NAG Library Routine Document
f07prf (zhptrf) computes the Bunch–Kaufman factorization of a complex Hermitian indefinite matrix, using packed storage.
|Integer, Intent (In)||:: ||n|
|Integer, Intent (Out)||:: ||ipiv(n), info|
|Complex (Kind=nag_wp), Intent (Inout)||:: ||ap(*)|
|Character (1), Intent (In)||:: ||uplo|
The routine may be called by its
f07prf (zhptrf) factorizes a complex Hermitian 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 an Hermitian 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 keeping the matrix Hermitian.
This method is suitable for Hermitian 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
- 1: – Character(1)Input
: 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: – Complex (Kind=nag_wp) arrayInput/Output
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
- 4: – Integer arrayOutput
: 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.
- 5: – IntegerOutput
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
f07prf (zhptrf) 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
unit diagonal blocks are not stored. The remaining elements of
are stored in the corresponding columns of the array ap
, but additional row interchanges must be applied to recover
explicitly (this is seldom necessary). If
(as is the case when
is positive definite), then
are stored explicitly in packed form (except for their unit diagonal elements which are equal to
The total number of real floating-point operations is approximately .
A call to f07prf (zhptrf)
may be followed by calls to the routines:
The real analogue of this routine is f07pdf (dsptrf)
This example computes the Bunch–Kaufman factorization of the matrix
using packed storage.
Program Text (f07prfe.f90)
Program Data (f07prfe.d)
Program Results (f07prfe.r)