NAG Toolbox: nag_lapack_dgbbrd (f08le)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

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

Indicates whether the matrices $Q$ and/or $P^{\mathrm{T}}$ are generated.

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

Neither $Q$ nor $P^{\mathrm{T}}$ is generated.

$\mathbf{vect}=\text{'Q'}$

$Q$ is generated.

$\mathbf{vect}=\text{'P'}$

$P^{\mathrm{T}}$ is generated.

$\mathbf{vect}=\text{'B'}$

Both $Q$ and $P^{\mathrm{T}}$ are generated.

Constraint: $\mathbf{vect}=\text{'N'}$, $\text{'Q'}$, $\text{'P'}$ or $\text{'B'}$.

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

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

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

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

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

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

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

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

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

5:     $\mathrm{ab}\left(\mathit{ldab},:\right)$ – double 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 $m$ by $n$ band matrix $A$.

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

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

6:     $\mathrm{c}\left(\mathit{ldc},:\right)$ – double array

The first dimension, $\mathit{ldc}$, of the array c must satisfy

• if $\mathbf{ncc}>0$, $\mathit{ldc}\ge \max \left(1,\mathbf{m}\right)$;
• if $\mathbf{ncc}=0$, $\mathit{ldc}\ge 1$.

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

An $m$ by $n_C$ matrix $C$.

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

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

Default: the second dimension of the array c.

$n_C$, the number of columns of the matrix $C$.

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

Output Parameters

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

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

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

ab stores values generated during the reduction.

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

The diagonal elements of the bidiagonal matrix $B$.

3:     $\mathrm{e}\left(\min \left(\mathbf{m},\mathbf{n}\right)-1\right)$ – double array

The superdiagonal elements of the bidiagonal matrix $B$.

4:     $\mathrm{q}\left(\mathit{ldq},:\right)$ – double array

The first dimension, $\mathit{ldq}$, of the array q will be

• if $\mathbf{vect}=\text{'Q'}$ or $\text{'B'}$, $\mathit{ldq}=\max \left(1,\mathbf{m}\right)$;
• otherwise $\mathit{ldq}=1$.

The second dimension of the array q will be $\max \left(1,\mathbf{m}\right)$ if $\mathbf{vect}=\text{'Q'}$ or $\text{'B'}$ and $1$ otherwise.

If $\mathbf{vect}=\text{'Q'}$ or $\text{'B'}$, contains the $m$ by $m$ orthogonal matrix $Q$.

If $\mathbf{vect}=\text{'N'}$ or $\text{'P'}$, q is not referenced.

5:     $\mathrm{pt}\left(\mathit{ldpt},:\right)$ – double array

The first dimension, $\mathit{ldpt}$, of the array pt will be

• if $\mathbf{vect}=\text{'P'}$ or $\text{'B'}$, $\mathit{ldpt}=\max \left(1,\mathbf{n}\right)$;
• otherwise $\mathit{ldpt}=1$.

The second dimension of the array pt will be $\max \left(1,\mathbf{n}\right)$ if $\mathbf{vect}=\text{'P'}$ or $\text{'B'}$ and $1$ otherwise.

The $n$ by $n$ orthogonal matrix $P^{\mathrm{T}}$, if $\mathbf{vect}=\text{'P'}$ or $\text{'B'}$. If $\mathbf{vect}=\text{'N'}$ or $\text{'Q'}$, pt is not referenced.

6:     $\mathrm{c}\left(\mathit{ldc},:\right)$ – double array

The first dimension, $\mathit{ldc}$, of the array c will be

• if $\mathbf{ncc}>0$, $\mathit{ldc}=\max \left(1,\mathbf{m}\right)$;
• if $\mathbf{ncc}=0$, $\mathit{ldc}=1$.

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

c stores $Q^{\mathrm{T}}C$. If $\mathbf{ncc}=0$, c is not referenced.

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

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

Error Indicators and Warnings

   $\mathbf{info}=-i$

If $\mathbf{info}=-i$, parameter $i$ had an illegal value on entry. The parameters are numbered as follows:

1: vect, 2: m, 3: n, 4: ncc, 5: kl, 6: ku, 7: ab, 8: ldab, 9: d, 10: e, 11: q, 12: ldq, 13: pt, 14: ldpt, 15: c, 16: ldc, 17: work, 18: info.

It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.

Accuracy

Further Comments

Example

Open in the MATLAB editor: f08le_example

function f08le_example


fprintf('f08le example results\n\n');

% Banded matrix A stored in banded format
m  = int64(6);
n  = int64(4);
kl = int64(2);
ku = int64(1);
ab = [ 0.00, -1.28, -0.31, -0.35;
      -0.57,  1.08,  0.40,  0.08;
      -1.93,  0.24, -0.66, -2.13;
       2.30,  0.64,  0.15,  0.50];

% Reduce A to bidiagonal form
c  = [];
vect = 'No Q or PT';
[abf, d, e, q, pt, c, info] = f08le( ...
                                     vect, m, kl, ku, ab, c);

fprintf('Diagonal:\n');
fprintf(' %8.4f',d);
fprintf('\nOff-diagonal (absolute values):\n');
fprintf(' %8.4f',abs(e));
fprintf('\n');

f08le example results

Diagonal:
   3.0561   1.5259   0.9690   1.5685
Off-diagonal (absolute values):
   0.6206   1.2353   1.1240