NAG Toolbox: nag_lapack_dggbal (f08wh)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

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

Specifies the operations to be performed on matrices $A$ and $B$.

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

No balancing is done. Initialize $\mathbf{ilo}=1$, $\mathbf{ihi}=\mathbf{n}$, $\mathbf{lscale}\left(\mathit{i}\right)=1.0$ and $\mathbf{rscale}\left(\mathit{i}\right)=1.0$, for $\mathit{i}=1,2,\dots ,n$.

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

Only permutations are used in balancing.

$\mathbf{job}=\text{'S'}$

Only scalings are are used in balancing.

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

Both permutations and scalings are used in balancing.

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

2:     $\mathrm{a}\left(\mathit{lda},:\right)$ – double array

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

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

The $n$ by $n$ matrix $A$.

3:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – double 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 $\max \left(1,\mathbf{n}\right)$.

The $n$ by $n$ matrix $B$.

Optional Input Parameters

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

Default: the first dimension of the arrays a, b and the second dimension of the arrays a, b.

$n$, the order of the matrices $A$ and $B$.

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

Output Parameters

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

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

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

a stores the balanced matrix. If $\mathbf{job}=\text{'N'}$, a is not referenced.

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

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

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

b stores the balanced matrix. If $\mathbf{job}=\text{'N'}$, b is not referenced.

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

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

$i_{\mathrm{lo}}$ and $i_{\mathrm{hi}}$ are set such that $\mathbf{a}\left(i,j\right)=0$ and $\mathbf{b}\left(i,j\right)=0$ if $i>j$ and $1\le j<i_{\mathrm{lo}}$ or $i_{\mathrm{hi}}<i\le n$.

If $\mathbf{job}=\text{'N'}$ or $\text{'S'}$, $i_{\mathrm{lo}}=1$ and $i_{\mathrm{hi}}=n$.

5:     $\mathrm{lscale}\left(\mathbf{n}\right)$ – double array

Details of the permutations and scaling factors applied to the left side of the matrices $A$ and $B$. If $P_i$ is the index of the row interchanged with row $i$ and $d_i$ is the scaling factor applied to row $i$, then

• $\mathbf{lscale}\left(\mathit{i}\right)=P_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathit{i}}_{\mathrm{lo}}-1$;
• $\mathbf{lscale}\left(\mathit{i}\right)={\mathit{d}}_{\mathit{i}}$, for $\mathit{i}={\mathit{i}}_{\mathrm{lo}},\dots ,{\mathit{i}}_{\mathrm{hi}}$;
• $\mathbf{lscale}\left(\mathit{i}\right)=P_{\mathit{i}}$, for $\mathit{i}={\mathit{i}}_{\mathrm{hi}}+1,\dots ,n$.

The order in which the interchanges are made is $n$ to $i_{\mathrm{hi}}+1$, then $1$ to $i_{\mathrm{lo}}-1$.

6:     $\mathrm{rscale}\left(\mathbf{n}\right)$ – double array

Details of the permutations and scaling factors applied to the right side of the matrices $A$ and $B$.

If $P_j$ is the index of the column interchanged with column $j$ and ${\hat{d}}_j$ is the scaling factor applied to column $j$, then

• $\mathbf{rscale}\left(\mathit{j}\right)=P_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,{\mathit{i}}_{\mathrm{lo}}-1$;
• $\mathbf{rscale}\left(\mathit{j}\right)={\hat{d}}_{\mathit{j}}$, for $\mathit{j}=i_{\mathrm{lo}},\dots ,i_{\mathrm{hi}}$;
• $\mathbf{rscale}\left(\mathit{j}\right)=P_{\mathit{j}}$, for $\mathit{j}=i_{\mathrm{hi}}+1,\dots ,n$.

The order in which the interchanges are made is $n$ to $i_{\mathrm{hi}}+1$, then $1$ to $i_{\mathrm{lo}}-1$.

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: job, 2: n, 3: a, 4: lda, 5: b, 6: ldb, 7: ilo, 8: ihi, 9: lscale, 10: rscale, 11: work, 12: 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: f08wh_example

function f08wh_example


fprintf('f08wh example results\n\n');

a = [1,  1,   1,    1,    1;
     2,  4,   8,   16,   32;
     3,  9,  27,   81,  243;
     4, 16,  64,  256, 1024;
     5, 25, 125,  625, 3125];
b = [1,  2,   3,    4,    5;
     1,  4,   9,   16,   25;
     1,  8,  27,   64,  125;
     1, 16,  81,  256,  625;
     1, 32, 243, 1024, 3125];

% Balance matrix pair (A,B)
job = 'Balance';
[a, b, ilo, ihi, lscale, rscale, info] = ...
  f08wh(job, a, b);

fprintf('Lower index for permuted central block matrix pair = %d\n', ilo);
fprintf('Upper index for permuted central block matrix pair = %d\n', ihi);
disp('Contents of left scaling vector:');
disp(lscale');
disp('Contents of right scaling vector:');
disp(rscale');
disp('Balanced matrix A');
disp(a);
disp('Balanced matrix B');
disp(b);

f08wh example results

Lower index for permuted central block matrix pair = 1
Upper index for permuted central block matrix pair = 5
Contents of left scaling vector:
    1.0000    1.0000    0.1000    0.1000    0.1000

Contents of right scaling vector:
    1.0000    1.0000    0.1000    0.1000    0.1000

Balanced matrix A
    1.0000    1.0000    0.1000    0.1000    0.1000
    2.0000    4.0000    0.8000    1.6000    3.2000
    0.3000    0.9000    0.2700    0.8100    2.4300
    0.4000    1.6000    0.6400    2.5600   10.2400
    0.5000    2.5000    1.2500    6.2500   31.2500

Balanced matrix B
    1.0000    2.0000    0.3000    0.4000    0.5000
    1.0000    4.0000    0.9000    1.6000    2.5000
    0.1000    0.8000    0.2700    0.6400    1.2500
    0.1000    1.6000    0.8100    2.5600    6.2500
    0.1000    3.2000    2.4300   10.2400   31.2500