nag_zpotrf (f07frc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_zpotrf (f07frc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_zpotrf (f07frc) computes the Cholesky factorization of a complex Hermitian positive definite matrix.

2  Specification

#include <nag.h>
#include <nagf07.h>
void  nag_zpotrf (Nag_OrderType order, Nag_UploType uplo, Integer n, Complex a[], Integer pda, NagError *fail)

3  Description

nag_zpotrf (f07frc) forms the Cholesky factorization of a complex Hermitian positive definite matrix A either as A=UHU if uplo=Nag_Upper or A=LLH if uplo=Nag_Lower, where U is an upper triangular matrix and L is lower triangular.

4  References

Demmel J W (1989) On floating-point errors in Cholesky LAPACK Working Note No. 14 University of Tennessee, Knoxville
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5  Arguments

1:     orderNag_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 order=Nag_RowMajor. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     uploNag_UploTypeInput
On entry: specifies whether the upper or lower triangular part of A is stored and how A is to be factorized.
uplo=Nag_Upper
The upper triangular part of A is stored and A is factorized as UHU, where U is upper triangular.
uplo=Nag_Lower
The lower triangular part of A is stored and A is factorized as LLH, where L is lower triangular.
Constraint: uplo=Nag_Upper or Nag_Lower.
3:     nIntegerInput
On entry: n, the order of the matrix A.
Constraint: n0.
4:     a[dim]ComplexInput/Output
Note: the dimension, dim, of the array a must be at least max1,pda×n.
On entry: the n by n Hermitian positive definite matrix A.
If order=Nag_ColMajor, Aij is stored in a[j-1×pda+i-1].
If order=Nag_RowMajor, Aij is stored in a[i-1×pda+j-1].
If uplo=Nag_Upper, the upper triangular part of A must be stored and the elements of the array below the diagonal are not referenced.
If uplo=Nag_Lower, the lower triangular part of A must be stored and the elements of the array above the diagonal are not referenced.
On exit: the upper or lower triangle of A is overwritten by the Cholesky factor U or L as specified by uplo.
5:     pdaIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) of the matrix A in the array a.
Constraint: pdamax1,n.
6:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, n=value.
Constraint: n0.
On entry, pda=value.
Constraint: pda>0.
NE_INT_2
On entry, pda=value and n=value.
Constraint: pdamax1,n.
NE_INTERNAL_ERROR
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.
NE_POS_DEF
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 matrix which is not positive definite, call nag_zhetrf (f07mrc) instead.

7  Accuracy

If uplo=Nag_Upper, the computed factor U is the exact factor of a perturbed matrix A+E, where
EcnεUHU ,
cn is a modest linear function of n, and ε is the machine precision. If uplo=Nag_Lower, a similar statement holds for the computed factor L. It follows that eijcnεaiiajj.

8  Further Comments

The total number of real floating point operations is approximately 43n3.
A call to nag_zpotrf (f07frc) may be followed by calls to the functions:
The real analogue of this function is nag_dpotrf (f07fdc).

9  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 .

9.1  Program Text

Program Text (f07frce.c)

9.2  Program Data

Program Data (f07frce.d)

9.3  Program Results

Program Results (f07frce.r)


nag_zpotrf (f07frc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012