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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_linsys_real_band_solve (f04bb)

## Purpose

nag_linsys_real_band_solve (f04bb) computes the solution to a real system of linear equations AX = B$AX=B$, where A$A$ is an n$n$ by n$n$ band matrix, with kl${k}_{l}$ subdiagonals and ku${k}_{u}$ superdiagonals, and X$X$ and B$B$ are n$n$ by r$r$ matrices. An estimate of the condition number of A$A$ and an error bound for the computed solution are also returned.

## Syntax

[ab, ipiv, b, rcond, errbnd, ifail] = f04bb(kl, ku, ab, b, 'n', n, 'nrhs_p', nrhs_p)
[ab, ipiv, b, rcond, errbnd, ifail] = nag_linsys_real_band_solve(kl, ku, ab, b, 'n', n, 'nrhs_p', nrhs_p)

## Description

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

## 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:     kl – int64int32nag_int scalar
The number of subdiagonals kl${k}_{l}$, within the band of A$A$.
Constraint: kl0${\mathbf{kl}}\ge 0$.
2:     ku – int64int32nag_int scalar
The number of superdiagonals ku${k}_{u}$, within the band of A$A$.
Constraint: ku0${\mathbf{ku}}\ge 0$.
3:     ab(ldab, : $:$) – double array
The first dimension of the array ab must be at least 2 × kl + ku + 1$2×{\mathbf{kl}}+{\mathbf{ku}}+1$
The second dimension of the array must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The n$n$ by n$n$ matrix A$A$.
The matrix is stored in rows kl + 1${k}_{l}+1$ to 2kl + ku + 1$2{k}_{l}+{k}_{u}+1$; the first kl${k}_{l}$ rows need not be set, more precisely, the element Aij${A}_{ij}$ must be stored in
 ab(kl + ku + 1 + i − j,j) = Aij  for ​max (1,j − ku) ≤ i ≤ min (n,j + kl).$abkl+ku+1+i-jj=Aij for ​max(1,j-ku)≤i≤min(n,j+kl).$
See Section [Further Comments] for further details.
4:     b(ldb, : $:$) – double array
The first dimension of the array b must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array must be at least max (1,nrhs)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$
The n$n$ by r$r$ matrix of right-hand sides B$B$.

### Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The first dimension of the array b The second dimension of the array ab.
The number of linear equations n$n$, i.e., the order of the matrix A$A$.
Constraint: n0${\mathbf{n}}\ge 0$.
2:     nrhs_p – int64int32nag_int scalar
Default: The second dimension of the array b.
The number of right-hand sides r$r$, i.e., the number of columns of the matrix B$B$.
Constraint: nrhs0${\mathbf{nrhs}}\ge 0$.

ldab ldb

### Output Parameters

1:     ab(ldab, : $:$) – double array
The first dimension of the array ab will be 2 × kl + ku + 1$2×{\mathbf{kl}}+{\mathbf{ku}}+1$
The second dimension of the array will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
ldab2 × kl + ku + 1$\mathit{ldab}\ge 2×{\mathbf{kl}}+{\mathbf{ku}}+1$.
If ${\mathbf{ifail}}\ge {\mathbf{0}}$, ab stores details of the factorization.
The upper triangular band matrix U$U$, with kl + ku${k}_{l}+{k}_{u}$ superdiagonals, is stored in rows 1$1$ to kl + ku + 1${k}_{l}+{k}_{u}+1$ of the array, and the multipliers used to form the matrix L$L$ are stored in rows kl + ku + 2${k}_{l}+{k}_{u}+2$ to 2kl + ku + 1$2{k}_{l}+{k}_{u}+1$.
2:     ipiv(n) – int64int32nag_int array
If ${\mathbf{ifail}}\ge {\mathbf{0}}$, the pivot indices that define the permutation matrix P$P$; at the i$i$th step row i$i$ of the matrix was interchanged with row ipiv(i)${\mathbf{ipiv}}\left(i\right)$. ipiv(i) = i${\mathbf{ipiv}}\left(i\right)=i$ indicates a row interchange was not required.
3:     b(ldb, : $:$) – double array
The first dimension of the array b will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array will be max (1,nrhs)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$
ldbmax (1,n)$\mathit{ldb}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
If ${\mathbf{ifail}}={\mathbf{0}}$ or n + 1$\mathbf{n}+{\mathbf{1}}$, the n$n$ by r$r$ solution matrix X$X$.
4:     rcond – double scalar
If no constraints are violated, an estimate of the reciprocal of the condition number of the matrix A$A$, computed as rcond = 1 / (A1A11)${\mathbf{rcond}}=1/\left({‖A‖}_{1}{‖{A}^{-1}‖}_{1}\right)$.
5:     errbnd – double scalar
If ${\mathbf{ifail}}={\mathbf{0}}$ or n + 1$\mathbf{n}+{\mathbf{1}}$, an estimate of the forward error bound for a computed solution $\stackrel{^}{x}$, such that x1 / x1errbnd${‖\stackrel{^}{x}-x‖}_{1}/{‖x‖}_{1}\le {\mathbf{errbnd}}$, where $\stackrel{^}{x}$ is a column of the computed solution returned in the array b and x$x$ is the corresponding column of the exact solution X$X$. If rcond is less than machine precision, then errbnd is returned as unity.
6:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ 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.

ifail < 0andifail999${\mathbf{ifail}}<0 \text{and} {\mathbf{ifail}}\ne -999$
If ifail = i${\mathbf{ifail}}=-i$, the i$i$th argument had an illegal value.
ifail = 999${\mathbf{ifail}}=-999$
Allocation of memory failed. The integer allocatable memory required is n, and the double allocatable memory required is 3 × n$3×{\mathbf{n}}$. In this case the factorization and the solution X$X$ have been computed, but rcond and errbnd have not been computed.
W ifail > 0andifailN${\mathbf{ifail}}>0 \text{and} {\mathbf{ifail}}\le {\mathbf{N}}$
If ifail = i${\mathbf{ifail}}=i$, uii${u}_{ii}$ is exactly zero. The factorization has been completed, but the factor U$U$ is exactly singular, so the solution could not be computed.
W ifail = N + 1${\mathbf{ifail}}={\mathbf{N}}+1$
rcond is less than machine precision, so that the matrix A$A$ is numerically singular. A solution to the equations AX = B$AX=B$ has nevertheless been computed.

## Accuracy

The computed solution for a single right-hand side, $\stackrel{^}{x}$, satisfies an equation of the form
 (A + E) x̂ = b, $(A+E) x^=b,$
where
 ‖E‖1 = O(ε) ‖A‖1 $‖E‖1 = O(ε) ‖A‖1$
and ε$\epsilon$ is the machine precision. An approximate error bound for the computed solution is given by
 (‖x̂ − x‖1)/(‖x‖1) ≤ κ(A) (‖E‖1)/(‖A‖1) , $‖x^-x‖1 ‖x‖1 ≤ κ(A) ‖E‖1 ‖A‖1 ,$
where κ(A) = A11 A1 $\kappa \left(A\right)={‖{A}^{-1}‖}_{1}{‖A‖}_{1}$, the condition number of A$A$ with respect to the solution of the linear equations. nag_linsys_real_band_solve (f04bb) uses the approximation E1 = εA1${‖E‖}_{1}=\epsilon {‖A‖}_{1}$ 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$n=6$, kl = 1${k}_{l}=1$, and ku = 2${k}_{u}=2$. Storage of the band matrix A$A$ in the array ab:
 * * * + + + * * a13 a24 a35 a46 * a12 a23 a34 a45 a56 a11 a22 a33 a44 a55 a66 a21 a32 a43 a54 a65 *
$* * * + + + * * a13 a24 a35 a46 * a12 a23 a34 a45 a56 a11 a22 a33 a44 a55 a66 a21 a32 a43 a54 a65 *$
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 u14${u}_{14}$, u25${u}_{25}$ and u36${u}_{36}$.
The total number of floating point operations required to solve the equations AX = B$AX=B$ depends upon the pivoting required, but if nkl + ku$n\gg {k}_{l}+{k}_{u}$ then it is approximately bounded by O(nkl(kl + ku)) $\mathit{O}\left(n{k}_{l}\left({k}_{l}+{k}_{u}\right)\right)$ for the factorization and O(n(2kl + ku)r) $\mathit{O}\left(n\left(2{k}_{l}+{k}_{u}\right)r\right)$ 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 complex analogue of nag_linsys_real_band_solve (f04bb) is nag_linsys_complex_band_solve (f04cb).

## Example

```function nag_linsys_real_band_solve_example
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];
b = [4.42, -36.01;
27.13, -31.67;
-6.14, -1.16;
10.5, -25.82];
[abOut, ipiv, bOut, rcond, errbnd, ifail] = nag_linsys_real_band_solve(kl, ku, ab, b)
```
```

0         0         0   -2.1300
0         0   -2.7300    4.0700
0    2.4600    2.4600   -3.8391
-6.9800    2.5600   -5.9329   -0.7269
0.0330    0.9605    0.8057         0

ipiv =

2
3
3
4

bOut =

-2.0000    1.0000
3.0000   -4.0000
1.0000    7.0000
-4.0000   -2.0000

rcond =

0.0177

errbnd =

6.2626e-15

ifail =

0

```
```function f04bb_example
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];
b = [4.42, -36.01;
27.13, -31.67;
-6.14, -1.16;
10.5, -25.82];
[abOut, ipiv, bOut, rcond, errbnd, ifail] = f04bb(kl, ku, ab, b)
```
```

0         0         0   -2.1300
0         0   -2.7300    4.0700
0    2.4600    2.4600   -3.8391
-6.9800    2.5600   -5.9329   -0.7269
0.0330    0.9605    0.8057         0

ipiv =

2
3
3
4

bOut =

-2.0000    1.0000
3.0000   -4.0000
1.0000    7.0000
-4.0000   -2.0000

rcond =

0.0177

errbnd =

6.2626e-15

ifail =

0

```