NAG Toolbox: nag_lapack_dgbtrf (f07bd)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

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

$m$, the number of rows of the matrix $A$.

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

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

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

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

The $m$ by $n$ matrix $A$.

The matrix is stored in rows $k_l+1$ to $2k_l+k_u+1$; the first $k_l$ rows need not be set, more precisely, the element $A_{ij}$ must be stored in

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

See Further Comments in nag_lapack_dgbsv (f07ba) for further details.

Optional Input Parameters

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

Default: the second dimension of the array ab.

$n$, the number of columns of the matrix $A$.

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

Output Parameters

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

The first dimension of the array ab will be $2\times \mathbf{kl}+\mathbf{ku}+1$.

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

If $\mathbf{info}\ge 0$, ab stores details of the factorization.

The upper triangular band matrix $U$, with $k_l+k_u$ superdiagonals, is stored in rows $1$ to $k_l+k_u+1$ of the array, and the multipliers used to form the matrix $L$ are stored in rows $k_l+k_u+2$ to $2k_l+k_u+1$.

2:     $\mathrm{ipiv}\left(\min \left(\mathbf{m},\mathbf{n}\right)\right)$ – int64int32nag_int array

The pivot indices that define the permutation matrix. At the $\mathit{i}$th step, if $\mathbf{ipiv}\left(\mathit{i}\right)>\mathit{i}$ then row $\mathit{i}$ of the matrix $A$ was interchanged with row $\mathbf{ipiv}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,\min \left(m,n\right)$. $\mathbf{ipiv}\left(i\right)\le i$ indicates that, at the $i$th step, a row interchange was not required.

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

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

Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_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.

W  $\mathbf{info}>0$

Element $\_$ of the diagonal is exactly zero. The factorization has been completed, but the factor $U$ is exactly singular, and division by zero will occur if it is used to solve a system of equations.

Accuracy

Further Comments

Example

Open in the MATLAB editor: f07bd_example

function f07bd_example


fprintf('f07bd example results\n\n');

m  = int64(4);
kl = int64(1);
ku = int64(2);
ab = [ 0,    0,     0,     0;
       0,    0,    -3.66, -2.13;
       0,    2.54, -2.73,  4.07;
      -0.23, 2.46,  2.46, -3.82;
      -6.98, 2.56, -4.78,  0];

[abf, ipiv, info] = f07bd( ...
                           m, kl, ku, ab);

mtitle = 'Details of factorization';
[ifail] = x04ce( ...
                 m, m, kl, kl+ku, abf, mtitle);
fprintf('\n');
disp('Pivot indices');
disp(double(ipiv'));

f07bd example results

 Details of factorization
             1          2          3          4
 1     -6.9800     2.4600    -2.7300    -2.1300
 2      0.0330     2.5600     2.4600     4.0700
 3                 0.9605    -5.9329    -3.8391
 4                            0.8057    -0.7269

Pivot indices
     2     3     3     4