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NAG Toolbox: nag_lapack_dpftri (f07wj)

Purpose

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. The RFP storage format is described in Section [Rectangular Full Packed (RFP) Storage] in the F07 Chapter Introduction.

Syntax

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

Description

nag_lapack_dpftri (f07wj) is used to compute the inverse of a real symmetric positive definite matrix AA, the function must be preceded by a call to nag_lapack_dpftrf (f07wd), which computes the Cholesky factorization of AA.
If uplo = 'U'uplo='U', A = UTUA=UTU and A1A-1 is computed by first inverting UU and then forming (U1)UT(U-1)U-T.
If uplo = 'L'uplo='L', A = LLTA=LLT and A1A-1 is computed by first inverting LL and then forming LT(L1)L-T(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 RFP representation of AA is normal or transposed.
transr = 'N'transr='N'
The matrix AA is stored in normal RFP format.
transr = 'T'transr='T'
The matrix AA is stored in transposed RFP format.
Constraint: transr = 'N'transr='N' or 'T''T'.
2:     uplo – string (length ≥ 1)
Specifies how AA has been factorized.
uplo = 'U'uplo='U'
A = UTUA=UTU, where UU is upper triangular.
uplo = 'L'uplo='L'
A = LLTA=LLT, 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) – double array
The Cholesky factorization of AA stored in RFP format, as returned by nag_lapack_dpftrf (f07wd).

Optional Input Parameters

None.

Input Parameters Omitted from the MATLAB Interface

None.

Output Parameters

1:     a(n × (n + 1) / 2n×(n+1)/2) – double 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

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

W 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 symmetric band matrix which is not positive definite; the matrix must be treated either as a nonsymmetric band matrix, by calling nag_lapack_dgbtrf (f07bd) or as a full symmetric matrix, by calling nag_lapack_dsytrf (f07md).

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 floating point operations is approximately (2/3)n323n3.
The complex analogue of this function is nag_lapack_zpftri (f07ww).

Example

function nag_lapack_dpftri_example
a = [0.76; 4.16; -3.12; 0.56; -0.10; 0.34; 1.18; 5.03; -0.83; 1.18];
transr = 'n';
uplo   = 'l';
n      = int64(4);
% Factorize a
[a, info] = nag_lapack_dpftrf(transr, uplo, n, a);

if info == 0
  % Compute inverse of a
  [a, info] = nag_lapack_dpftri(transr, uplo, n, a);
  % Convert inverse to full array form, and print it
  [f, info] = nag_matop_dtfttr(transr, uplo, n, a);
  fprintf('\n');
  [ifail] = nag_file_print_matrix_real_gen(uplo, 'n', f, 'Inverse');
else
  fprintf('\na is not positive definite.\n');
end
 

 Inverse
             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

function f07wj_example
a = [0.76; 4.16; -3.12; 0.56; -0.10; 0.34; 1.18; 5.03; -0.83; 1.18];
transr = 'n';
uplo   = 'l';
n      = int64(4);
% Factorize a
[a, info] = f07wd(transr, uplo, n, a);

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

 Inverse
             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


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

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