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_zpftrf (f07wr)

Purpose

nag_lapack_zpftrf (f07wr) computes the Cholesky factorization of a complex Hermitian positive definite matrix stored in Rectangular Full Packed (RFP) format. The RFP storage format is described in Section [Rectangular Full Packed (RFP) Storage] in the F07 Chapter Introduction.

Syntax

[a, info] = f07wr(transr, uplo, n, a)
[a, info] = nag_lapack_zpftrf(transr, uplo, n, a)

Description

nag_lapack_zpftrf (f07wr) 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 an lower triangular, stored using RFP format.

References

Demmel J W (1989) On floating-point errors in Cholesky LAPACK Working Note No. 14 University of Tennessee, Knoxville
Gustavson F G, Waśniewski J, Dongarra J J and Langou J (2010) Rectangular full packed format for Cholesky's algorithm: factorization, solution, and inversion ACM Trans. Math. Software 37, 2

Parameters

Compulsory Input Parameters

1:     transr – string (length ≥ 1)
Specifies whether the normal RFP representation of AA or its conjugate transpose is stored.
transr = 'N'transr='N'
The matrix AA is stored in normal RFP format.
transr = 'C'transr='C'
The conjugate transpose of the RFP representation of the matrix AA is stored.
Constraint: transr = 'N'transr='N' or 'C''C'.
2:     uplo – string (length ≥ 1)
Specifies whether the upper or lower triangular part of AA is stored.
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'.
3:     n – int64int32nag_int scalar
nn, the order of the matrix AA.
Constraint: n0n0.
4:     a(n × (n + 1) / 2n×(n+1)/2) – complex array
The nn by nn Hermitian matrix AA, stored in RFP format, as described in Section [Rectangular Full Packed (RFP) Storage] in the F07 Chapter Introduction.

Optional Input Parameters

None.

Input Parameters Omitted from the MATLAB Interface

None.

Output Parameters

1:     a(n × (n + 1) / 2n×(n+1)/2) – complex array
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 > 0INFO>0
The leading minor of order __ 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. There is no function specifically designed to factorize a Hermitian band matrix which is not positive definite; the matrix must be treated either as a nonsymmetric band matrix, by calling nag_lapack_zgbtrf (f07br) or as a full Hermitian matrix, by calling nag_lapack_zhetrf (f07mr).

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)n243n2.
A call to nag_lapack_zpftrf (f07wr) may be followed by calls to the functions:
The real analogue of this function is nag_lapack_dpftrf (f07wd).

Example

function nag_lapack_zpftrf_example
a = [ 4.09 + 0.00i;
      3.23 + 0.00i;
      1.51 + 1.92i;
      1.90 - 0.84i;
      0.42 - 2.50i;
      2.33 + 0.14i;
      4.29 + 0.00i;
      3.58 + 0.00i;
      -0.23 - 1.11i;
      -1.18 - 1.37i];
transr = 'n';
uplo   = 'l';
n      = int64(4);

% Factorize a
[aOut, info] = nag_lapack_zpftrf(transr, uplo, n, a);

if info == 0
  % Convert factor to full array form, and print it
  [f, info] = nag_matop_ztfttr(transr, uplo, n, aOut);
  fprintf('\n');
  [ifail] = ...
    nag_file_print_matrix_complex_gen_comp(uplo, 'n', f, 'b', 'f7.4', 'Factor', 'i', 'i', int64(80), int64(0));
else
  fprintf('\na is not positive definite.\n');
end
 

 Factor
                    1                 2                 3                 4
 1  ( 1.7972, 0.0000)
 2  ( 0.8402, 1.0683) ( 1.3164, 0.0000)
 3  ( 1.0572,-0.4674) (-0.4702, 0.3131) ( 1.5604, 0.0000)
 4  ( 0.2337,-1.3910) ( 0.0834, 0.0368) ( 0.9360, 0.8105) ( 0.8713, 0.0000)

function f07wr_example
a = [ 4.09 + 0.00i;
      3.23 + 0.00i;
      1.51 + 1.92i;
      1.90 - 0.84i;
      0.42 - 2.50i;
      2.33 + 0.14i;
      4.29 + 0.00i;
      3.58 + 0.00i;
      -0.23 - 1.11i;
      -1.18 - 1.37i];
transr = 'n';
uplo   = 'l';
n      = int64(4);

% Factorize a
[aOut, info] = f07wr(transr, uplo, n, a);

if info == 0
  % Convert factor to full array form, and print it
  [f, info] = f01vh(transr, uplo, n, aOut);
  fprintf('\n');
  [ifail] = x04db(uplo, 'n', f, 'b', 'f7.4', 'Factor', 'i', 'i', int64(80), int64(0));
else
  fprintf('\na is not positive definite.\n');
end
 

 Factor
                    1                 2                 3                 4
 1  ( 1.7972, 0.0000)
 2  ( 0.8402, 1.0683) ( 1.3164, 0.0000)
 3  ( 1.0572,-0.4674) (-0.4702, 0.3131) ( 1.5604, 0.0000)
 4  ( 0.2337,-1.3910) ( 0.0834, 0.0368) ( 0.9360, 0.8105) ( 0.8713, 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. 2009–2013