NAG Library Routine Document
F08VNF (ZGGSVD)
1 Purpose
F08VNF (ZGGSVD) computes the generalized singular value decomposition (GSVD) of an m by n complex matrix A and a p by n complex matrix B.
2 Specification
| SUBROUTINE F08VNF ( |
JOBU, JOBV, JOBQ, M, N, P, K, L, A, LDA, B, LDB, ALPHA, BETA, U, LDU, V, LDV, Q, LDQ, WORK, RWORK, IWORK, INFO) |
| INTEGER |
M, N, P, K, L, LDA, LDB, LDU, LDV, LDQ, IWORK(N), INFO |
| REAL (KIND=nag_wp) |
ALPHA(N), BETA(N), RWORK(2*N) |
| COMPLEX (KIND=nag_wp) |
A(LDA,*), B(LDB,*), U(LDU,*), V(LDV,*), Q(LDQ,*), WORK(max(3*N,M,P)+N) |
| CHARACTER(1) |
JOBU, JOBV, JOBQ |
|
The routine may be called by its
LAPACK
name zggsvd.
3 Description
The generalized singular value decomposition is given by
where
U,
V and
Q are unitary matrices. Let
k+l be the effective numerical rank of the matrix
A
B
, 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 structured as follows:
if
m-k-l≥0,
where
and
R is stored as a submatrix of
A with elements
Rij stored as
Ai,n-k-l+j on exit.
If
m-k-l<0
,
where
and
R11
R12
R13
0
R22
R23
is stored as a submatrix of
A with
Rij stored as
Ai,n-k-l+j, and
R33
is stored as a submatrix of
B with
R33ij stored as
Bm-k+i,n+m-k-l+j.
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×B-1:
If
A
B
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:
In some literature, the GSVD of
A and
B is presented in the form
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
4 References
Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999)
LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia
http://www.netlib.org/lapack/lugGolub G H and Van Loan C F (1996)
Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
5 Parameters
- 1: JOBU – CHARACTER(1)Input
-
On entry: if
JOBU='U', the
unitary
matrix
U is computed.
If JOBU='N', U is not computed.
Constraint:
JOBU='U' or 'N'.
- 2: JOBV – CHARACTER(1)Input
-
On entry: if
JOBV='V', the
unitary
matrix
V is computed.
If JOBV='N', V is not computed.
Constraint:
JOBV='V' or 'N'.
- 3: JOBQ – CHARACTER(1)Input
-
On entry: if
JOBQ='Q', the
unitary
matrix
Q is computed.
If JOBQ='N', Q is not computed.
Constraint:
JOBQ='Q' or 'N'.
- 4: M – INTEGERInput
-
On entry:
m, the number of rows of the matrix A.
Constraint:
M≥0.
- 5: N – INTEGERInput
-
On entry:
n, the number of columns of the matrices A and B.
Constraint:
N≥0.
- 6: P – INTEGERInput
-
On entry:
p, the number of rows of the matrix B.
Constraint:
P≥0.
- 7: K – INTEGEROutput
- 8: L – INTEGEROutput
-
On exit:
K and
L specify the dimension of the subblocks
k and
l as described in
Section 3;
k+l is the effective numerical rank of
AB.
- 9: A(LDA,*) – COMPLEX (KIND=nag_wp) arrayInput/Output
-
Note: the second dimension of the array
A
must be at least
max1,N.
On entry: the m by n matrix A.
On exit: contains the triangular matrix
R, or part of
R. See
Section 3 for details.
- 10: LDA – INTEGERInput
-
On entry: the first dimension of the array
A as declared in the (sub)program from which F08VNF (ZGGSVD) is called.
Constraint:
LDA≥max1,M.
- 11: B(LDB,*) – COMPLEX (KIND=nag_wp) arrayInput/Output
-
Note: the second dimension of the array
B
must be at least
max1,N.
On entry: the p by n matrix B.
On exit:
contains the triangular matrix
R if
m-k-l<0. See
Section 3 for details.
- 12: LDB – INTEGERInput
-
On entry: the first dimension of the array
B as declared in the (sub)program from which F08VNF (ZGGSVD) is called.
Constraint:
LDB≥max1,P.
- 13: ALPHA(N) – REAL (KIND=nag_wp) arrayOutput
On exit: see the description of
BETA.
- 14: BETA(N) – REAL (KIND=nag_wp) arrayOutput
On exit:
ALPHA and
BETA contain the generalized singular value pairs of
A and
B,
αi
and
βi
;
-
ALPHA1:K
=
1
,
-
BETA1:K
=
0
,
and if
m-k-l≥0
,
-
ALPHAK+1:K+L
=
C
,
-
BETAK+1:K+L
=
S
,
or if
m-k-l<0
,
-
ALPHAK+1:M
=
C
,
-
ALPHAM+1:K+L
=
0
,
-
BETAK+1:M
=
S
,
-
BETAM+1:K+L
=
1
, and
-
ALPHAK+L+1:N
=
0
,
-
BETAK+L+1:N
=
0
.
The notation ALPHAK:N above refers to consecutive elements ALPHAi, for i=K,…,N.
- 15: U(LDU,*) – COMPLEX (KIND=nag_wp) arrayOutput
-
Note: the second dimension of the array
U
must be at least
max1,M if
JOBU='U', and at least
1 otherwise.
On exit: if
JOBU='U',
U contains the
m by
m unitary
matrix
U.
If
JOBU='N',
U is not referenced.
- 16: LDU – INTEGERInput
-
On entry: the first dimension of the array
U as declared in the (sub)program from which F08VNF (ZGGSVD) is called.
Constraints:
- if JOBU='U', LDU≥max1,M;
- otherwise LDU≥1.
- 17: V(LDV,*) – COMPLEX (KIND=nag_wp) arrayOutput
-
Note: the second dimension of the array
V
must be at least
max1,P if
JOBV='V', and at least
1 otherwise.
On exit: if
JOBV='V',
V contains the
p by
p unitary
matrix
V.
If
JOBV='N',
V is not referenced.
- 18: LDV – INTEGERInput
-
On entry: the first dimension of the array
V as declared in the (sub)program from which F08VNF (ZGGSVD) is called.
Constraints:
- if JOBV='V', LDV≥max1,P;
- otherwise LDV≥1.
- 19: Q(LDQ,*) – COMPLEX (KIND=nag_wp) arrayOutput
-
Note: the second dimension of the array
Q
must be at least
max1,N if
JOBQ='Q', and at least
1 otherwise.
On exit: if
JOBQ='Q',
Q contains the
n by
n unitary
matrix
Q.
If
JOBQ='N',
Q is not referenced.
- 20: LDQ – INTEGERInput
-
On entry: the first dimension of the array
Q as declared in the (sub)program from which F08VNF (ZGGSVD) is called.
Constraints:
- if JOBQ='Q', LDQ≥max1,N;
- otherwise LDQ≥1.
- 21: WORK(max3×N,M,P+N) – COMPLEX (KIND=nag_wp) arrayWorkspace
- 22: RWORK(2×N) – REAL (KIND=nag_wp) arrayWorkspace
- 23: IWORK(N) – INTEGER arrayOutput
On exit: 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
ALPHA1≥ALPHA2≥⋯≥ALPHAN.
- 24: INFO – INTEGEROutput
On exit:
INFO=0 unless the routine detects an error (see
Section 6).
6 Error Indicators and Warnings
Errors or warnings detected by the routine:
- INFO<0
If INFO=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
- INFO=1
If INFO=1, the Jacobi-type procedure failed to converge.
7 Accuracy
The computed generalized singular value decomposition is nearly the exact generalized singular value decomposition for nearby matrices
A+E
and
B+F
, where
and
ε
is the
machine precision. See Section 4.12 of
Anderson et al. (1999) for further details.
8 Further Comments
The diagonal elements of the matrix R are real.
The real analogue of this routine is
F08VAF (DGGSVD).
9 Example
This example finds the generalized singular value decomposition
where
and
together with estimates for the condition number of
R and the error bound for the computed generalized singular values.
The example program assumes that m≥n, and would need slight modification if this is not the case.
9.1 Program Text
Program Text (f08vnfe.f90)
9.2 Program Data
Program Data (f08vnfe.d)
9.3 Program Results
Program Results (f08vnfe.r)