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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_zgbrfs (f07bv)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_zgbrfs (f07bv) returns error bounds for the solution of a complex band system of linear equations with multiple right-hand sides, AX=B, ATX=B or AHX=B. It improves the solution by iterative refinement, in order to reduce the backward error as much as possible.

Syntax

[x, ferr, berr, info] = f07bv(trans, kl, ku, ab, afb, ipiv, b, x, 'n', n, 'nrhs_p', nrhs_p)
[x, ferr, berr, info] = nag_lapack_zgbrfs(trans, kl, ku, ab, afb, ipiv, b, x, 'n', n, 'nrhs_p', nrhs_p)

Description

nag_lapack_zgbrfs (f07bv) returns the backward errors and estimated bounds on the forward errors for the solution of a complex band system of linear equations with multiple right-hand sides AX=B, ATX=B or AHX=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_zgbrfs (f07bv) 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:     trans – string (length ≥ 1)
Indicates the form of the linear equations for which X is the computed solution as follows:
trans='N'
The linear equations are of the form AX=B.
trans='T'
The linear equations are of the form ATX=B.
trans='C'
The linear equations are of the form AHX=B.
Constraint: trans='N', 'T' or 'C'.
2:     kl int64int32nag_int scalar
kl, the number of subdiagonals within the band of the matrix A.
Constraint: kl0.
3:     ku int64int32nag_int scalar
ku, the number of superdiagonals within the band of the matrix A.
Constraint: ku0.
4:     abldab: – complex array
The first dimension of the array ab must be at least kl+ku+1.
The second dimension of the array ab must be at least max1,n.
The original n by n band matrix A as supplied to nag_lapack_zgbtrf (f07br).
The matrix is stored in rows 1 to kl+ku+1, more precisely, the element Aij must be stored in
abku+1+i-jj  for ​max1,j-kuiminn,j+kl. 
See Further Comments in nag_lapack_zgbsv (f07bn) for further details.
5:     afbldafb: – complex array
The first dimension of the array afb must be at least 2×kl+ku+1.
The second dimension of the array afb must be at least max1,n.
The LU factorization of A, as returned by nag_lapack_zgbtrf (f07br).
6:     ipiv: int64int32nag_int array
The dimension of the array ipiv must be at least max1,n
The pivot indices, as returned by nag_lapack_zgbtrf (f07br).
7:     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.
8:     xldx: – complex 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_zgbtrs (f07bs).

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:     xldx: – complex array
The first dimension of the array x will be max1,n.
The second dimension of the array x will be max1,nrhs_p.
The improved solution matrix X.
2:     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.
3:     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.
4:     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

For each right-hand side, computation of the backward error involves a minimum of 16nkl+ku real floating-point operations. Each step of iterative refinement involves an additional 8n4kl+3ku real operations. This assumes nkl and nku. At most five steps of iterative refinement are performed, but usually only one or two steps are required.
Estimating the forward error involves solving a number of systems of linear equations of the form Ax=b or AHx=b; the number is usually 5 and never more than 11. Each solution involves approximately 8n2kl+ku real operations.
The real analogue of this function is nag_lapack_dgbrfs (f07bh).

Example

This example solves the system of equations AX=B using iterative refinement and to compute the forward and backward error bounds, where
A= -1.65+2.26i -2.05-0.85i 0.97-2.84i 0.00+0.00i 0.00+6.30i -1.48-1.75i -3.99+4.01i 0.59-0.48i 0.00+0.00i -0.77+2.83i -1.06+1.94i 3.33-1.04i 0.00+0.00i 0.00+0.00i 4.48-1.09i -0.46-1.72i  
and
B= -1.06+21.50i 12.85+02.84i -22.72-53.90i -70.22+21.57i 28.24-38.60i -20.73-01.23i -34.56+16.73i 26.01+31.97i .  
Here A is nonsymmetric and is treated as a band matrix, which must first be factorized by nag_lapack_zgbtrf (f07br).
function f07bv_example


fprintf('f07bv example results\n\n');

m  = int64(4);
kl = int64(1);
ku = int64(2);
ab = [ 0    + 0i,     0    + 0i,     0.97 - 2.84i,  0.59 - 0.48i;
       0    + 0i,    -2.05 - 0.85i, -3.99 + 4.01i,  3.33 - 1.04i;
      -1.65 + 2.26i, -1.48 - 1.75i, -1.06 + 1.94i, -0.46 - 1.72i;
       0    + 6.3i,  -0.77 + 2.83i,  4.48 - 1.09i,  0    + 0i];

% Convert ab to full representation a
[a, ab, ifail] = f01zd( ...
                        'u', kl, ku, complex(zeros(m, m)), ab);

% Exact Solution
nrhs = 2;
y = [ -3 + 2i,   1 + 6i;
       1 - 7i,  -7 - 4i;
      -5 + 4i,   3 + 5i;
       6 - 8i,  -8 + 2i];

% Evaluate RHS
b = a*y;

% Factorize
afb = [complex(zeros(kl,m)); ab];
[afb, ipiv, info]     = f07br( ...
                               m, kl, ku, afb);
% Solve
trans = 'N';
[x, info]             = f07bs( ...
                               trans, kl, ku, afb, ipiv, b);

% Iterative refinement
[x, ferr, berr, info] = f07bv( ...
                               trans, kl, ku, ab, afb, ipiv, b, x);

fprintf('Refined solution:\n');
disp(x);

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


f07bv example results

Refined solution:
  -3.0000 + 2.0000i   1.0000 + 6.0000i
   1.0000 - 7.0000i  -7.0000 - 4.0000i
  -5.0000 + 4.0000i   3.0000 + 5.0000i
   6.0000 - 8.0000i  -8.0000 + 2.0000i

Backward errors (machine dependent)
    5.4e-17    8.4e-17
Estimated forward error bounds
    3.6e-14    4.4e-14

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Chapter Contents
Chapter Introduction
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