LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ sggsvd3()

 subroutine sggsvd3 ( character JOBU, character JOBV, character JOBQ, integer M, integer N, integer P, integer K, integer L, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, real, dimension( * ) ALPHA, real, dimension( * ) BETA, real, dimension( ldu, * ) U, integer LDU, real, dimension( ldv, * ) V, integer LDV, real, dimension( ldq, * ) Q, integer LDQ, real, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer INFO )

SGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices

Purpose:
``` SGGSVD3 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 REAL 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 REAL 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 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 REAL 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 REAL 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 REAL 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 REAL 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 STGSJA.```
Internal Parameters:
```  TOLA    REAL
TOLB    REAL
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.```
Contributors:
Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA
Further Details:
SGGSVD3 replaces the deprecated subroutine SGGSVD.

Definition at line 346 of file sggsvd3.f.

349 *
350 * -- LAPACK driver routine --
351 * -- LAPACK is a software package provided by Univ. of Tennessee, --
352 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
353 *
354 * .. Scalar Arguments ..
355  CHARACTER JOBQ, JOBU, JOBV
356  INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P,
357  \$ LWORK
358 * ..
359 * .. Array Arguments ..
360  INTEGER IWORK( * )
361  REAL A( LDA, * ), ALPHA( * ), B( LDB, * ),
362  \$ BETA( * ), Q( LDQ, * ), U( LDU, * ),
363  \$ V( LDV, * ), WORK( * )
364 * ..
365 *
366 * =====================================================================
367 *
368 * .. Local Scalars ..
369  LOGICAL WANTQ, WANTU, WANTV, LQUERY
370  INTEGER I, IBND, ISUB, J, NCYCLE, LWKOPT
371  REAL ANORM, BNORM, SMAX, TEMP, TOLA, TOLB, ULP, UNFL
372 * ..
373 * .. External Functions ..
374  LOGICAL LSAME
375  REAL SLAMCH, SLANGE
376  EXTERNAL lsame, slamch, slange
377 * ..
378 * .. External Subroutines ..
379  EXTERNAL scopy, sggsvp3, stgsja, xerbla
380 * ..
381 * .. Intrinsic Functions ..
382  INTRINSIC max, min
383 * ..
384 * .. Executable Statements ..
385 *
386 * Decode and test the input parameters
387 *
388  wantu = lsame( jobu, 'U' )
389  wantv = lsame( jobv, 'V' )
390  wantq = lsame( jobq, 'Q' )
391  lquery = ( lwork.EQ.-1 )
392  lwkopt = 1
393 *
394 * Test the input arguments
395 *
396  info = 0
397  IF( .NOT.( wantu .OR. lsame( jobu, 'N' ) ) ) THEN
398  info = -1
399  ELSE IF( .NOT.( wantv .OR. lsame( jobv, 'N' ) ) ) THEN
400  info = -2
401  ELSE IF( .NOT.( wantq .OR. lsame( jobq, 'N' ) ) ) THEN
402  info = -3
403  ELSE IF( m.LT.0 ) THEN
404  info = -4
405  ELSE IF( n.LT.0 ) THEN
406  info = -5
407  ELSE IF( p.LT.0 ) THEN
408  info = -6
409  ELSE IF( lda.LT.max( 1, m ) ) THEN
410  info = -10
411  ELSE IF( ldb.LT.max( 1, p ) ) THEN
412  info = -12
413  ELSE IF( ldu.LT.1 .OR. ( wantu .AND. ldu.LT.m ) ) THEN
414  info = -16
415  ELSE IF( ldv.LT.1 .OR. ( wantv .AND. ldv.LT.p ) ) THEN
416  info = -18
417  ELSE IF( ldq.LT.1 .OR. ( wantq .AND. ldq.LT.n ) ) THEN
418  info = -20
419  ELSE IF( lwork.LT.1 .AND. .NOT.lquery ) THEN
420  info = -24
421  END IF
422 *
423 * Compute workspace
424 *
425  IF( info.EQ.0 ) THEN
426  CALL sggsvp3( jobu, jobv, jobq, m, p, n, a, lda, b, ldb, tola,
427  \$ tolb, k, l, u, ldu, v, ldv, q, ldq, iwork, work,
428  \$ work, -1, info )
429  lwkopt = n + int( work( 1 ) )
430  lwkopt = max( 2*n, lwkopt )
431  lwkopt = max( 1, lwkopt )
432  work( 1 ) = real( lwkopt )
433  END IF
434 *
435  IF( info.NE.0 ) THEN
436  CALL xerbla( 'SGGSVD3', -info )
437  RETURN
438  END IF
439  IF( lquery ) THEN
440  RETURN
441  ENDIF
442 *
443 * Compute the Frobenius norm of matrices A and B
444 *
445  anorm = slange( '1', m, n, a, lda, work )
446  bnorm = slange( '1', p, n, b, ldb, work )
447 *
448 * Get machine precision and set up threshold for determining
449 * the effective numerical rank of the matrices A and B.
450 *
451  ulp = slamch( 'Precision' )
452  unfl = slamch( 'Safe Minimum' )
453  tola = max( m, n )*max( anorm, unfl )*ulp
454  tolb = max( p, n )*max( bnorm, unfl )*ulp
455 *
456 * Preprocessing
457 *
458  CALL sggsvp3( jobu, jobv, jobq, m, p, n, a, lda, b, ldb, tola,
459  \$ tolb, k, l, u, ldu, v, ldv, q, ldq, iwork, work,
460  \$ work( n+1 ), lwork-n, info )
461 *
462 * Compute the GSVD of two upper "triangular" matrices
463 *
464  CALL stgsja( jobu, jobv, jobq, m, p, n, k, l, a, lda, b, ldb,
465  \$ tola, tolb, alpha, beta, u, ldu, v, ldv, q, ldq,
466  \$ work, ncycle, info )
467 *
468 * Sort the singular values and store the pivot indices in IWORK
469 * Copy ALPHA to WORK, then sort ALPHA in WORK
470 *
471  CALL scopy( n, alpha, 1, work, 1 )
472  ibnd = min( l, m-k )
473  DO 20 i = 1, ibnd
474 *
475 * Scan for largest ALPHA(K+I)
476 *
477  isub = i
478  smax = work( k+i )
479  DO 10 j = i + 1, ibnd
480  temp = work( k+j )
481  IF( temp.GT.smax ) THEN
482  isub = j
483  smax = temp
484  END IF
485  10 CONTINUE
486  IF( isub.NE.i ) THEN
487  work( k+isub ) = work( k+i )
488  work( k+i ) = smax
489  iwork( k+i ) = k + isub
490  ELSE
491  iwork( k+i ) = k + i
492  END IF
493  20 CONTINUE
494 *
495  work( 1 ) = real( lwkopt )
496  RETURN
497 *
498 * End of SGGSVD3
499 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
real function slange(NORM, M, N, A, LDA, WORK)
SLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value ...
Definition: slange.f:114
subroutine stgsja(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)
STGSJA
Definition: stgsja.f:378
subroutine sggsvp3(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)
SGGSVP3
Definition: sggsvp3.f:272
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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