 LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ zggsvd3()

 subroutine zggsvd3 ( 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, integer LWORK, double precision, dimension( * ) RWORK, integer, dimension( * ) IWORK, integer INFO )

ZGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices

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Purpose:
``` ZGGSVD3 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*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(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 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)*MACHEPS,
TOLB = MAX(P,N)*norm(B)*MACHEPS.
The size of TOLA and TOLB may affect the size of backward
errors of the decomposition.```
Contributors:
Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA
Further Details:
ZGGSVD3 replaces the deprecated subroutine ZGGSVD.

Definition at line 350 of file zggsvd3.f.

353 *
354 * -- LAPACK driver routine --
355 * -- LAPACK is a software package provided by Univ. of Tennessee, --
356 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
357 *
358 * .. Scalar Arguments ..
359  CHARACTER JOBQ, JOBU, JOBV
360  INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P,
361  \$ LWORK
362 * ..
363 * .. Array Arguments ..
364  INTEGER IWORK( * )
365  DOUBLE PRECISION ALPHA( * ), BETA( * ), RWORK( * )
366  COMPLEX*16 A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
367  \$ U( LDU, * ), V( LDV, * ), WORK( * )
368 * ..
369 *
370 * =====================================================================
371 *
372 * .. Local Scalars ..
373  LOGICAL WANTQ, WANTU, WANTV, LQUERY
374  INTEGER I, IBND, ISUB, J, NCYCLE, LWKOPT
375  DOUBLE PRECISION ANORM, BNORM, SMAX, TEMP, TOLA, TOLB, ULP, UNFL
376 * ..
377 * .. External Functions ..
378  LOGICAL LSAME
379  DOUBLE PRECISION DLAMCH, ZLANGE
380  EXTERNAL lsame, dlamch, zlange
381 * ..
382 * .. External Subroutines ..
383  EXTERNAL dcopy, xerbla, zggsvp3, ztgsja
384 * ..
385 * .. Intrinsic Functions ..
386  INTRINSIC max, min
387 * ..
388 * .. Executable Statements ..
389 *
390 * Decode and test the input parameters
391 *
392  wantu = lsame( jobu, 'U' )
393  wantv = lsame( jobv, 'V' )
394  wantq = lsame( jobq, 'Q' )
395  lquery = ( lwork.EQ.-1 )
396  lwkopt = 1
397 *
398 * Test the input arguments
399 *
400  info = 0
401  IF( .NOT.( wantu .OR. lsame( jobu, 'N' ) ) ) THEN
402  info = -1
403  ELSE IF( .NOT.( wantv .OR. lsame( jobv, 'N' ) ) ) THEN
404  info = -2
405  ELSE IF( .NOT.( wantq .OR. lsame( jobq, 'N' ) ) ) THEN
406  info = -3
407  ELSE IF( m.LT.0 ) THEN
408  info = -4
409  ELSE IF( n.LT.0 ) THEN
410  info = -5
411  ELSE IF( p.LT.0 ) THEN
412  info = -6
413  ELSE IF( lda.LT.max( 1, m ) ) THEN
414  info = -10
415  ELSE IF( ldb.LT.max( 1, p ) ) THEN
416  info = -12
417  ELSE IF( ldu.LT.1 .OR. ( wantu .AND. ldu.LT.m ) ) THEN
418  info = -16
419  ELSE IF( ldv.LT.1 .OR. ( wantv .AND. ldv.LT.p ) ) THEN
420  info = -18
421  ELSE IF( ldq.LT.1 .OR. ( wantq .AND. ldq.LT.n ) ) THEN
422  info = -20
423  ELSE IF( lwork.LT.1 .AND. .NOT.lquery ) THEN
424  info = -24
425  END IF
426 *
427 * Compute workspace
428 *
429  IF( info.EQ.0 ) THEN
430  CALL zggsvp3( jobu, jobv, jobq, m, p, n, a, lda, b, ldb, tola,
431  \$ tolb, k, l, u, ldu, v, ldv, q, ldq, iwork, rwork,
432  \$ work, work, -1, info )
433  lwkopt = n + int( work( 1 ) )
434  lwkopt = max( 2*n, lwkopt )
435  lwkopt = max( 1, lwkopt )
436  work( 1 ) = dcmplx( lwkopt )
437  END IF
438 *
439  IF( info.NE.0 ) THEN
440  CALL xerbla( 'ZGGSVD3', -info )
441  RETURN
442  END IF
443  IF( lquery ) THEN
444  RETURN
445  ENDIF
446 *
447 * Compute the Frobenius norm of matrices A and B
448 *
449  anorm = zlange( '1', m, n, a, lda, rwork )
450  bnorm = zlange( '1', p, n, b, ldb, rwork )
451 *
452 * Get machine precision and set up threshold for determining
453 * the effective numerical rank of the matrices A and B.
454 *
455  ulp = dlamch( 'Precision' )
456  unfl = dlamch( 'Safe Minimum' )
457  tola = max( m, n )*max( anorm, unfl )*ulp
458  tolb = max( p, n )*max( bnorm, unfl )*ulp
459 *
460  CALL zggsvp3( jobu, jobv, jobq, m, p, n, a, lda, b, ldb, tola,
461  \$ tolb, k, l, u, ldu, v, ldv, q, ldq, iwork, rwork,
462  \$ work, work( n+1 ), lwork-n, info )
463 *
464 * Compute the GSVD of two upper "triangular" matrices
465 *
466  CALL ztgsja( jobu, jobv, jobq, m, p, n, k, l, a, lda, b, ldb,
467  \$ tola, tolb, alpha, beta, u, ldu, v, ldv, q, ldq,
468  \$ work, ncycle, info )
469 *
470 * Sort the singular values and store the pivot indices in IWORK
471 * Copy ALPHA to RWORK, then sort ALPHA in RWORK
472 *
473  CALL dcopy( n, alpha, 1, rwork, 1 )
474  ibnd = min( l, m-k )
475  DO 20 i = 1, ibnd
476 *
477 * Scan for largest ALPHA(K+I)
478 *
479  isub = i
480  smax = rwork( k+i )
481  DO 10 j = i + 1, ibnd
482  temp = rwork( k+j )
483  IF( temp.GT.smax ) THEN
484  isub = j
485  smax = temp
486  END IF
487  10 CONTINUE
488  IF( isub.NE.i ) THEN
489  rwork( k+isub ) = rwork( k+i )
490  rwork( k+i ) = smax
491  iwork( k+i ) = k + isub
492  ELSE
493  iwork( k+i ) = k + i
494  END IF
495  20 CONTINUE
496 *
497  work( 1 ) = dcmplx( lwkopt )
498  RETURN
499 *
500 * End of ZGGSVD3
501 *
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
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:115
subroutine zggsvp3(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)
ZGGSVP3
Definition: zggsvp3.f:278
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:379
subroutine dcopy(N, DX, INCX, DY, INCY)
DCOPY
Definition: dcopy.f:82
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