LAPACK
3.4.2
LAPACK: Linear Algebra PACKage

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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 
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
Download SGGSVD + dependencies [TGZ] [ZIP] [TXT]SGGSVD computes the generalized singular value decomposition (GSVD) of an MbyN real matrix A and PbyN 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+LbyK+L nonsingular upper triangular matrix, D1 and D2 are Mby(K+L) and Pby(K+L) "diagonal" matrices and of the following structures, respectively: If MKL >= 0, K L D1 = K ( I 0 ) L ( 0 C ) MKL ( 0 0 ) K L D2 = L ( 0 S ) PL ( 0 0 ) NKL 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,NKL+1:N) on exit. If MKL < 0, K MK K+LM D1 = K ( I 0 0 ) MK ( 0 C 0 ) K MK K+LM D2 = MK ( 0 S 0 ) K+LM ( 0 0 I ) PL ( 0 0 0 ) NKL K MK K+LM ( 0 R ) = K ( 0 R11 R12 R13 ) MK ( 0 0 R22 R23 ) K+LM ( 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, NKL+1:N), and R33 is stored ( 0 R22 R23 ) in B(MK+1:L,N+MKL+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 NbyN 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) ).
[in]  JOBU  JOBU is CHARACTER*1 = 'U': Orthogonal matrix U is computed; = 'N': U is not computed. 
[in]  JOBV  JOBV is CHARACTER*1 = 'V': Orthogonal matrix V is computed; = 'N': V is not computed. 
[in]  JOBQ  JOBQ is CHARACTER*1 = 'Q': Orthogonal matrix Q is computed; = 'N': Q is not computed. 
[in]  M  M is INTEGER The number of rows of the matrix A. M >= 0. 
[in]  N  N is INTEGER The number of columns of the matrices A and B. N >= 0. 
[in]  P  P is INTEGER The number of rows of the matrix B. P >= 0. 
[out]  K  K is INTEGER 
[out]  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. 
[in,out]  A  A is REAL array, dimension (LDA,N) On entry, the MbyN matrix A. On exit, A contains the triangular matrix R, or part of R. See Purpose for details. 
[in]  LDA  LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M). 
[in,out]  B  B is REAL array, dimension (LDB,N) On entry, the PbyN matrix B. On exit, B contains the triangular matrix R if MKL < 0. See Purpose for details. 
[in]  LDB  LDB is INTEGER The leading dimension of the array B. LDB >= max(1,P). 
[out]  ALPHA  ALPHA is REAL array, dimension (N) 
[out]  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 MKL >= 0, ALPHA(K+1:K+L) = C, BETA(K+1:K+L) = S, or if MKL < 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 
[out]  U  U is REAL array, dimension (LDU,M) If JOBU = 'U', U contains the MbyM orthogonal matrix U. If JOBU = 'N', U is not referenced. 
[in]  LDU  LDU is INTEGER The leading dimension of the array U. LDU >= max(1,M) if JOBU = 'U'; LDU >= 1 otherwise. 
[out]  V  V is REAL array, dimension (LDV,P) If JOBV = 'V', V contains the PbyP orthogonal matrix V. If JOBV = 'N', V is not referenced. 
[in]  LDV  LDV is INTEGER The leading dimension of the array V. LDV >= max(1,P) if JOBV = 'V'; LDV >= 1 otherwise. 
[out]  Q  Q is REAL array, dimension (LDQ,N) If JOBQ = 'Q', Q contains the NbyN orthogonal matrix Q. If JOBQ = 'N', Q is not referenced. 
[in]  LDQ  LDQ is INTEGER The leading dimension of the array Q. LDQ >= max(1,N) if JOBQ = 'Q'; LDQ >= 1 otherwise. 
[out]  WORK  WORK is REAL array, dimension (max(3*N,M,P)+N) 
[out]  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). 
[out]  INFO  INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value. > 0: if INFO = 1, the Jacobitype procedure failed to converge. For further details, see subroutine STGSJA. 
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.
Definition at line 331 of file sggsvd.f.