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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_dtbrfs (f07vh)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_dtbrfs (f07vh) returns error bounds for the solution of a real triangular band system of linear equations with multiple right-hand sides, AX=B or ATX=B.

Syntax

[ferr, berr, info] = f07vh(uplo, trans, diag, kd, ab, b, x, 'n', n, 'nrhs_p', nrhs_p)
[ferr, berr, info] = nag_lapack_dtbrfs(uplo, trans, diag, kd, ab, b, x, 'n', n, 'nrhs_p', nrhs_p)

Description

nag_lapack_dtbrfs (f07vh) returns the backward errors and estimated bounds on the forward errors for the solution of a real triangular band 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_dtbrfs (f07vh) 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:     kd int64int32nag_int scalar
kd, the number of superdiagonals of the matrix A if uplo='U', or the number of subdiagonals if uplo='L'.
Constraint: kd0.
5:     abldab: – double array
The first dimension of the array ab must be at least kd+1.
The second dimension of the array ab must be at least max1,n.
The n by n triangular band matrix A.
The matrix is stored in rows 1 to kd+1, more precisely,
  • if uplo='U', the elements of the upper triangle of A within the band must be stored with element Aij in abkd+1+i-jj​ for ​max1,j-kdij;
  • if uplo='L', the elements of the lower triangle of A within the band must be stored with element Aij in ab1+i-jj​ for ​jiminn,j+kd.
If diag='U', the diagonal elements of A are assumed to be 1, and are not referenced.
6:     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.
7:     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_dtbtrs (f07ve).

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the second dimension of the array ab.
n, the order of the matrix A.
Constraint: n0.
2:     nrhs_p int64int32nag_int scalar
Default: the second dimension of the arrays b, x.
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_dtbrfs (f07vh), 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 2nk floating-point operations (assuming nk).
The complex analogue of this function is nag_lapack_ztbrfs (f07vv).

Example

This example solves the system of equations AX=B and to compute forward and backward error bounds, where
A= -4.16 0.00 0.00 0.00 -2.25 4.78 0.00 0.00 0.00 5.86 6.32 0.00 0.00 0.00 -4.82 0.16   and   B= -16.64 -4.16 -13.78 -16.59 13.10 -4.94 -14.14 -9.96 .  
function f07vh_example


fprintf('f07vh example results\n\n');

% Solve AX=B and compute error bounds, where A is lower triangular banded
% and stored in triangular/symmetric banded format 
kd = int64(1);
ab = [-4.16, 4.78,  6.32, 0.16;
      -2.25, 5.86, -4.82, 0.00];
b = [-16.64,  -4.16;
     -13.78, -16.59;
      13.10,  -4.94;
     -14.14,  -9.96];

% Solve
uplo  = 'L';
trans = 'N';
diag  = 'N';
[x, info] = f07ve( ...
                   uplo, trans, diag, kd, ab, b);

% Compute error bounds
[ferr, berr, info] = f07vh( ...
                            uplo, trans, diag, kd, ab, 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');


f07vh example results

 Solution(s)
             1          2
 1      4.0000     1.0000
 2     -1.0000    -3.0000
 3      3.0000     2.0000
 4      2.0000    -2.0000

Backward errors (machine-dependent)
       4.7e-17    2.5e-17
Estimated forward error bounds (machine-dependent)
       5.4e-14    5.8e-14

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