The function may be called by the names: f07mrc, nag_lapacklin_zhetrf or nag_zhetrf.
f07mrc 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: – Nag_OrderTypeInput
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.
2: – Nag_UploTypeInput
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.
3: – IntegerInput
On entry: , the order of the matrix .
4: – ComplexInput/Output
Note: the dimension, dim, of the array a
must be at least
On entry: the Hermitian indefinite matrix .
If , is stored in .
If , is stored in .
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.
5: – IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix in the array
6: – IntegerOutput
Note: the dimension, dim, 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.
7: – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument had an illegal value.
On entry, .
On entry, . Constraint: .
On entry, and .
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.
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.
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
Background information to multithreading can be found in the Multithreading documentation.
f07mrc 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 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 f07mrc may be followed by calls to the functions: