# sggsvd.f man page

sggsvd.f —

## Synopsis

### Functions/Subroutines

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

## Function/Subroutine Documentation

### subroutine sggsvd (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, dimension( * ) IWORK, integer INFO)

**SGGSVD computes the singular value decomposition (SVD) for OTHER matrices**

**Purpose:**

This routine is deprecated and has been replaced by routine SGGSVD3. SGGSVD 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:**-
*JOBU*JOBU is CHARACTER*1 = 'U': Orthogonal matrix U is computed; = 'N': U is not computed.

*JOBV*JOBV is CHARACTER*1 = 'V': Orthogonal matrix V is computed; = 'N': V is not computed.

*JOBQ*JOBQ is CHARACTER*1 = 'Q': Orthogonal matrix Q is computed; = 'N': Q is not computed.

*M*M is INTEGER The number of rows of the matrix A. M >= 0.

*N*N is INTEGER The number of columns of the matrices A and B. N >= 0.

*P*P is INTEGER The number of rows of the matrix B. P >= 0.

*K*K is INTEGER

*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.

*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.

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

*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.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= max(1,P).

*ALPHA*ALPHA is REAL array, dimension (N)

*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

*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.

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

*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.

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

*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.

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

*WORK*WORK is REAL array, dimension (max(3*N,M,P)+N)

*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).

*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.

**Author:**-
Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date:**December 2016

**Contributors:**Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA

Definition at line 336 of file sggsvd.f.

## Author

Generated automatically by Doxygen for LAPACK from the source code.

## Referenced By

The man page sggsvd(3) is an alias of sggsvd.f(3).

Sat Jun 24 2017 Version 3.7.1 LAPACK