LAPACK
3.4.2
LAPACK: Linear Algebra PACKage
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Go to the source code of this file.
Functions/Subroutines | |
subroutine | dggsvd (JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B, LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, IWORK, INFO) |
DGGSVD computes the singular value decomposition (SVD) for OTHER matrices |
subroutine dggsvd | ( | character | JOBU, |
character | JOBV, | ||
character | JOBQ, | ||
integer | M, | ||
integer | N, | ||
integer | P, | ||
integer | K, | ||
integer | L, | ||
double precision, dimension( lda, * ) | A, | ||
integer | LDA, | ||
double precision, dimension( ldb, * ) | B, | ||
integer | LDB, | ||
double precision, dimension( * ) | ALPHA, | ||
double precision, dimension( * ) | BETA, | ||
double precision, dimension( ldu, * ) | U, | ||
integer | LDU, | ||
double precision, dimension( ldv, * ) | V, | ||
integer | LDV, | ||
double precision, dimension( ldq, * ) | Q, | ||
integer | LDQ, | ||
double precision, dimension( * ) | WORK, | ||
integer, dimension( * ) | IWORK, | ||
integer | INFO | ||
) |
DGGSVD computes the singular value decomposition (SVD) for OTHER matrices
Download DGGSVD + dependencies [TGZ] [ZIP] [TXT]DGGSVD 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) ).
[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 DOUBLE PRECISION 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. |
[in] | LDA | LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M). |
[in,out] | B | B is DOUBLE PRECISION 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. |
[in] | LDB | LDB is INTEGER The leading dimension of the array B. LDB >= max(1,P). |
[out] | ALPHA | ALPHA is DOUBLE PRECISION array, dimension (N) |
[out] | BETA | BETA is 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 |
[out] | U | U is DOUBLE PRECISION array, dimension (LDU,M) If JOBU = 'U', U contains the M-by-M 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 DOUBLE PRECISION array, dimension (LDV,P) If JOBV = 'V', V contains the P-by-P 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 DOUBLE PRECISION array, dimension (LDQ,N) If JOBQ = 'Q', Q contains the N-by-N 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 DOUBLE PRECISION 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 i-th argument had an illegal value. > 0: if INFO = 1, the Jacobi-type procedure failed to converge. For further details, see subroutine DTGSJA. |
TOLA DOUBLE PRECISION TOLB DOUBLE PRECISION TOLA and TOLB are the thresholds to determine the effective rank of (A',B')**T. 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.
Definition at line 331 of file dggsvd.f.