LAPACK  3.8.0
LAPACK: Linear Algebra PACKage

◆ dorcsd()

recursive subroutine dorcsd ( character  JOBU1,
character  JOBU2,
character  JOBV1T,
character  JOBV2T,
character  TRANS,
character  SIGNS,
integer  M,
integer  P,
integer  Q,
double precision, dimension( ldx11, * )  X11,
integer  LDX11,
double precision, dimension( ldx12, * )  X12,
integer  LDX12,
double precision, dimension( ldx21, * )  X21,
integer  LDX21,
double precision, dimension( ldx22, * )  X22,
integer  LDX22,
double precision, dimension( * )  THETA,
double precision, dimension( ldu1, * )  U1,
integer  LDU1,
double precision, dimension( ldu2, * )  U2,
integer  LDU2,
double precision, dimension( ldv1t, * )  V1T,
integer  LDV1T,
double precision, dimension( ldv2t, * )  V2T,
integer  LDV2T,
double precision, dimension( * )  WORK,
integer  LWORK,
integer, dimension( * )  IWORK,
integer  INFO 
)

DORCSD

Download DORCSD + dependencies [TGZ] [ZIP] [TXT]

Purpose:
 DORCSD computes the CS decomposition of an M-by-M partitioned
 orthogonal matrix X:

                                 [  I  0  0 |  0  0  0 ]
                                 [  0  C  0 |  0 -S  0 ]
     [ X11 | X12 ]   [ U1 |    ] [  0  0  0 |  0  0 -I ] [ V1 |    ]**T
 X = [-----------] = [---------] [---------------------] [---------]   .
     [ X21 | X22 ]   [    | U2 ] [  0  0  0 |  I  0  0 ] [    | V2 ]
                                 [  0  S  0 |  0  C  0 ]
                                 [  0  0  I |  0  0  0 ]

 X11 is P-by-Q. The orthogonal matrices U1, U2, V1, and V2 are P-by-P,
 (M-P)-by-(M-P), Q-by-Q, and (M-Q)-by-(M-Q), respectively. C and S are
 R-by-R nonnegative diagonal matrices satisfying C^2 + S^2 = I, in
 which R = MIN(P,M-P,Q,M-Q).
Parameters
[in]JOBU1
          JOBU1 is CHARACTER
          = 'Y':      U1 is computed;
          otherwise:  U1 is not computed.
[in]JOBU2
          JOBU2 is CHARACTER
          = 'Y':      U2 is computed;
          otherwise:  U2 is not computed.
[in]JOBV1T
          JOBV1T is CHARACTER
          = 'Y':      V1T is computed;
          otherwise:  V1T is not computed.
[in]JOBV2T
          JOBV2T is CHARACTER
          = 'Y':      V2T is computed;
          otherwise:  V2T is not computed.
[in]TRANS
          TRANS is CHARACTER
          = 'T':      X, U1, U2, V1T, and V2T are stored in row-major
                      order;
          otherwise:  X, U1, U2, V1T, and V2T are stored in column-
                      major order.
[in]SIGNS
          SIGNS is CHARACTER
          = 'O':      The lower-left block is made nonpositive (the
                      "other" convention);
          otherwise:  The upper-right block is made nonpositive (the
                      "default" convention).
[in]M
          M is INTEGER
          The number of rows and columns in X.
[in]P
          P is INTEGER
          The number of rows in X11 and X12. 0 <= P <= M.
[in]Q
          Q is INTEGER
          The number of columns in X11 and X21. 0 <= Q <= M.
[in,out]X11
          X11 is DOUBLE PRECISION array, dimension (LDX11,Q)
          On entry, part of the orthogonal matrix whose CSD is desired.
[in]LDX11
          LDX11 is INTEGER
          The leading dimension of X11. LDX11 >= MAX(1,P).
[in,out]X12
          X12 is DOUBLE PRECISION array, dimension (LDX12,M-Q)
          On entry, part of the orthogonal matrix whose CSD is desired.
[in]LDX12
          LDX12 is INTEGER
          The leading dimension of X12. LDX12 >= MAX(1,P).
[in,out]X21
          X21 is DOUBLE PRECISION array, dimension (LDX21,Q)
          On entry, part of the orthogonal matrix whose CSD is desired.
[in]LDX21
          LDX21 is INTEGER
          The leading dimension of X11. LDX21 >= MAX(1,M-P).
[in,out]X22
          X22 is DOUBLE PRECISION array, dimension (LDX22,M-Q)
          On entry, part of the orthogonal matrix whose CSD is desired.
[in]LDX22
          LDX22 is INTEGER
          The leading dimension of X11. LDX22 >= MAX(1,M-P).
[out]THETA
          THETA is DOUBLE PRECISION array, dimension (R), in which R =
          MIN(P,M-P,Q,M-Q).
          C = DIAG( COS(THETA(1)), ... , COS(THETA(R)) ) and
          S = DIAG( SIN(THETA(1)), ... , SIN(THETA(R)) ).
[out]U1
          U1 is DOUBLE PRECISION array, dimension (LDU1,P)
          If JOBU1 = 'Y', U1 contains the P-by-P orthogonal matrix U1.
[in]LDU1
          LDU1 is INTEGER
          The leading dimension of U1. If JOBU1 = 'Y', LDU1 >=
          MAX(1,P).
[out]U2
          U2 is DOUBLE PRECISION array, dimension (LDU2,M-P)
          If JOBU2 = 'Y', U2 contains the (M-P)-by-(M-P) orthogonal
          matrix U2.
[in]LDU2
          LDU2 is INTEGER
          The leading dimension of U2. If JOBU2 = 'Y', LDU2 >=
          MAX(1,M-P).
[out]V1T
          V1T is DOUBLE PRECISION array, dimension (LDV1T,Q)
          If JOBV1T = 'Y', V1T contains the Q-by-Q matrix orthogonal
          matrix V1**T.
[in]LDV1T
          LDV1T is INTEGER
          The leading dimension of V1T. If JOBV1T = 'Y', LDV1T >=
          MAX(1,Q).
[out]V2T
          V2T is DOUBLE PRECISION array, dimension (LDV2T,M-Q)
          If JOBV2T = 'Y', V2T contains the (M-Q)-by-(M-Q) orthogonal
          matrix V2**T.
[in]LDV2T
          LDV2T is INTEGER
          The leading dimension of V2T. If JOBV2T = 'Y', LDV2T >=
          MAX(1,M-Q).
[out]WORK
          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
          If INFO > 0 on exit, WORK(2:R) contains the values PHI(1),
          ..., PHI(R-1) that, together with THETA(1), ..., THETA(R),
          define the matrix in intermediate bidiagonal-block form
          remaining after nonconvergence. INFO specifies the number
          of nonzero PHI's.
[in]LWORK
          LWORK is INTEGER
          The dimension of the array WORK.

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the work array, and no error
          message related to LWORK is issued by XERBLA.
[out]IWORK
          IWORK is INTEGER array, dimension (M-MIN(P, M-P, Q, M-Q))
[out]INFO
          INFO is INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  DBBCSD did not converge. See the description of WORK
                above for details.
References:
[1] Brian D. Sutton. Computing the complete CS decomposition. Numer. Algorithms, 50(1):33-65, 2009.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
June 2017

Definition at line 302 of file dorcsd.f.

302 *
303 * -- LAPACK computational routine (version 3.7.1) --
304 * -- LAPACK is a software package provided by Univ. of Tennessee, --
305 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
306 * June 2017
307 *
308 * .. Scalar Arguments ..
309  CHARACTER jobu1, jobu2, jobv1t, jobv2t, signs, trans
310  INTEGER info, ldu1, ldu2, ldv1t, ldv2t, ldx11, ldx12,
311  $ ldx21, ldx22, lwork, m, p, q
312 * ..
313 * .. Array Arguments ..
314  INTEGER iwork( * )
315  DOUBLE PRECISION theta( * )
316  DOUBLE PRECISION u1( ldu1, * ), u2( ldu2, * ), v1t( ldv1t, * ),
317  $ v2t( ldv2t, * ), work( * ), x11( ldx11, * ),
318  $ x12( ldx12, * ), x21( ldx21, * ), x22( ldx22,
319  $ * )
320 * ..
321 *
322 * ===================================================================
323 *
324 * .. Parameters ..
325  DOUBLE PRECISION one, zero
326  parameter( one = 1.0d0,
327  $ zero = 0.0d0 )
328 * ..
329 * .. Local Scalars ..
330  CHARACTER transt, signst
331  INTEGER childinfo, i, ib11d, ib11e, ib12d, ib12e,
332  $ ib21d, ib21e, ib22d, ib22e, ibbcsd, iorbdb,
333  $ iorglq, iorgqr, iphi, itaup1, itaup2, itauq1,
334  $ itauq2, j, lbbcsdwork, lbbcsdworkmin,
335  $ lbbcsdworkopt, lorbdbwork, lorbdbworkmin,
336  $ lorbdbworkopt, lorglqwork, lorglqworkmin,
337  $ lorglqworkopt, lorgqrwork, lorgqrworkmin,
338  $ lorgqrworkopt, lworkmin, lworkopt
339  LOGICAL colmajor, defaultsigns, lquery, wantu1, wantu2,
340  $ wantv1t, wantv2t
341 * ..
342 * .. External Subroutines ..
343  EXTERNAL dbbcsd, dlacpy, dlapmr, dlapmt,
345 * ..
346 * .. External Functions ..
347  LOGICAL lsame
348  EXTERNAL lsame
349 * ..
350 * .. Intrinsic Functions
351  INTRINSIC int, max, min
352 * ..
353 * .. Executable Statements ..
354 *
355 * Test input arguments
356 *
357  info = 0
358  wantu1 = lsame( jobu1, 'Y' )
359  wantu2 = lsame( jobu2, 'Y' )
360  wantv1t = lsame( jobv1t, 'Y' )
361  wantv2t = lsame( jobv2t, 'Y' )
362  colmajor = .NOT. lsame( trans, 'T' )
363  defaultsigns = .NOT. lsame( signs, 'O' )
364  lquery = lwork .EQ. -1
365  IF( m .LT. 0 ) THEN
366  info = -7
367  ELSE IF( p .LT. 0 .OR. p .GT. m ) THEN
368  info = -8
369  ELSE IF( q .LT. 0 .OR. q .GT. m ) THEN
370  info = -9
371  ELSE IF ( colmajor .AND. ldx11 .LT. max( 1, p ) ) THEN
372  info = -11
373  ELSE IF (.NOT. colmajor .AND. ldx11 .LT. max( 1, q ) ) THEN
374  info = -11
375  ELSE IF (colmajor .AND. ldx12 .LT. max( 1, p ) ) THEN
376  info = -13
377  ELSE IF (.NOT. colmajor .AND. ldx12 .LT. max( 1, m-q ) ) THEN
378  info = -13
379  ELSE IF (colmajor .AND. ldx21 .LT. max( 1, m-p ) ) THEN
380  info = -15
381  ELSE IF (.NOT. colmajor .AND. ldx21 .LT. max( 1, q ) ) THEN
382  info = -15
383  ELSE IF (colmajor .AND. ldx22 .LT. max( 1, m-p ) ) THEN
384  info = -17
385  ELSE IF (.NOT. colmajor .AND. ldx22 .LT. max( 1, m-q ) ) THEN
386  info = -17
387  ELSE IF( wantu1 .AND. ldu1 .LT. p ) THEN
388  info = -20
389  ELSE IF( wantu2 .AND. ldu2 .LT. m-p ) THEN
390  info = -22
391  ELSE IF( wantv1t .AND. ldv1t .LT. q ) THEN
392  info = -24
393  ELSE IF( wantv2t .AND. ldv2t .LT. m-q ) THEN
394  info = -26
395  END IF
396 *
397 * Work with transpose if convenient
398 *
399  IF( info .EQ. 0 .AND. min( p, m-p ) .LT. min( q, m-q ) ) THEN
400  IF( colmajor ) THEN
401  transt = 'T'
402  ELSE
403  transt = 'N'
404  END IF
405  IF( defaultsigns ) THEN
406  signst = 'O'
407  ELSE
408  signst = 'D'
409  END IF
410  CALL dorcsd( jobv1t, jobv2t, jobu1, jobu2, transt, signst, m,
411  $ q, p, x11, ldx11, x21, ldx21, x12, ldx12, x22,
412  $ ldx22, theta, v1t, ldv1t, v2t, ldv2t, u1, ldu1,
413  $ u2, ldu2, work, lwork, iwork, info )
414  RETURN
415  END IF
416 *
417 * Work with permutation [ 0 I; I 0 ] * X * [ 0 I; I 0 ] if
418 * convenient
419 *
420  IF( info .EQ. 0 .AND. m-q .LT. q ) THEN
421  IF( defaultsigns ) THEN
422  signst = 'O'
423  ELSE
424  signst = 'D'
425  END IF
426  CALL dorcsd( jobu2, jobu1, jobv2t, jobv1t, trans, signst, m,
427  $ m-p, m-q, x22, ldx22, x21, ldx21, x12, ldx12, x11,
428  $ ldx11, theta, u2, ldu2, u1, ldu1, v2t, ldv2t, v1t,
429  $ ldv1t, work, lwork, iwork, info )
430  RETURN
431  END IF
432 *
433 * Compute workspace
434 *
435  IF( info .EQ. 0 ) THEN
436 *
437  iphi = 2
438  itaup1 = iphi + max( 1, q - 1 )
439  itaup2 = itaup1 + max( 1, p )
440  itauq1 = itaup2 + max( 1, m - p )
441  itauq2 = itauq1 + max( 1, q )
442  iorgqr = itauq2 + max( 1, m - q )
443  CALL dorgqr( m-q, m-q, m-q, u1, max(1,m-q), u1, work, -1,
444  $ childinfo )
445  lorgqrworkopt = int( work(1) )
446  lorgqrworkmin = max( 1, m - q )
447  iorglq = itauq2 + max( 1, m - q )
448  CALL dorglq( m-q, m-q, m-q, u1, max(1,m-q), u1, work, -1,
449  $ childinfo )
450  lorglqworkopt = int( work(1) )
451  lorglqworkmin = max( 1, m - q )
452  iorbdb = itauq2 + max( 1, m - q )
453  CALL dorbdb( trans, signs, m, p, q, x11, ldx11, x12, ldx12,
454  $ x21, ldx21, x22, ldx22, theta, v1t, u1, u2, v1t,
455  $ v2t, work, -1, childinfo )
456  lorbdbworkopt = int( work(1) )
457  lorbdbworkmin = lorbdbworkopt
458  ib11d = itauq2 + max( 1, m - q )
459  ib11e = ib11d + max( 1, q )
460  ib12d = ib11e + max( 1, q - 1 )
461  ib12e = ib12d + max( 1, q )
462  ib21d = ib12e + max( 1, q - 1 )
463  ib21e = ib21d + max( 1, q )
464  ib22d = ib21e + max( 1, q - 1 )
465  ib22e = ib22d + max( 1, q )
466  ibbcsd = ib22e + max( 1, q - 1 )
467  CALL dbbcsd( jobu1, jobu2, jobv1t, jobv2t, trans, m, p, q,
468  $ theta, theta, u1, ldu1, u2, ldu2, v1t, ldv1t, v2t,
469  $ ldv2t, u1, u1, u1, u1, u1, u1, u1, u1, work, -1,
470  $ childinfo )
471  lbbcsdworkopt = int( work(1) )
472  lbbcsdworkmin = lbbcsdworkopt
473  lworkopt = max( iorgqr + lorgqrworkopt, iorglq + lorglqworkopt,
474  $ iorbdb + lorbdbworkopt, ibbcsd + lbbcsdworkopt ) - 1
475  lworkmin = max( iorgqr + lorgqrworkmin, iorglq + lorglqworkmin,
476  $ iorbdb + lorbdbworkopt, ibbcsd + lbbcsdworkmin ) - 1
477  work(1) = max(lworkopt,lworkmin)
478 *
479  IF( lwork .LT. lworkmin .AND. .NOT. lquery ) THEN
480  info = -22
481  ELSE
482  lorgqrwork = lwork - iorgqr + 1
483  lorglqwork = lwork - iorglq + 1
484  lorbdbwork = lwork - iorbdb + 1
485  lbbcsdwork = lwork - ibbcsd + 1
486  END IF
487  END IF
488 *
489 * Abort if any illegal arguments
490 *
491  IF( info .NE. 0 ) THEN
492  CALL xerbla( 'DORCSD', -info )
493  RETURN
494  ELSE IF( lquery ) THEN
495  RETURN
496  END IF
497 *
498 * Transform to bidiagonal block form
499 *
500  CALL dorbdb( trans, signs, m, p, q, x11, ldx11, x12, ldx12, x21,
501  $ ldx21, x22, ldx22, theta, work(iphi), work(itaup1),
502  $ work(itaup2), work(itauq1), work(itauq2),
503  $ work(iorbdb), lorbdbwork, childinfo )
504 *
505 * Accumulate Householder reflectors
506 *
507  IF( colmajor ) THEN
508  IF( wantu1 .AND. p .GT. 0 ) THEN
509  CALL dlacpy( 'L', p, q, x11, ldx11, u1, ldu1 )
510  CALL dorgqr( p, p, q, u1, ldu1, work(itaup1), work(iorgqr),
511  $ lorgqrwork, info)
512  END IF
513  IF( wantu2 .AND. m-p .GT. 0 ) THEN
514  CALL dlacpy( 'L', m-p, q, x21, ldx21, u2, ldu2 )
515  CALL dorgqr( m-p, m-p, q, u2, ldu2, work(itaup2),
516  $ work(iorgqr), lorgqrwork, info )
517  END IF
518  IF( wantv1t .AND. q .GT. 0 ) THEN
519  CALL dlacpy( 'U', q-1, q-1, x11(1,2), ldx11, v1t(2,2),
520  $ ldv1t )
521  v1t(1, 1) = one
522  DO j = 2, q
523  v1t(1,j) = zero
524  v1t(j,1) = zero
525  END DO
526  CALL dorglq( q-1, q-1, q-1, v1t(2,2), ldv1t, work(itauq1),
527  $ work(iorglq), lorglqwork, info )
528  END IF
529  IF( wantv2t .AND. m-q .GT. 0 ) THEN
530  CALL dlacpy( 'U', p, m-q, x12, ldx12, v2t, ldv2t )
531  IF (m-p .GT. q) Then
532  CALL dlacpy( 'U', m-p-q, m-p-q, x22(q+1,p+1), ldx22,
533  $ v2t(p+1,p+1), ldv2t )
534  END IF
535  IF (m .GT. q) THEN
536  CALL dorglq( m-q, m-q, m-q, v2t, ldv2t, work(itauq2),
537  $ work(iorglq), lorglqwork, info )
538  END IF
539  END IF
540  ELSE
541  IF( wantu1 .AND. p .GT. 0 ) THEN
542  CALL dlacpy( 'U', q, p, x11, ldx11, u1, ldu1 )
543  CALL dorglq( p, p, q, u1, ldu1, work(itaup1), work(iorglq),
544  $ lorglqwork, info)
545  END IF
546  IF( wantu2 .AND. m-p .GT. 0 ) THEN
547  CALL dlacpy( 'U', q, m-p, x21, ldx21, u2, ldu2 )
548  CALL dorglq( m-p, m-p, q, u2, ldu2, work(itaup2),
549  $ work(iorglq), lorglqwork, info )
550  END IF
551  IF( wantv1t .AND. q .GT. 0 ) THEN
552  CALL dlacpy( 'L', q-1, q-1, x11(2,1), ldx11, v1t(2,2),
553  $ ldv1t )
554  v1t(1, 1) = one
555  DO j = 2, q
556  v1t(1,j) = zero
557  v1t(j,1) = zero
558  END DO
559  CALL dorgqr( q-1, q-1, q-1, v1t(2,2), ldv1t, work(itauq1),
560  $ work(iorgqr), lorgqrwork, info )
561  END IF
562  IF( wantv2t .AND. m-q .GT. 0 ) THEN
563  CALL dlacpy( 'L', m-q, p, x12, ldx12, v2t, ldv2t )
564  CALL dlacpy( 'L', m-p-q, m-p-q, x22(p+1,q+1), ldx22,
565  $ v2t(p+1,p+1), ldv2t )
566  CALL dorgqr( m-q, m-q, m-q, v2t, ldv2t, work(itauq2),
567  $ work(iorgqr), lorgqrwork, info )
568  END IF
569  END IF
570 *
571 * Compute the CSD of the matrix in bidiagonal-block form
572 *
573  CALL dbbcsd( jobu1, jobu2, jobv1t, jobv2t, trans, m, p, q, theta,
574  $ work(iphi), u1, ldu1, u2, ldu2, v1t, ldv1t, v2t,
575  $ ldv2t, work(ib11d), work(ib11e), work(ib12d),
576  $ work(ib12e), work(ib21d), work(ib21e), work(ib22d),
577  $ work(ib22e), work(ibbcsd), lbbcsdwork, info )
578 *
579 * Permute rows and columns to place identity submatrices in top-
580 * left corner of (1,1)-block and/or bottom-right corner of (1,2)-
581 * block and/or bottom-right corner of (2,1)-block and/or top-left
582 * corner of (2,2)-block
583 *
584  IF( q .GT. 0 .AND. wantu2 ) THEN
585  DO i = 1, q
586  iwork(i) = m - p - q + i
587  END DO
588  DO i = q + 1, m - p
589  iwork(i) = i - q
590  END DO
591  IF( colmajor ) THEN
592  CALL dlapmt( .false., m-p, m-p, u2, ldu2, iwork )
593  ELSE
594  CALL dlapmr( .false., m-p, m-p, u2, ldu2, iwork )
595  END IF
596  END IF
597  IF( m .GT. 0 .AND. wantv2t ) THEN
598  DO i = 1, p
599  iwork(i) = m - p - q + i
600  END DO
601  DO i = p + 1, m - q
602  iwork(i) = i - p
603  END DO
604  IF( .NOT. colmajor ) THEN
605  CALL dlapmt( .false., m-q, m-q, v2t, ldv2t, iwork )
606  ELSE
607  CALL dlapmr( .false., m-q, m-q, v2t, ldv2t, iwork )
608  END IF
609  END IF
610 *
611  RETURN
612 *
613 * End DORCSD
614 *
subroutine dlacpy(UPLO, M, N, A, LDA, B, LDB)
DLACPY copies all or part of one two-dimensional array to another.
Definition: dlacpy.f:105
subroutine dbbcsd(JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS, M, P, Q, THETA, PHI, U1, LDU1, U2, LDU2, V1T, LDV1T, V2T, LDV2T, B11D, B11E, B12D, B12E, B21D, B21E, B22D, B22E, WORK, LWORK, INFO)
DBBCSD
Definition: dbbcsd.f:334
subroutine dorglq(M, N, K, A, LDA, TAU, WORK, LWORK, INFO)
DORGLQ
Definition: dorglq.f:129
subroutine dlapmr(FORWRD, M, N, X, LDX, K)
DLAPMR rearranges rows of a matrix as specified by a permutation vector.
Definition: dlapmr.f:106
recursive subroutine dorcsd(JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS, SIGNS, M, P, Q, X11, LDX11, X12, LDX12, X21, LDX21, X22, LDX22, THETA, U1, LDU1, U2, LDU2, V1T, LDV1T, V2T, LDV2T, WORK, LWORK, IWORK, INFO)
DORCSD
Definition: dorcsd.f:302
subroutine dorgqr(M, N, K, A, LDA, TAU, WORK, LWORK, INFO)
DORGQR
Definition: dorgqr.f:130
subroutine dorbdb(TRANS, SIGNS, M, P, Q, X11, LDX11, X12, LDX12, X21, LDX21, X22, LDX22, THETA, PHI, TAUP1, TAUP2, TAUQ1, TAUQ2, WORK, LWORK, INFO)
DORBDB
Definition: dorbdb.f:289
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
subroutine dlapmt(FORWRD, M, N, X, LDX, K)
DLAPMT performs a forward or backward permutation of the columns of a matrix.
Definition: dlapmt.f:106
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