The routine may be called by the names f07mrf, nagf_lapacklin_zhetrf or its LAPACK name zhetrf.
f07mrf factorizes a complex Hermitian matrix , using the Bunch–Kaufman diagonal pivoting method. is factorized either as 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 and diagonal blocks; (or ) has unit diagonal blocks corresponding to the 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 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: – Complex (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array a
must be at least
On entry: the Hermitian indefinite matrix .
If , the upper triangular part of must be stored and the elements of the array below the diagonal are not referenced.
If , the lower triangular part of must be stored and the elements of the array above the diagonal are not referenced.
On exit: the upper or lower triangle of is overwritten by details of the block diagonal matrix and the multipliers used to obtain the factor or as specified by uplo.
4: – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f07mrf is called.
5: – Integer arrayOutput
Note: the dimension of the array ipiv
must be at least
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.
6: – Complex (Kind=nag_wp) arrayWorkspace
On exit: if , contains the minimum value of lwork required for optimum performance.
7: – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f07mrf is called, unless , in which case a workspace query is assumed and the routine only calculates the optimal dimension of work (using the formula given below).
for optimum performance lwork should be at least , where is the block size.
8: – 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
f07mrf 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 are stored in the corresponding columns of the array a, 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 is stored explicitly (except for its unit diagonal elements which are equal to ).
The total number of real floating-point operations is approximately .
A call to f07mrf may be followed by calls to the routines: