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_zpotrf (f07fr)

Purpose

nag_lapack_zpotrf (f07fr) computes the Cholesky factorization of a complex Hermitian positive definite matrix.

Syntax

[a, info] = f07fr(uplo, a, 'n', n)
[a, info] = nag_lapack_zpotrf(uplo, a, 'n', n)

Description

nag_lapack_zpotrf (f07fr) 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.

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:     a(lda, : :) – complex array
The first dimension of the array a must be at least max (1,n)max(1,n)
The second dimension of the array must be at least max (1,n)max(1,n)
The nn by nn Hermitian positive definite matrix AA.
  • If uplo = 'U'uplo='U', the upper triangular part of aa must be stored and the elements of the array below the diagonal are not referenced.
  • If uplo = 'L'uplo='L', the lower triangular part of aa 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 The second dimension of the array a.
nn, the order of the matrix AA.
Constraint: n0n0.

Input Parameters Omitted from the MATLAB Interface

lda

Output Parameters

1:     a(lda, : :) – complex array
The first dimension of the array a will be max (1,n)max(1,n)
The second dimension of the array will be max (1,n)max(1,n)
ldamax (1,n)ldamax(1,n).
The upper or lower triangle of AA stores the Cholesky factor UU or LL as specified by uplo.
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: a, 4: lda, 5: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.
  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_zhetrf (f07mr) 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_zpotrf (f07fr) may be followed by calls to the functions:
The real analogue of this function is nag_lapack_dpotrf (f07fd).

Example

function nag_lapack_zpotrf_example
uplo = 'L';
a = [complex(3.23),  0 + 0i,  0 + 0i,  0 + 0i;
      1.51 + 1.92i,  3.58 + 0i,  0 + 0i,  0 + 0i;
      1.9 - 0.84i,  -0.23 - 1.11i,  4.09 + 0i,  0 + 0i;
      0.42 - 2.5i,  -1.18 - 1.37i,  2.33 + 0.14i,  4.29 + 0i];
[aOut, info] = nag_lapack_zpotrf(uplo, a)
 

aOut =

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


info =

                    0


function f07fr_example
uplo = 'L';
a = [complex(3.23),  0 + 0i,  0 + 0i,  0 + 0i;
      1.51 + 1.92i,  3.58 + 0i,  0 + 0i,  0 + 0i;
      1.9 - 0.84i,  -0.23 - 1.11i,  4.09 + 0i,  0 + 0i;
      0.42 - 2.5i,  -1.18 - 1.37i,  2.33 + 0.14i,  4.29 + 0i];
[aOut, info] = f07fr(uplo, a)
 

aOut =

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


info =

                    0



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

© The Numerical Algorithms Group Ltd, Oxford, UK. 2009–2013