 LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ clals0()

 subroutine clals0 ( integer ICOMPQ, integer NL, integer NR, integer SQRE, integer NRHS, complex, dimension( ldb, * ) B, integer LDB, complex, dimension( ldbx, * ) BX, integer LDBX, integer, dimension( * ) PERM, integer GIVPTR, integer, dimension( ldgcol, * ) GIVCOL, integer LDGCOL, real, dimension( ldgnum, * ) GIVNUM, integer LDGNUM, real, dimension( ldgnum, * ) POLES, real, dimension( * ) DIFL, real, dimension( ldgnum, * ) DIFR, real, dimension( * ) Z, integer K, real C, real S, real, dimension( * ) RWORK, integer INFO )

CLALS0 applies back multiplying factors in solving the least squares problem using divide and conquer SVD approach. Used by sgelsd.

Purpose:
``` CLALS0 applies back the multiplying factors of either the left or the
right singular vector matrix of a diagonal matrix appended by a row
to the right hand side matrix B in solving the least squares problem
using the divide-and-conquer SVD approach.

For the left singular vector matrix, three types of orthogonal
matrices are involved:

(1L) Givens rotations: the number of such rotations is GIVPTR; the
pairs of columns/rows they were applied to are stored in GIVCOL;
and the C- and S-values of these rotations are stored in GIVNUM.

(2L) Permutation. The (NL+1)-st row of B is to be moved to the first
row, and for J=2:N, PERM(J)-th row of B is to be moved to the
J-th row.

(3L) The left singular vector matrix of the remaining matrix.

For the right singular vector matrix, four types of orthogonal
matrices are involved:

(1R) The right singular vector matrix of the remaining matrix.

(2R) If SQRE = 1, one extra Givens rotation to generate the right
null space.

(3R) The inverse transformation of (2L).

(4R) The inverse transformation of (1L).```
Parameters
 [in] ICOMPQ ``` ICOMPQ is INTEGER Specifies whether singular vectors are to be computed in factored form: = 0: Left singular vector matrix. = 1: Right singular vector matrix.``` [in] NL ``` NL is INTEGER The row dimension of the upper block. NL >= 1.``` [in] NR ``` NR is INTEGER The row dimension of the lower block. NR >= 1.``` [in] SQRE ``` SQRE is INTEGER = 0: the lower block is an NR-by-NR square matrix. = 1: the lower block is an NR-by-(NR+1) rectangular matrix. The bidiagonal matrix has row dimension N = NL + NR + 1, and column dimension M = N + SQRE.``` [in] NRHS ``` NRHS is INTEGER The number of columns of B and BX. NRHS must be at least 1.``` [in,out] B ``` B is COMPLEX array, dimension ( LDB, NRHS ) On input, B contains the right hand sides of the least squares problem in rows 1 through M. On output, B contains the solution X in rows 1 through N.``` [in] LDB ``` LDB is INTEGER The leading dimension of B. LDB must be at least max(1,MAX( M, N ) ).``` [out] BX ` BX is COMPLEX array, dimension ( LDBX, NRHS )` [in] LDBX ``` LDBX is INTEGER The leading dimension of BX.``` [in] PERM ``` PERM is INTEGER array, dimension ( N ) The permutations (from deflation and sorting) applied to the two blocks.``` [in] GIVPTR ``` GIVPTR is INTEGER The number of Givens rotations which took place in this subproblem.``` [in] GIVCOL ``` GIVCOL is INTEGER array, dimension ( LDGCOL, 2 ) Each pair of numbers indicates a pair of rows/columns involved in a Givens rotation.``` [in] LDGCOL ``` LDGCOL is INTEGER The leading dimension of GIVCOL, must be at least N.``` [in] GIVNUM ``` GIVNUM is REAL array, dimension ( LDGNUM, 2 ) Each number indicates the C or S value used in the corresponding Givens rotation.``` [in] LDGNUM ``` LDGNUM is INTEGER The leading dimension of arrays DIFR, POLES and GIVNUM, must be at least K.``` [in] POLES ``` POLES is REAL array, dimension ( LDGNUM, 2 ) On entry, POLES(1:K, 1) contains the new singular values obtained from solving the secular equation, and POLES(1:K, 2) is an array containing the poles in the secular equation.``` [in] DIFL ``` DIFL is REAL array, dimension ( K ). On entry, DIFL(I) is the distance between I-th updated (undeflated) singular value and the I-th (undeflated) old singular value.``` [in] DIFR ``` DIFR is REAL array, dimension ( LDGNUM, 2 ). On entry, DIFR(I, 1) contains the distances between I-th updated (undeflated) singular value and the I+1-th (undeflated) old singular value. And DIFR(I, 2) is the normalizing factor for the I-th right singular vector.``` [in] Z ``` Z is REAL array, dimension ( K ) Contain the components of the deflation-adjusted updating row vector.``` [in] K ``` K is INTEGER Contains the dimension of the non-deflated matrix, This is the order of the related secular equation. 1 <= K <=N.``` [in] C ``` C is REAL C contains garbage if SQRE =0 and the C-value of a Givens rotation related to the right null space if SQRE = 1.``` [in] S ``` S is REAL S contains garbage if SQRE =0 and the S-value of a Givens rotation related to the right null space if SQRE = 1.``` [out] RWORK ``` RWORK is REAL array, dimension ( K*(1+NRHS) + 2*NRHS )``` [out] INFO ``` INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value.```
Contributors:
Ming Gu and Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA
Osni Marques, LBNL/NERSC, USA

Definition at line 267 of file clals0.f.

270*
271* -- LAPACK computational routine --
272* -- LAPACK is a software package provided by Univ. of Tennessee, --
273* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
274*
275* .. Scalar Arguments ..
276 INTEGER GIVPTR, ICOMPQ, INFO, K, LDB, LDBX, LDGCOL,
277 \$ LDGNUM, NL, NR, NRHS, SQRE
278 REAL C, S
279* ..
280* .. Array Arguments ..
281 INTEGER GIVCOL( LDGCOL, * ), PERM( * )
282 REAL DIFL( * ), DIFR( LDGNUM, * ),
283 \$ GIVNUM( LDGNUM, * ), POLES( LDGNUM, * ),
284 \$ RWORK( * ), Z( * )
285 COMPLEX B( LDB, * ), BX( LDBX, * )
286* ..
287*
288* =====================================================================
289*
290* .. Parameters ..
291 REAL ONE, ZERO, NEGONE
292 parameter( one = 1.0e0, zero = 0.0e0, negone = -1.0e0 )
293* ..
294* .. Local Scalars ..
295 INTEGER I, J, JCOL, JROW, M, N, NLP1
296 REAL DIFLJ, DIFRJ, DJ, DSIGJ, DSIGJP, TEMP
297* ..
298* .. External Subroutines ..
299 EXTERNAL ccopy, clacpy, clascl, csrot, csscal, sgemv,
300 \$ xerbla
301* ..
302* .. External Functions ..
303 REAL SLAMC3, SNRM2
304 EXTERNAL slamc3, snrm2
305* ..
306* .. Intrinsic Functions ..
307 INTRINSIC aimag, cmplx, max, real
308* ..
309* .. Executable Statements ..
310*
311* Test the input parameters.
312*
313 info = 0
314 n = nl + nr + 1
315*
316 IF( ( icompq.LT.0 ) .OR. ( icompq.GT.1 ) ) THEN
317 info = -1
318 ELSE IF( nl.LT.1 ) THEN
319 info = -2
320 ELSE IF( nr.LT.1 ) THEN
321 info = -3
322 ELSE IF( ( sqre.LT.0 ) .OR. ( sqre.GT.1 ) ) THEN
323 info = -4
324 ELSE IF( nrhs.LT.1 ) THEN
325 info = -5
326 ELSE IF( ldb.LT.n ) THEN
327 info = -7
328 ELSE IF( ldbx.LT.n ) THEN
329 info = -9
330 ELSE IF( givptr.LT.0 ) THEN
331 info = -11
332 ELSE IF( ldgcol.LT.n ) THEN
333 info = -13
334 ELSE IF( ldgnum.LT.n ) THEN
335 info = -15
336 ELSE IF( k.LT.1 ) THEN
337 info = -20
338 END IF
339 IF( info.NE.0 ) THEN
340 CALL xerbla( 'CLALS0', -info )
341 RETURN
342 END IF
343*
344 m = n + sqre
345 nlp1 = nl + 1
346*
347 IF( icompq.EQ.0 ) THEN
348*
349* Apply back orthogonal transformations from the left.
350*
351* Step (1L): apply back the Givens rotations performed.
352*
353 DO 10 i = 1, givptr
354 CALL csrot( nrhs, b( givcol( i, 2 ), 1 ), ldb,
355 \$ b( givcol( i, 1 ), 1 ), ldb, givnum( i, 2 ),
356 \$ givnum( i, 1 ) )
357 10 CONTINUE
358*
359* Step (2L): permute rows of B.
360*
361 CALL ccopy( nrhs, b( nlp1, 1 ), ldb, bx( 1, 1 ), ldbx )
362 DO 20 i = 2, n
363 CALL ccopy( nrhs, b( perm( i ), 1 ), ldb, bx( i, 1 ), ldbx )
364 20 CONTINUE
365*
366* Step (3L): apply the inverse of the left singular vector
367* matrix to BX.
368*
369 IF( k.EQ.1 ) THEN
370 CALL ccopy( nrhs, bx, ldbx, b, ldb )
371 IF( z( 1 ).LT.zero ) THEN
372 CALL csscal( nrhs, negone, b, ldb )
373 END IF
374 ELSE
375 DO 100 j = 1, k
376 diflj = difl( j )
377 dj = poles( j, 1 )
378 dsigj = -poles( j, 2 )
379 IF( j.LT.k ) THEN
380 difrj = -difr( j, 1 )
381 dsigjp = -poles( j+1, 2 )
382 END IF
383 IF( ( z( j ).EQ.zero ) .OR. ( poles( j, 2 ).EQ.zero ) )
384 \$ THEN
385 rwork( j ) = zero
386 ELSE
387 rwork( j ) = -poles( j, 2 )*z( j ) / diflj /
388 \$ ( poles( j, 2 )+dj )
389 END IF
390 DO 30 i = 1, j - 1
391 IF( ( z( i ).EQ.zero ) .OR.
392 \$ ( poles( i, 2 ).EQ.zero ) ) THEN
393 rwork( i ) = zero
394 ELSE
395 rwork( i ) = poles( i, 2 )*z( i ) /
396 \$ ( slamc3( poles( i, 2 ), dsigj )-
397 \$ diflj ) / ( poles( i, 2 )+dj )
398 END IF
399 30 CONTINUE
400 DO 40 i = j + 1, k
401 IF( ( z( i ).EQ.zero ) .OR.
402 \$ ( poles( i, 2 ).EQ.zero ) ) THEN
403 rwork( i ) = zero
404 ELSE
405 rwork( i ) = poles( i, 2 )*z( i ) /
406 \$ ( slamc3( poles( i, 2 ), dsigjp )+
407 \$ difrj ) / ( poles( i, 2 )+dj )
408 END IF
409 40 CONTINUE
410 rwork( 1 ) = negone
411 temp = snrm2( k, rwork, 1 )
412*
413* Since B and BX are complex, the following call to SGEMV
414* is performed in two steps (real and imaginary parts).
415*
416* CALL SGEMV( 'T', K, NRHS, ONE, BX, LDBX, WORK, 1, ZERO,
417* \$ B( J, 1 ), LDB )
418*
419 i = k + nrhs*2
420 DO 60 jcol = 1, nrhs
421 DO 50 jrow = 1, k
422 i = i + 1
423 rwork( i ) = real( bx( jrow, jcol ) )
424 50 CONTINUE
425 60 CONTINUE
426 CALL sgemv( 'T', k, nrhs, one, rwork( 1+k+nrhs*2 ), k,
427 \$ rwork( 1 ), 1, zero, rwork( 1+k ), 1 )
428 i = k + nrhs*2
429 DO 80 jcol = 1, nrhs
430 DO 70 jrow = 1, k
431 i = i + 1
432 rwork( i ) = aimag( bx( jrow, jcol ) )
433 70 CONTINUE
434 80 CONTINUE
435 CALL sgemv( 'T', k, nrhs, one, rwork( 1+k+nrhs*2 ), k,
436 \$ rwork( 1 ), 1, zero, rwork( 1+k+nrhs ), 1 )
437 DO 90 jcol = 1, nrhs
438 b( j, jcol ) = cmplx( rwork( jcol+k ),
439 \$ rwork( jcol+k+nrhs ) )
440 90 CONTINUE
441 CALL clascl( 'G', 0, 0, temp, one, 1, nrhs, b( j, 1 ),
442 \$ ldb, info )
443 100 CONTINUE
444 END IF
445*
446* Move the deflated rows of BX to B also.
447*
448 IF( k.LT.max( m, n ) )
449 \$ CALL clacpy( 'A', n-k, nrhs, bx( k+1, 1 ), ldbx,
450 \$ b( k+1, 1 ), ldb )
451 ELSE
452*
453* Apply back the right orthogonal transformations.
454*
455* Step (1R): apply back the new right singular vector matrix
456* to B.
457*
458 IF( k.EQ.1 ) THEN
459 CALL ccopy( nrhs, b, ldb, bx, ldbx )
460 ELSE
461 DO 180 j = 1, k
462 dsigj = poles( j, 2 )
463 IF( z( j ).EQ.zero ) THEN
464 rwork( j ) = zero
465 ELSE
466 rwork( j ) = -z( j ) / difl( j ) /
467 \$ ( dsigj+poles( j, 1 ) ) / difr( j, 2 )
468 END IF
469 DO 110 i = 1, j - 1
470 IF( z( j ).EQ.zero ) THEN
471 rwork( i ) = zero
472 ELSE
473 rwork( i ) = z( j ) / ( slamc3( dsigj, -poles( i+1,
474 \$ 2 ) )-difr( i, 1 ) ) /
475 \$ ( dsigj+poles( i, 1 ) ) / difr( i, 2 )
476 END IF
477 110 CONTINUE
478 DO 120 i = j + 1, k
479 IF( z( j ).EQ.zero ) THEN
480 rwork( i ) = zero
481 ELSE
482 rwork( i ) = z( j ) / ( slamc3( dsigj, -poles( i,
483 \$ 2 ) )-difl( i ) ) /
484 \$ ( dsigj+poles( i, 1 ) ) / difr( i, 2 )
485 END IF
486 120 CONTINUE
487*
488* Since B and BX are complex, the following call to SGEMV
489* is performed in two steps (real and imaginary parts).
490*
491* CALL SGEMV( 'T', K, NRHS, ONE, B, LDB, WORK, 1, ZERO,
492* \$ BX( J, 1 ), LDBX )
493*
494 i = k + nrhs*2
495 DO 140 jcol = 1, nrhs
496 DO 130 jrow = 1, k
497 i = i + 1
498 rwork( i ) = real( b( jrow, jcol ) )
499 130 CONTINUE
500 140 CONTINUE
501 CALL sgemv( 'T', k, nrhs, one, rwork( 1+k+nrhs*2 ), k,
502 \$ rwork( 1 ), 1, zero, rwork( 1+k ), 1 )
503 i = k + nrhs*2
504 DO 160 jcol = 1, nrhs
505 DO 150 jrow = 1, k
506 i = i + 1
507 rwork( i ) = aimag( b( jrow, jcol ) )
508 150 CONTINUE
509 160 CONTINUE
510 CALL sgemv( 'T', k, nrhs, one, rwork( 1+k+nrhs*2 ), k,
511 \$ rwork( 1 ), 1, zero, rwork( 1+k+nrhs ), 1 )
512 DO 170 jcol = 1, nrhs
513 bx( j, jcol ) = cmplx( rwork( jcol+k ),
514 \$ rwork( jcol+k+nrhs ) )
515 170 CONTINUE
516 180 CONTINUE
517 END IF
518*
519* Step (2R): if SQRE = 1, apply back the rotation that is
520* related to the right null space of the subproblem.
521*
522 IF( sqre.EQ.1 ) THEN
523 CALL ccopy( nrhs, b( m, 1 ), ldb, bx( m, 1 ), ldbx )
524 CALL csrot( nrhs, bx( 1, 1 ), ldbx, bx( m, 1 ), ldbx, c, s )
525 END IF
526 IF( k.LT.max( m, n ) )
527 \$ CALL clacpy( 'A', n-k, nrhs, b( k+1, 1 ), ldb,
528 \$ bx( k+1, 1 ), ldbx )
529*
530* Step (3R): permute rows of B.
531*
532 CALL ccopy( nrhs, bx( 1, 1 ), ldbx, b( nlp1, 1 ), ldb )
533 IF( sqre.EQ.1 ) THEN
534 CALL ccopy( nrhs, bx( m, 1 ), ldbx, b( m, 1 ), ldb )
535 END IF
536 DO 190 i = 2, n
537 CALL ccopy( nrhs, bx( i, 1 ), ldbx, b( perm( i ), 1 ), ldb )
538 190 CONTINUE
539*
540* Step (4R): apply back the Givens rotations performed.
541*
542 DO 200 i = givptr, 1, -1
543 CALL csrot( nrhs, b( givcol( i, 2 ), 1 ), ldb,
544 \$ b( givcol( i, 1 ), 1 ), ldb, givnum( i, 2 ),
545 \$ -givnum( i, 1 ) )
546 200 CONTINUE
547 END IF
548*
549 RETURN
550*
551* End of CLALS0
552*
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine ccopy(N, CX, INCX, CY, INCY)
CCOPY
Definition: ccopy.f:81
subroutine csscal(N, SA, CX, INCX)
CSSCAL
Definition: csscal.f:78
subroutine csrot(N, CX, INCX, CY, INCY, C, S)
CSROT
Definition: csrot.f:98
subroutine clascl(TYPE, KL, KU, CFROM, CTO, M, N, A, LDA, INFO)
CLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.
Definition: clascl.f:143
subroutine clacpy(UPLO, M, N, A, LDA, B, LDB)
CLACPY copies all or part of one two-dimensional array to another.
Definition: clacpy.f:103
real(wp) function snrm2(n, x, incx)
SNRM2
Definition: snrm2.f90:89
subroutine sgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
SGEMV
Definition: sgemv.f:156
real function slamc3(A, B)
SLAMC3
Definition: slamch.f:169
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