LAPACK  3.4.2 LAPACK: Linear Algebra PACKage
zggsvd.f File Reference

Go to the source code of this file.

## Functions/Subroutines

subroutine zggsvd (JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B, LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, RWORK, IWORK, INFO)
ZGGSVD computes the singular value decomposition (SVD) for OTHER matrices

## Function/Subroutine Documentation

 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

Purpose:
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.
Date:
November 2011
Contributors:
Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA

Definition at line 334 of file zggsvd.f.

Here is the call graph for this function:

Here is the caller graph for this function: