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 and diagonal blocks; (or ) has unit diagonal blocks corresponding to the 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 blocks occur in .
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
1: – Character(1)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.
2: – IntegerInput
On entry: , the order of the matrix .
3: – Real (Kind=nag_wp) arrayInput/Output
Note: the dimension of the array ap
must be at least
On entry: the symmetric matrix , 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 .
On exit: is overwritten by details of the block diagonal matrix and the multipliers used to obtain the factor or as specified by uplo.
4: – Integer arrayOutput
On exit: details of the interchanges and the block structure of . More precisely,
if , is a pivot block and the th row and column of were interchanged with the th row and column;
if and , is a pivot block and the th row and column of were interchanged with the th row and column;
if and , is a pivot block and the th row and column of were interchanged with the th row and column.
5: – IntegerOutput
On exit: unless the routine detects an error (see Section 6).
6Error 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.
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 .
8Parallelism 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 ; if has blocks, only the upper or lower triangle is stored, as specified by uplo.
The unit diagonal elements of or and the 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: