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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_dtprfs (f07uh)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_dtprfs (f07uh) returns error bounds for the solution of a real triangular system of linear equations with multiple right-hand sides, AX=B or ATX=B, using packed storage.

Syntax

[ferr, berr, info] = f07uh(uplo, trans, diag, ap, b, x, 'n', n, 'nrhs_p', nrhs_p)
[ferr, berr, info] = nag_lapack_dtprfs(uplo, trans, diag, ap, b, x, 'n', n, 'nrhs_p', nrhs_p)

Description

nag_lapack_dtprfs (f07uh) returns the backward errors and estimated bounds on the forward errors for the solution of a real triangular system of linear equations with multiple right-hand sides AX=B or ATX=B, using packed storage. The function handles each right-hand side vector (stored as a column of the matrix B) independently, so we describe the function of nag_lapack_dtprfs (f07uh) in terms of a single right-hand side b and solution x.
Given a computed solution x, the function computes the component-wise backward error β. This is the size of the smallest relative perturbation in each element of A and b such that x is the exact solution of a perturbed system
A+δAx=b+δb δaijβaij   and   δbiβbi .  
Then the function estimates a bound for the component-wise forward error in the computed solution, defined by:
maxixi-x^i/maxixi  
where x^ is the true solution.
For details of the method, see the F07 Chapter Introduction.

References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

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' or 'C'
The equations are of the form ATX=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: – double 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: – double 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.
6:     xldx: – double array
The first dimension of the array x must be at least max1,n.
The second dimension of the array x must be at least max1,nrhs_p.
The n by r solution matrix X, as returned by nag_lapack_dtptrs (f07ue).

Optional Input Parameters

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

Output Parameters

1:     ferrnrhs_p – double array
ferrj contains an estimated error bound for the jth solution vector, that is, the jth column of X, for j=1,2,,r.
2:     berrnrhs_p – double array
berrj contains the component-wise backward error bound β for the jth solution vector, that is, the jth column of X, for j=1,2,,r.
3:     info int64int32nag_int scalar
info=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

   info<0
If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.

Accuracy

The bounds returned in ferr are not rigorous, because they are estimated, not computed exactly; but in practice they almost always overestimate the actual error.

Further Comments

A call to nag_lapack_dtprfs (f07uh), for each right-hand side, involves solving a number of systems of linear equations of the form Ax=b or ATx=b; the number is usually 4 or 5 and never more than 11. Each solution involves approximately n2 floating-point operations.
The complex analogue of this function is nag_lapack_ztprfs (f07uv).

Example

This example solves the system of equations AX=B and to compute forward and backward error bounds, where
A= 4.30 0.00 0.00 0.00 -3.96 -4.87 0.00 0.00 0.40 0.31 -8.02 0.00 -0.27 0.07 -5.95 0.12   and   B= -12.90 -21.50 16.75 14.93 -17.55 6.33 -11.04 8.09 ,  
using packed storage for A.
function f07uh_example


fprintf('f07uh example results\n\n');

% Solve AX=B and estimate errors, where A is Lower triangular and packed
n = int64(4);
ap = [ 4.30; -3.96;  0.40; -0.27;
             -4.87;  0.31;  0.07;
                    -8.02; -5.95;
                            0.12];
b = [-12.90, -21.50;
      16.75,  14.93;
     -17.55,   6.33;
     -11.04,   8.09];

uplo = 'L';
trans = 'N';
diag = 'N';
% Solve
[x, info] = f07ue( ...
                   uplo, trans, diag, ap, b);

% Estimate error bounds
[ferr, berr, info] = f07uh( ...
                            uplo, trans, diag, ap, b, x);

% Display solution
disp('Solution(s)');
disp(x);

fprintf('Backward errors (machine-dependent)\n   ')
fprintf('%11.1e', berr);
fprintf('\nEstimated forward error bounds (machine-dependent)\n   ')
fprintf('%11.1e', ferr);
fprintf('\n');


f07uh example results

Solution(s)
   -3.0000   -5.0000
   -1.0000    1.0000
    2.0000   -1.0000
    1.0000    6.0000

Backward errors (machine-dependent)
       1.0e-16    0.0e+00
Estimated forward error bounds (machine-dependent)
       9.1e-14    2.6e-14

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