---
name: "zggsvd"
section: "3lapack"
platform: "linux"
package: "liblapack-doc"
version: "3.2.1-8"
release: "Debian 6.0 (Squeeze)"
platforms: ["linux"]
url: "https://new.carta.tech/zggsvd/3lapack"
---

# zggsvd(3lapack)

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

## NAME

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

## SYNOPSIS

- 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 )

- CHARACTER

  JOBQ, JOBU, JOBV

- INTEGER

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

- INTEGER

  IWORK( \* )

- DOUBLE

  PRECISION ALPHA( \* ), BETA( \* ), RWORK( \* )

- COMPLEX\*16

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

## 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'\*A\*Q = D1\*( 0 R ),    V'\*B\*Q = D2\*( 0 R )\
 where U, V and Q are unitary matrices, and Z' means the conjugate 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 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'.\
 If ( A',B')' 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'\*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, 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) )\

## ARGUMENTS

- JOBU (input) CHARACTER\*1

  \= 'U':  Unitary matrix U is computed;\
   \= 'N':  U is not computed.

- JOBV (input) CHARACTER\*1

  \= 'V':  Unitary matrix V is computed;\
   \= 'N':  V is not computed.

- JOBQ (input) CHARACTER\*1

  \= 'Q':  Unitary 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 Purpose. K + L = effective numerical rank of (A',B')'.

- A (input/output) 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.

- LDA (input) INTEGER

  The leading dimension of the array A. LDA \>= max(1,M).

- B (input/output) 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.

- 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) 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.

- LDU (input) INTEGER

  The leading dimension of the array U. LDU \>= max(1,M) if JOBU = 'U'; LDU \>= 1 otherwise.

- V (output) 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.

- LDV (input) INTEGER

  The leading dimension of the array V. LDV \>= max(1,P) if JOBV = 'V'; LDV \>= 1 otherwise.

- Q (output) 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.

- LDQ (input) INTEGER

  The leading dimension of the array Q. LDQ \>= max(1,N) if JOBQ = 'Q'; LDQ \>= 1 otherwise.

- WORK (workspace) COMPLEX\*16 array, dimension (max(3\*N,M,P)+N)

  - RWORK (workspace) DOUBLE PRECISION 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 ZTGSJA.

## 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