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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_dpftri (f07wj)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_lapack_dpftri (f07wj) computes the inverse of a real symmetric positive definite matrix using the Cholesky factorization computed by nag_lapack_dpftrf (f07wd) stored in Rectangular Full Packed (RFP) format.


[ar, info] = f07wj(transr, uplo, n, ar)
[ar, info] = nag_lapack_dpftri(transr, uplo, n, ar)


nag_lapack_dpftri (f07wj) is used to compute the inverse of a real symmetric 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_dpftrf (f07wd), which computes the Cholesky factorization of A.
If uplo='U', A=UTU and A-1 is computed by first inverting U and then forming U-1U-T.
If uplo='L', A=LLT and A-1 is computed by first inverting L and then forming L-TL-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 RFP representation of A is normal or transposed.
The matrix A is stored in normal RFP format.
The matrix A is stored in transposed RFP format.
Constraint: transr='N' or 'T'.
2:     uplo – string (length ≥ 1)
Specifies how A has been factorized.
A=UTU, where U is upper triangular.
A=LLT, 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 – double array
The Cholesky factorization of A stored in RFP format, as returned by nag_lapack_dpftrf (f07wd).

Optional Input Parameters


Output Parameters

1:     arn×n+1/2 – double 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

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
W  info>0
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 symmetric matrix stored in RFP format which is not positive definite; the matrix must be treated as a full symmetric matrix, by calling nag_lapack_dsytri (f07mj).


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 floating-point operations is approximately 23n3.
The complex analogue of this function is nag_lapack_zpftri (f07ww).


This example computes the inverse of the matrix A, where
A= 4.16 -3.12 0.56 -0.10 -3.12 5.03 -0.83 1.18 0.56 -0.83 0.76 0.34 -0.10 1.18 0.34 1.18 .  
Here A is symmetric positive definite, stored in RFP format, and must first be factorized by nag_lapack_dpftrf (f07wd).
function f07wj_example

fprintf('f07wj example results\n\n');

% Symmetric matrix in RFP format
transr = 'n';
uplo   = 'l';
ar = [ 0.76   0.34; 
       4.16   1.18;
      -3.12   5.03;
       0.56  -0.83;
      -0.10   1.18];
n  = int64(4);
n2 = (n*(n+1))/2;
ar  = reshape(ar,[n2,1]);

% Factorize a
[ar, info] = f07wd(transr, uplo, n, ar);

if info == 0
  % Compute inverse of a
  [ar, info] = f07wj(transr, uplo, n, ar);
  % Convert inverse to full array form, and print it
  [a, info] = f01vg(transr, uplo, n, ar);
  [ifail] = x04ca(uplo, 'n', a, 'Inverse');
  fprintf('\na is not positive definite.\n');

f07wj example results

             1          2          3          4
 1      0.6995
 2      0.7769     1.4239
 3      0.7508     1.8255     4.0688
 4     -0.9340    -1.8841    -2.9342     3.4978

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Chapter Contents
Chapter Introduction
NAG Toolbox

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