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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_ztbrfs (f07vv)

## Purpose

nag_lapack_ztbrfs (f07vv) returns error bounds for the solution of a complex triangular band system of linear equations with multiple right-hand sides, AX = B$AX=B$, ATX = B${A}^{\mathrm{T}}X=B$ or AHX = B${A}^{\mathrm{H}}X=B$.

## Syntax

[ferr, berr, info] = f07vv(uplo, trans, diag, kd, ab, b, x, 'n', n, 'nrhs_p', nrhs_p)
[ferr, berr, info] = nag_lapack_ztbrfs(uplo, trans, diag, kd, ab, b, x, 'n', n, 'nrhs_p', nrhs_p)

## Description

nag_lapack_ztbrfs (f07vv) returns the backward errors and estimated bounds on the forward errors for the solution of a complex triangular band system of linear equations with multiple right-hand sides AX = B$AX=B$, ATX = B${A}^{\mathrm{T}}X=B$ or AHX = B${A}^{\mathrm{H}}X=B$. 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_ztbrfs (f07vv) 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'${\mathbf{trans}}=\text{'T'}$
The equations are of the form ATX = B${A}^{\mathrm{T}}X=B$.
trans = 'C'${\mathbf{trans}}=\text{'C'}$
The equations are of the form AHX = B${A}^{\mathrm{H}}X=B$.
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, : $:$) – complex 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, : $:$) – complex 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, : $:$) – complex 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_ztbtrs (f07vs).

### 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 rwork

### 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: rwork, 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_ztbrfs (f07vv), for each right-hand side, involves solving a number of systems of linear equations of the form Ax = b$Ax=b$ or AHx = b${A}^{\mathrm{H}}x=b$; the number is usually 5$5$ and never more than 11$11$. Each solution involves approximately 8nk$8nk$ real floating point operations (assuming nk$n\gg k$).
The real analogue of this function is nag_lapack_dtbrfs (f07vh).

## Example

```function nag_lapack_ztbrfs_example
uplo = 'L';
trans = 'N';
diag = 'N';
kd = int64(2);
ab = [ -1.94 + 4.43i,  4.12 - 4.27i,  0.43 - 2.66i,  0.44 + 0.1i;
-3.39 + 3.44i,  -1.84 + 5.53i,  1.74 - 0.04i,  0 + 0i;
1.62 + 3.68i,  -2.77 - 1.93i,  0 + 0i,  0 + 0i];
b = [ -8.86 - 3.88i,  -24.09 - 5.27i;
-15.57 - 23.41i,  -57.97 + 8.14i;
-7.63 + 22.78i,  19.09 - 29.51i;
-14.74 - 2.4i,  19.17 + 21.33i];
[x, info] = nag_lapack_ztbtrs(uplo, trans, diag, kd, ab, b);
[ferr, berr, info] = nag_lapack_ztbrfs(uplo, trans, diag, kd, ab, b, x)
```
```

ferr =

1.0e-13 *

0.1803
0.2199

berr =

1.0e-16 *

0.0904
0.6571

info =

0

```
```function f07vv_example
uplo = 'L';
trans = 'N';
diag = 'N';
kd = int64(2);
ab = [ -1.94 + 4.43i,  4.12 - 4.27i,  0.43 - 2.66i,  0.44 + 0.1i;
-3.39 + 3.44i,  -1.84 + 5.53i,  1.74 - 0.04i,  0 + 0i;
1.62 + 3.68i,  -2.77 - 1.93i,  0 + 0i,  0 + 0i];
b = [ -8.86 - 3.88i,  -24.09 - 5.27i;
-15.57 - 23.41i,  -57.97 + 8.14i;
-7.63 + 22.78i,  19.09 - 29.51i;
-14.74 - 2.4i,  19.17 + 21.33i];
[x, info] = f07vs(uplo, trans, diag, kd, ab, b);
[ferr, berr, info] = f07vv(uplo, trans, diag, kd, ab, b, x)
```
```

ferr =

1.0e-13 *

0.1803
0.2199

berr =

1.0e-16 *

0.0904
0.6571

info =

0

```