 DGGSVD(3) computes the generalized singular value decomposition (GSVD) of an M-by-N real matrix A and P-by-N real matrix B

## SYNOPSIS

SUBROUTINE DGGSVD(
JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B, LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, IWORK, INFO )

CHARACTER JOBQ, JOBU, JOBV

INTEGER INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N, P

INTEGER IWORK( * )

DOUBLE PRECISION A( LDA, * ), ALPHA( * ), B( LDB, * ), BETA( * ), Q( LDQ, * ), U( LDU, * ), V( LDV, * ), WORK( * )

## PURPOSE

DGGSVD computes the generalized singular value decomposition (GSVD) of an M-by-N real matrix A and P-by-N real matrix B:
U'*A*Q = D1*( 0 R ),    V'*B*Q = D2*( 0 R )
where U, V and Q are orthogonal matrices, and Z' is the transpose of Z. Let K+L = the effective numerical rank of the matrix (A',B')', 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'.
If ( A',B')' 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'*A x = lambda* B'*B x.
In some literature, the GSVD of A and B is presented in the form
U'*A*X = ( 0 D1 ),   V'*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) ).

## ARGUMENTS

JOBU (input) CHARACTER*1
= 'U': Orthogonal matrix U is computed;
= 'N': U is not computed.
JOBV (input) CHARACTER*1

= 'V': Orthogonal matrix V is computed;
= 'N': V is not computed.
JOBQ (input) CHARACTER*1

= 'Q': Orthogonal matrix Q is computed;
= 'N': Q is not computed.
M (input) INTEGER
The number of rows of the matrix A. M >= 0.
N (input) INTEGER
The number of columns of the matrices A and B. N >= 0.
P (input) INTEGER
The number of rows of the matrix B. P >= 0.
K (output) INTEGER
L (output) INTEGER On exit, K and L specify the dimension of the subblocks described in the Purpose section. K + L = effective numerical rank of (A',B')'.
A (input/output) 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.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,M).
B (input/output) 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.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,P).
ALPHA (output) DOUBLE PRECISION array, dimension (N)
BETA (output) 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
U (output) 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.
LDU (input) INTEGER
The leading dimension of the array U. LDU >= max(1,M) if JOBU = 'U'; LDU >= 1 otherwise.
V (output) 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.
LDV (input) INTEGER
The leading dimension of the array V. LDV >= max(1,P) if JOBV = 'V'; LDV >= 1 otherwise.
Q (output) 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.
LDQ (input) INTEGER
The leading dimension of the array Q. LDQ >= max(1,N) if JOBQ = 'Q'; LDQ >= 1 otherwise.
WORK (workspace) DOUBLE PRECISION array,
dimension (max(3*N,M,P)+N)
IWORK (workspace/output) 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).
INFO (output) 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.

## PARAMETERS

TOLA DOUBLE PRECISION
TOLB DOUBLE PRECISION TOLA and TOLB are the thresholds to determine the effective rank of (A',B')'. 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. Further Details =============== 2-96 Based on modifications by Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA