nag_zpptrf (f07grc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG Library Manual

NAG Library Function Document

nag_zpptrf (f07grc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_zpptrf (f07grc) computes the Cholesky factorization of a complex Hermitian positive definite matrix, using packed storage.

2  Specification

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

3  Description

nag_zpptrf (f07grc) 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, using packed storage.

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 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.
The upper triangular part of A is stored and A is factorized as UHU, where U is upper triangular.
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:     ap[dim]ComplexInput/Output
Note: the dimension, dim, of the array ap must be at least max1,n×n+1/2.
On entry: the n by n Hermitian matrix A, packed by rows or columns.
The storage of elements Aij depends on the order and uplo arguments as follows:
  • if order=Nag_ColMajor and uplo=Nag_Upper,
              Aij is stored in ap[j-1×j/2+i-1], for ij;
  • if order=Nag_ColMajor and uplo=Nag_Lower,
              Aij is stored in ap[2n-j×j-1/2+i-1], for ij;
  • if order=Nag_RowMajor and uplo=Nag_Upper,
              Aij is stored in ap[2n-i×i-1/2+j-1], for ij;
  • if order=Nag_RowMajor and uplo=Nag_Lower,
              Aij is stored in ap[i-1×i/2+j-1], for ij.
On exit: if fail.code= NE_NOERROR, the factor U or L from the Cholesky factorization A=UHU or A=LLH, in the same storage format as A.
5:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

On entry, argument value had an illegal value.
On entry, n=value.
Constraint: n0.
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.
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_zhptrf (f07prc) 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_zpptrf (f07grc) is not threaded by NAG in any implementation.
nag_zpptrf (f07grc) 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_zpptrf (f07grc) may be followed by calls to the functions:
The real analogue of this function is nag_dpptrf (f07gdc).

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 (f07grce.c)

10.2  Program Data

Program Data (f07grce.d)

10.3  Program Results

Program Results (f07grce.r)

nag_zpptrf (f07grc) (PDF version)
f07 Chapter Contents
f07 Chapter Introduction
NAG Library Manual

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