NAG CL Interface
f08kbc (dgesvd)
1
Purpose
f08kbc computes the singular value decomposition (SVD) of a real by matrix , optionally computing the left and/or right singular vectors.
2
Specification
void |
f08kbc (Nag_OrderType order,
Nag_ComputeUType jobu,
Nag_ComputeVTType jobvt,
Integer m,
Integer n,
double a[],
Integer pda,
double s[],
double u[],
Integer pdu,
double vt[],
Integer pdvt,
double work[],
NagError *fail) |
|
The function may be called by the names: f08kbc, nag_lapackeig_dgesvd or nag_dgesvd.
3
Description
The SVD is written as
where
is an
by
matrix which is zero except for its
diagonal elements,
is an
by
orthogonal matrix, and
is an
by
orthogonal matrix. The diagonal elements of
are the singular values of
; they are real and non-negative, and are returned in descending order. The first
columns of
and
are the left and right singular vectors of
.
Note that the function returns , not .
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:
– Nag_OrderType
Input
-
On entry: the
order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by
. See
Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint:
or .
-
2:
– Nag_ComputeUType
Input
-
On entry: specifies options for computing all or part of the matrix
.
- All columns of are returned in array u.
- The first columns of (the left singular vectors) are returned in the array u.
- The first columns of (the left singular vectors) are overwritten on the array a.
- No columns of (no left singular vectors) are computed.
Constraint:
, , or .
-
3:
– Nag_ComputeVTType
Input
-
On entry: specifies options for computing all or part of the matrix
.
- All rows of are returned in the array vt.
- The first rows of (the right singular vectors) are returned in the array vt.
- The first rows of (the right singular vectors) are overwritten on the array a.
- No rows of (no right singular vectors) are computed.
Constraints:
- , , or ;
- If , jobvt cannot be .
-
4:
– Integer
Input
-
On entry: , the number of rows of the matrix .
Constraint:
.
-
5:
– Integer
Input
-
On entry: , the number of columns of the matrix .
Constraint:
.
-
6:
– double
Input/Output
-
Note: the dimension,
dim, of the array
a
must be at least
- when
;
- when
.
The
th element of the matrix
is stored in
- when ;
- when .
On entry: the by matrix .
On exit: if
,
a is overwritten with the first
columns of
(the left singular vectors, stored column-wise).
If
,
a is overwritten with the first
rows of
(the right singular vectors, stored row-wise).
If
and
, the contents of
a are destroyed.
-
7:
– Integer
Input
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
a.
Constraints:
- if ,
;
- if , .
-
8:
– double
Output
-
Note: the dimension,
dim, of the array
s
must be at least
.
On exit: the singular values of , sorted so that .
-
9:
– double
Output
-
Note: the dimension,
dim, of the array
u
must be at least
- when
;
- when
and
;
- when
and
;
- otherwise.
The
th element of the matrix
is stored in
- when ;
- when .
On exit: if
,
u contains the
by
orthogonal matrix
.
If
,
u contains the first
columns of
(the left singular vectors, stored column-wise).
If
or
,
u is not referenced.
-
10:
– Integer
Input
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
u.
Constraints:
- if ,
- if , ;
- if , ;
- otherwise ;
- if ,
- if ,
;
- if ,
;
- otherwise .
-
11:
– double
Output
-
Note: the dimension,
dim, of the array
vt
must be at least
- when
;
- when
and
;
- when
and
;
- otherwise.
The
th element of the matrix is stored in
- when ;
- when .
On exit: if
,
vt contains the
by
orthogonal matrix
.
If
,
vt contains the first
rows of
(the right singular vectors, stored row-wise).
If
or
,
vt is not referenced.
-
12:
– Integer
Input
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
vt.
Constraints:
- if ,
- if , ;
- if , ;
- otherwise ;
- if ,
- if ,
;
- if ,
;
- otherwise .
-
13:
– double
Output
-
On exit: if
NE_CONVERGENCE,
(using the notation described in
Section 3.1.4 in the Introduction to the NAG Library CL Interface) contains the unconverged superdiagonal elements of an upper bidiagonal matrix
whose diagonal is in
s (not necessarily sorted).
satisfies
, so it has the same singular values as
, and singular vectors related by
and
.
-
14:
– NagError *
Input/Output
-
The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_CONVERGENCE
-
If f08kbc did not converge, specifies how many superdiagonals of an intermediate bidiagonal form did not converge to zero.
- NE_ENUM_INT_2
-
On entry, , and .
Constraint: if , ;
if , ;
otherwise .
On entry, , and .
Constraint: if ,
;
if ,
;
otherwise .
- NE_ENUM_INT_3
-
On entry, , , and .
Constraint: if ,
;
if ,
;
otherwise .
On entry, , , and .
Constraint: if , ;
if , ;
otherwise .
- NE_INT
-
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
- NE_INT_2
-
On entry, and .
Constraint: .
On entry, and .
Constraint: .
- NE_INTERNAL_ERROR
-
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact
NAG for assistance.
See
Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library CL Interface for further information.
7
Accuracy
The computed singular value decomposition is nearly the exact singular value decomposition for a nearby matrix
, where
and
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
Parallelism and Performance
f08kbc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08kbc 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 function. Please also consult the
Users' Note for your implementation for any additional implementation-specific information.
The total number of floating-point operations is approximately proportional to when and otherwise.
The singular values are returned in descending order.
The complex analogue of this function is
f08kpc.
10
Example
This example finds the singular values and left and right singular vectors of the
by
matrix
together with approximate error bounds for the computed singular values and vectors.
The example program for
f08kdc illustrates finding a singular value decomposition for the case
.
10.1
Program Text
10.2
Program Data
10.3
Program Results