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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_linsys_complex_band_solve (f04cb)

## Purpose

nag_linsys_complex_band_solve (f04cb) computes the solution to a complex system of linear equations $AX=B$, where $A$ is an $n$ by $n$ band matrix, with ${k}_{l}$ subdiagonals and ${k}_{u}$ superdiagonals, and $X$ and $B$ are $n$ by $r$ matrices. An estimate of the condition number of $A$ and an error bound for the computed solution are also returned.

## Syntax

[ab, ipiv, b, rcond, errbnd, ifail] = f04cb(kl, ku, ab, b, 'n', n, 'nrhs_p', nrhs_p)
[ab, ipiv, b, rcond, errbnd, ifail] = nag_linsys_complex_band_solve(kl, ku, ab, b, 'n', n, 'nrhs_p', nrhs_p)

## Description

The $LU$ decomposition with partial pivoting and row interchanges is used to factor $A$ as $A=PLU$, where $P$ is a permutation matrix, $L$ is the product of permutation matrices and unit lower triangular matrices with ${k}_{l}$ subdiagonals, and $U$ is upper triangular with $\left({k}_{l}+{k}_{u}\right)$ superdiagonals. The factored form of $A$ is then used to solve the system of equations $AX=B$.

## References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug
Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{kl}$int64int32nag_int scalar
The number of subdiagonals ${k}_{l}$, within the band of $A$.
Constraint: ${\mathbf{kl}}\ge 0$.
2:     $\mathrm{ku}$int64int32nag_int scalar
The number of superdiagonals ${k}_{u}$, within the band of $A$.
Constraint: ${\mathbf{ku}}\ge 0$.
3:     $\mathrm{ab}\left(\mathit{ldab},:\right)$ – complex array
The first dimension of the array ab must be at least $2×{\mathbf{kl}}+{\mathbf{ku}}+1$.
The second dimension of the array ab must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The $n$ by $n$ matrix $A$.
The matrix is stored in rows ${k}_{l}+1$ to $2{k}_{l}+{k}_{u}+1$; the first ${k}_{l}$ rows need not be set, more precisely, the element ${A}_{ij}$ must be stored in
 $abkl+ku+1+i-jj=Aij for ​max1,j-ku≤i≤minn,j+kl.$
See Further Comments for further details.
4:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – complex array
The first dimension of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs_p}}\right)$.
The $n$ by $r$ matrix of right-hand sides $B$.

### Optional Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
Default: the first dimension of the array b and the second dimension of the array ab.
The number of linear equations $n$, i.e., the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
2:     $\mathrm{nrhs_p}$int64int32nag_int scalar
Default: the second dimension of the array b.
The number of right-hand sides $r$, i.e., the number of columns of the matrix $B$.
Constraint: ${\mathbf{nrhs_p}}\ge 0$.

### Output Parameters

1:     $\mathrm{ab}\left(\mathit{ldab},:\right)$ – complex array
The first dimension of the array ab will be $2×{\mathbf{kl}}+{\mathbf{ku}}+1$.
The second dimension of the array ab will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
If ${\mathbf{ifail}}\ge {\mathbf{0}}$, 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 $2{k}_{l}+{k}_{u}+1$.
2:     $\mathrm{ipiv}\left({\mathbf{n}}\right)$int64int32nag_int array
If ${\mathbf{ifail}}\ge {\mathbf{0}}$, the pivot indices that define the permutation matrix $P$; at the $i$th step row $i$ of the matrix was interchanged with row ${\mathbf{ipiv}}\left(i\right)$. ${\mathbf{ipiv}}\left(i\right)=i$ indicates a row interchange was not required.
3:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – complex array
The first dimension of the array b will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array b will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs_p}}\right)$.
If ${\mathbf{ifail}}={\mathbf{0}}$ or $\mathbf{n}+{\mathbf{1}}$, the $n$ by $r$ solution matrix $X$.
4:     $\mathrm{rcond}$ – double scalar
If ${\mathbf{ifail}}\ge {\mathbf{0}}$, an estimate of the reciprocal of the condition number of the matrix $A$, computed as ${\mathbf{rcond}}=\left({‖A‖}_{1}{‖{A}^{-1}‖}_{1}\right)$.
5:     $\mathrm{errbnd}$ – double scalar
If ${\mathbf{ifail}}={\mathbf{0}}$ or $\mathbf{n}+{\mathbf{1}}$, an estimate of the forward error bound for a computed solution $\stackrel{^}{x}$, such that ${‖\stackrel{^}{x}-x‖}_{1}/{‖x‖}_{1}\le {\mathbf{errbnd}}$, where $\stackrel{^}{x}$ is a column of the computed solution returned in the array b and $x$ is the corresponding column of the exact solution $X$. If rcond is less than machine precision, then errbnd is returned as unity.
6:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Errors or warnings detected by the function:

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

${\mathbf{ifail}}>0 \text{and} {\mathbf{ifail}}\le {\mathbf{n}}$
Diagonal element $_$ of the upper triangular factor is zero. The factorization has been completed, but the solution could not be computed.
W  ${\mathbf{ifail}}={\mathbf{n}}+1$
A solution has been computed, but rcond is less than machine precision so that the matrix $A$ is numerically singular.
${\mathbf{ifail}}=-1$
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=-2$
Constraint: ${\mathbf{kl}}\ge 0$.
${\mathbf{ifail}}=-3$
Constraint: ${\mathbf{ku}}\ge 0$.
${\mathbf{ifail}}=-4$
Constraint: ${\mathbf{nrhs_p}}\ge 0$.
${\mathbf{ifail}}=-6$
Constraint: $\mathit{ldab}\ge 2×{\mathbf{kl}}+{\mathbf{ku}}+1$.
${\mathbf{ifail}}=-9$
Constraint: $\mathit{ldb}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
The double allocatable memory required is n, and the complex allocatable memory required is $2×{\mathbf{n}}$. In this case the factorization and the solution $X$ have been computed, but rcond and errbnd have not been computed.

## Accuracy

The computed solution for a single right-hand side, $\stackrel{^}{x}$, satisfies an equation of the form
 $A+E x^=b,$
where
 $E1=Oε A1$
and $\epsilon$ is the machine precision. An approximate error bound for the computed solution is given by
 $x^-x1 x1 ≤ κA E1 A1 ,$
where $\kappa \left(A\right)={‖{A}^{-1}‖}_{1}{‖A‖}_{1}$, the condition number of $A$ with respect to the solution of the linear equations. nag_linsys_complex_band_solve (f04cb) uses the approximation ${‖E‖}_{1}=\epsilon {‖A‖}_{1}$ to estimate errbnd. See Section 4.4 of Anderson et al. (1999) for further details.

The band storage scheme for the array ab is illustrated by the following example, when $n=6$, ${k}_{l}=1$, and ${k}_{u}=2$. Storage of the band matrix $A$ in the array ab:
 $* * * + + + * * a13 a24 a35 a46 * a12 a23 a34 a45 a56 a11 a22 a33 a44 a55 a66 a21 a32 a43 a54 a65 *$
Array elements marked $*$ need not be set and are not referenced by the function. Array elements marked + need not be set, but are defined on exit from the function and contain the elements ${u}_{14}$, ${u}_{25}$ and ${u}_{36}$.
The total number of floating-point operations required to solve the equations $AX=B$ depends upon the pivoting required, but if $n\gg {k}_{l}+{k}_{u}$ then it is approximately bounded by $\mathit{O}\left(n{k}_{l}\left({k}_{l}+{k}_{u}\right)\right)$ for the factorization and $\mathit{O}\left(n\left(2{k}_{l}+{k}_{u}\right),r\right)$ for the solution following the factorization. The condition number estimation typically requires between four and five solves and never more than eleven solves, following the factorization.
In practice the condition number estimator is very reliable, but it can underestimate the true condition number; see Section 15.3 of Higham (2002) for further details.
The real analogue of nag_linsys_complex_band_solve (f04cb) is nag_linsys_real_band_solve (f04bb).

## Example

This example solves the equations
 $AX=B,$
where $A$ is the band matrix
 $A= -1.65+2.26i -2.05-0.85i 0.97-2.84i 0.00i+0.00 0.00+6.30i -1.48-1.75i -3.99+4.01i 0.59-0.48i 0.00i+0.00 -0.77+2.83i -1.06+1.94i 3.33-1.04i 0.00i+0.00 0.00i+0.00 4.48-1.09i -0.46-1.72i$
and
 $B= -1.06+21.50i 12.85+02.84i -22.72-53.90i -70.22+21.57i 28.24-38.60i -20.73-01.23i -34.56+16.73i 26.01+31.97i .$
An estimate of the condition number of $A$ and an approximate error bound for the computed solutions are also printed.
```function f04cb_example

fprintf('f04cb example results\n\n');

% Solve complex Ax = b for banded A with error bound and condition number
kl = int64(1);
ku = int64(2);
cz = complex(0);
ab = [ cz,              cz,             cz,            cz;
cz,              cz,             0.97 - 2.84i,  0.59 - 0.48i;
cz,             -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    +  6.30i,  -0.77 +  2.83i,  4.48 - 1.09i,  cz          ];
b = [ -1.06 + 21.50i,  12.85 +  2.84i;
-22.72 - 53.90i, -70.22 + 21.57i;
28.24 - 38.60i, -20.73 -  1.23i;
-34.56 + 16.73i,  26.01 + 31.97i];

[ab, ipiv, x, rcond, errbnd, ifail] = ...
f04cb(kl, ku, ab, b);

disp('Solution');
disp(x);
disp('Estimate of condition number');
fprintf('%10.1f\n\n',1/rcond);
disp('Estimate of error bound for computed solutions');
fprintf('%10.1e\n\n',errbnd);

```
```f04cb example results

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

Estimate of condition number
104.2

Estimate of error bound for computed solutions
1.2e-14

```