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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_ztptri (f07uw)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_lapack_ztptri (f07uw) computes the inverse of a complex triangular matrix, using packed storage.


[ap, info] = f07uw(uplo, diag, n, ap)
[ap, info] = nag_lapack_ztptri(uplo, diag, n, ap)


nag_lapack_ztptri (f07uw) forms the inverse of a complex triangular matrix A, using packed storage. Note that the inverse of an upper (lower) triangular matrix is also upper (lower) triangular.


Du Croz J J and Higham N J (1992) Stability of methods for matrix inversion IMA J. Numer. Anal. 12 1–19


Compulsory Input Parameters

1:     uplo – string (length ≥ 1)
Specifies whether A is upper or lower triangular.
A is upper triangular.
A is lower triangular.
Constraint: uplo='U' or 'L'.
2:     diag – string (length ≥ 1)
Indicates whether A is a nonunit or unit triangular matrix.
A is a nonunit triangular matrix.
A is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be 1.
Constraint: diag='N' or 'U'.
3:     n int64int32nag_int scalar
n, the order of the matrix A.
Constraint: n0.
4:     ap: – complex array
The dimension of the array ap must be at least max1,n×n+1/2
The n by n triangular matrix A, packed by columns.
More precisely,
  • if uplo='U', the upper triangle of A must be stored with element Aij in api+jj-1/2 for ij;
  • if uplo='L', the lower triangle of A must be stored with element Aij in api+2n-jj-1/2 for ij.
If diag='U', the diagonal elements of A are assumed to be 1, and are not referenced; the same storage scheme is used whether diag='N' or ‘U’.

Optional Input Parameters


Output Parameters

1:     ap: – complex array
The dimension of the array ap will be max1,n×n+1/2
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.

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.


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_dtptri (f07uj).


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 ,  
using packed storage.
function f07uw_example

fprintf('f07uw example results\n\n');

% Invert A, where A is Complex Lower triangular and packed.
n = int64(4);
ap = [ 4.78 + 4.56i;  2.00 - 0.30i;  2.89 - 1.34i; -1.89 + 1.15i;
                     -4.11 + 1.25i;  2.36 - 4.25i;  0.04 - 3.69i;
                                     4.15 + 0.80i; -0.02 + 0.46i;
                                                    0.33 - 0.26i];

% Invert
uplo = 'L';
diag = 'N';
[ainv, info] = f07uw(uplo, diag, n, ap);

[ifail] = x04dc( ...
                 uplo, 'Non-unit', n, ainv, 'Inverse');

f07uw example results

             1          2          3          4
 1      0.1095

 2      0.0582    -0.2227
       -0.0411    -0.0677

 3      0.0032     0.1538     0.2323
        0.1905    -0.2192    -0.0448

 4      0.7602     1.6184     0.1289     1.8697
        0.2814    -1.4346    -0.2250     1.4731

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