NAG FL Interface
f08mdf (dbdsdc)
1
Purpose
f08mdf computes the singular values and, optionally, the left and right singular vectors of a real by (upper or lower) bidiagonal matrix .
2
Specification
Fortran Interface
Subroutine f08mdf ( |
uplo, compq, n, d, e, u, ldu, vt, ldvt, q, iq, work, iwork, info) |
Integer, Intent (In) |
:: |
n, ldu, ldvt |
Integer, Intent (Inout) |
:: |
iq(*) |
Integer, Intent (Out) |
:: |
iwork(8*n), info |
Real (Kind=nag_wp), Intent (Inout) |
:: |
d(*), e(*), u(ldu,*), vt(ldvt,*), q(*), work(*) |
Character (1), Intent (In) |
:: |
uplo, compq |
|
C Header Interface
#include <nag.h>
void |
f08mdf_ (const char *uplo, const char *compq, const Integer *n, double d[], double e[], double u[], const Integer *ldu, double vt[], const Integer *ldvt, double q[], Integer iq[], double work[], Integer iwork[], Integer *info, const Charlen length_uplo, const Charlen length_compq) |
|
C++ Header Interface
#include <nag.h> extern "C" {
void |
f08mdf_ (const char *uplo, const char *compq, const Integer &n, double d[], double e[], double u[], const Integer &ldu, double vt[], const Integer &ldvt, double q[], Integer iq[], double work[], Integer iwork[], Integer &info, const Charlen length_uplo, const Charlen length_compq) |
}
|
The routine may be called by the names f08mdf, nagf_lapackeig_dbdsdc or its LAPACK name dbdsdc.
3
Description
f08mdf computes the singular value decomposition (SVD) of the (upper or lower) bidiagonal matrix
as
where
is a diagonal matrix with non-negative diagonal elements
, such that
and
and
are orthogonal matrices. The diagonal elements of
are the singular values of
and the columns of
and
are respectively the corresponding left and right singular vectors of
.
When only singular values are required the routine uses the algorithm, but when singular vectors are required a divide and conquer method is used. The singular values can optionally be returned in compact form, although currently no routine is available to apply or when stored in compact form.
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
https://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
5
Arguments
-
1:
– Character(1)
Input
-
On entry: indicates whether
is upper or lower bidiagonal.
- is upper bidiagonal.
- is lower bidiagonal.
Constraint:
or .
-
2:
– Character(1)
Input
-
On entry: specifies whether singular vectors are to be computed.
- Compute singular values only.
- Compute singular values and compute singular vectors in compact form.
- Compute singular values and singular vectors.
Constraint:
, or .
-
3:
– Integer
Input
-
On entry: , the order of the matrix .
Constraint:
.
-
4:
– Real (Kind=nag_wp) array
Input/Output
-
Note: the dimension of the array
d
must be at least
.
On entry: the diagonal elements of the bidiagonal matrix .
On exit: if , the singular values of .
-
5:
– Real (Kind=nag_wp) array
Input/Output
-
Note: the dimension of the array
e
must be at least
.
On entry: the off-diagonal elements of the bidiagonal matrix .
On exit: the contents of
e are destroyed.
-
6:
– Real (Kind=nag_wp) array
Output
-
Note: the second dimension of the array
u
must be at least
if
.
On exit: if
, then if
,
u contains the left singular vectors of the bidiagonal matrix
.
If
,
u is not referenced.
-
7:
– Integer
Input
-
On entry: the first dimension of the array
u as declared in the (sub)program from which
f08mdf is called.
Constraints:
- if , ;
- otherwise .
-
8:
– Real (Kind=nag_wp) array
Output
-
Note: the second dimension of the array
vt
must be at least
if
.
On exit: if
, then if
, the rows of
vt contain the right singular vectors of the bidiagonal matrix
.
If
,
vt is not referenced.
-
9:
– Integer
Input
-
On entry: the first dimension of the array
vt as declared in the (sub)program from which
f08mdf is called.
Constraints:
- if , ;
- otherwise .
-
10:
– Real (Kind=nag_wp) array
Output
-
Note: the dimension of the array
q
must be at least
.
On exit: if
, then if
,
q and
iq contain the left and right singular vectors in a compact form, requiring
space instead of
. In particular,
q contains all the real data in the first
elements of
q, where
is equal to the maximum size of the subproblems at the bottom of the computation tree (usually about
).
If
,
q is not referenced.
-
11:
– Integer array
Output
-
Note: the dimension of the array
iq
must be at least
.
On exit: if
, then if
,
q and
iq contain the left and right singular vectors in a compact form, requiring
space instead of
. In particular,
iq contains all integer data in the first
elements of
iq, where
is equal to the maximum size of the subproblems at the bottom of the computation tree (usually about
).
If
,
iq is not referenced.
-
12:
– Real (Kind=nag_wp) array
Workspace
-
Note: the dimension of the array
work
must be at least
if
,
if
,
if
, and at least
otherwise.
-
13:
– Integer array
Workspace
-
-
14:
– Integer
Output
On exit:
unless the routine detects an error (see
Section 6).
6
Error Indicators and Warnings
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
-
The algorithm failed to compute a singular value. The update process of divide-and-conquer failed.
7
Accuracy
Each computed singular value of
is accurate to nearly full relative precision, no matter how tiny the singular value. The
th computed singular value,
, satisfies the bound
where
is the
machine precision and
is a modest function of
.
For bounds on the computed singular values, see Section 4.9.1 of
Anderson et al. (1999). See also
f08flf.
8
Parallelism and Performance
f08mdf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08mdf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the
X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the
Users' Note for your implementation for any additional implementation-specific information.
If only singular values are required, the total number of floating-point operations is approximately proportional to
. When singular vectors are required the number of operations is bounded above by approximately the same number of operations as
f08mef, but for large matrices
f08mdf is usually much faster.
There is no complex analogue of f08mdf.
10
Example
This example computes the singular value decomposition of the upper bidiagonal matrix
10.1
Program Text
10.2
Program Data
10.3
Program Results