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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dgbtrs (f07be)

## Purpose

nag_lapack_dgbtrs (f07be) solves a real band system of linear equations with multiple right-hand sides,
 AX = B   or   ATX = B , $AX=B or ATX=B ,$
where A$A$ has been factorized by nag_lapack_dgbtrf (f07bd).

## Syntax

[b, info] = f07be(trans, kl, ku, ab, ipiv, b, 'n', n, 'nrhs_p', nrhs_p)
[b, info] = nag_lapack_dgbtrs(trans, kl, ku, ab, ipiv, b, 'n', n, 'nrhs_p', nrhs_p)

## Description

nag_lapack_dgbtrs (f07be) is used to solve a real band system of linear equations AX = B$AX=B$ or ATX = B${A}^{\mathrm{T}}X=B$, the function must be preceded by a call to nag_lapack_dgbtrf (f07bd) which computes the LU$LU$ factorization of A$A$ as A = PLU$A=PLU$. The solution is computed by forward and backward substitution.
If trans = 'N'${\mathbf{trans}}=\text{'N'}$, the solution is computed by solving PLY = B$PLY=B$ and then UX = Y$UX=Y$.
If trans = 'T'${\mathbf{trans}}=\text{'T'}$ or 'C'$\text{'C'}$, the solution is computed by solving UTY = B${U}^{\mathrm{T}}Y=B$ and then LTPTX = Y${L}^{\mathrm{T}}{P}^{\mathrm{T}}X=Y$.

## References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## Parameters

### Compulsory Input Parameters

1:     trans – string (length ≥ 1)
Indicates the form of the equations.
trans = 'N'${\mathbf{trans}}=\text{'N'}$
AX = B$AX=B$ is solved for X$X$.
trans = 'T'${\mathbf{trans}}=\text{'T'}$ or 'C'$\text{'C'}$
ATX = B${A}^{\mathrm{T}}X=B$ is solved for X$X$.
Constraint: trans = 'N'${\mathbf{trans}}=\text{'N'}$, 'T'$\text{'T'}$ or 'C'$\text{'C'}$.
2:     kl – int64int32nag_int scalar
kl${k}_{l}$, the number of subdiagonals within the band of the matrix A$A$.
Constraint: kl0${\mathbf{kl}}\ge 0$.
3:     ku – int64int32nag_int scalar
ku${k}_{u}$, the number of superdiagonals within the band of the matrix A$A$.
Constraint: ku0${\mathbf{ku}}\ge 0$.
4:     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 LU$LU$ factorization of A$A$, as returned by nag_lapack_dgbtrf (f07bd).
5:     ipiv( : $:$) – int64int32nag_int array
Note: the dimension of the array ipiv must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The pivot indices, as returned by nag_lapack_dgbtrf (f07bd).
6:     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 second dimension of the array ab.
n$n$, 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.
Constraint: nrhs0${\mathbf{nrhs}}\ge 0$.

ldab ldb

### Output Parameters

1:     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)$\mathit{ldb}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The n$n$ by r$r$ solution matrix X$X$.
2:     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

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: trans, 2: n, 3: kl, 4: ku, 5: nrhs_p, 6: ab, 7: ldab, 8: ipiv, 9: b, 10: ldb, 11: 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

For each right-hand side vector b$b$, the computed solution x$x$ is the exact solution of a perturbed system of equations (A + E)x = b$\left(A+E\right)x=b$, where
 |E| ≤ c(k)εP|L||U| , $|E|≤c(k)εP|L||U| ,$
c(k)$c\left(k\right)$ is a modest linear function of k = kl + ku + 1$k={k}_{l}+{k}_{u}+1$, and ε$\epsilon$ is the machine precision. This assumes kn$k\ll n$.
If $\stackrel{^}{x}$ is the true solution, then the computed solution x$x$ satisfies a forward error bound of the form
 (‖x − x̂‖∞)/(‖x‖∞) ≤ c(k)cond(A,x)ε $‖x-x^‖∞ ‖x‖∞ ≤c(k)cond(A,x)ε$
where cond(A,x) = |A1||A||x| / xcond(A) = |A1||A|κ(A)$\mathrm{cond}\left(A,x\right)={‖|{A}^{-1}||A||x|‖}_{\infty }/{‖x‖}_{\infty }\le \mathrm{cond}\left(A\right)={‖|{A}^{-1}||A|‖}_{\infty }\le {\kappa }_{\infty }\left(A\right)$.
Note that cond(A,x)$\mathrm{cond}\left(A,x\right)$ can be much smaller than cond(A)$\mathrm{cond}\left(A\right)$, and cond(AT)$\mathrm{cond}\left({A}^{\mathrm{T}}\right)$ can be much larger (or smaller) than cond(A)$\mathrm{cond}\left(A\right)$.
Forward and backward error bounds can be computed by calling nag_lapack_dgbrfs (f07bh), and an estimate for κ(A)${\kappa }_{\infty }\left(A\right)$ can be obtained by calling nag_lapack_dgbcon (f07bg) with norm = 'I'${\mathbf{norm}}=\text{'I'}$.

The total number of floating point operations is approximately 2n(2kl + ku)r$2n\left(2{k}_{l}+{k}_{u}\right)r$, assuming nkl$n\gg {k}_{l}$ and nku$n\gg {k}_{u}$.
This function may be followed by a call to nag_lapack_dgbrfs (f07bh) to refine the solution and return an error estimate.
The complex analogue of this function is nag_lapack_zgbtrs (f07bs).

## Example

```function nag_lapack_dgbtrs_example
trans = 'N';
m = int64(4);
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];
% Factorize A
[ab, ipiv, info] = nag_lapack_dgbtrf(m, kl, ku, ab);

% Compute Solution
[bOut, info] = nag_lapack_dgbtrs(trans, kl, ku, ab, ipiv, b)
```
```

bOut =

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

info =

0

```
```function f07be_example
trans = 'N';
m = int64(4);
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];
% Factorize A
[ab, ipiv, info] = f07bd(m, kl, ku, ab);

% Compute Solution
[bOut, info] = f07be(trans, kl, ku, ab, ipiv, b)
```
```

bOut =

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

info =

0

```