NAG FL Interface
f07pdf computes the Bunch–Kaufman factorization of a real symmetric indefinite matrix, using packed storage.
|Integer, Intent (In)
|Integer, Intent (Out)
|Real (Kind=nag_wp), Intent (Inout)
|Character (1), Intent (In)
The routine may be called by the names f07pdf, nagf_lapacklin_dsptrf or its LAPACK name dsptrf.
f07pdf 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 .
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
: 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.
On entry: , the order of the matrix .
– Real (Kind=nag_wp) array
the dimension of the array ap
must be at least
, packed by columns.
- if , the upper triangle of must be stored with element in for ;
- if , the lower triangle of must be stored with element in for .
is overwritten by details of the block diagonal matrix
and the multipliers used to obtain the factor
as specified by uplo
– Integer array
: 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.
unless the routine detects an error (see Section 6
Error Indicators and Warnings
If , argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
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.
, the computed factors
are the exact factors of a perturbed matrix
is a modest linear function of
is the machine precision
If , a similar statement holds for the computed factors and .
Parallelism and Performance
f07pdf 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.
The elements of
overwrite the corresponding elements of
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 f07pdf
may be followed by calls to the routines:
- f07pef to solve ;
- f07pgf to estimate the condition number of ;
- f07pjf to compute the inverse of .
The complex analogues of this routine are f07prf
for Hermitian matrices and f07qrf
for symmetric matrices.
This example computes the Bunch–Kaufman factorization of the matrix
using packed storage.