 LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ 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

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.```
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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