# sggsvd3.f - Man Page

## Synopsis

### Functions/Subroutines

subroutine **sggsvd3** (JOBU, JOBV, JOBQ, M, **N**, P, K, L, A, **LDA**, B, **LDB**, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, LWORK, IWORK, INFO)

**SGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices**

## Function/Subroutine Documentation

### subroutine sggsvd3 (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 LWORK, integer, dimension( * ) IWORK, integer INFO)

**SGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices**

**Purpose:**

SGGSVD3 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(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

*LWORK*LWORK is INTEGER The dimension of the array WORK. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.

*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:**August 2015

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

**Further Details:**SGGSVD3 replaces the deprecated subroutine SGGSVD.

Definition at line 351 of file sggsvd3.f.

## Author

Generated automatically by Doxygen for LAPACK from the source code.

## Referenced By

The man page sggsvd3(3) is an alias of sggsvd3.f(3).