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LAPACK
3.4.2
LAPACK: Linear Algebra PACKage
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Go to the source code of this file.
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 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 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. |
| [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 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 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 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 REAL 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 REAL 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 REAL 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 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 i-th argument had an illegal value.
> 0: if INFO = 1, the Jacobi-type 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.
1.8.1.1