NAG FL Interface
f07grf (zpptrf)

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1 Purpose

f07grf computes the Cholesky factorization of a complex Hermitian positive definite matrix, using packed storage.

2 Specification

Fortran Interface
Subroutine f07grf ( uplo, n, ap, info)
Integer, Intent (In) :: n
Integer, Intent (Out) :: info
Complex (Kind=nag_wp), Intent (Inout) :: ap(*)
Character (1), Intent (In) :: uplo
C Header Interface
#include <nag.h>
void  f07grf_ (const char *uplo, const Integer *n, Complex ap[], Integer *info, const Charlen length_uplo)
The routine may be called by the names f07grf, nagf_lapacklin_zpptrf or its LAPACK name zpptrf.

3 Description

f07grf forms the Cholesky factorization of a complex Hermitian positive definite matrix A either as A=UHU if uplo='U' or A=LLH if uplo='L', where U is an upper triangular matrix and L is lower triangular, using packed storage.

4 References

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

5 Arguments

1: uplo Character(1) Input
On entry: specifies whether the upper or lower triangular part of A is stored and how A is to be factorized.
uplo='U'
The upper triangular part of A is stored and A is factorized as UHU, where U is upper triangular.
uplo='L'
The lower triangular part of A is stored and A is factorized as LLH, where L is lower triangular.
Constraint: uplo='U' or 'L'.
2: n Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
3: ap(*) Complex (Kind=nag_wp) array Input/Output
Note: the dimension of the array ap must be at least max(1,n×(n+1)/2).
On entry: the n×n Hermitian matrix A, packed by columns.
More precisely,
  • if uplo='U', the upper triangle of A must be stored with element Aij in ap(i+j(j-1)/2) for ij;
  • if uplo='L', the lower triangle of A must be stored with element Aij in ap(i+(2n-j)(j-1)/2) for ij.
On exit: if info=0, the factor U or L from the Cholesky factorization A=UHU or A=LLH, in the same storage format as A.
4: info Integer Output
On exit: info=0 unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

info<0
If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
info>0
The leading minor of order value is not positive definite and the factorization could not be completed. Hence A itself is not positive definite. This may indicate an error in forming the matrix A. To factorize a Hermitian matrix which is not positive definite, call f07prf instead.

7 Accuracy

If uplo='U', the computed factor U is the exact factor of a perturbed matrix A+E, where
|E|c(n)ε|UH||U| ,  
c(n) is a modest linear function of n, and ε is the machine precision.
If uplo='L', a similar statement holds for the computed factor L. It follows that |eij|c(n)εaiiajj.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
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.

9 Further Comments

The total number of real floating-point operations is approximately 43n3.
A call to f07grf may be followed by calls to the routines:
The real analogue of this routine is f07gdf.

10 Example

This example computes the Cholesky factorization of the matrix A, where
A= ( 3.23+0.00i 1.51-1.92i 1.90+0.84i 0.42+2.50i 1.51+1.92i 3.58+0.00i -0.23+1.11i -1.18+1.37i 1.90-0.84i -0.23-1.11i 4.09+0.00i 2.33-0.14i 0.42-2.50i -1.18-1.37i 2.33+0.14i 4.29+0.00i ) .  
using packed storage.

10.1 Program Text

Program Text (f07grfe.f90)

10.2 Program Data

Program Data (f07grfe.d)

10.3 Program Results

Program Results (f07grfe.r)