NAG Toolbox: nag_lapack_dpbequ (f07hf)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

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

Indicates whether the upper or lower triangular part of $A$ is stored in the array ab, as follows:

$\mathbf{uplo}=\text{'U'}$

The upper triangle of $A$ is stored.

$\mathbf{uplo}=\text{'L'}$

The lower triangle of $A$ is stored.

Constraint: $\mathbf{uplo}=\text{'U'}$ or $\text{'L'}$.

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

$k_d$, the number of superdiagonals of the matrix $A$ if $\mathbf{uplo}=\text{'U'}$, or the number of subdiagonals if $\mathbf{uplo}=\text{'L'}$.

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

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

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

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

The upper or lower triangle of the symmetric positive definite band matrix $A$ whose scaling factors are to be computed.

The matrix is stored in rows $1$ to $k_d+1$, more precisely,

• if $\mathbf{uplo}=\text{'U'}$, the elements of the upper triangle of $A$ within the band must be stored with element $A_{ij}$ in $\mathbf{ab}\left(k_d+1+i-j,j\right)\text{ for }\max \left(1,j-k_d\right)\le i\le j$;
• if $\mathbf{uplo}=\text{'L'}$, the elements of the lower triangle of $A$ within the band must be stored with element $A_{ij}$ in $\mathbf{ab}\left(1+i-j,j\right)\text{ for }j\le i\le \min \left(n,j+k_d\right)\text{.}$

Only the elements of the array ab corresponding to the diagonal elements of $A$ are referenced. (Row $\left(k_d+1\right)$ of ab when $\mathbf{uplo}=\text{'U'}$, row $1$ of ab when $\mathbf{uplo}=\text{'L'}$.)

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

Output Parameters

1:     $\mathrm{s}\left(\mathbf{n}\right)$ – double array

If $\mathbf{info}=\mathbf{0}$, s contains the diagonal elements of the scaling matrix $S$.

2:     $\mathrm{scond}$ – double scalar

If $\mathbf{info}=\mathbf{0}$, scond contains the ratio of the smallest value of s to the largest value of s. If $\mathbf{scond}\ge 0.1$ and amax is neither too large nor too small, it is not worth scaling by $S$.

3:     $\mathrm{amax}$ – double scalar

$\max \left|a_{ij}\right|$. If amax is very close to overflow or underflow, the matrix $A$ should be scaled.

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.

   $\mathbf{info}>0$

The $\_$th diagonal element of $A$ is not positive (and hence $A$ cannot be positive definite).

Accuracy

Further Comments

Example

Open in the MATLAB editor: f07hf_example

function f07hf_example


fprintf('f07hf example results\n\n');

% Symmetric A (one lower/upper off-diagonal) in banded form 
uplo = 'U';
kd = int64(1);
n  = int64(4);
ab = [0,    2.68e10, -2.39e10, -2.22;
      5.49, 5.63e20,  2.60,     5.17];

% Scale A
[s, scond, amax, info] = f07hf( ...
                                uplo, kd, ab);

fprintf('scond = %8.1e, amax = %8.1e\n\n', scond, amax);
disp('Diagonal scaling factors');
fprintf('%10.1e',s);
fprintf('\n\n');

% Apply scalings
asp = ab*diag(s);
for i = 1:n
  for j = 0:min(kd,n-i)
    asp(kd+1-j,i+j) = s(i)*asp(kd+1-j,i+j);
  end
end

kl = int64(0);
[ifail] = x04ce( ...
                 n, n, kl, kd, asp, 'Scaled matrix');

f07hf example results

scond =  6.8e-11, amax =  5.6e+20

Diagonal scaling factors
   4.3e-01   4.2e-11   6.2e-01   4.4e-01

 Scaled matrix
             1          2          3          4
 1      1.0000     0.4821
 2                 1.0000    -0.6247
 3                            1.0000    -0.6055
 4                                       1.0000