LAPACK  3.8.0
LAPACK: Linear Algebra PACKage

◆ zggsvd()

subroutine zggsvd ( character  JOBU,
character  JOBV,
character  JOBQ,
integer  M,
integer  N,
integer  P,
integer  K,
integer  L,
complex*16, dimension( lda, * )  A,
integer  LDA,
complex*16, dimension( ldb, * )  B,
integer  LDB,
double precision, dimension( * )  ALPHA,
double precision, dimension( * )  BETA,
complex*16, dimension( ldu, * )  U,
integer  LDU,
complex*16, dimension( ldv, * )  V,
integer  LDV,
complex*16, dimension( ldq, * )  Q,
integer  LDQ,
complex*16, dimension( * )  WORK,
double precision, dimension( * )  RWORK,
integer, dimension( * )  IWORK,
integer  INFO 
)

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

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

Purpose:
 This routine is deprecated and has been replaced by routine ZGGSVD3.

 ZGGSVD 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 orthnormal 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*16 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*16 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 DOUBLE PRECISION array, dimension (N)
[out]BETA
          BETA is DOUBLE PRECISION 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*16 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*16 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*16 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*16 array, dimension (max(3*N,M,P)+N)
[out]RWORK
          RWORK is DOUBLE PRECISION 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 ZTGSJA.
Internal Parameters:
  TOLA    DOUBLE PRECISION
  TOLB    DOUBLE PRECISION
          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)*MAZHEPS,
                   TOLB = MAX(P,N)*norm(B)*MAZHEPS.
          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.
Date
December 2016
Contributors:
Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA

Definition at line 339 of file zggsvd.f.

339 *
340 * -- LAPACK driver routine (version 3.7.0) --
341 * -- LAPACK is a software package provided by Univ. of Tennessee, --
342 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
343 * December 2016
344 *
345 * .. Scalar Arguments ..
346  CHARACTER jobq, jobu, jobv
347  INTEGER info, k, l, lda, ldb, ldq, ldu, ldv, m, n, p
348 * ..
349 * .. Array Arguments ..
350  INTEGER iwork( * )
351  DOUBLE PRECISION alpha( * ), beta( * ), rwork( * )
352  COMPLEX*16 a( lda, * ), b( ldb, * ), q( ldq, * ),
353  $ u( ldu, * ), v( ldv, * ), work( * )
354 * ..
355 *
356 * =====================================================================
357 *
358 * .. Local Scalars ..
359  LOGICAL wantq, wantu, wantv
360  INTEGER i, ibnd, isub, j, ncycle
361  DOUBLE PRECISION anorm, bnorm, smax, temp, tola, tolb, ulp, unfl
362 * ..
363 * .. External Functions ..
364  LOGICAL lsame
365  DOUBLE PRECISION dlamch, zlange
366  EXTERNAL lsame, dlamch, zlange
367 * ..
368 * .. External Subroutines ..
369  EXTERNAL dcopy, xerbla, zggsvp, ztgsja
370 * ..
371 * .. Intrinsic Functions ..
372  INTRINSIC max, min
373 * ..
374 * .. Executable Statements ..
375 *
376 * Decode and test the input parameters
377 *
378  wantu = lsame( jobu, 'U' )
379  wantv = lsame( jobv, 'V' )
380  wantq = lsame( jobq, 'Q' )
381 *
382  info = 0
383  IF( .NOT.( wantu .OR. lsame( jobu, 'N' ) ) ) THEN
384  info = -1
385  ELSE IF( .NOT.( wantv .OR. lsame( jobv, 'N' ) ) ) THEN
386  info = -2
387  ELSE IF( .NOT.( wantq .OR. lsame( jobq, 'N' ) ) ) THEN
388  info = -3
389  ELSE IF( m.LT.0 ) THEN
390  info = -4
391  ELSE IF( n.LT.0 ) THEN
392  info = -5
393  ELSE IF( p.LT.0 ) THEN
394  info = -6
395  ELSE IF( lda.LT.max( 1, m ) ) THEN
396  info = -10
397  ELSE IF( ldb.LT.max( 1, p ) ) THEN
398  info = -12
399  ELSE IF( ldu.LT.1 .OR. ( wantu .AND. ldu.LT.m ) ) THEN
400  info = -16
401  ELSE IF( ldv.LT.1 .OR. ( wantv .AND. ldv.LT.p ) ) THEN
402  info = -18
403  ELSE IF( ldq.LT.1 .OR. ( wantq .AND. ldq.LT.n ) ) THEN
404  info = -20
405  END IF
406  IF( info.NE.0 ) THEN
407  CALL xerbla( 'ZGGSVD', -info )
408  RETURN
409  END IF
410 *
411 * Compute the Frobenius norm of matrices A and B
412 *
413  anorm = zlange( '1', m, n, a, lda, rwork )
414  bnorm = zlange( '1', p, n, b, ldb, rwork )
415 *
416 * Get machine precision and set up threshold for determining
417 * the effective numerical rank of the matrices A and B.
418 *
419  ulp = dlamch( 'Precision' )
420  unfl = dlamch( 'Safe Minimum' )
421  tola = max( m, n )*max( anorm, unfl )*ulp
422  tolb = max( p, n )*max( bnorm, unfl )*ulp
423 *
424  CALL zggsvp( jobu, jobv, jobq, m, p, n, a, lda, b, ldb, tola,
425  $ tolb, k, l, u, ldu, v, ldv, q, ldq, iwork, rwork,
426  $ work, work( n+1 ), info )
427 *
428 * Compute the GSVD of two upper "triangular" matrices
429 *
430  CALL ztgsja( jobu, jobv, jobq, m, p, n, k, l, a, lda, b, ldb,
431  $ tola, tolb, alpha, beta, u, ldu, v, ldv, q, ldq,
432  $ work, ncycle, info )
433 *
434 * Sort the singular values and store the pivot indices in IWORK
435 * Copy ALPHA to RWORK, then sort ALPHA in RWORK
436 *
437  CALL dcopy( n, alpha, 1, rwork, 1 )
438  ibnd = min( l, m-k )
439  DO 20 i = 1, ibnd
440 *
441 * Scan for largest ALPHA(K+I)
442 *
443  isub = i
444  smax = rwork( k+i )
445  DO 10 j = i + 1, ibnd
446  temp = rwork( k+j )
447  IF( temp.GT.smax ) THEN
448  isub = j
449  smax = temp
450  END IF
451  10 CONTINUE
452  IF( isub.NE.i ) THEN
453  rwork( k+isub ) = rwork( k+i )
454  rwork( k+i ) = smax
455  iwork( k+i ) = k + isub
456  ELSE
457  iwork( k+i ) = k + i
458  END IF
459  20 CONTINUE
460 *
461  RETURN
462 *
463 * End of ZGGSVD
464 *
double precision function zlange(NORM, M, N, A, LDA, WORK)
ZLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value ...
Definition: zlange.f:117
subroutine ztgsja(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)
ZTGSJA
Definition: ztgsja.f:381
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
subroutine dcopy(N, DX, INCX, DY, INCY)
DCOPY
Definition: dcopy.f:84
subroutine zggsvp(JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ, IWORK, RWORK, TAU, WORK, INFO)
ZGGSVP
Definition: zggsvp.f:267
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:65
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