NAG FL Interface
f07frf computes the Cholesky factorization of a complex Hermitian positive definite matrix.
|Integer, Intent (In)
|Integer, Intent (Out)
|Complex (Kind=nag_wp), Intent (Inout)
|Character (1), Intent (In)
|C Header Interface
f07frf_ (const char *uplo, const Integer *n, Complex a, const Integer *lda, Integer *info, const Charlen length_uplo)
The routine may be called by the names f07frf, nagf_lapacklin_zpotrf or its LAPACK name zpotrf.
f07frf forms the Cholesky factorization of a complex Hermitian positive definite matrix either as if or if , where is an upper triangular matrix and is lower triangular.
Demmel J W (1989) On floating-point errors in Cholesky LAPACK Working Note No. 14
University of Tennessee, Knoxville https://www.netlib.org/lapack/lawnspdf/lawn14.pdf
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 .
– Complex (Kind=nag_wp) array
the second dimension of the array a
must be at least
Hermitian positive definite 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.
: the upper or lower triangle of
is overwritten by the Cholesky factor
as specified by uplo
: the first dimension of the array a
as declared in the (sub)program from which f07frf
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.
The leading minor of order
is not positive definite
and the factorization could not be completed. Hence
is not positive definite. This may indicate an error in forming the
. To factorize a Hermitian matrix which is not
positive definite, call f07mrf
, the computed factor
is the exact factor of a perturbed matrix
is a modest linear function of
is the machine precision
. If ,
a similar statement holds for the computed factor
. It follows that
Parallelism and Performance
f07frf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07frf 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 total number of real floating-point operations is approximately .
A call to f07frf
may be followed by calls to the routines:
- f07fsf to solve ;
- f07fuf to estimate the condition number of ;
- f07fwf to compute the inverse of .
The real analogue of this routine is f07fdf
This example computes the Cholesky factorization of the matrix