# cggsvd (l) - Linux Man Pages

## NAME

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

## SYNOPSIS

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

CHARACTER JOBQ, JOBU, JOBV

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

INTEGER IWORK( * )

REAL ALPHA( * ), BETA( * ), RWORK( * )

COMPLEX A( LDA, * ), B( LDB, * ), Q( LDQ, * ), U( LDU, * ), V( LDV, * ), WORK( * )

## PURPOSE

CGGSVD computes the generalized singular value decomposition (GSVD) of an M-by-N complex matrix A and P-by-N complex matrix B:
Uaq*A*Q D1*( 0 R ),    Vaq*B*Q D2*( 0 R )
where U, V and Q are unitary matrices, and Zaq means the conjugate transpose of Z. Let K+L = the effective numerical rank of the matrix (Aaq,Baq)aq, 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,

L

D1      )

)

M-K-L  )

L

D2    )

P-L  )

N-K-L     L

0 R    R11  R12 )

R22 )
where

diag( ALPHA(K+1), ... , ALPHA(K+L) ),

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         )

M-K       )

K M-K K+L-M

D2   M-K      )

K+L-M      )

P-L      )

N-K-L    M-K  K+L-M

0 R        R11  R12  R13  )

M-K       R22  R23  )

K+L-M          R33  )
where

diag( ALPHA(K+1), ... , ALPHA(M) ),

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
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))*Vaq.
If ( Aaq,Baq)aq 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:
Aaq*A x lambda* Baq*B x.
In some literature, the GSVD of A and B is presented in the form
Uaq*A*X 0 D1 ),   Vaq*B*X 0 D2 )
where U and V are orthogonal and X is nonsingular, and D1 and D2 are ``diagonalaqaq. The former GSVD form can be converted to the latter form by taking the nonsingular matrix X as

Q*(       )

0 inv(R) )

## ARGUMENTS

JOBU (input) CHARACTER*1
= aqUaq: Unitary matrix U is computed;
= aqNaq: U is not computed.
JOBV (input) CHARACTER*1

= aqVaq: Unitary matrix V is computed;
= aqNaq: V is not computed.
JOBQ (input) CHARACTER*1

= aqQaq: Unitary matrix Q is computed;
= aqNaq: 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 Purpose. K + L = effective numerical rank of (Aaq,Baq)aq.
A (input/output) 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.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,M).
B (input/output) 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.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,P).
ALPHA (output) REAL array, dimension (N)
BETA (output) 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
U (output) COMPLEX array, dimension (LDU,M)
If JOBU = aqUaq, U contains the M-by-M unitary matrix U. If JOBU = aqNaq, U is not referenced.
LDU (input) INTEGER
The leading dimension of the array U. LDU >= max(1,M) if JOBU = aqUaq; LDU >= 1 otherwise.
V (output) COMPLEX array, dimension (LDV,P)
If JOBV = aqVaq, V contains the P-by-P unitary matrix V. If JOBV = aqNaq, V is not referenced.
LDV (input) INTEGER
The leading dimension of the array V. LDV >= max(1,P) if JOBV = aqVaq; LDV >= 1 otherwise.
Q (output) COMPLEX array, dimension (LDQ,N)
If JOBQ = aqQaq, Q contains the N-by-N unitary matrix Q. If JOBQ = aqNaq, Q is not referenced.
LDQ (input) INTEGER
The leading dimension of the array Q. LDQ >= max(1,N) if JOBQ = aqQaq; LDQ >= 1 otherwise.
WORK (workspace) COMPLEX array, dimension (max(3*N,M,P)+N)
RWORK (workspace) REAL array, dimension (2*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 CTGSJA.

## PARAMETERS

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