NAG Library Routine Document
F08KBF (DGESVD)
1 Purpose
F08KBF (DGESVD) computes the singular value decomposition (SVD) of a real m by n matrix A, optionally computing the left and/or right singular vectors.
2 Specification
SUBROUTINE F08KBF ( |
JOBU, JOBVT, M, N, A, LDA, S, U, LDU, VT, LDVT, WORK, LWORK, INFO) |
INTEGER |
M, N, LDA, LDU, LDVT, LWORK, INFO |
REAL (KIND=nag_wp) |
A(LDA,*), S(*), U(LDU,*), VT(LDVT,*), WORK(max(1,LWORK)) |
CHARACTER(1) |
JOBU, JOBVT |
|
The routine may be called by its
LAPACK
name dgesvd.
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 orthogonal matrix, and
V is an
n by
n orthogonal 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 VT, 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: JOBU – CHARACTER(1)Input
On entry: specifies options for computing all or part of the matrix
U.
- JOBU='A'
- All m columns of U are returned in array U.
- JOBU='S'
- The first minm,n columns of U (the left singular vectors) are returned in the array U.
- JOBU='O'
- The first minm,n columns of U (the left singular vectors) are overwritten on the array A.
- JOBU='N'
- No columns of U (no left singular vectors) are computed.
Constraint:
JOBU='A', 'S', 'O' or 'N'.
- 2: JOBVT – CHARACTER(1)Input
On entry: specifies options for computing all or part of the matrix
VT.
- JOBVT='A'
- All n rows of VT are returned in the array VT.
- JOBVT='S'
- The first minm,n rows of VT (the right singular vectors) are returned in the array VT.
- JOBVT='O'
- The first minm,n rows of VT (the right singular vectors) are overwritten on the array A.
- JOBVT='N'
- No rows of VT (no right singular vectors) are computed.
Constraints:
- JOBVT='A', 'S', 'O' or 'N';
- JOBVT and JOBU cannot both be 'O'.
- 3: M – INTEGERInput
On entry: m, the number of rows of the matrix A.
Constraint:
M≥0.
- 4: N – INTEGERInput
On entry: n, the number of columns of the matrix A.
Constraint:
N≥0.
- 5: A(LDA,*) – REAL (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
JOBU='O',
A is overwritten with the first
minm,n columns of
U (the left singular vectors, stored column-wise).
If
JOBVT='O',
A is overwritten with the first
minm,n rows of
VT (the right singular vectors, stored row-wise).
If
JOBU≠'O' and
JOBVT≠'O', the contents of
A are destroyed.
- 6: LDA – INTEGERInput
On entry: the first dimension of the array
A as declared in the (sub)program from which F08KBF (DGESVD) is called.
Constraint:
LDA≥max1,M.
- 7: S(*) – REAL (KIND=nag_wp) arrayOutput
-
Note: the dimension of the array
S
must be at least
max1,minM,N
.
On exit: the singular values of A, sorted so that Si≥Si+1.
- 8: U(LDU,*) – REAL (KIND=nag_wp) arrayOutput
-
Note: the second dimension of the array
U
must be at least
max1,M if
JOBU='A',
max1,minM,N if
JOBU='S', and at least
1 otherwise.
On exit: if
JOBU='A',
U contains the
m by
m orthogonal matrix
U.
If
JOBU='S',
U contains the first
minm,n columns of
U (the left singular vectors, stored column-wise).
If
JOBU='N' or
'O',
U is not referenced.
- 9: LDU – INTEGERInput
On entry: the first dimension of the array
U as declared in the (sub)program from which F08KBF (DGESVD) is called.
Constraints:
- if JOBU='A' or 'S', LDU≥ max1,M ;
- otherwise LDU≥1.
- 10: VT(LDVT,*) – REAL (KIND=nag_wp) arrayOutput
-
Note: the second dimension of the array
VT
must be at least
max1,N if
JOBVT='A' or
'S', and at least
1 otherwise.
On exit: if
JOBVT='A',
VT contains the
n by
n orthogonal matrix
VT.
If
JOBVT='S',
VT contains the first
minm,n rows of
VT (the right singular vectors, stored row-wise).
If
JOBVT='N' or
'O',
VT is not referenced.
- 11: LDVT – INTEGERInput
On entry: the first dimension of the array
VT as declared in the (sub)program from which F08KBF (DGESVD) is called.
Constraints:
- if JOBVT='A', LDVT≥ max1,N ;
- if JOBVT='S', LDVT≥ max1,minM,N ;
- otherwise LDVT≥1.
- 12: WORK(max1,LWORK) – REAL (KIND=nag_wp) arrayWorkspace
On exit: if
INFO=0,
WORK1 returns the optimal
LWORK.
If
INFO>0,
WORK2:minM,N contains the unconverged superdiagonal elements of an upper bidiagonal matrix
B whose diagonal is in
S (not necessarily sorted).
B satisfies
A=UBVT, so it has the same singular values as
A, and singular vectors related by
U and
VT.
- 13: LWORK – INTEGERInput
On entry: the dimension of the array
WORK as declared in the (sub)program from which F08KBF (DGESVD) 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.
Constraint:
LWORK ≥ max1, 3 × minM,N + maxM,N , 5 × minM,N .
- 14: 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
If F08KBF (DGESVD) did not converge,
INFO specifies how many superdiagonals of an intermediate bidiagonal form did not converge to zero. See the description of
WORK
above for details.
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 complex analogue of this routine is
F08KPF (ZGESVD).
9 Example
This example finds the singular values and left and right singular vectors of the
6 by
4 matrix
together with approximate error bounds for the computed singular values and vectors.
The example program for
F08KDF (DGESDD) illustrates finding a singular value decomposition for the case
m≤n.
9.1 Program Text
Program Text (f08kbfe.f90)
9.2 Program Data
Program Data (f08kbfe.d)
9.3 Program Results
Program Results (f08kbfe.r)