NAG Library Function Document
nag_dggsvd3 (f08vcc)
1
Purpose
nag_dggsvd3 (f08vcc) computes the generalized singular value decomposition (GSVD) of an by real matrix and a by real matrix .
2
Specification
#include <nag.h> |
#include <nagf08.h> |
void |
nag_dggsvd3 (Nag_OrderType order,
Nag_ComputeUType jobu,
Nag_ComputeVType jobv,
Nag_ComputeQType jobq,
Integer m,
Integer n,
Integer p,
Integer *k,
Integer *l,
double a[],
Integer pda,
double b[],
Integer pdb,
double alpha[],
double beta[],
double u[],
Integer pdu,
double v[],
Integer pdv,
double q[],
Integer pdq,
Integer iwork[],
NagError *fail) |
|
3
Description
Given an
by
real matrix
and a
by
real matrix
, the generalized singular value decomposition is given by
where
,
and
are orthogonal matrices. Let
be the effective numerical rank of
and
be the effective numerical rank of the matrix
, then the first
generalized singular values are infinite and the remaining
are finite.
is a
by
nonsingular upper triangular matrix,
and
are
by
and
by
‘diagonal’ matrices structured as follows:
if
,
where
and
is stored as a submatrix of
with elements
stored as
on exit.
If
,
where
and
is stored as a submatrix of
with
stored as
, and
is stored as a submatrix of
with
stored as
.
The function computes , , and, optionally, the orthogonal transformation matrices , and .
In particular, if
is an
by
nonsingular matrix, then the GSVD of
and
implicitly gives the SVD of
:
If
has orthonormal columns, then the GSVD of
and
is also equal to the CS decomposition of
and
. Furthermore, the GSVD can be used to derive the solution of the eigenvalue problem:
In some literature, the GSVD of
and
is presented in the form
where
and
are orthogonal and
is nonsingular, and
and
are ‘diagonal’. The former GSVD form can be converted to the latter form by setting
A two stage process is used to compute the GSVD of the matrix pair
. The pair is first reduced to upper triangular form by orthogonal transformations using
nag_dggsvp3 (f08vgc). The GSVD of the resulting upper triangular matrix pair is then performed by
nag_dtgsja (f08yec) which uses a variant of the Kogbetliantz algorithm (a cyclic Jacobi method).
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 (2012) Matrix Computations (4th Edition) Johns Hopkins University Press, Baltimore
5
Arguments
- 1:
– Nag_OrderTypeInput
-
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 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint:
or .
- 2:
– Nag_ComputeUTypeInput
-
On entry: if
, the orthogonal matrix
is computed.
If , is not computed.
Constraint:
or .
- 3:
– Nag_ComputeVTypeInput
-
On entry: if
, the orthogonal matrix
is computed.
If , is not computed.
Constraint:
or .
- 4:
– Nag_ComputeQTypeInput
-
On entry: if
, the orthogonal matrix
is computed.
If , is not computed.
Constraint:
or .
- 5:
– IntegerInput
-
On entry: , the number of rows of the matrix .
Constraint:
.
- 6:
– IntegerInput
-
On entry: , the number of columns of the matrices and .
Constraint:
.
- 7:
– IntegerInput
-
On entry: , the number of rows of the matrix .
Constraint:
.
- 8:
– Integer *Output
- 9:
– Integer *Output
-
On exit:
k and
l specify the dimension of the subblocks
and
as described in
Section 3;
is the effective numerical rank of
.
- 10:
– doubleInput/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: contains the triangular matrix
, or part of
. See
Section 3 for details.
- 11:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
a.
Constraints:
- if ,
;
- if , .
- 12:
– doubleInput/Output
-
Note: the dimension,
dim, of the array
b
must be at least
- when
;
- when
.
The
th element of the matrix
is stored in
- when ;
- when .
On entry: the by matrix .
On exit: contains the triangular matrix
if
. See
Section 3 for details.
- 13:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
b.
Constraints:
- if ,
;
- if , .
- 14:
– doubleOutput
-
On exit: see the description of
beta.
- 15:
– doubleOutput
-
On exit:
alpha and
beta contain the generalized singular value pairs of
and
,
and
;
- ,
- ,
and if
,
- ,
- ,
or if
,
- ,
- ,
- ,
- , and
- ,
- .
The notation above refers to consecutive elements
, for .
- 16:
– doubleOutput
-
Note: the dimension,
dim, of the array
u
must be at least
- when
;
- otherwise.
The
th element of the matrix
is stored in
- when ;
- when .
On exit: if
,
u contains the
by
orthogonal matrix
.
If
,
u is not referenced.
- 17:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
u.
Constraints:
- if , ;
- otherwise .
- 18:
– doubleOutput
-
Note: the dimension,
dim, of the array
v
must be at least
- when
;
- otherwise.
The
th element of the matrix
is stored in
- when ;
- when .
On exit: if
,
v contains the
by
orthogonal matrix
.
If
,
v is not referenced.
- 19:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
v.
Constraints:
- if , ;
- otherwise .
- 20:
– doubleOutput
-
Note: the dimension,
dim, of the array
q
must be at least
- when
;
- otherwise.
The
th element of the matrix
is stored in
- when ;
- when .
On exit: if
,
q contains the
by
orthogonal matrix
.
If
,
q is not referenced.
- 21:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
q.
Constraints:
- if , ;
- otherwise .
- 22:
– IntegerOutput
-
On exit: stores the sorting information. More precisely, if
is the ordered set of indices of
alpha containing
(denote as
, see
beta), then the corresponding elements
contain the swap pivots,
, that sorts
such that
is in descending numerical order.
The following pseudocode sorts the set :
- 23:
– NagError *Input/Output
-
The NAG error argument (see
Section 3.7 in How to Use the NAG Library and its Documentation).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_CONVERGENCE
-
The Jacobi-type procedure failed to converge.
- NE_ENUM_INT_2
-
On entry, , and .
Constraint: if , ;
otherwise .
On entry, , and .
Constraint: if , ;
otherwise .
On entry, , and .
Constraint: if , ;
otherwise .
- NE_INT
-
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
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: .
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
7
Accuracy
The computed generalized singular value decomposition is nearly the exact generalized singular value decomposition for nearby matrices
and
, where
and
is the
machine precision. See Section 4.12 of
Anderson et al. (1999) for further details.
8
Parallelism and Performance
nag_dggsvd3 (f08vcc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_dggsvd3 (f08vcc) 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.
This function replaces the deprecated function
nag_dggsvd (f08vac) which used an unblocked algorithm and therefore did not make best use of level 3 BLAS functions.
The complex analogue of this function is
nag_zggsvd3 (f08vqc).
10
Example
This example finds the generalized singular value decomposition
where
together with estimates for the condition number of
and the error bound for the computed generalized singular values.
The example program assumes that , and would need slight modification if this is not the case.
10.1
Program Text
Program Text (f08vcce.c)
10.2
Program Data
Program Data (f08vcce.d)
10.3
Program Results
Program Results (f08vcce.r)