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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_dtbrfs (f07vh)

Purpose

nag_lapack_dtbrfs (f07vh) returns error bounds for the solution of a real triangular band system of linear equations with multiple right-hand sides, AX = B$AX=B$ or ATX = B${A}^{\mathrm{T}}X=B$.

Syntax

[ferr, berr, info] = f07vh(uplo, trans, diag, kd, ab, b, x, 'n', n, 'nrhs_p', nrhs_p)
[ferr, berr, info] = nag_lapack_dtbrfs(uplo, trans, diag, kd, ab, b, x, 'n', n, 'nrhs_p', nrhs_p)

Description

nag_lapack_dtbrfs (f07vh) returns the backward errors and estimated bounds on the forward errors for the solution of a real triangular 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_dtbrfs (f07vh) 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:     uplo – string (length ≥ 1)
Specifies whether A$A$ is upper or lower triangular.
uplo = 'U'${\mathbf{uplo}}=\text{'U'}$
A$A$ is upper triangular.
uplo = 'L'${\mathbf{uplo}}=\text{'L'}$
A$A$ is lower triangular.
Constraint: uplo = 'U'${\mathbf{uplo}}=\text{'U'}$ or 'L'$\text{'L'}$.
2:     trans – string (length ≥ 1)
Indicates the form of the equations.
trans = 'N'${\mathbf{trans}}=\text{'N'}$
The equations are of the form AX = B$AX=B$.
trans = 'T'${\mathbf{trans}}=\text{'T'}$ or 'C'$\text{'C'}$
The equations are of the form ATX = B${A}^{\mathrm{T}}X=B$.
Constraint: trans = 'N'${\mathbf{trans}}=\text{'N'}$, 'T'$\text{'T'}$ or 'C'$\text{'C'}$.
3:     diag – string (length ≥ 1)
Indicates whether A$A$ is a nonunit or unit triangular matrix.
diag = 'N'${\mathbf{diag}}=\text{'N'}$
A$A$ is a nonunit triangular matrix.
diag = 'U'${\mathbf{diag}}=\text{'U'}$
A$A$ is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be 1$1$.
Constraint: diag = 'N'${\mathbf{diag}}=\text{'N'}$ or 'U'$\text{'U'}$.
4:     kd – int64int32nag_int scalar
kd${k}_{d}$, the number of superdiagonals of the matrix A$A$ if uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, or the number of subdiagonals if uplo = 'L'${\mathbf{uplo}}=\text{'L'}$.
Constraint: kd0${\mathbf{kd}}\ge 0$.
5:     ab(ldab, : $:$) – double array
The first dimension of the array ab must be at least kd + 1${\mathbf{kd}}+1$
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$n$ triangular band matrix A$A$.
The matrix is stored in rows 1$1$ to kd + 1${k}_{d}+1$, more precisely,
• if uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, the elements of the upper triangle of A$A$ within the band must be stored with element Aij${A}_{ij}$ in ab(kd + 1 + ij,j)​ for ​max (1,jkd)ij${\mathbf{ab}}\left({k}_{d}+1+i-j,j\right)\text{​ for ​}\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,j-{k}_{d}\right)\le i\le j$;
• if uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, the elements of the lower triangle of A$A$ within the band must be stored with element Aij${A}_{ij}$ in ab(1 + ij,j)​ for ​jimin (n,j + kd).${\mathbf{ab}}\left(1+i-j,j\right)\text{​ for ​}j\le i\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,j+{k}_{d}\right)\text{.}$
If diag = 'U'${\mathbf{diag}}=\text{'U'}$, the diagonal elements of A$A$ are assumed to be 1$1$, and are not referenced.
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$B$.
7:     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_dtbtrs (f07ve).

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 ldb ldx work iwork

Output Parameters

1:     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$\mathit{j}=1,2,\dots ,r$.
2:     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$\mathit{j}=1,2,\dots ,r$.
3:     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: uplo, 2: trans, 3: diag, 4: n, 5: kd, 6: nrhs_p, 7: ab, 8: ldab, 9: b, 10: ldb, 11: x, 12: ldx, 13: ferr, 14: berr, 15: work, 16: iwork, 17: 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.

A call to nag_lapack_dtbrfs (f07vh), for each right-hand side, 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 2nk$2nk$ floating point operations (assuming nk$n\gg k$).
The complex analogue of this function is nag_lapack_ztbrfs (f07vv).

Example

```function nag_lapack_dtbrfs_example
uplo = 'L';
trans = 'N';
diag = 'N';
kd = int64(1);
ab = [-4.16, 4.78, 6.32, 0.16;
-2.25, 5.86, -4.82, 0];
b = [-16.64, -4.16;
-13.78, -16.59;
13.1, -4.94;
-14.14, -9.96];
x = [4, 1;
-1, -3;
3, 2;
2, -2];
[ferr, berr, info] = nag_lapack_dtbrfs(uplo, trans, diag, kd, ab, b, x)
```
```

ferr =

1.0e-13 *

0.5698
0.5829

berr =

1.0e-16 *

0.6445
0.2526

info =

0

```
```function f07vh_example
uplo = 'L';
trans = 'N';
diag = 'N';
kd = int64(1);
ab = [-4.16, 4.78, 6.32, 0.16;
-2.25, 5.86, -4.82, 0];
b = [-16.64, -4.16;
-13.78, -16.59;
13.1, -4.94;
-14.14, -9.96];
x = [4, 1;
-1, -3;
3, 2;
2, -2];
[ferr, berr, info] = f07vh(uplo, trans, diag, kd, ab, b, x)
```
```

ferr =

1.0e-13 *

0.5698
0.5829

berr =

1.0e-16 *

0.6445
0.2526

info =

0

```