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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_zpftri (f07ww)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_lapack_zpftri (f07ww) computes the inverse of a complex Hermitian positive definite matrix using the Cholesky factorization computed by nag_lapack_zpftrf (f07wr) stored in Rectangular Full Packed (RFP) format.


[ar, info] = f07ww(transr, uplo, n, ar)
[ar, info] = nag_lapack_zpftri(transr, uplo, n, ar)


nag_lapack_zpftri (f07ww) is used to compute the inverse of a complex Hermitian positive definite matrix A, stored in RFP format. The RFP storage format is described in Rectangular Full Packed (RFP) Storage in the F07 Chapter Introduction. The function must be preceded by a call to nag_lapack_zpftrf (f07wr), which computes the Cholesky factorization of A.
If uplo='U', A=UHU and A-1 is computed by first inverting U and then forming U-1U-H.
If uplo='L', A=LLH and A-1 is computed by first inverting L and then forming L-HL-1.


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


Compulsory Input Parameters

1:     transr – string (length ≥ 1)
Specifies whether the normal RFP representation of A or its conjugate transpose is stored.
The matrix A is stored in normal RFP format.
The conjugate transpose of the RFP representation of the matrix A is stored.
Constraint: transr='N' or 'C'.
2:     uplo – string (length ≥ 1)
Specifies how A has been factorized.
A=UHU, where U is upper triangular.
A=LLH, where L is lower triangular.
Constraint: uplo='U' or 'L'.
3:     n int64int32nag_int scalar
n, the order of the matrix A.
Constraint: n0.
4:     arn×n+1/2 – complex array
The Cholesky factorization of A stored in RFP format, as returned by nag_lapack_zpftrf (f07wr).

Optional Input Parameters


Output Parameters

1:     arn×n+1/2 – complex array
The factorization stores the n by n matrix A-1 stored in RFP format.
2:     info int64int32nag_int scalar
info=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
The leading minor of order _ is not positive definite and the factorization could not be completed. Hence A itself is not positive definite. This may indicate an error in forming the matrix A. There is no function specifically designed to invert a Hermitian matrix stored in RFP format which is not positive definite; the matrix must be treated as a full Hermitian matrix, by calling nag_lapack_zhetri (f07mw).


The computed inverse X satisfies
XA-I2cnεκ2A   and   AX-I2cnεκ2A ,  
where cn is a modest function of n, ε is the machine precision and κ2A is the condition number of A defined by
κ2A=A2A-12 .  

Further Comments

The total number of real floating-point operations is approximately 83n3.
The real analogue of this function is nag_lapack_dpftri (f07wj).


This example computes the inverse of the matrix A, where
A= 3.23+0.00i 1.51-1.92i 1.90+0.84i 0.42+2.50i 1.51+1.92i 3.58+0.00i -0.23+1.11i -1.18+1.37i 1.90-0.84i -0.23-1.11i 4.09+0.00i 2.33-0.14i 0.42-2.50i -1.18-1.37i 2.33+0.14i 4.29+0.00i .  
Here A is Hermitian positive definite, stored in RFP format, and must first be factorized by nag_lapack_zpftrf (f07wr).
function f07ww_example

fprintf('f07ww example results\n\n');

% Symmetric matrix in RFP format
transr = 'n';
uplo   = 'l';
ar = [ 4.09 + 0.00i  2.33 - 0.14i;
       3.23 + 0.00i  4.29 + 0.00i;
       1.51 + 1.92i  3.58 + 0.00i;
       1.90 - 0.84i -0.23 - 1.11i;
       0.42 - 2.50i -1.18 - 1.37i];
n  = int64(4);
n2 = (n*(n+1))/2;
ar  = reshape(ar,[n2,1]);

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

if info == 0
  % Compute inverse of a
  [ar, info] = f07ww( ...
                      transr, uplo, n, ar);
  % Convert inverse to full array form for display
  [a, info] = f01vh( ...
                     transr, uplo, n, ar);
  ncols  = int64(80);
  indent = int64(0);
  form   = 'f7.4';
  title  = 'Inverse, lower triangle:';
  diag   = 'n';
  [ifail] = x04db( ...
                   uplo, diag, a, 'brackets', form, title, ...
                   'int', 'int', ncols, indent);
  fprintf('\na is not positive definite.\n');

f07ww example results

 Inverse, lower triangle:
                    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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