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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dgbsv (f07ba)

## Purpose

nag_lapack_dgbsv (f07ba) 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.

## Syntax

[ab, ipiv, b, info] = f07ba(kl, ku, ab, b, 'n', n, 'nrhs_p', nrhs_p)
[ab, ipiv, b, info] = nag_lapack_dgbsv(kl, ku, ab, b, 'n', n, 'nrhs_p', nrhs_p)

## Description

nag_lapack_dgbsv (f07ba) uses the LU$LU$ decomposition with partial pivoting and row interchanges to factor A$A$ as A = PLU$A=PLU$, where P$P$ is a permutation matrix, L$L$ is a product of permutation 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
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## Parameters

### Compulsory Input Parameters

1:     kl – int64int32nag_int scalar
kl${k}_{l}$, the number of subdiagonals within the band of the matrix A$A$.
Constraint: kl0${\mathbf{kl}}\ge 0$.
2:     ku – int64int32nag_int scalar
ku${k}_{u}$, the number of superdiagonals within the band of the matrix 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$n$ coefficient 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$r$ right-hand side matrix B$B$.

### Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The first dimension of the array b.
n$n$, the number of linear equations, 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.
r$r$, the number of right-hand sides, 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 info0${\mathbf{info}}\ge 0$, 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 no constraints are violated, 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 ${\mathbf{INFO}}={\mathbf{0}}$, the n$n$ by r$r$ solution matrix X$X$.
4:     info – int64int32nag_int scalar
info = 0${\mathbf{info}}=0$ unless the function detects an error (see Section [Error Indicators and Warnings]).

## Error Indicators and Warnings

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

info = i${\mathbf{info}}=-i$
If info = i${\mathbf{info}}=-i$, parameter i$i$ had an illegal value on entry. The parameters are numbered as follows:
1: n, 2: kl, 3: ku, 4: nrhs_p, 5: ab, 6: ldab, 7: ipiv, 8: b, 9: ldb, 10: 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.
W INFO > 0${\mathbf{INFO}}>0$
If info = i${\mathbf{info}}=i$, 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.

## 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. See Section 4.4 of Anderson et al. (1999) for further details.
Following the use of nag_lapack_dgbsv (f07ba), nag_lapack_dgbcon (f07bg) can be used to estimate the condition number of A $A$ and nag_lapack_dgbrfs (f07bh) can be used to obtain approximate error bounds. Alternatives to nag_lapack_dgbsv (f07ba), which return condition and error estimates directly are nag_linsys_real_band_solve (f04bb) and nag_lapack_dgbsvx (f07bb).

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 ( 2 kl + ku ) r ) $\mathit{O}\left(n\left(2{k}_{l}+{k}_{u}\right)r\right)$ for the solution following the factorization.
The complex analogue of this function is nag_lapack_zgbsv (f07bn).

## Example

```function nag_lapack_dgbsv_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;
27.13;
-6.14;
10.5];
[abOut, ipiv, bOut, info] = nag_lapack_dgbsv(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
3.0000
1.0000
-4.0000

info =

0

```
```function f07ba_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;
27.13;
-6.14;
10.5];
[abOut, ipiv, bOut, info] = f07ba(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
3.0000
1.0000
-4.0000

info =

0

```