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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_ztrtri (f07tw)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_ztrtri (f07tw) computes the inverse of a complex triangular matrix.

Syntax

[a, info] = f07tw(uplo, diag, a, 'n', n)
[a, info] = nag_lapack_ztrtri(uplo, diag, a, 'n', n)

Description

nag_lapack_ztrtri (f07tw) forms the inverse of a complex triangular matrix A. 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

Parameters

Compulsory Input Parameters

1:     uplo – string (length ≥ 1)
Specifies whether A is upper or lower triangular.
uplo='U'
A is upper triangular.
uplo='L'
A is lower triangular.
Constraint: uplo='U' or 'L'.
2:     diag – string (length ≥ 1)
Indicates whether A is a nonunit or unit triangular matrix.
diag='N'
A is a nonunit triangular matrix.
diag='U'
A is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be 1.
Constraint: diag='N' or 'U'.
3:     alda: – complex array
The first dimension of the array a must be at least max1,n.
The second dimension of the array a must be at least max1,n.
The n by n triangular matrix A.
  • If uplo='U', a is upper triangular and the elements of the array below the diagonal are not referenced.
  • If uplo='L', a is lower triangular and the elements of the array above the diagonal are not referenced.
  • If diag='U', the diagonal elements of a are assumed to be 1, and are not referenced.

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the first dimension of the array a and the second dimension of the array a.
n, the order of the matrix A.
Constraint: n0.

Output Parameters

1:     alda: – complex array
The first dimension of the array a will be max1,n.
The second dimension of the array a will be max1,n.
A stores A-1, using the same storage format as described above.
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.

   info<0
If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
W  info>0
Element _ of the diagonal is exactly zero. A is singular its inverse cannot be computed.

Accuracy

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

Further Comments

The total number of real floating-point operations is approximately 43n3.
The real analogue of this function is nag_lapack_dtrtri (f07tj).

Example

This example computes the inverse of the matrix A, where
A= 4.78+4.56i 0.00+0.00i 0.00+0.00i 0.00+0.00i 2.00-0.30i -4.11+1.25i 0.00+0.00i 0.00+0.00i 2.89-1.34i 2.36-4.25i 4.15+0.80i 0.00+0.00i -1.89+1.15i 0.04-3.69i -0.02+0.46i 0.33-0.26i .  
function f07tw_example


fprintf('f07tw example results\n\n');

% Invert A, where A is Lower triangular
a = [  4.78 +  4.56i,   0    +  0i,     0    + 0i,    0    + 0i;
       2.00 -  0.30i,  -4.11 +  1.25i,  0    + 0i,    0    + 0i;
       2.89 -  1.34i,   2.36 -  4.25i,  4.15 + 0.8i,  0    + 0i;
      -1.89 +  1.15i,   0.04 -  3.69i, -0.02 + 0.46i, 0.33 - 0.26i];

% Invert
uplo = 'L';
diag = 'N';
[ainv, info] = f07tw(uplo, diag, a);

% Display inverse
disp('Inverse');
disp(ainv);


f07tw example results

Inverse
   0.1095 - 0.1045i   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i
   0.0582 - 0.0411i  -0.2227 - 0.0677i   0.0000 + 0.0000i   0.0000 + 0.0000i
   0.0032 + 0.1905i   0.1538 - 0.2192i   0.2323 - 0.0448i   0.0000 + 0.0000i
   0.7602 + 0.2814i   1.6184 - 1.4346i   0.1289 - 0.2250i   1.8697 + 1.4731i


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