NAG Toolbox: nag_lapack_zgbrfs (f07bv)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{trans}$ – string (length ≥ 1)

Indicates the form of the linear equations for which $X$ is the computed solution as follows:

$\mathbf{trans}=\text{'N'}$

The linear equations are of the form $AX=B$.

$\mathbf{trans}=\text{'T'}$

The linear equations are of the form $A^{\mathrm{T}}X=B$.

$\mathbf{trans}=\text{'C'}$

The linear equations are of the form $A^{\mathrm{H}}X=B$.

Constraint: $\mathbf{trans}=\text{'N'}$, $\text{'T'}$ or $\text{'C'}$.

2:     $\mathrm{kl}$ – int64int32nag_int scalar

$k_l$, the number of subdiagonals within the band of the matrix $A$.

Constraint: $\mathbf{kl}\ge 0$.

3:     $\mathrm{ku}$ – int64int32nag_int scalar

$k_u$, the number of superdiagonals within the band of the matrix $A$.

Constraint: $\mathbf{ku}\ge 0$.

4:     $\mathrm{ab}\left(\mathit{ldab},:\right)$ – complex array

The first dimension of the array ab must be at least $\mathbf{kl}+\mathbf{ku}+1$.

The second dimension of the array ab must be at least $\max \left(1,\mathbf{n}\right)$.

The original $n$ by $n$ band matrix $A$ as supplied to nag_lapack_zgbtrf (f07br).

The matrix is stored in rows $1$ to $k_l+k_u+1$, more precisely, the element $A_{ij}$ must be stored in

$$\mathbf{ab}\left(k_u+1+i-j,j\right)\text{\hspace{1em} for }\max \left(1,j-k_u\right)\le i\le \min \left(n,j+k_l\right)\text{.}$$

See Further Comments in nag_lapack_zgbsv (f07bn) for further details.

5:     $\mathrm{afb}\left(\mathit{ldafb},:\right)$ – complex array

The first dimension of the array afb must be at least $2\times \mathbf{kl}+\mathbf{ku}+1$.

The second dimension of the array afb must be at least $\max \left(1,\mathbf{n}\right)$.

The $LU$ factorization of $A$, as returned by nag_lapack_zgbtrf (f07br).

6:     $\mathrm{ipiv}\left(:\right)$ – int64int32nag_int array

The dimension of the array ipiv must be at least $\max \left(1,\mathbf{n}\right)$

The pivot indices, as returned by nag_lapack_zgbtrf (f07br).

7:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – complex array

The first dimension of the array b must be at least $\max \left(1,\mathbf{n}\right)$.

The second dimension of the array b must be at least $\max \left(1,\mathbf{nrhs\_p}\right)$.

The $n$ by $r$ right-hand side matrix $B$.

8:     $\mathrm{x}\left(\mathit{ldx},:\right)$ – complex array

The first dimension of the array x must be at least $\max \left(1,\mathbf{n}\right)$.

The second dimension of the array x must be at least $\max \left(1,\mathbf{nrhs\_p}\right)$.

The $n$ by $r$ solution matrix $X$, as returned by nag_lapack_zgbtrs (f07bs).

Optional Input Parameters

1:     $\mathrm{n}$ – int64int32nag_int scalar

Default: the second dimension of the array ab.

$n$, the order of the matrix $A$.

Constraint: $\mathbf{n}\ge 0$.

2:     $\mathrm{nrhs\_p}$ – int64int32nag_int scalar

Default: the second dimension of the arrays b, x.

$r$, the number of right-hand sides.

Constraint: $\mathbf{nrhs\_p}\ge 0$.

Output Parameters

1:     $\mathrm{x}\left(\mathit{ldx},:\right)$ – complex array

The first dimension of the array x will be $\max \left(1,\mathbf{n}\right)$.

The second dimension of the array x will be $\max \left(1,\mathbf{nrhs\_p}\right)$.

The improved solution matrix $X$.

2:     $\mathrm{ferr}\left(\mathbf{nrhs\_p}\right)$ – double array

$\mathbf{ferr}\left(\mathit{j}\right)$ contains an estimated error bound for the $\mathit{j}$th solution vector, that is, the $\mathit{j}$th column of $X$, for $\mathit{j}=1,2,\dots ,r$.

3:     $\mathrm{berr}\left(\mathbf{nrhs\_p}\right)$ – double array

$\mathbf{berr}\left(\mathit{j}\right)$ contains the component-wise backward error bound $\beta$ for the $\mathit{j}$th solution vector, that is, the $\mathit{j}$th column of $X$, for $\mathit{j}=1,2,\dots ,r$.

4:     $\mathrm{info}$ – int64int32nag_int scalar

$\mathbf{info}=0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

   $\mathbf{info}<0$

If $\mathbf{info}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

Accuracy

Further Comments

Example

Open in the MATLAB editor: f07bv_example

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