NAG CL Interface
f07pdc (dsptrf)
1
Purpose
f07pdc computes the Bunch–Kaufman factorization of a real symmetric indefinite matrix, using packed storage.
2
Specification
void |
f07pdc (Nag_OrderType order,
Nag_UploType uplo,
Integer n,
double ap[],
Integer ipiv[],
NagError *fail) |
|
The function may be called by the names: f07pdc, nag_lapacklin_dsptrf or nag_dsptrf.
3
Description
f07pdc factorizes a real symmetric matrix , using the Bunch–Kaufman diagonal pivoting method and packed storage. is factorized as either if or if , where is a permutation matrix, (or ) is a unit upper (or lower) triangular matrix and is a symmetric block diagonal matrix with by and by diagonal blocks; (or ) has by unit diagonal blocks corresponding to the by blocks of . Row and column interchanges are performed to ensure numerical stability while preserving symmetry.
This method is suitable for symmetric matrices which are not known to be positive definite. If is in fact positive definite, no interchanges are performed and no by blocks occur in .
4
References
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_UploType
Input
-
On entry: specifies whether the upper or lower triangular part of
is stored and how
is to be factorized.
- The upper triangular part of is stored and is factorized as , where is upper triangular.
- The lower triangular part of is stored and is factorized as , where is lower triangular.
Constraint:
or .
-
3:
– Integer
Input
-
On entry: , the order of the matrix .
Constraint:
.
-
4:
– double
Input/Output
-
Note: the dimension,
dim, of the array
ap
must be at least
.
On entry: the
by
symmetric matrix
, packed by rows or columns.
The storage of elements
depends on the
order and
uplo arguments as follows:
- if and ,
- is stored in , for ;
- if and ,
- is stored in , for ;
- if and ,
- is stored in , for ;
- if and ,
- is stored in , for .
On exit:
is overwritten by details of the block diagonal matrix
and the multipliers used to obtain the factor
or
as specified by
uplo.
-
5:
– Integer
Output
-
On exit: details of the interchanges and the block structure of
. More precisely,
- if , is a by pivot block and the th row and column of were interchanged with the th row and column;
- if and , is a by pivot block and the th row and column of were interchanged with the th row and column;
- if and , is a by pivot block and the th row and column of were interchanged with the th row and column.
-
6:
– 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_INT
-
On entry, .
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.
- NE_SINGULAR
-
Element of the diagonal is exactly zero.
The factorization has been completed, but the block diagonal matrix
is exactly singular, and division by zero will occur if it is
used to solve a system of equations.
7
Accuracy
If
, the computed factors
and
are the exact factors of a perturbed matrix
, where
is a modest linear function of
, and
is the
machine precision.
If , a similar statement holds for the computed factors and .
8
Parallelism and Performance
f07pdc 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 elements of
overwrite the corresponding elements of
; if
has
by
blocks, only the upper or lower triangle is stored, as specified by
uplo.
The unit diagonal elements of or and the by unit diagonal blocks are not stored. The remaining elements of or overwrite elements in the corresponding columns of , but additional row interchanges must be applied to recover or explicitly (this is seldom necessary). If , for (as is the case when is positive definite), then or are stored explicitly in packed form (except for their unit diagonal elements which are equal to ).
The total number of floating-point operations is approximately .
A call to
f07pdc may be followed by calls to the functions:
- f07pec to solve ;
- f07pgc to estimate the condition number of ;
- f07pjc to compute the inverse of .
The complex analogues of this function are
f07prc for Hermitian matrices and
f07qrc for symmetric matrices.
10
Example
This example computes the Bunch–Kaufman factorization of the matrix
, where
using packed storage.
10.1
Program Text
10.2
Program Data
10.3
Program Results