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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_ztptrs (f07us)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_ztptrs (f07us) solves a complex triangular system of linear equations with multiple right-hand sides, AX=B, ATX=B or AHX=B, using packed storage.

Syntax

[b, info] = f07us(uplo, trans, diag, ap, b, 'n', n, 'nrhs_p', nrhs_p)
[b, info] = nag_lapack_ztptrs(uplo, trans, diag, ap, b, 'n', n, 'nrhs_p', nrhs_p)

Description

nag_lapack_ztptrs (f07us) solves a complex triangular system of linear equations AX=B, ATX=B or AHX=B, using packed storage.

References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Higham N J (1989) The accuracy of solutions to triangular systems SIAM J. Numer. Anal. 26 1252–1265

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:     trans – string (length ≥ 1)
Indicates the form of the equations.
trans='N'
The equations are of the form AX=B.
trans='T'
The equations are of the form ATX=B.
trans='C'
The equations are of the form AHX=B.
Constraint: trans='N', 'T' or 'C'.
3:     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'.
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’.
5:     bldb: – complex array
The first dimension of the array b must be at least max1,n.
The second dimension of the array b must be at least max1,nrhs_p.
The n by r right-hand side matrix B.

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the first dimension of the array ap and the second dimension of the array ap. (An error is raised if these dimensions are not equal.)
n, the order of the matrix A.
Constraint: n0.
2:     nrhs_p int64int32nag_int scalar
Default: the second dimension of the array b.
r, the number of right-hand sides.
Constraint: nrhs_p0.

Output Parameters

1:     bldb: – complex array
The first dimension of the array b will be max1,n.
The second dimension of the array b will be max1,nrhs_p.
The n by r solution matrix X.
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 and the solution has not been computed.

Accuracy

The solutions of triangular systems of equations are usually computed to high accuracy. See Higham (1989).
For each right-hand side vector b, the computed solution x is the exact solution of a perturbed system of equations A+Ex=b, where
EcnεA ,  
cn is a modest linear function of n, and ε is the machine precision.
If x^ is the true solution, then the computed solution x satisfies a forward error bound of the form
x-x^ x cncondA,xε ,   provided   cncondA,xε<1 ,  
where condA,x=A-1Ax/x.
Note that condA,xcondA=A-1AκA; condA,x can be much smaller than condA and it is also possible for condAH, which is the same as condAT, to be much larger (or smaller) than condA.
Forward and backward error bounds can be computed by calling nag_lapack_ztprfs (f07uv), and an estimate for κA can be obtained by calling nag_lapack_ztpcon (f07uu) with norm_p='I'.

Further Comments

The total number of real floating-point operations is approximately 4n2r.
The real analogue of this function is nag_lapack_dtptrs (f07ue).

Example

This example solves the system of equations AX=B, 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  
and
B= -14.78-32.36i -18.02+28.46i 2.98-02.14i 14.22+15.42i -20.96+17.06i 5.62+35.89i 9.54+09.91i -16.46-01.73i ,  
using packed storage for A.
function f07us_example


fprintf('f07us example results\n\n');

% Solve AX=B where A is Complex Lower triangular and packed.
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];
b = [-14.78 - 32.36i, -18.02 + 28.46i;
       2.98 -  2.14i,  14.22 + 15.42i;
     -20.96 + 17.06i,   5.62 + 35.89i;
       9.54 +  9.91i, -16.46 -  1.73i];

% Solve
uplo = 'L';
trans = 'N';
diag = 'N';
[x, info] = f07us( ...
                   uplo, trans, diag, ap, b);

disp('Solution(s)');
disp(x);


f07us example results

Solution(s)
  -5.0000 - 2.0000i   1.0000 + 5.0000i
  -3.0000 - 1.0000i  -2.0000 - 2.0000i
   2.0000 + 1.0000i   3.0000 + 4.0000i
   4.0000 + 3.0000i   4.0000 - 3.0000i


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