED = 0 00269 * 00270 DO 2000 ibr = 1, NBL 00271 00272 igl = ( ibr-1 )*KBL + 1 00273 * 00274 DO 1002 ir1 = 0, MIN0( LKAHEAD, NBL-ibr ) 00275 * 00276 igl = igl + ir1*KBL 00277 * 00278 DO 2001 p = igl, MIN0( igl+KBL-1, N-1 ) 00279 00280 * .. de Rijk's pivoting 00281 q = ISAMAX( N-p+1, SVA( p ), 1 ) + p - 1 00282 IF( p.NE.q ) THEN 00283 CALL SSWAP( M, A( 1, p ), 1, A( 1, q ), 1 ) 00284 IF( RSVEC )CALL SSWAP( MVL, V( 1, p ), 1, 00285 $ V( 1, q ), 1 ) 00286 TEMP1 = SVA( p ) 00287 SVA( p ) = SVA( q ) 00288 SVA( q ) = TEMP1 00289 TEMP1 = D( p ) 00290 D( p ) = D( q ) 00291 D( q ) = TEMP1 00292 END IF 00293 * 00294 IF( ir1.EQ.0 ) THEN 00295 * 00296 * Column norms are periodically updated by explicit 00297 * norm computation. 00298 * Caveat: 00299 * Some BLAS implementations compute SNRM2(M,A(1,p),1) 00300 * as SQRT(SDOT(M,A(1,p),1,A(1,p),1)), which may result in 00301 * overflow for ||A(:,p)||_2 > SQRT(overflow_threshold), and 00302 * undeflow for ||A(:,p)||_2 < SQRT(underflow_threshold). 00303 * Hence, SNRM2 cannot be trusted, not even in the case when 00304 * the true norm is far from the under(over)flow boundaries. 00305 * If properly implemented SNRM2 is available, the IF-THEN-ELSE 00306 * below should read "AAPP = SNRM2( M, A(1,p), 1 ) * D(p)". 00307 * 00308 IF( ( SVA( p ).LT.ROOTBIG ) .AND. 00309 $ ( SVA( p ).GT.ROOTSFMIN ) ) THEN 00310 SVA( p ) = SNRM2( M, A( 1, p ), 1 )*D( p ) 00311 ELSE 00312 TEMP1 = ZERO 00313 AAPP = ONE 00314 CALL SLASSQ( M, A( 1, p ), 1, TEMP1, AAPP ) 00315 SVA( p ) = TEMP1*SQRT( AAPP )*D( p ) 00316 END IF 00317 AAPP = SVA( p ) 00318 ELSE 00319 AAPP = SVA( p ) 00320 END IF 00321 00322 * 00323 IF( AAPP.GT.ZERO ) THEN 00324 * 00325 PSKIPPED = 0 00326 * 00327 DO 2002 q = p + 1, MIN0( igl+KBL-1, N ) 00328 * 00329 AAQQ = SVA( q ) 00330 00331 IF( AAQQ.GT.ZERO ) THEN 00332 * 00333 AAPP0 = AAPP 00334 IF( AAQQ.GE.ONE ) THEN 00335 ROTOK = ( SMALL*AAPP ).LE.AAQQ 00336 IF( AAPP.LT.( BIG / AAQQ ) ) THEN 00337 AAPQ = ( SDOT( M, A( 1, p ), 1, A( 1, 00338 $ q ), 1 )*D( p )*D( q ) / AAQQ ) 00339 $ / AAPP 00340 ELSE 00341 CALL SCOPY( M, A( 1, p ), 1, WORK, 1 ) 00342 CALL SLASCL( 'G', 0, 0, AAPP, D( p ), 00343 $ M, 1, WORK, LDA, IERR ) 00344 AAPQ = SDOT( M, WORK, 1, A( 1, q ), 00345 $ 1 )*D( q ) / AAQQ 00346 END IF 00347 ELSE 00348 ROTOK = AAPP.LE.( AAQQ / SMALL ) 00349 IF( AAPP.GT.( SMALL / AAQQ ) ) THEN 00350 AAPQ = ( SDOT( M, A( 1, p ), 1, A( 1, 00351 $ q ), 1 )*D( p )*D( q ) / AAQQ ) 00352 $ / AAPP 00353 ELSE 00354 CALL SCOPY( M, A( 1, q ), 1, WORK, 1 ) 00355 CALL SLASCL( 'G', 0, 0, AAQQ, D( q ), 00356 $ M, 1, WORK, LDA, IERR ) 00357 AAPQ = SDOT( M, WORK, 1, A( 1, p ), 00358 $ 1 )*D( p ) / AAPP 00359 END IF 00360 END IF 00361 * 00362 MXAAPQ = AMAX1( MXAAPQ, ABS( AAPQ ) ) 00363 * 00364 * TO rotate or NOT to rotate, THAT is the question ... 00365 * 00366 IF( ABS( AAPQ ).GT.TOL ) THEN 00367 * 00368 * .. rotate 00369 * ROTATED = ROTATED + ONE 00370 * 00371 IF( ir1.EQ.0 ) THEN 00372 NOTROT = 0 00373 PSKIPPED = 0 00374 ISWROT = ISWROT + 1 00375 END IF 00376 * 00377 IF( ROTOK ) THEN 00378 * 00379 AQOAP = AAQQ / AAPP 00380 APOAQ = AAPP / AAQQ 00381 THETA = -HALF*ABS( AQOAP-APOAQ ) / AAPQ 00382 * 00383 IF( ABS( THETA ).GT.BIGTHETA ) THEN 00384 * 00385 T = HALF / THETA 00386 FASTR( 3 ) = T*D( p ) / D( q ) 00387 FASTR( 4 ) = -T*D( q ) / D( p ) 00388 CALL SROTM( M, A( 1, p ), 1, 00389 $ A( 1, q ), 1, FASTR ) 00390 IF( RSVEC )CALL SROTM( MVL, 00391 $ V( 1, p ), 1, 00392 $ V( 1, q ), 1, 00393 $ FASTR ) 00394 SVA( q ) = AAQQ*SQRT( AMAX1( ZERO, 00395 $ ONE+T*APOAQ*AAPQ ) ) 00396 AAPP = AAPP*SQRT( AMAX1( ZERO, 00397 $ ONE-T*AQOAP*AAPQ ) ) 00398 MXSINJ = AMAX1( MXSINJ, ABS( T ) ) 00399 * 00400 ELSE 00401 * 00402 * .. choose correct signum for THETA and rotate 00403 * 00404 THSIGN = -SIGN( ONE, AAPQ ) 00405 T = ONE / ( THETA+THSIGN* 00406 $ SQRT( ONE+THETA*THETA ) ) 00407 CS = SQRT( ONE / ( ONE+T*T ) ) 00408 SN = T*CS 00409 * 00410 MXSINJ = AMAX1( MXSINJ, ABS( SN ) ) 00411 SVA( q ) = AAQQ*SQRT( AMAX1( ZERO, 00412 $ ONE+T*APOAQ*AAPQ ) ) 00413 AAPP = AAPP*SQRT( AMAX1( ZERO, 00414 $ ONE-T*AQOAP*AAPQ ) ) 00415 * 00416 APOAQ = D( p ) / D( q ) 00417 AQOAP = D( q ) / D( p ) 00418 IF( D( p ).GE.ONE ) THEN 00419 IF( D( q ).GE.ONE ) THEN 00420 FASTR( 3 ) = T*APOAQ 00421 FASTR( 4 ) = -T*AQOAP 00422 D( p ) = D( p )*CS 00423 D( q ) = D( q )*CS 00424 CALL SROTM( M, A( 1, p ), 1, 00425 $ A( 1, q ), 1, 00426 $ FASTR ) 00427 IF( RSVEC )CALL SROTM( MVL, 00428 $ V( 1, p ), 1, V( 1, q ), 00429 $ 1, FASTR ) 00430 ELSE 00431 CALL SAXPY( M, -T*AQOAP, 00432 $ A( 1, q ), 1, 00433 $ A( 1, p ), 1 ) 00434 CALL SAXPY( M, CS*SN*APOAQ, 00435 $ A( 1, p ), 1, 00436 $ A( 1, q ), 1 ) 00437 D( p ) = D( p )*CS 00438 D( q ) = D( q ) / CS 00439 IF( RSVEC ) THEN 00440 CALL SAXPY( MVL, -T*AQOAP, 00441 $ V( 1, q ), 1, 00442 $ V( 1, p ), 1 ) 00443 CALL SAXPY( MVL, 00444 $ CS*SN*APOAQ, 00445 $ V( 1, p ), 1, 00446 $ V( 1, q ), 1 ) 00447 END IF 00448 END IF 00449 ELSE 00450 IF( D( q ).GE.ONE ) THEN 00451 CALL SAXPY( M, T*APOAQ, 00452 $ A( 1, p ), 1, 00453 $ A( 1, q ), 1 ) 00454 CALL SAXPY( M, -CS*SN*AQOAP, 00455 $ A( 1, q ), 1, 00456 $ A( 1, p ), 1 ) 00457 D( p ) = D( p ) / CS 00458 D( q ) = D( q )*CS 00459 IF( RSVEC ) THEN 00460 CALL SAXPY( MVL, T*APOAQ, 00461 $ V( 1, p ), 1, 00462 $ V( 1, q ), 1 ) 00463 CALL SAXPY( MVL, 00464 $ -CS*SN*AQOAP, 00465 $ V( 1, q ), 1, 00466 $ V( 1, p ), 1 ) 00467 END IF 00468 ELSE 00469 IF( D( p ).GE.D( q ) ) THEN 00470 CALL SAXPY( M, -T*AQOAP, 00471 $ A( 1, q ), 1, 00472 $ A( 1, p ), 1 ) 00473 CALL SAXPY( M, CS*SN*APOAQ, 00474 $ A( 1, p ), 1, 00475 $ A( 1, q ), 1 ) 00476 D( p ) = D( p )*CS 00477 D( q ) = D( q ) / CS 00478 IF( RSVEC ) THEN 00479 CALL SAXPY( MVL, 00480 $ -T*AQOAP, 00481 $ V( 1, q ), 1, 00482 $ V( 1, p ), 1 ) 00483 CALL SAXPY( MVL, 00484 $ CS*SN*APOAQ, 00485 $ V( 1, p ), 1, 00486 $ V( 1, q ), 1 ) 00487 END IF 00488 ELSE 00489 CALL SAXPY( M, T*APOAQ, 00490 $ A( 1, p ), 1, 00491 $ A( 1, q ), 1 ) 00492 CALL SAXPY( M, 00493 $ -CS*SN*AQOAP, 00494 $ A( 1, q ), 1, 00495 $ A( 1, p ), 1 ) 00496 D( p ) = D( p ) / CS 00497 D( q ) = D( q )*CS 00498 IF( RSVEC ) THEN 00499 CALL SAXPY( MVL, 00500 $ T*APOAQ, V( 1, p ), 00501 $ 1, V( 1, q ), 1 ) 00502 CALL SAXPY( MVL, 00503 $ -CS*SN*AQOAP, 00504 $ V( 1, q ), 1, 00505 $ V( 1, p ), 1 ) 00506 END IF 00507 END IF 00508 END IF 00509 END IF 00510 END IF 00511 * 00512 ELSE 00513 * .. have to use modified Gram-Schmidt like transformation 00514 CALL SCOPY( M, A( 1, p ), 1, WORK, 1 ) 00515 CALL SLASCL( 'G', 0, 0, AAPP, ONE, M, 00516 $ 1, WORK, LDA, IERR ) 00517 CALL SLASCL( 'G', 0, 0, AAQQ, ONE, M, 00518 $ 1, A( 1, q ), LDA, IERR ) 00519 TEMP1 = -AAPQ*D( p ) / D( q ) 00520 CALL SAXPY( M, TEMP1, WORK, 1, 00521 $ A( 1, q ), 1 ) 00522 CALL SLASCL( 'G', 0, 0, ONE, AAQQ, M, 00523 $ 1, A( 1, q ), LDA, IERR ) 00524 SVA( q ) = AAQQ*SQRT( AMAX1( ZERO, 00525 $ ONE-AAPQ*AAPQ ) ) 00526 MXSINJ = AMAX1( MXSINJ, SFMIN ) 00527 END IF 00528 * END IF ROTOK THEN ... ELSE 00529 * 00530 * In the case of cancellation in updating SVA(q), SVA(p) 00531 * recompute SVA(q), SVA(p). 00532 IF( ( SVA( q ) / AAQQ )**2.LE.ROOTEPS ) 00533 $ THEN 00534 IF( ( AAQQ.LT.ROOTBIG ) .AND. 00535 $ ( AAQQ.GT.ROOTSFMIN ) ) THEN 00536 SVA( q ) = SNRM2( M, A( 1, q ), 1 )* 00537 $ D( q ) 00538 ELSE 00539 T = ZERO 00540 AAQQ = ONE 00541 CALL SLASSQ( M, A( 1, q ), 1, T, 00542 $ AAQQ ) 00543 SVA( q ) = T*SQRT( AAQQ )*D( q ) 00544 END IF 00545 END IF 00546 IF( ( AAPP / AAPP0 ).LE.ROOTEPS ) THEN 00547 IF( ( AAPP.LT.ROOTBIG ) .AND. 00548 $ ( AAPP.GT.ROOTSFMIN ) ) THEN 00549 AAPP = SNRM2( M, A( 1, p ), 1 )* 00550 $ D( p ) 00551 ELSE 00552 T = ZERO 00553 AAPP = ONE 00554 CALL SLASSQ( M, A( 1, p ), 1, T, 00555 $ AAPP ) 00556 AAPP = T*SQRT( AAPP )*D( p ) 00557 END IF 00558 SVA( p ) = AAPP 00559 END IF 00560 * 00561 ELSE 00562 * A(:,p) and A(:,q) already numerically orthogonal 00563 IF( ir1.EQ.0 )NOTROT = NOTROT + 1 00564 PSKIPPED = PSKIPPED + 1 00565 END IF 00566 ELSE 00567 * A(:,q) is zero column 00568 IF( ir1.EQ.0 )NOTROT = NOTROT + 1 00569 PSKIPPED = PSKIPPED + 1 00570 END IF 00571 * 00572 IF( ( i.LE.SWBAND ) .AND. 00573 $ ( PSKIPPED.GT.ROWSKIP ) ) THEN 00574 IF( ir1.EQ.0 )AAPP = -AAPP 00575 NOTROT = 0 00576 GO TO 2103 00577 END IF 00578 * 00579 2002 CONTINUE 00580 * END q-LOOP 00581 * 00582 2103 CONTINUE 00583 * bailed out of q-loop 00584 00585 SVA( p ) = AAPP 00586 00587 ELSE 00588 SVA( p ) = AAPP 00589 IF( ( ir1.EQ.0 ) .AND. ( AAPP.EQ.ZERO ) ) 00590 $ NOTROT = NOTROT + MIN0( igl+KBL-1, N ) - p 00591 END IF 00592 * 00593 2001 CONTINUE 00594 * end of the p-loop 00595 * end of doing the block ( ibr, ibr ) 00596 1002 CONTINUE 00597 * end of ir1-loop 00598 * 00599 *........................................................ 00600 * ... go to the off diagonal blocks 00601 * 00602 igl = ( ibr-1 )*KBL + 1 00603 * 00604 DO 2010 jbc = ibr + 1, NBL 00605 * 00606 jgl = ( jbc-1 )*KBL + 1 00607 * 00608 * doing the block at ( ibr, jbc ) 00609 * 00610 IJBLSK = 0 00611 DO 2100 p = igl, MIN0( igl+KBL-1, N ) 00612 * 00613 AAPP = SVA( p ) 00614 * 00615 IF( AAPP.GT.ZERO ) THEN 00616 * 00617 PSKIPPED = 0 00618 * 00619 DO 2200 q = jgl, MIN0( jgl+KBL-1, N ) 00620 * 00621 AAQQ = SVA( q ) 00622 * 00623 IF( AAQQ.GT.ZERO ) THEN 00624 AAPP0 = AAPP 00625 * 00626 * .. M x 2 Jacobi SVD .. 00627 * 00628 * .. Safe Gram matrix computation .. 00629 * 00630 IF( AAQQ.GE.ONE ) THEN 00631 IF( AAPP.GE.AAQQ ) THEN 00632 ROTOK = ( SMALL*AAPP ).LE.AAQQ 00633 ELSE 00634 ROTOK = ( SMALL*AAQQ ).LE.AAPP 00635 END IF 00636 IF( AAPP.LT.( BIG / AAQQ ) ) THEN 00637 AAPQ = ( SDOT( M, A( 1, p ), 1, A( 1, 00638 $ q ), 1 )*D( p )*D( q ) / AAQQ ) 00639 $ / AAPP 00640 ELSE 00641 CALL SCOPY( M, A( 1, p ), 1, WORK, 1 ) 00642 CALL SLASCL( 'G', 0, 0, AAPP, D( p ), 00643 $ M, 1, WORK, LDA, IERR ) 00644 AAPQ = SDOT( M, WORK, 1, A( 1, q ), 00645 $ 1 )*D( q ) / AAQQ 00646 END IF 00647 ELSE 00648 IF( AAPP.GE.AAQQ ) THEN 00649 ROTOK = AAPP.LE.( AAQQ / SMALL ) 00650 ELSE 00651 ROTOK = AAQQ.LE.( AAPP / SMALL ) 00652 END IF 00653 IF( AAPP.GT.( SMALL / AAQQ ) ) THEN 00654 AAPQ = ( SDOT( M, A( 1, p ), 1, A( 1, 00655 $ q ), 1 )*D( p )*D( q ) / AAQQ ) 00656 $ / AAPP 00657 ELSE 00658 CALL SCOPY( M, A( 1, q ), 1, WORK, 1 ) 00659 CALL SLASCL( 'G', 0, 0, AAQQ, D( q ), 00660 $ M, 1, WORK, LDA, IERR ) 00661 AAPQ = SDOT( M, WORK, 1, A( 1, p ), 00662 $ 1 )*D( p ) / AAPP 00663 END IF 00664 END IF 00665 * 00666 MXAAPQ = AMAX1( MXAAPQ, ABS( AAPQ ) ) 00667 * 00668 * TO rotate or NOT to rotate, THAT is the question ... 00669 * 00670 IF( ABS( AAPQ ).GT.TOL ) THEN 00671 NOTROT = 0 00672 * ROTATED = ROTATED + 1 00673 PSKIPPED = 0 00674 ISWROT = ISWROT + 1 00675 * 00676 IF( ROTOK ) THEN 00677 * 00678 AQOAP = AAQQ / AAPP 00679 APOAQ = AAPP / AAQQ 00680 THETA = -HALF*ABS( AQOAP-APOAQ ) / AAPQ 00681 IF( AAQQ.GT.AAPP0 )THETA = -THETA 00682 * 00683 IF( ABS( 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 SROTM( M, A( 1, p ), 1, 00688 $ A( 1, q ), 1, FASTR ) 00689 IF( RSVEC )CALL SROTM( MVL, 00690 $ V( 1, p ), 1, 00691 $ V( 1, q ), 1, 00692 $ FASTR ) 00693 SVA( q ) = AAQQ*SQRT( AMAX1( ZERO, 00694 $ ONE+T*APOAQ*AAPQ ) ) 00695 AAPP = AAPP*SQRT( AMAX1( ZERO, 00696 $ ONE-T*AQOAP*AAPQ ) ) 00697 MXSINJ = AMAX1( MXSINJ, ABS( T ) ) 00698 ELSE 00699 * 00700 * .. choose correct signum for THETA and rotate 00701 * 00702 THSIGN = -SIGN( ONE, AAPQ ) 00703 IF( AAQQ.GT.AAPP0 )THSIGN = -THSIGN 00704 T = ONE / ( THETA+THSIGN* 00705 $ SQRT( ONE+THETA*THETA ) ) 00706 CS = SQRT( ONE / ( ONE+T*T ) ) 00707 SN = T*CS 00708 MXSINJ = AMAX1( MXSINJ, ABS( SN ) ) 00709 SVA( q ) = AAQQ*SQRT( AMAX1( ZERO, 00710 $ ONE+T*APOAQ*AAPQ ) ) 00711 AAPP = AAPP*SQRT( AMAX1( 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 SROTM( M, A( 1, p ), 1, 00724 $ A( 1, q ), 1, 00725 $ FASTR ) 00726 IF( RSVEC )CALL SROTM( MVL, 00727 $ V( 1, p ), 1, V( 1, q ), 00728 $ 1, FASTR ) 00729 ELSE 00730 CALL SAXPY( M, -T*AQOAP, 00731 $ A( 1, q ), 1, 00732 $ A( 1, p ), 1 ) 00733 CALL SAXPY( M, CS*SN*APOAQ, 00734 $ A( 1, p ), 1, 00735 $ A( 1, q ), 1 ) 00736 IF( RSVEC ) THEN 00737 CALL SAXPY( MVL, -T*AQOAP, 00738 $ V( 1, q ), 1, 00739 $ V( 1, p ), 1 ) 00740 CALL SAXPY( 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 SAXPY( M, T*APOAQ, 00751 $ A( 1, p ), 1, 00752 $ A( 1, q ), 1 ) 00753 CALL SAXPY( M, -CS*SN*AQOAP, 00754 $ A( 1, q ), 1, 00755 $ A( 1, p ), 1 ) 00756 IF( RSVEC ) THEN 00757 CALL SAXPY( MVL, T*APOAQ, 00758 $ V( 1, p ), 1, 00759 $ V( 1, q ), 1 ) 00760 CALL SAXPY( 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 SAXPY( M, -T*AQOAP, 00770 $ A( 1, q ), 1, 00771 $ A( 1, p ), 1 ) 00772 CALL SAXPY( 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 SAXPY( MVL, 00779 $ -T*AQOAP, 00780 $ V( 1, q ), 1, 00781 $ V( 1, p ), 1 ) 00782 CALL SAXPY( MVL, 00783 $ CS*SN*APOAQ, 00784 $ V( 1, p ), 1, 00785 $ V( 1, q ), 1 ) 00786 END IF 00787 ELSE 00788 CALL SAXPY( M, T*APOAQ, 00789 $ A( 1, p ), 1, 00790 $ A( 1, q ), 1 ) 00791 CALL SAXPY( 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 SAXPY( MVL, 00799 $ T*APOAQ, V( 1, p ), 00800 $ 1, V( 1, q ), 1 ) 00801 CALL SAXPY( 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 SCOPY( M, A( 1, p ), 1, WORK, 00814 $ 1 ) 00815 CALL SLASCL( 'G', 0, 0, AAPP, ONE, 00816 $ M, 1, WORK, LDA, IERR ) 00817 CALL SLASCL( 'G', 0, 0, AAQQ, ONE, 00818 $ M, 1, A( 1, q ), LDA, 00819 $ IERR ) 00820 TEMP1 = -AAPQ*D( p ) / D( q ) 00821 CALL SAXPY( M, TEMP1, WORK, 1, 00822 $ A( 1, q ), 1 ) 00823 CALL SLASCL( 'G', 0, 0, ONE, AAQQ, 00824 $ M, 1, A( 1, q ), LDA, 00825 $ IERR ) 00826 SVA( q ) = AAQQ*SQRT( AMAX1( ZERO, 00827 $ ONE-AAPQ*AAPQ ) ) 00828 MXSINJ = AMAX1( MXSINJ, SFMIN ) 00829 ELSE 00830 CALL SCOPY( M, A( 1, q ), 1, WORK, 00831 $ 1 ) 00832 CALL SLASCL( 'G', 0, 0, AAQQ, ONE, 00833 $ M, 1, WORK, LDA, IERR ) 00834 CALL SLASCL( 'G', 0, 0, AAPP, ONE, 00835 $ M, 1, A( 1, p ), LDA, 00836 $ IERR ) 00837 TEMP1 = -AAPQ*D( q ) / D( p ) 00838 CALL SAXPY( M, TEMP1, WORK, 1, 00839 $ A( 1, p ), 1 ) 00840 CALL SLASCL( 'G', 0, 0, ONE, AAPP, 00841 $ M, 1, A( 1, p ), LDA, 00842 $ IERR ) 00843 SVA( p ) = AAPP*SQRT( AMAX1( ZERO, 00844 $ ONE-AAPQ*AAPQ ) ) 00845 MXSINJ = AMAX1( 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 ) = SNRM2( M, A( 1, q ), 1 )* 00857 $ D( q ) 00858 ELSE 00859 T = ZERO 00860 AAQQ = ONE 00861 CALL SLASSQ( M, A( 1, q ), 1, T, 00862 $ AAQQ ) 00863 SVA( q ) = T*SQRT( 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 = SNRM2( M, A( 1, p ), 1 )* 00870 $ D( p ) 00871 ELSE 00872 T = ZERO 00873 AAPP = ONE 00874 CALL SLASSQ( M, A( 1, p ), 1, T, 00875 $ AAPP ) 00876 AAPP = T*SQRT( 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 ) = ABS( 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 ) = SNRM2( M, A( 1, N ), 1 )*D( N ) 00934 ELSE 00935 T = ZERO 00936 AAPP = ONE 00937 CALL SLASSQ( M, A( 1, N ), 1, T, AAPP ) 00938 SVA( N ) = T*SQRT( 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.FLOAT( N )*TOL ) .AND. 00947 $ ( FLOAT( 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 = ISAMAX( 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 SSWAP( M, A( 1, p ), 1, A( 1, q ), 1 ) 00978 IF( RSVEC )CALL SSWAP( MVL, V( 1, p ), 1, V( 1, q ), 1 ) 00979 END IF 00980 5991 CONTINUE 00981 * 00982 RETURN 00983 * .. 00984 * .. END OF SGSVJ0 00985 * .. 00986 END
1.7.3 
