NAG FL Interface
f07grf computes the Cholesky factorization of a complex Hermitian positive definite matrix, using packed storage.
|Integer, Intent (In)
|Integer, Intent (Out)
|Complex (Kind=nag_wp), Intent (Inout)
|Character (1), Intent (In)
The routine may be called by the names f07grf, nagf_lapacklin_zpptrf or its LAPACK name zpptrf.
f07grf 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, using packed storage.
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 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 .
On exit: if , the factor or from the Cholesky factorization or , in the same storage format as .
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 f07prf
, 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
f07grf 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 f07grf
may be followed by calls to the routines:
- f07gsf to solve ;
- f07guf to estimate the condition number of ;
- f07gwf to compute the inverse of .
The real analogue of this routine is f07gdf
This example computes the Cholesky factorization of the matrix
using packed storage.