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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_dtrrfs (f07th)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_dtrrfs (f07th) returns error bounds for the solution of a real triangular system of linear equations with multiple right-hand sides, AX=B or ATX=B.

Syntax

[ferr, berr, info] = f07th(uplo, trans, diag, a, b, x, 'n', n, 'nrhs_p', nrhs_p)
[ferr, berr, info] = nag_lapack_dtrrfs(uplo, trans, diag, a, b, x, 'n', n, 'nrhs_p', nrhs_p)

Description

nag_lapack_dtrrfs (f07th) 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. 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_dtrrfs (f07th) 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:     alda: – double 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.
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_dtrtrs (f07te).

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the first dimension of the arrays a, b, x and the second dimension of the array a.
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_dtrrfs (f07th), 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_ztrrfs (f07tv).

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 .  
function f07th_example


fprintf('f07th example results\n\n');

% Solve AX=B and compute error bounds, where A is Lower triangular
a = [ 4.30,  0,     0,    0;
     -3.96, -4.87,  0,    0;
      0.40,  0.31, -8.02, 0;
     -0.27,  0.07, -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] = f07te( ...
                   uplo, trans, diag, a, b);

% Error bounds
[ferr, berr, info] = f07th( ...
                            uplo, trans, diag, a, b, x);

% Display solution
[ifail] = x04ca( ...
                 'Gen', diag, x, 'Solution(s)');

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


f07th example results

 Solution(s)
             1          2
 1     -3.0000    -5.0000
 2     -1.0000     1.0000
 3      2.0000    -1.0000
 4      1.0000     6.0000

Backward errors (machine-dependent)
       6.9e-17    0.0e+00
Estimated forward error bounds (machine-dependent)
       8.3e-14    2.6e-14

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