hide long namesshow long names
hide short namesshow short names
Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_zpptrf (f07gr)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_zpptrf (f07gr) computes the Cholesky factorization of a complex Hermitian positive definite matrix, using packed storage.

Syntax

[ap, info] = f07gr(uplo, n, ap)
[ap, info] = nag_lapack_zpptrf(uplo, n, ap)

Description

nag_lapack_zpptrf (f07gr) 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.

References

Demmel J W (1989) On floating-point errors in Cholesky LAPACK Working Note No. 14 University of Tennessee, Knoxville http://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

Parameters

Compulsory Input Parameters

1:     uplo – string (length ≥ 1)
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 int64int32nag_int scalar
n, the order of the matrix A.
Constraint: n0.
3:     ap: – complex array
The dimension of the array ap must be at least max1,n×n+1/2
The n by n Hermitian matrix A, packed by columns.
More precisely,
  • if uplo='U', the upper triangle of A must be stored with element Aij in api+jj-1/2 for ij;
  • if uplo='L', the lower triangle of A must be stored with element Aij in api+2n-jj-1/2 for ij.

Optional Input Parameters

None.

Output Parameters

1:     ap: – complex array
The dimension of the array ap will be max1,n×n+1/2
If info=0, the factor U or L from the Cholesky factorization A=UHU or A=LLH, in the same storage format as A.
2:     info int64int32nag_int scalar
info=0 unless the function detects an error (see Error Indicators and Warnings).

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 _ 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_lapack_zhptrf (f07pr) instead.

Accuracy

If uplo='U', 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='L', a similar statement holds for the computed factor L. It follows that eijcnεaiiajj.

Further Comments

The total number of real floating-point operations is approximately 43n3.
A call to nag_lapack_zpptrf (f07gr) may be followed by calls to the functions:
The real analogue of this function is nag_lapack_dpptrf (f07gd).

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.
function f07gr_example


fprintf('f07gr example results\n\n');

uplo = 'L';
n = int64(4);
ap = [3.23 + 0i    1.51 + 1.92i    1.90 - 0.84i    0.42 - 2.50i ...
                   3.58 + 0i      -0.23 - 1.11i   -1.18 - 1.37i ...
                                   4.09 + 0.00i    2.33 + 0.14i ...
                                                   4.29 + 0.00i];

[L, info] = f07gr( ...
                     uplo, n, ap);

[ifail] = x04dc( ...
                 uplo, 'Non-unit', n, L, 'factor');


f07gr example results

 factor
             1          2          3          4
 1      1.7972
        0.0000

 2      0.8402     1.3164
        1.0683     0.0000

 3      1.0572    -0.4702     1.5604
       -0.4674     0.3131     0.0000

 4      0.2337     0.0834     0.9360     0.6603
       -1.3910     0.0368     0.9900     0.0000

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

© The Numerical Algorithms Group Ltd, Oxford, UK. 2009–2015