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NAG Toolbox: nag_lapack_dpptrf (f07gd)

Purpose

nag_lapack_dpptrf (f07gd) computes the Cholesky factorization of a real symmetric positive definite matrix, using packed storage.

Syntax

[ap, info] = f07gd(uplo, n, ap)
[ap, info] = nag_lapack_dpptrf(uplo, n, ap)

Description

nag_lapack_dpptrf (f07gd) forms the Cholesky factorization of a real symmetric positive definite matrix AA either as A = UTUA=UTU if uplo = 'U'uplo='U' or A = LLTA=LLT if uplo = 'L'uplo='L', where UU is an upper triangular matrix and LL 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
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 AA is stored and how AA is to be factorized.
uplo = 'U'uplo='U'
The upper triangular part of AA is stored and AA is factorized as UTUUTU, where UU is upper triangular.
uplo = 'L'uplo='L'
The lower triangular part of AA is stored and AA is factorized as LLTLLT, where LL is lower triangular.
Constraint: uplo = 'U'uplo='U' or 'L''L'.
2:     n – int64int32nag_int scalar
nn, the order of the matrix AA.
Constraint: n0n0.
3:     ap( : :) – double array
Note: the dimension of the array ap must be at least max (1,n × (n + 1) / 2)max(1,n×(n+1)/2).
The nn by nn symmetric matrix AA, packed by columns.
More precisely,
  • if uplo = 'U'uplo='U', the upper triangle of AA must be stored with element AijAij in ap(i + j(j1) / 2)api+j(j-1)/2 for ijij;
  • if uplo = 'L'uplo='L', the lower triangle of AA must be stored with element AijAij in ap(i + (2nj)(j1) / 2)api+(2n-j)(j-1)/2 for ijij.

Optional Input Parameters

None.

Input Parameters Omitted from the MATLAB Interface

None.

Output Parameters

1:     ap( : :) – double array
Note: the dimension of the array ap must be at least max (1,n × (n + 1) / 2)max(1,n×(n+1)/2).
If INFO = 0INFO=0, the factor UU or LL from the Cholesky factorization A = UTUA=UTU or A = LLTA=LLT, in the same storage format as AA.
2:     info – int64int32nag_int scalar
info = 0info=0 unless the function detects an error (see Section [Error Indicators and Warnings]).

Error Indicators and Warnings

  info = iinfo=-i
If info = iinfo=-i, parameter ii had an illegal value on entry. The parameters are numbered as follows:
1: uplo, 2: n, 3: ap, 4: info.
  INFO > 0INFO>0
If info = iinfo=i, the leading minor of order ii is not positive definite and the factorization could not be completed. Hence AA itself is not positive definite. This may indicate an error in forming the matrix AA. To factorize a symmetric matrix which is not positive definite, call nag_lapack_dsptrf (f07pd) instead.

Accuracy

If uplo = 'U'uplo='U', the computed factor UU is the exact factor of a perturbed matrix A + EA+E, where
|E|c(n)ε|UT||U| ,
|E|c(n)ε|UT||U| ,
c(n)c(n) is a modest linear function of nn, and εε is the machine precision.
If uplo = 'L'uplo='L', a similar statement holds for the computed factor LL. It follows that |eij|c(n)ε×sqrt(aiiajj)|eij|c(n)εaiiajj.

Further Comments

The total number of floating point operations is approximately (1/3)n313n3.
A call to nag_lapack_dpptrf (f07gd) may be followed by calls to the functions:
The complex analogue of this function is nag_lapack_zpptrf (f07gr).

Example

function nag_lapack_dpptrf_example
uplo = 'L';
n = int64(4);
ap = [4.16;
     -3.12;
     0.56;
     -0.1;
     5.03;
     -0.83;
     1.18;
     0.76;
     0.34;
     1.18];
[apOut, info] = nag_lapack_dpptrf(uplo, n, ap)
 

apOut =

    2.0396
   -1.5297
    0.2746
   -0.0490
    1.6401
   -0.2500
    0.6737
    0.7887
    0.6617
    0.5347


info =

                    0


function f07gd_example
uplo = 'L';
n = int64(4);
ap = [4.16;
     -3.12;
     0.56;
     -0.1;
     5.03;
     -0.83;
     1.18;
     0.76;
     0.34;
     1.18];
[apOut, info] = f07gd(uplo, n, ap)
 

apOut =

    2.0396
   -1.5297
    0.2746
   -0.0490
    1.6401
   -0.2500
    0.6737
    0.7887
    0.6617
    0.5347


info =

                    0



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