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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_dgbrfs (f07bh)

Purpose

nag_lapack_dgbrfs (f07bh) returns error bounds for the solution of a real band system of linear equations with multiple right-hand sides, AX = B$AX=B$ or ATX = B${A}^{\mathrm{T}}X=B$. It improves the solution by iterative refinement, in order to reduce the backward error as much as possible.

Syntax

[x, ferr, berr, info] = f07bh(trans, kl, ku, ab, afb, ipiv, b, x, 'n', n, 'nrhs_p', nrhs_p)
[x, ferr, berr, info] = nag_lapack_dgbrfs(trans, kl, ku, ab, afb, ipiv, b, x, 'n', n, 'nrhs_p', nrhs_p)

Description

nag_lapack_dgbrfs (f07bh) returns the backward errors and estimated bounds on the forward errors for the solution of a real band system of linear equations with multiple right-hand sides AX = B$AX=B$ or ATX = B${A}^{\mathrm{T}}X=B$. The function handles each right-hand side vector (stored as a column of the matrix B$B$) independently, so we describe the function of nag_lapack_dgbrfs (f07bh) in terms of a single right-hand side b$b$ and solution x$x$.
Given a computed solution x$x$, the function computes the component-wise backward error β$\beta$. This is the size of the smallest relative perturbation in each element of A$A$ and b$b$ such that x$x$ is the exact solution of a perturbed system
 (A + δA)x = b + δb |δaij| ≤ β|aij|   and   |δbi| ≤ β|bi| .
$(A+δA)x=b+δb |δaij|≤β|aij| and |δbi|≤β|bi| .$
Then the function estimates a bound for the component-wise forward error in the computed solution, defined by:
 max |xi − x̂i| / max |xi| i i
$maxi | xi - x^i | / maxi |xi|$
where $\stackrel{^}{x}$ is the true solution.
For details of the method, see the F07 Chapter Introduction.

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 linear equations for which X$X$ is the computed solution.
trans = 'N'${\mathbf{trans}}=\text{'N'}$
The linear equations are of the form AX = B$AX=B$.
trans = 'T'${\mathbf{trans}}=\text{'T'}$ or 'C'$\text{'C'}$
The linear equations are of the form ATX = B${A}^{\mathrm{T}}X=B$.
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 kl + ku + 1${\mathbf{kl}}+{\mathbf{ku}}+1$
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 original n$n$ by n$n$ band matrix A$A$ as supplied to nag_lapack_dgbtrf (f07bd).
The matrix is stored in rows 1$1$ to kl + ku + 1${k}_{l}+{k}_{u}+1$, more precisely, the element Aij${A}_{ij}$ must be stored in
 ab(ku + 1 + i − j,j)  for ​max (1,j − ku) ≤ i ≤ min (n,j + kl).$abku+1+i-jj for ​max(1,j-ku)≤i≤min(n,j+kl).$
See Section [Further Comments] in (f07ba) for further details.
5:     afb(ldafb, : $:$) – double array
The first dimension of the array afb 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).
6:     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).
7:     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$B$.
8:     x(ldx, : $:$) – double array
The first dimension of the array x 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$r$ solution matrix X$X$, as returned by nag_lapack_dgbtrs (f07be).

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 arrays b, x.
r$r$, the number of right-hand sides.
Constraint: nrhs0${\mathbf{nrhs}}\ge 0$.

Input Parameters Omitted from the MATLAB Interface

ldab ldafb ldb ldx work iwork

Output Parameters

1:     x(ldx, : $:$) – double array
The first dimension of the array x 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)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$
ldxmax (1,n)$\mathit{ldx}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The improved solution matrix X$X$.
2:     ferr(nrhs_p) – double array
ferr(j)${\mathbf{ferr}}\left(\mathit{j}\right)$ contains an estimated error bound for the j$\mathit{j}$th solution vector, that is, the j$\mathit{j}$th column of X$X$, for j = 1,2,,r$\mathit{j}=1,2,\dots ,r$.
3:     berr(nrhs_p) – double array
berr(j)${\mathbf{berr}}\left(\mathit{j}\right)$ contains the component-wise backward error bound β$\beta$ for the j$\mathit{j}$th solution vector, that is, the j$\mathit{j}$th column of X$X$, for j = 1,2,,r$\mathit{j}=1,2,\dots ,r$.
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

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: afb, 9: ldafb, 10: ipiv, 11: b, 12: ldb, 13: x, 14: ldx, 15: ferr, 16: berr, 17: work, 18: iwork, 19: 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

The bounds returned in ferr are not rigorous, because they are estimated, not computed exactly; but in practice they almost always overestimate the actual error.

For each right-hand side, computation of the backward error involves a minimum of 4n(kl + ku)$4n\left({k}_{l}+{k}_{u}\right)$ floating point operations. Each step of iterative refinement involves an additional 2n(4kl + 3ku)$2n\left(4{k}_{l}+3{k}_{u}\right)$ operations. This assumes nkl$n\gg {k}_{l}$ and nku$n\gg {k}_{u}$. At most five steps of iterative refinement are performed, but usually only one or two steps are required.
Estimating the forward error involves solving a number of systems of linear equations of the form Ax = b$Ax=b$ or ATx = b${A}^{\mathrm{T}}x=b$; the number is usually 4$4$ or 5$5$ and never more than 11$11$. Each solution involves approximately 2n(2kl + ku)$2n\left(2{k}_{l}+{k}_{u}\right)$ operations.
The complex analogue of this function is nag_lapack_zgbrfs (f07bv).

Example

```function nag_lapack_dgbrfs_example
trans = 'N';
kl = int64(1);
ku = int64(2);
ab = [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];
afb = [0, -0.01423481108172949, -0.01423851186822769, -2.13;
0, 0, -2.73, 4.07;
0, 2.46, 2.46, -3.839143870881089;
-6.98, 2.56, -5.932930470988539, -0.7269066639923109;
0.0329512893982808, 0.9605233703438396, 0.8056726812110376, 0];
ipiv = [int64(2);3;3;4];
b = [4.42, -36.01;
27.13, -31.67;
-6.14, -1.16;
10.5, -25.82];
x = [-2, 1;
3, -4;
1, 7;
-4, -2];
[xOut, ferr, berr, info] = nag_lapack_dgbrfs(trans, kl, ku, ab, afb, ipiv, b, x)
```
```

xOut =

-2     1
3    -4
1     7
-4    -2

ferr =

1.0e-13 *

0.1426
0.1740

berr =

1.0e-16 *

0.5813
0.4832

info =

0

```
```function f07bh_example
trans = 'N';
kl = int64(1);
ku = int64(2);
ab = [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];
afb = [0, -0.01423481108172949, -0.01423851186822769, -2.13;
0, 0, -2.73, 4.07;
0, 2.46, 2.46, -3.839143870881089;
-6.98, 2.56, -5.932930470988539, -0.7269066639923109;
0.0329512893982808, 0.9605233703438396, 0.8056726812110376, 0];
ipiv = [int64(2);3;3;4];
b = [4.42, -36.01;
27.13, -31.67;
-6.14, -1.16;
10.5, -25.82];
x = [-2, 1;
3, -4;
1, 7;
-4, -2];
[xOut, ferr, berr, info] = f07bh(trans, kl, ku, ab, afb, ipiv, b, x)
```
```

xOut =

-2     1
3    -4
1     7
-4    -2

ferr =

1.0e-13 *

0.1426
0.1740

berr =

1.0e-16 *

0.5813
0.4832

info =

0

```