hide long namesshow long names
hide short namesshow short names
Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int

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

NAG Toolbox: nag_lapack_ztftri (f07wx)

Purpose

nag_lapack_ztftri (f07wx) computes the inverse of a complex 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] = f07wx(transr, uplo, diag, n, a)
[a, info] = nag_lapack_ztftri(transr, uplo, diag, n, a)

Description

nag_lapack_ztftri (f07wx) forms the inverse of a complex 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 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 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) – complex 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) – complex 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 real floating point operations is approximately (4/3)n343n3.
The real analogue of this function is nag_lapack_dtftri (f07wk).

Example

function nag_lapack_ztftri_example
a = [ 4.15 - 0.80i;
      4.78 + 4.56i;
      2.00 - 0.30i;
      2.89 - 1.34i;
      -1.89 + 1.15i;
      -0.02 - 0.46i;
      0.33 + 0.26i;
      -4.11 + 1.25i;
      2.36 - 4.25i;
      0.04 - 3.69i];
transr = 'n';
uplo   = 'l';
diag   = 'n';
n      = int64(4);
% Compute inverse of a
[a, info] = nag_lapack_ztftri(transr, uplo, diag, n, a);

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

 Inverse
                    1                 2                 3                 4
 1  ( 0.1095,-0.1045)
 2  ( 0.0582,-0.0411) (-0.2227,-0.0677)
 3  ( 0.0032, 0.1905) ( 0.1538,-0.2192) ( 0.2323,-0.0448)
 4  ( 0.7602, 0.2814) ( 1.6184,-1.4346) ( 0.1289,-0.2250) ( 1.8697, 1.4731)

function f07wx_example
a = [ 4.15 - 0.80i;
      4.78 + 4.56i;
      2.00 - 0.30i;
      2.89 - 1.34i;
      -1.89 + 1.15i;
      -0.02 - 0.46i;
      0.33 + 0.26i;
      -4.11 + 1.25i;
      2.36 - 4.25i;
      0.04 - 3.69i];
transr = 'n';
uplo   = 'l';
diag   = 'n';
n      = int64(4);
% Compute inverse of a
[a, info] = f07wx(transr, uplo, diag, n, a);

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

 Inverse
                    1                 2                 3                 4
 1  ( 0.1095,-0.1045)
 2  ( 0.0582,-0.0411) (-0.2227,-0.0677)
 3  ( 0.0032, 0.1905) ( 0.1538,-0.2192) ( 0.2323,-0.0448)
 4  ( 0.7602, 0.2814) ( 1.6184,-1.4346) ( 0.1289,-0.2250) ( 1.8697, 1.4731)


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

© The Numerical Algorithms Group Ltd, Oxford, UK. 2009–2013