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NAG Toolbox: nag_lapack_dpotrf (f07fd)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_dpotrf (f07fd) computes the Cholesky factorization of a real symmetric positive definite matrix.

Syntax

[a, info] = f07fd(uplo, a, 'n', n)
[a, info] = nag_lapack_dpotrf(uplo, a, 'n', n)

Description

nag_lapack_dpotrf (f07fd) forms the Cholesky factorization of a real symmetric positive definite matrix A either as A=UTU if uplo='U' or A=LLT if uplo='L', where U is an upper triangular matrix and L is lower triangular.

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 UTU, where U is upper triangular.
uplo='L'
The lower triangular part of A is stored and A is factorized as LLT, where L is lower triangular.
Constraint: uplo='U' or 'L'.
2:     alda: – double array
The first dimension of the array a must be at least max1,n.
The second dimension of the array a must be at least max1,n.
The n by n symmetric positive definite matrix A.
  • If uplo='U', the upper triangular part of a must be stored and the elements of the array below the diagonal are not referenced.
  • If uplo='L', the lower triangular part of a must be stored and the elements of the array above the diagonal are not referenced.

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the first dimension of the array a and the second dimension of the array a.
n, the order of the matrix A.
Constraint: n0.

Output Parameters

1:     alda: – double array
The first dimension of the array a will be max1,n.
The second dimension of the array a will be max1,n.
The upper or lower triangle of A stores the Cholesky factor U or L as specified by uplo.
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 symmetric matrix which is not positive definite, call nag_lapack_dsytrf (f07md) instead.

Accuracy

If uplo='U', the computed factor U is the exact factor of a perturbed matrix A+E, where
EcnεUTU ,  
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 floating-point operations is approximately 13n3.
A call to nag_lapack_dpotrf (f07fd) may be followed by calls to the functions:
The complex analogue of this function is nag_lapack_zpotrf (f07fr).

Example

This example computes the Cholesky factorization of the matrix A, where
A= 4.16 -3.12 0.56 -0.10 -3.12 5.03 -0.83 1.18 0.56 -0.83 0.76 0.34 -0.10 1.18 0.34 1.18 .  
function f07fd_example


fprintf('f07fd example results\n\n');

% Lower triangular part of symmetric matrix A
uplo = 'Lower';
a = [ 4.16,  0,    0,    0;
     -3.12,  5.03, 0,    0;
      0.56, -0.83, 0.76, 0;
     -0.10,  1.18, 0.34, 1.18];

[L, info] = f07fd( ...
                   uplo, a);

[ifail] = x04ca( ...
                 uplo, 'Non-unit', L, 'Cholesky factor');


f07fd example results

 Cholesky factor
             1          2          3          4
 1      2.0396
 2     -1.5297     1.6401
 3      0.2746    -0.2500     0.7887
 4     -0.0490     0.6737     0.6617     0.5347

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

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