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NAG Toolbox: nag_lapack_dtftri (f07wk)

Purpose

nag_lapack_dtftri (f07wk) computes the inverse of a real triangular 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] = f07wk(transr, uplo, diag, n, a)
[a, info] = nag_lapack_dtftri(transr, uplo, diag, n, a)

Description

nag_lapack_dtftri (f07wk) forms the inverse of a real triangular matrix AA, stored using RFP format. Note that the inverse of an upper (lower) triangular matrix is also upper (lower) triangular.

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 whether AA is upper or lower triangular.
uplo = 'U'uplo='U'
AA is upper triangular.
uplo = 'L'uplo='L'
AA is lower triangular.
Constraint: uplo = 'U'uplo='U' or 'L''L'.
3:     diag – string (length ≥ 1)
Indicates whether AA is a nonunit or unit triangular matrix.
diag = 'N'diag='N'
AA is a nonunit triangular matrix.
diag = 'U'diag='U'
AA is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be 11.
Constraint: diag = 'N'diag='N' or 'U''U'.
4:     n – int64int32nag_int scalar
nn, the order of the matrix AA.
Constraint: n0n0.
5:     a(n × (n + 1) / 2n×(n+1)/2) – double array
The nn by nn triangular matrix AA, stored in RFP format.

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
AA stores A1A-1, 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

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
a(_,_)a(_,_) is exactly zero. AA is singular its inverse cannot be computed.

Accuracy

The computed inverse XX satisfies
|XAI|c(n)ε|X||A| ,
|XA-I|c(n)ε|X||A| ,
where c(n)c(n) is a modest linear function of nn, and εε is the machine precision.
Note that a similar bound for |AXI||AX-I| cannot be guaranteed, although it is almost always satisfied.
The computed inverse satisfies the forward error bound
|XA1|c(n)ε|A1||A||X| .
|X-A-1|c(n)ε|A-1||A||X| .
See Du Croz and Higham (1992).

Further Comments

The total number of floating point operations is approximately (1/3)n313n3.
The complex analogue of this function is nag_lapack_ztftri (f07wx).

Example

function nag_lapack_dtftri_example
a = [-8.02; 4.30; -3.96; 0.40; -0.27; -5.95; 0.12; -4.87; 0.31; 0.07];
transr = 'n';
uplo   = 'l';
diag   = 'n';
n      = int64(4);
% Compute inverse of a
[a, info] = nag_lapack_dtftri(transr, uplo, diag, n, a);

if info == 0
  % 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 singular.\n');
end
 

 Inverse
             1          2          3          4
 1      0.2326
 2     -0.1891    -0.2053
 3      0.0043    -0.0079    -0.1247
 4      0.8463    -0.2738    -6.1825     8.3333

function f07wk_example
a = [-8.02; 4.30; -3.96; 0.40; -0.27; -5.95; 0.12; -4.87; 0.31; 0.07];
transr = 'n';
uplo   = 'l';
diag   = 'n';
n      = int64(4);
% Compute inverse of a
[a, info] = f07wk(transr, uplo, diag, n, a);

if info == 0
  % 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 singular.\n');
end
 

 Inverse
             1          2          3          4
 1      0.2326
 2     -0.1891    -0.2053
 3      0.0043    -0.0079    -0.1247
 4      0.8463    -0.2738    -6.1825     8.3333


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

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