LAPACK  3.10.1
LAPACK: Linear Algebra PACKage

◆ cggsvd3()

subroutine cggsvd3 ( character  JOBU,
character  JOBV,
character  JOBQ,
integer  M,
integer  N,
integer  P,
integer  K,
integer  L,
complex, dimension( lda, * )  A,
integer  LDA,
complex, dimension( ldb, * )  B,
integer  LDB,
real, dimension( * )  ALPHA,
real, dimension( * )  BETA,
complex, dimension( ldu, * )  U,
integer  LDU,
complex, dimension( ldv, * )  V,
integer  LDV,
complex, dimension( ldq, * )  Q,
integer  LDQ,
complex, dimension( * )  WORK,
integer  LWORK,
real, dimension( * )  RWORK,
integer, dimension( * )  IWORK,
integer  INFO 
)

CGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices

Download CGGSVD3 + dependencies [TGZ] [ZIP] [TXT]

Purpose:
 CGGSVD3 computes the generalized singular value decomposition (GSVD)
 of an M-by-N complex matrix A and P-by-N complex matrix B:

       U**H*A*Q = D1*( 0 R ),    V**H*B*Q = D2*( 0 R )

 where U, V and Q are unitary matrices.
 Let K+L = the effective numerical rank of the
 matrix (A**H,B**H)**H, then R is a (K+L)-by-(K+L) nonsingular upper
 triangular matrix, D1 and D2 are M-by-(K+L) and P-by-(K+L) "diagonal"
 matrices and of the following structures, respectively:

 If M-K-L >= 0,

                     K  L
        D1 =     K ( I  0 )
                 L ( 0  C )
             M-K-L ( 0  0 )

                   K  L
        D2 =   L ( 0  S )
             P-L ( 0  0 )

                 N-K-L  K    L
   ( 0 R ) = K (  0   R11  R12 )
             L (  0    0   R22 )

 where

   C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
   S = diag( BETA(K+1),  ... , BETA(K+L) ),
   C**2 + S**2 = I.

   R is stored in A(1:K+L,N-K-L+1:N) on exit.

 If M-K-L < 0,

                   K M-K K+L-M
        D1 =   K ( I  0    0   )
             M-K ( 0  C    0   )

                     K M-K K+L-M
        D2 =   M-K ( 0  S    0  )
             K+L-M ( 0  0    I  )
               P-L ( 0  0    0  )

                    N-K-L  K   M-K  K+L-M
   ( 0 R ) =     K ( 0    R11  R12  R13  )
               M-K ( 0     0   R22  R23  )
             K+L-M ( 0     0    0   R33  )

 where

   C = diag( ALPHA(K+1), ... , ALPHA(M) ),
   S = diag( BETA(K+1),  ... , BETA(M) ),
   C**2 + S**2 = I.

   (R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N), and R33 is stored
   ( 0  R22 R23 )
   in B(M-K+1:L,N+M-K-L+1:N) on exit.

 The routine computes C, S, R, and optionally the unitary
 transformation matrices U, V and Q.

 In particular, if B is an N-by-N nonsingular matrix, then the GSVD of
 A and B implicitly gives the SVD of A*inv(B):
                      A*inv(B) = U*(D1*inv(D2))*V**H.
 If ( A**H,B**H)**H has orthonormal columns, then the GSVD of A and B is also
 equal to the CS decomposition of A and B. Furthermore, the GSVD can
 be used to derive the solution of the eigenvalue problem:
                      A**H*A x = lambda* B**H*B x.
 In some literature, the GSVD of A and B is presented in the form
                  U**H*A*X = ( 0 D1 ),   V**H*B*X = ( 0 D2 )
 where U and V are orthogonal and X is nonsingular, and D1 and D2 are
 ``diagonal''.  The former GSVD form can be converted to the latter
 form by taking the nonsingular matrix X as

                       X = Q*(  I   0    )
                             (  0 inv(R) )
Parameters
[in]JOBU
          JOBU is CHARACTER*1
          = 'U':  Unitary matrix U is computed;
          = 'N':  U is not computed.
[in]JOBV
          JOBV is CHARACTER*1
          = 'V':  Unitary matrix V is computed;
          = 'N':  V is not computed.
[in]JOBQ
          JOBQ is CHARACTER*1
          = 'Q':  Unitary matrix Q is computed;
          = 'N':  Q is not computed.
[in]M
          M is INTEGER
          The number of rows of the matrix A.  M >= 0.
[in]N
          N is INTEGER
          The number of columns of the matrices A and B.  N >= 0.
[in]P
          P is INTEGER
          The number of rows of the matrix B.  P >= 0.
[out]K
          K is INTEGER
[out]L
          L is INTEGER

          On exit, K and L specify the dimension of the subblocks
          described in Purpose.
          K + L = effective numerical rank of (A**H,B**H)**H.
[in,out]A
          A is COMPLEX array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit, A contains the triangular matrix R, or part of R.
          See Purpose for details.
[in]LDA
          LDA is INTEGER
          The leading dimension of the array A. LDA >= max(1,M).
[in,out]B
          B is COMPLEX array, dimension (LDB,N)
          On entry, the P-by-N matrix B.
          On exit, B contains part of the triangular matrix R if
          M-K-L < 0.  See Purpose for details.
[in]LDB
          LDB is INTEGER
          The leading dimension of the array B. LDB >= max(1,P).
[out]ALPHA
          ALPHA is REAL array, dimension (N)
[out]BETA
          BETA is REAL array, dimension (N)

          On exit, ALPHA and BETA contain the generalized singular
          value pairs of A and B;
            ALPHA(1:K) = 1,
            BETA(1:K)  = 0,
          and if M-K-L >= 0,
            ALPHA(K+1:K+L) = C,
            BETA(K+1:K+L)  = S,
          or if M-K-L < 0,
            ALPHA(K+1:M)=C, ALPHA(M+1:K+L)=0
            BETA(K+1:M) =S, BETA(M+1:K+L) =1
          and
            ALPHA(K+L+1:N) = 0
            BETA(K+L+1:N)  = 0
[out]U
          U is COMPLEX array, dimension (LDU,M)
          If JOBU = 'U', U contains the M-by-M unitary matrix U.
          If JOBU = 'N', U is not referenced.
[in]LDU
          LDU is INTEGER
          The leading dimension of the array U. LDU >= max(1,M) if
          JOBU = 'U'; LDU >= 1 otherwise.
[out]V
          V is COMPLEX array, dimension (LDV,P)
          If JOBV = 'V', V contains the P-by-P unitary matrix V.
          If JOBV = 'N', V is not referenced.
[in]LDV
          LDV is INTEGER
          The leading dimension of the array V. LDV >= max(1,P) if
          JOBV = 'V'; LDV >= 1 otherwise.
[out]Q
          Q is COMPLEX array, dimension (LDQ,N)
          If JOBQ = 'Q', Q contains the N-by-N unitary matrix Q.
          If JOBQ = 'N', Q is not referenced.
[in]LDQ
          LDQ is INTEGER
          The leading dimension of the array Q. LDQ >= max(1,N) if
          JOBQ = 'Q'; LDQ >= 1 otherwise.
[out]WORK
          WORK is COMPLEX array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
[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]RWORK
          RWORK is REAL array, dimension (2*N)
[out]IWORK
          IWORK is INTEGER array, dimension (N)
          On exit, IWORK stores the sorting information. More
          precisely, the following loop will sort ALPHA
             for I = K+1, min(M,K+L)
                 swap ALPHA(I) and ALPHA(IWORK(I))
             endfor
          such that ALPHA(1) >= ALPHA(2) >= ... >= ALPHA(N).
[out]INFO
          INFO is INTEGER
          = 0:  successful exit.
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          > 0:  if INFO = 1, the Jacobi-type procedure failed to
                converge.  For further details, see subroutine CTGSJA.
Internal Parameters:
  TOLA    REAL
  TOLB    REAL
          TOLA and TOLB are the thresholds to determine the effective
          rank of (A**H,B**H)**H. Generally, they are set to
                   TOLA = MAX(M,N)*norm(A)*MACHEPS,
                   TOLB = MAX(P,N)*norm(B)*MACHEPS.
          The size of TOLA and TOLB may affect the size of backward
          errors of the decomposition.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA
Further Details:
CGGSVD3 replaces the deprecated subroutine CGGSVD.

Definition at line 351 of file cggsvd3.f.

354 *
355 * -- LAPACK driver routine --
356 * -- LAPACK is a software package provided by Univ. of Tennessee, --
357 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
358 *
359 * .. Scalar Arguments ..
360  CHARACTER JOBQ, JOBU, JOBV
361  INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P,
362  $ LWORK
363 * ..
364 * .. Array Arguments ..
365  INTEGER IWORK( * )
366  REAL ALPHA( * ), BETA( * ), RWORK( * )
367  COMPLEX A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
368  $ U( LDU, * ), V( LDV, * ), WORK( * )
369 * ..
370 *
371 * =====================================================================
372 *
373 * .. Local Scalars ..
374  LOGICAL WANTQ, WANTU, WANTV, LQUERY
375  INTEGER I, IBND, ISUB, J, NCYCLE, LWKOPT
376  REAL ANORM, BNORM, SMAX, TEMP, TOLA, TOLB, ULP, UNFL
377 * ..
378 * .. External Functions ..
379  LOGICAL LSAME
380  REAL CLANGE, SLAMCH
381  EXTERNAL lsame, clange, slamch
382 * ..
383 * .. External Subroutines ..
384  EXTERNAL cggsvp3, ctgsja, scopy, xerbla
385 * ..
386 * .. Intrinsic Functions ..
387  INTRINSIC max, min
388 * ..
389 * .. Executable Statements ..
390 *
391 * Decode and test the input parameters
392 *
393  wantu = lsame( jobu, 'U' )
394  wantv = lsame( jobv, 'V' )
395  wantq = lsame( jobq, 'Q' )
396  lquery = ( lwork.EQ.-1 )
397  lwkopt = 1
398 *
399 * Test the input arguments
400 *
401  info = 0
402  IF( .NOT.( wantu .OR. lsame( jobu, 'N' ) ) ) THEN
403  info = -1
404  ELSE IF( .NOT.( wantv .OR. lsame( jobv, 'N' ) ) ) THEN
405  info = -2
406  ELSE IF( .NOT.( wantq .OR. lsame( jobq, 'N' ) ) ) THEN
407  info = -3
408  ELSE IF( m.LT.0 ) THEN
409  info = -4
410  ELSE IF( n.LT.0 ) THEN
411  info = -5
412  ELSE IF( p.LT.0 ) THEN
413  info = -6
414  ELSE IF( lda.LT.max( 1, m ) ) THEN
415  info = -10
416  ELSE IF( ldb.LT.max( 1, p ) ) THEN
417  info = -12
418  ELSE IF( ldu.LT.1 .OR. ( wantu .AND. ldu.LT.m ) ) THEN
419  info = -16
420  ELSE IF( ldv.LT.1 .OR. ( wantv .AND. ldv.LT.p ) ) THEN
421  info = -18
422  ELSE IF( ldq.LT.1 .OR. ( wantq .AND. ldq.LT.n ) ) THEN
423  info = -20
424  ELSE IF( lwork.LT.1 .AND. .NOT.lquery ) THEN
425  info = -24
426  END IF
427 *
428 * Compute workspace
429 *
430  IF( info.EQ.0 ) THEN
431  CALL cggsvp3( jobu, jobv, jobq, m, p, n, a, lda, b, ldb, tola,
432  $ tolb, k, l, u, ldu, v, ldv, q, ldq, iwork, rwork,
433  $ work, work, -1, info )
434  lwkopt = n + int( work( 1 ) )
435  lwkopt = max( 2*n, lwkopt )
436  lwkopt = max( 1, lwkopt )
437  work( 1 ) = cmplx( lwkopt )
438  END IF
439 *
440  IF( info.NE.0 ) THEN
441  CALL xerbla( 'CGGSVD3', -info )
442  RETURN
443  END IF
444  IF( lquery ) THEN
445  RETURN
446  ENDIF
447 *
448 * Compute the Frobenius norm of matrices A and B
449 *
450  anorm = clange( '1', m, n, a, lda, rwork )
451  bnorm = clange( '1', p, n, b, ldb, rwork )
452 *
453 * Get machine precision and set up threshold for determining
454 * the effective numerical rank of the matrices A and B.
455 *
456  ulp = slamch( 'Precision' )
457  unfl = slamch( 'Safe Minimum' )
458  tola = max( m, n )*max( anorm, unfl )*ulp
459  tolb = max( p, n )*max( bnorm, unfl )*ulp
460 *
461  CALL cggsvp3( jobu, jobv, jobq, m, p, n, a, lda, b, ldb, tola,
462  $ tolb, k, l, u, ldu, v, ldv, q, ldq, iwork, rwork,
463  $ work, work( n+1 ), lwork-n, info )
464 *
465 * Compute the GSVD of two upper "triangular" matrices
466 *
467  CALL ctgsja( jobu, jobv, jobq, m, p, n, k, l, a, lda, b, ldb,
468  $ tola, tolb, alpha, beta, u, ldu, v, ldv, q, ldq,
469  $ work, ncycle, info )
470 *
471 * Sort the singular values and store the pivot indices in IWORK
472 * Copy ALPHA to RWORK, then sort ALPHA in RWORK
473 *
474  CALL scopy( n, alpha, 1, rwork, 1 )
475  ibnd = min( l, m-k )
476  DO 20 i = 1, ibnd
477 *
478 * Scan for largest ALPHA(K+I)
479 *
480  isub = i
481  smax = rwork( k+i )
482  DO 10 j = i + 1, ibnd
483  temp = rwork( k+j )
484  IF( temp.GT.smax ) THEN
485  isub = j
486  smax = temp
487  END IF
488  10 CONTINUE
489  IF( isub.NE.i ) THEN
490  rwork( k+isub ) = rwork( k+i )
491  rwork( k+i ) = smax
492  iwork( k+i ) = k + isub
493  ELSE
494  iwork( k+i ) = k + i
495  END IF
496  20 CONTINUE
497 *
498  work( 1 ) = cmplx( lwkopt )
499  RETURN
500 *
501 * End of CGGSVD3
502 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
real function clange(NORM, M, N, A, LDA, WORK)
CLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value ...
Definition: clange.f:115
subroutine cggsvp3(JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ, IWORK, RWORK, TAU, WORK, LWORK, INFO)
CGGSVP3
Definition: cggsvp3.f:278
subroutine ctgsja(JOBU, JOBV, JOBQ, M, P, N, K, L, A, LDA, B, LDB, TOLA, TOLB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, NCYCLE, INFO)
CTGSJA
Definition: ctgsja.f:379
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:82
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68
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