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NAG Toolbox

NAG Toolbox: nag_lapack_zpftri (f07ww)

Purpose

nag_lapack_zpftri (f07ww) computes the inverse of a complex Hermitian positive definite matrix using the Cholesky factorization computed by nag_lapack_zpftrf (f07wr) and 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] = f07ww(transr, uplo, n, a)
[a, info] = nag_lapack_zpftri(transr, uplo, n, a)

Description

nag_lapack_zpftri (f07ww) is used to compute the inverse of a complex Hermitian positive definite matrix AA, the function must be preceded by a call to nag_lapack_zpftrf (f07wr), which computes the Cholesky factorization of AA.
If uplo = 'U'uplo='U', A = UHUA=UHU and A1A-1 is computed by first inverting UU and then forming (U1)UH(U-1)U-H.
If uplo = 'L'uplo='L', A = LLHA=LLH and A1A-1 is computed by first inverting LL and then forming LH(L1)L-H(L-1).

References

Du Croz J J and Higham N J (1992) Stability of methods for matrix inversion IMA J. Numer. Anal. 12 1–19
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 how AA has been factorized.
uplo = 'U'uplo='U'
A = UHUA=UHU, where UU is upper triangular.
uplo = 'L'uplo='L'
A = LLHA=LLH, 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 Cholesky factorization of AA stored in RFP format, as returned by nag_lapack_zpftrf (f07wr).

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
The factorization stores the nn by nn matrix A1A-1 stored in RFP format.
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

The computed inverse XX satisfies
XAI2c(n)εκ2(A)   and   AXI2c(n)εκ2(A) ,
XA-I2c(n)εκ2(A)   and   AX-I2c(n)εκ2(A) ,
where c(n)c(n) is a modest function of nn, εε is the machine precision and κ2(A)κ2(A) is the condition number of AA defined by
κ2(A) = A2A12 .
κ2(A)=A2A-12 .

Further Comments

The total number of real floating point operations is approximately (8/3)n383n3.
The real analogue of this function is nag_lapack_dpftri (f07wj).

Example

function nag_lapack_zpftri_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
[a, info] = nag_lapack_zpftrf(transr, uplo, n, a);

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

 Inverse
                    1                 2                 3                 4
 1  ( 5.4691, 0.0000)
 2  (-1.2624,-1.5491) ( 1.1024, 0.0000)
 3  (-2.9746,-0.9616) ( 0.8989,-0.5672) ( 2.1589, 0.0000)
 4  ( 1.1962, 2.9772) (-0.9826,-0.2566) (-1.3756,-1.4550) ( 2.2934, 0.0000)

function f07ww_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
[a, info] = f07wr(transr, uplo, n, a);

if info == 0
  % Compute inverse of a
  [a, info] = f07ww(transr, uplo, n, a);
  % Convert inverse to full array form, and print it
  [f, info] = f01vh(transr, uplo, n, a);
  fprintf('\n');
  [ifail] = x04db(uplo, 'n', f, 'b', 'f7.4', 'Inverse', 'i', 'i', int64(80), int64(0));
else
  fprintf('\na is not positive definite.\n');
end
 

 Inverse
                    1                 2                 3                 4
 1  ( 5.4691, 0.0000)
 2  (-1.2624,-1.5491) ( 1.1024, 0.0000)
 3  (-2.9746,-0.9616) ( 0.8989,-0.5672) ( 2.1589, 0.0000)
 4  ( 1.1962, 2.9772) (-0.9826,-0.2566) (-1.3756,-1.4550) ( 2.2934, 0.0000)


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

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