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NAG Toolbox: nag_lapack_zpptrf (f07gr)

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 AA either as A = UHUA=UHU if uplo = 'U'uplo='U' or A = LLHA=LLH 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 UHUUHU, where UU is upper triangular.
uplo = 'L'uplo='L'
The lower triangular part of AA is stored and AA is factorized as LLHLLH, 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( : :) – complex 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 Hermitian 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( : :) – complex 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 = UHUA=UHU or A = LLHA=LLH, 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 Hermitian matrix which is not positive definite, call nag_lapack_zhptrf (f07pr) 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|UH||U| ,
|E|c(n)ε|UH||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 real floating point operations is approximately (4/3)n343n3.
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

function nag_lapack_zpptrf_example
uplo = 'L';
n = int64(4);
ap = [3.23;
      1.51 + 1.92i;
      1.9 - 0.84i;
      0.42 - 2.5i;
      3.58 + 0i;
      -0.23 - 1.11i;
      -1.18 - 1.37i;
      4.09 + 0i;
      2.33 + 0.14i;
      4.29 + 0i];
[apOut, info] = nag_lapack_zpptrf(uplo, n, ap)
 

apOut =

   1.7972 + 0.0000i
   0.8402 + 1.0683i
   1.0572 - 0.4674i
   0.2337 - 1.3910i
   1.3164 + 0.0000i
  -0.4702 + 0.3131i
   0.0834 + 0.0368i
   1.5604 + 0.0000i
   0.9360 + 0.9900i
   0.6603 + 0.0000i


info =

                    0


function f07gr_example
uplo = 'L';
n = int64(4);
ap = [3.23;
      1.51 + 1.92i;
      1.9 - 0.84i;
      0.42 - 2.5i;
      3.58 + 0i;
      -0.23 - 1.11i;
      -1.18 - 1.37i;
      4.09 + 0i;
      2.33 + 0.14i;
      4.29 + 0i];
[apOut, info] = f07gr(uplo, n, ap)
 

apOut =

   1.7972 + 0.0000i
   0.8402 + 1.0683i
   1.0572 - 0.4674i
   0.2337 - 1.3910i
   1.3164 + 0.0000i
  -0.4702 + 0.3131i
   0.0834 + 0.0368i
   1.5604 + 0.0000i
   0.9360 + 0.9900i
   0.6603 + 0.0000i


info =

                    0



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