LAPACK  3.6.1 LAPACK: Linear Algebra PACKage
 subroutine dggsvd3 ( character JOBU, character JOBV, character JOBQ, integer M, integer N, integer P, integer K, integer L, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( * ) ALPHA, double precision, dimension( * ) BETA, double precision, dimension( ldu, * ) U, integer LDU, double precision, dimension( ldv, * ) V, integer LDV, double precision, dimension( ldq, * ) Q, integer LDQ, double precision, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer INFO )

DGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices

Purpose:
``` DGGSVD3 computes the generalized singular value decomposition (GSVD)
of an M-by-N real matrix A and P-by-N real matrix B:

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

where U, V and Q are orthogonal matrices.
Let K+L = the effective numerical rank of the matrix (A**T,B**T)**T,
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 orthogonal
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**T.
If ( A**T,B**T)**T  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**T*A x = lambda* B**T*B x.
In some literature, the GSVD of A and B is presented in the form
U**T*A*X = ( 0 D1 ),   V**T*B*X = ( 0 D2 )
where U and V are orthogonal and X is nonsingular, 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': Orthogonal matrix U is computed; = 'N': U is not computed.``` [in] JOBV ``` JOBV is CHARACTER*1 = 'V': Orthogonal matrix V is computed; = 'N': V is not computed.``` [in] JOBQ ``` JOBQ is CHARACTER*1 = 'Q': Orthogonal 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**T,B**T)**T.``` [in,out] A ``` A is DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (LDB,N) On entry, the P-by-N matrix B. On exit, B contains 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 DOUBLE PRECISION array, dimension (LDU,M) If JOBU = 'U', U contains the M-by-M orthogonal 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 DOUBLE PRECISION array, dimension (LDV,P) If JOBV = 'V', V contains the P-by-P orthogonal 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 DOUBLE PRECISION array, dimension (LDQ,N) If JOBQ = 'Q', Q contains the N-by-N orthogonal 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 DOUBLE PRECISION 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] 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 DTGSJA.```
Internal Parameters:
```  TOLA    DOUBLE PRECISION
TOLB    DOUBLE PRECISION
TOLA and TOLB are the thresholds to determine the effective
rank of (A**T,B**T)**T. 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.```
Date
August 2015
Contributors:
Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA
Further Details:
DGGSVD3 replaces the deprecated subroutine DGGSVD.

Definition at line 351 of file dggsvd3.f.

351 *
352 * -- LAPACK driver routine (version 3.6.0) --
353 * -- LAPACK is a software package provided by Univ. of Tennessee, --
354 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
355 * August 2015
356 *
357 * .. Scalar Arguments ..
358  CHARACTER jobq, jobu, jobv
359  INTEGER info, k, l, lda, ldb, ldq, ldu, ldv, m, n, p,
360  \$ lwork
361 * ..
362 * .. Array Arguments ..
363  INTEGER iwork( * )
364  DOUBLE PRECISION a( lda, * ), alpha( * ), b( ldb, * ),
365  \$ beta( * ), q( ldq, * ), u( ldu, * ),
366  \$ v( ldv, * ), work( * )
367 * ..
368 *
369 * =====================================================================
370 *
371 * .. Local Scalars ..
372  LOGICAL wantq, wantu, wantv, lquery
373  INTEGER i, ibnd, isub, j, ncycle, lwkopt
374  DOUBLE PRECISION anorm, bnorm, smax, temp, tola, tolb, ulp, unfl
375 * ..
376 * .. External Functions ..
377  LOGICAL lsame
378  DOUBLE PRECISION dlamch, dlange
379  EXTERNAL lsame, dlamch, dlange
380 * ..
381 * .. External Subroutines ..
382  EXTERNAL dcopy, dggsvp3, dtgsja, xerbla
383 * ..
384 * .. Intrinsic Functions ..
385  INTRINSIC max, min
386 * ..
387 * .. Executable Statements ..
388 *
389 * Decode and test the input parameters
390 *
391  wantu = lsame( jobu, 'U' )
392  wantv = lsame( jobv, 'V' )
393  wantq = lsame( jobq, 'Q' )
394  lquery = ( lwork.EQ.-1 )
395  lwkopt = 1
396 *
397 * Test the input arguments
398 *
399  info = 0
400  IF( .NOT.( wantu .OR. lsame( jobu, 'N' ) ) ) THEN
401  info = -1
402  ELSE IF( .NOT.( wantv .OR. lsame( jobv, 'N' ) ) ) THEN
403  info = -2
404  ELSE IF( .NOT.( wantq .OR. lsame( jobq, 'N' ) ) ) THEN
405  info = -3
406  ELSE IF( m.LT.0 ) THEN
407  info = -4
408  ELSE IF( n.LT.0 ) THEN
409  info = -5
410  ELSE IF( p.LT.0 ) THEN
411  info = -6
412  ELSE IF( lda.LT.max( 1, m ) ) THEN
413  info = -10
414  ELSE IF( ldb.LT.max( 1, p ) ) THEN
415  info = -12
416  ELSE IF( ldu.LT.1 .OR. ( wantu .AND. ldu.LT.m ) ) THEN
417  info = -16
418  ELSE IF( ldv.LT.1 .OR. ( wantv .AND. ldv.LT.p ) ) THEN
419  info = -18
420  ELSE IF( ldq.LT.1 .OR. ( wantq .AND. ldq.LT.n ) ) THEN
421  info = -20
422  ELSE IF( lwork.LT.1 .AND. .NOT.lquery ) THEN
423  info = -24
424  END IF
425 *
426 * Compute workspace
427 *
428  IF( info.EQ.0 ) THEN
429  CALL dggsvp3( jobu, jobv, jobq, m, p, n, a, lda, b, ldb, tola,
430  \$ tolb, k, l, u, ldu, v, ldv, q, ldq, iwork, work,
431  \$ work, -1, info )
432  lwkopt = n + int( work( 1 ) )
433  lwkopt = max( 2*n, lwkopt )
434  lwkopt = max( 1, lwkopt )
435  work( 1 ) = dble( lwkopt )
436  END IF
437 *
438  IF( info.NE.0 ) THEN
439  CALL xerbla( 'DGGSVD3', -info )
440  RETURN
441  END IF
442  IF( lquery ) THEN
443  RETURN
444  ENDIF
445 *
446 * Compute the Frobenius norm of matrices A and B
447 *
448  anorm = dlange( '1', m, n, a, lda, work )
449  bnorm = dlange( '1', p, n, b, ldb, work )
450 *
451 * Get machine precision and set up threshold for determining
452 * the effective numerical rank of the matrices A and B.
453 *
454  ulp = dlamch( 'Precision' )
455  unfl = dlamch( 'Safe Minimum' )
456  tola = max( m, n )*max( anorm, unfl )*ulp
457  tolb = max( p, n )*max( bnorm, unfl )*ulp
458 *
459 * Preprocessing
460 *
461  CALL dggsvp3( jobu, jobv, jobq, m, p, n, a, lda, b, ldb, tola,
462  \$ tolb, k, l, u, ldu, v, ldv, q, ldq, iwork, work,
463  \$ work( n+1 ), lwork-n, info )
464 *
465 * Compute the GSVD of two upper "triangular" matrices
466 *
467  CALL dtgsja( 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 WORK, then sort ALPHA in WORK
473 *
474  CALL dcopy( n, alpha, 1, work, 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 = work( k+i )
482  DO 10 j = i + 1, ibnd
483  temp = work( 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  work( k+isub ) = work( k+i )
491  work( 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 ) = dble( lwkopt )
499  RETURN
500 *
501 * End of DGGSVD3
502 *
subroutine dcopy(N, DX, INCX, DY, INCY)
DCOPY
Definition: dcopy.f:53
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:65
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
double precision function dlange(NORM, M, N, A, LDA, WORK)
DLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value ...
Definition: dlange.f:116
subroutine dtgsja(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)
DTGSJA
Definition: dtgsja.f:380
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
subroutine dggsvp3(JOBU, JOBV, JOBQ, M, P, N, A, LDA, B, LDB, TOLA, TOLB, K, L, U, LDU, V, LDV, Q, LDQ, IWORK, TAU, WORK, LWORK, INFO)
DGGSVP3
Definition: dggsvp3.f:274

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