nag_zpotrf (f07frc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG 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 Hermitian 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  Parallelism and Performance

nag_zpotrf (f07frc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_zpotrf (f07frc) 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 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 nag_zpotrf (f07frc) may be followed by calls to the functions:
The real analogue of this function is nag_dpotrf (f07fdc).

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 .

10.1  Program Text

Program Text (f07frce.c)

10.2  Program Data

Program Data (f07frce.d)

10.3  Program Results

Program Results (f07frce.r)


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

© The Numerical Algorithms Group Ltd, Oxford, UK. 2014