NAG Library Routine Document
F08KRF (ZGESDD)
1 Purpose
F08KRF (ZGESDD) computes the singular value decomposition (SVD) of a complex m by n matrix A, optionally computing the left and/or right singular vectors, by using a divide-and-conquer method.
2 Specification
SUBROUTINE F08KRF ( |
JOBZ, M, N, A, LDA, S, U, LDU, VT, LDVT, WORK, LWORK, RWORK, IWORK, INFO) |
INTEGER |
M, N, LDA, LDU, LDVT, LWORK, IWORK(8*min(M,N)), INFO |
REAL (KIND=nag_wp) |
S(min(M,N)), RWORK(*) |
COMPLEX (KIND=nag_wp) |
A(LDA,*), U(LDU,*), VT(LDVT,*), WORK(max(1,LWORK)) |
CHARACTER(1) |
JOBZ |
|
The routine may be called by its
LAPACK
name zgesdd.
3 Description
The SVD is written as
where
Σ is an
m by
n matrix which is zero except for its
minm,n diagonal elements,
U is an
m by
m unitary matrix, and
V is an
n by
n unitary matrix. The diagonal elements of
Σ are the singular values of
A; they are real and non-negative, and are returned in descending order. The first
minm,n columns of
U and
V are the left and right singular vectors of
A.
Note that the routine returns VH, not V.
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/lug
Golub G H and Van Loan C F (1996)
Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
5 Parameters
- 1: JOBZ – CHARACTER(1)Input
On entry: specifies options for computing all or part of the matrix
U.
- JOBZ='A'
- All m columns of U and all n rows of VH are returned in the arrays U and VT.
- JOBZ='S'
- The first minm,n columns of U and the first minm,n rows of VH are returned in the arrays U and VT.
- JOBZ='O'
- If M≥N, the first n columns of U are overwritten on the array A and all rows of VH are returned in the array VT. Otherwise, all columns of U are returned in the array U and the first m rows of VH are overwritten in the array VT.
- JOBZ='N'
- No columns of U or rows of VH are computed.
Constraint:
JOBZ='A', 'S', 'O' or 'N'.
- 2: M – INTEGERInput
On entry: m, the number of rows of the matrix A.
Constraint:
M≥0.
- 3: N – INTEGERInput
On entry: n, the number of columns of the matrix A.
Constraint:
N≥0.
- 4: 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: if
JOBZ='O',
A is overwritten with the first
n columns of
U (the left singular vectors, stored column-wise) if
M≥N;
A is overwritten with the first
m rows of
VH (the right singular vectors, stored row-wise) otherwise.
If
JOBZ≠'O', the contents of
A are destroyed.
- 5: LDA – INTEGERInput
On entry: the first dimension of the array
A as declared in the (sub)program from which F08KRF (ZGESDD) is called.
Constraint:
LDA≥max1,M.
- 6: S(minM,N) – REAL (KIND=nag_wp) arrayOutput
On exit: the singular values of A, sorted so that Si≥Si+1.
- 7: U(LDU,*) – COMPLEX (KIND=nag_wp) arrayOutput
-
Note: the second dimension of the array
U
must be at least
max1,M if
JOBZ='A' or
JOBZ='O' and
M<N,
max1,minM,N if
JOBZ='S', and at least
1 otherwise.
On exit:
If
JOBZ='A' or
JOBZ='O' and
M<N,
U contains the
m by
m unitary matrix
U.
If
JOBZ='S',
U contains the first
minm,n columns of
U (the left singular vectors, stored column-wise).
If
JOBZ='O' and
M≥N, or
JOBZ='N',
U is not referenced.
- 8: LDU – INTEGERInput
On entry: the first dimension of the array
U as declared in the (sub)program from which F08KRF (ZGESDD) is called.
Constraints:
- if JOBZ='S' or 'A' or JOBZ='O' and M<N, LDU≥ max1,M ;
- otherwise LDU≥1.
- 9: VT(LDVT,*) – COMPLEX (KIND=nag_wp) arrayOutput
-
Note: the second dimension of the array
VT
must be at least
max1,N if
JOBZ='A' or
'S' or
JOBZ='O' and
M≥N, and at least
1 otherwise.
On exit: if
JOBZ='A' or
JOBZ='O' and
M≥N,
VT contains the
n by
n unitary matrix
VH.
If
JOBZ='S',
VT contains the first
minm,n rows of
VH (the right singular vectors, stored row-wise).
If
JOBZ='O' and
M<N, or
JOBZ='N',
VT is not referenced.
- 10: LDVT – INTEGERInput
On entry: the first dimension of the array
VT as declared in the (sub)program from which F08KRF (ZGESDD) is called.
Constraints:
- if JOBZ='A' or JOBZ='O' and M≥N, LDVT≥ max1,N ;
- if JOBZ='S', LDVT≥ max1,minM,N ;
- otherwise LDVT≥1.
- 11: WORK(max1,LWORK) – COMPLEX (KIND=nag_wp) arrayWorkspace
On exit: if
INFO=0, the real part of
WORK1 contains the minimum value of
LWORK required for optimal performance.
- 12: LWORK – INTEGERInput
On entry: the dimension of the array
WORK as declared in the (sub)program from which F08KRF (ZGESDD) is called.
If
LWORK=-1, 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.
Suggested value:
for optimal performance,
LWORK should generally be larger. Consider increasing
LWORK by at least
nb×minM,N , where
nb is the optimal
block size.
Constraints:
- if JOBZ='N', LWORK≥2× minM,N+ max1,M,N ;
- if JOBZ='O', LWORK≥2× minM,N× minM,N+2× minM,N+ max1,M,N ;
- if JOBZ='S' or 'A', LWORK≥ minM,N× minM,N+2× minM,N+ max1,M,N;
- otherwise LWORK≥1.
- 13: RWORK(*) – REAL (KIND=nag_wp) arrayWorkspace
-
Note: the dimension of the array
RWORK
must be at least
max1,5×minM,N if
JOBZ='N', and at least
max1,
minM,N×
max
5×
minM,N+7
,
2×
maxM,N+
2×
minM,N+
1
otherwise.
- 14: IWORK(8×minM,N) – INTEGER arrayWorkspace
- 15: 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>0
F08KRF (ZGESDD) did not converge, the updating process failed.
7 Accuracy
The computed singular value decomposition is nearly the exact singular value decomposition for a nearby matrix
A+E
, where
and
ε
is the
machine precision. In addition, the computed singular vectors are nearly orthogonal to working precision. See Section 4.9 of
Anderson et al. (1999) for further details.
8 Further Comments
The total number of floating point operations is approximately proportional to
mn2
when m>n and
m2n
otherwise.
The singular values are returned in descending order.
The real analogue of this routine is
F08KDF (DGESDD).
9 Example
This example finds the singular values and left and right singular vectors of the
4 by
6 matrix
together with approximate error bounds for the computed singular values and vectors.
The example program for
F08KPF (ZGESVD) illustrates finding a singular value decomposition for the case
m≥n.
9.1 Program Text
Program Text (f08krfe.f90)
9.2 Program Data
Program Data (f08krfe.d)
9.3 Program Results
Program Results (f08krfe.r)