table of contents
| ggsvd3(3) | LAPACK | ggsvd3(3) |
NAME¶
ggsvd3 - ggsvd3: SVD, QR iteration
SYNOPSIS¶
Functions¶
subroutine cggsvd3 (jobu, jobv, jobq, m, n, p, k, l, a,
lda, b, ldb, alpha, beta, u, ldu, v, ldv, q, ldq, work, lwork, rwork, iwork,
info)
CGGSVD3 computes the singular value decomposition (SVD) for OTHER
matrices subroutine dggsvd3 (jobu, jobv, jobq, m, n, p, k, l, a,
lda, b, ldb, alpha, beta, u, ldu, v, ldv, q, ldq, work, lwork, iwork, info)
DGGSVD3 computes the singular value decomposition (SVD) for OTHER
matrices 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 subroutine zggsvd3 (jobu, jobv, jobq, m, n, p, k, l, a,
lda, b, ldb, alpha, beta, u, ldu, v, ldv, q, ldq, work, lwork, rwork, iwork,
info)
ZGGSVD3 computes the singular value decomposition (SVD) for OTHER
matrices
Detailed Description¶
Function Documentation¶
subroutine cggsvd3 (character jobu, character jobv, character jobq, integer m, integer n, integer p, integer k, integer l, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, real, dimension( * ) alpha, real, dimension( * ) beta, complex, dimension( ldu, * ) u, integer ldu, complex, dimension( ldv, * ) v, integer ldv, complex, dimension( ldq, * ) q, integer ldq, complex, dimension( * ) work, integer lwork, real, dimension( * ) rwork, integer, dimension( * ) iwork, integer info)¶
CGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices
Purpose:
CGGSVD3 computes the generalized singular value decomposition (GSVD)
of an M-by-N complex matrix A and P-by-N complex matrix B:
U**H*A*Q = D1*( 0 R ), V**H*B*Q = D2*( 0 R )
where U, V and Q are unitary matrices.
Let K+L = the effective numerical rank of the
matrix (A**H,B**H)**H, 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 unitary
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**H.
If ( A**H,B**H)**H 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**H*A x = lambda* B**H*B x.
In some literature, the GSVD of A and B is presented in the form
U**H*A*X = ( 0 D1 ), V**H*B*X = ( 0 D2 )
where U and V are orthogonal and X is nonsingular, and 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': Unitary matrix U is computed;
= 'N': U is not computed.
JOBV
JOBV is CHARACTER*1
= 'V': Unitary matrix V is computed;
= 'N': V is not computed.
JOBQ
JOBQ is CHARACTER*1
= 'Q': Unitary 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**H,B**H)**H.
A
A is COMPLEX 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 COMPLEX array, dimension (LDB,N)
On entry, the P-by-N matrix B.
On exit, B contains part of 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 COMPLEX array, dimension (LDU,M)
If JOBU = 'U', U contains the M-by-M unitary 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 COMPLEX array, dimension (LDV,P)
If JOBV = 'V', V contains the P-by-P unitary 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 COMPLEX array, dimension (LDQ,N)
If JOBQ = 'Q', Q contains the N-by-N unitary 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 COMPLEX 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.
RWORK
RWORK is REAL array, dimension (2*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 CTGSJA.
Internal Parameters:
TOLA REAL
TOLB REAL
TOLA and TOLB are the thresholds to determine the effective
rank of (A**H,B**H)**H. 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.
Contributors:
Ming Gu and Huan Ren, Computer Science Division,
University of California at Berkeley, USA
Further Details:
CGGSVD3 replaces the deprecated subroutine CGGSVD.
subroutine dggsvd3 (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 lwork, integer, dimension( * ) iwork, integer info)¶
DGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices
Purpose:
DGGSVD3 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 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.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
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.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,P).
ALPHA
ALPHA is DOUBLE PRECISION array, dimension (N)
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
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.
LDU
LDU is INTEGER
The leading dimension of the array U. LDU >= max(1,M) if
JOBU = 'U'; LDU >= 1 otherwise.
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.
LDV
LDV is INTEGER
The leading dimension of the array V. LDV >= max(1,P) if
JOBV = 'V'; LDV >= 1 otherwise.
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.
LDQ
LDQ is INTEGER
The leading dimension of the array Q. LDQ >= max(1,N) if
JOBQ = 'Q'; LDQ >= 1 otherwise.
WORK
WORK is DOUBLE PRECISION 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 DTGSJA.
Internal Parameters:
TOLA DOUBLE PRECISION
TOLB DOUBLE PRECISION
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.
Contributors:
Ming Gu and Huan Ren, Computer Science Division,
University of California at Berkeley, USA
Further Details:
DGGSVD3 replaces the deprecated subroutine DGGSVD.
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.
Contributors:
Ming Gu and Huan Ren, Computer Science Division,
University of California at Berkeley, USA
Further Details:
SGGSVD3 replaces the deprecated subroutine SGGSVD.
subroutine zggsvd3 (character jobu, character jobv, character jobq, integer m, integer n, integer p, integer k, integer l, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb, double precision, dimension( * ) alpha, double precision, dimension( * ) beta, complex*16, dimension( ldu, * ) u, integer ldu, complex*16, dimension( ldv, * ) v, integer ldv, complex*16, dimension( ldq, * ) q, integer ldq, complex*16, dimension( * ) work, integer lwork, double precision, dimension( * ) rwork, integer, dimension( * ) iwork, integer info)¶
ZGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices
Purpose:
ZGGSVD3 computes the generalized singular value decomposition (GSVD)
of an M-by-N complex matrix A and P-by-N complex matrix B:
U**H*A*Q = D1*( 0 R ), V**H*B*Q = D2*( 0 R )
where U, V and Q are unitary matrices.
Let K+L = the effective numerical rank of the
matrix (A**H,B**H)**H, 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 unitary
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**H.
If ( A**H,B**H)**H 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**H*A x = lambda* B**H*B x.
In some literature, the GSVD of A and B is presented in the form
U**H*A*X = ( 0 D1 ), V**H*B*X = ( 0 D2 )
where U and V are orthogonal and X is nonsingular, and 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': Unitary matrix U is computed;
= 'N': U is not computed.
JOBV
JOBV is CHARACTER*1
= 'V': Unitary matrix V is computed;
= 'N': V is not computed.
JOBQ
JOBQ is CHARACTER*1
= 'Q': Unitary 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**H,B**H)**H.
A
A is COMPLEX*16 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 COMPLEX*16 array, dimension (LDB,N)
On entry, the P-by-N matrix B.
On exit, B contains part of 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 DOUBLE PRECISION array, dimension (N)
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
U
U is COMPLEX*16 array, dimension (LDU,M)
If JOBU = 'U', U contains the M-by-M unitary 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 COMPLEX*16 array, dimension (LDV,P)
If JOBV = 'V', V contains the P-by-P unitary 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 COMPLEX*16 array, dimension (LDQ,N)
If JOBQ = 'Q', Q contains the N-by-N unitary 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 COMPLEX*16 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.
RWORK
RWORK is DOUBLE PRECISION array, dimension (2*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 ZTGSJA.
Internal Parameters:
TOLA DOUBLE PRECISION
TOLB DOUBLE PRECISION
TOLA and TOLB are the thresholds to determine the effective
rank of (A**H,B**H)**H. 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.
Contributors:
Ming Gu and Huan Ren, Computer Science Division,
University of California at Berkeley, USA
Further Details:
ZGGSVD3 replaces the deprecated subroutine ZGGSVD.
Author¶
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| Wed Feb 7 2024 11:30:40 | Version 3.12.0 |