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# NAG Toolbox: nag_lapack_zpbrfs (f07hv)

## Purpose

nag_lapack_zpbrfs (f07hv) returns error bounds for the solution of a complex Hermitian positive definite band system of linear equations with multiple right-hand sides, AX = B$AX=B$. It improves the solution by iterative refinement, in order to reduce the backward error as much as possible.

## Syntax

[x, ferr, berr, info] = f07hv(uplo, kd, ab, afb, b, x, 'n', n, 'nrhs_p', nrhs_p)
[x, ferr, berr, info] = nag_lapack_zpbrfs(uplo, kd, ab, afb, b, x, 'n', n, 'nrhs_p', nrhs_p)

## Description

nag_lapack_zpbrfs (f07hv) returns the backward errors and estimated bounds on the forward errors for the solution of a complex Hermitian positive definite band system of linear equations with multiple right-hand sides AX = B$AX=B$. 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_zpbrfs (f07hv) 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 the upper or lower triangular part of A$A$ is stored and how A$A$ is to be factorized.
uplo = 'U'${\mathbf{uplo}}=\text{'U'}$
The upper triangular part of A$A$ is stored and A$A$ is factorized as UHU${U}^{\mathrm{H}}U$, where U$U$ is upper triangular.
uplo = 'L'${\mathbf{uplo}}=\text{'L'}$
The lower triangular part of A$A$ is stored and A$A$ is factorized as LLH$L{L}^{\mathrm{H}}$, where L$L$ is lower triangular.
Constraint: uplo = 'U'${\mathbf{uplo}}=\text{'U'}$ or 'L'$\text{'L'}$.
2:     kd – int64int32nag_int scalar
kd${k}_{d}$, the number of superdiagonals or subdiagonals of the matrix A$A$.
Constraint: kd0${\mathbf{kd}}\ge 0$.
3:     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$n$ original Hermitian positive definite band matrix A$A$ as supplied to nag_lapack_zpbtrf (f07hr).
4:     afb(ldafb, : $:$) – complex array
The first dimension of the array afb 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 Cholesky factor of A$A$, as returned by nag_lapack_zpbtrf (f07hr).
5:     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$B$.
6:     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_zpbtrs (f07hs).

### 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 rwork

### Output Parameters

1:     x(ldx, : $:$) – complex 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: uplo, 2: n, 3: kd, 4: nrhs_p, 5: ab, 6: ldab, 7: afb, 8: ldafb, 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.

## Further Comments

For each right-hand side, computation of the backward error involves a minimum of 32nk$32nk$ real floating point operations. Each step of iterative refinement involves an additional 48nk$48nk$ real operations. This assumes nk$n\gg k$. 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$Ax=b$; the number is usually 5$5$ and never more than 11$11$. Each solution involves approximately 16nk$16nk$ real operations.
The real analogue of this function is nag_lapack_dpbrfs (f07hh).

## Example

```function nag_lapack_zpbrfs_example
uplo = 'L';
kd = int64(1);
ab = [complex(9.39),  1.69 + 0i,  2.65 + 0i,  2.17 + 0i;
1.08 + 1.73i,  -0.04 - 0.29i,  -0.33 - 2.24i,  0 + 0i];
afb = [complex(3.064310689208912),  1.116713953189507 + 0i,  1.606635558731136 + 0i,  0.4289150674026451 + 0i;
0.3524446799090123 + 0.5645641631875845i, ...
-0.03581937870996763 - 0.2596904956472653i,  -0.2053981677466558 - 1.394217865916694i,  0 + 0i];
b = [ -12.42 + 68.42i,  54.3 - 56.56i;
-9.93 + 0.88i,  18.32 + 4.76i;
-27.3 - 0.01i,  -4.4 + 9.97i;
5.31 + 23.63i,  9.43 + 1.41i];
x = [ -1 + 8i,  5 - 6i;
2 - 3i,  2 + 3i;
-4 - 5i,  -8 + 4i;
7 + 6i,  -1 - 7i];
[xOut, ferr, berr, info] = nag_lapack_zpbrfs(uplo, kd, ab, afb, b, x)
```
```

xOut =

-1.0000 + 8.0000i   5.0000 - 6.0000i
2.0000 - 3.0000i   2.0000 + 3.0000i
-4.0000 - 5.0000i  -8.0000 + 4.0000i
7.0000 + 6.0000i  -1.0000 - 7.0000i

ferr =

1.0e-13 *

0.3448
0.3219

berr =

1.0e-16 *

0.4425
0.6017

info =

0

```
```function f07hv_example
uplo = 'L';
kd = int64(1);
ab = [complex(9.39),  1.69 + 0i,  2.65 + 0i,  2.17 + 0i;
1.08 + 1.73i,  -0.04 - 0.29i,  -0.33 - 2.24i,  0 + 0i];
afb = [complex(3.064310689208912), 1.116713953189507 + 0i, 1.606635558731136 + 0i,  ...
0.4289150674026451 + 0i;
0.3524446799090123 + 0.5645641631875845i, ...
-0.03581937870996763 - 0.2596904956472653i,  ...
-0.2053981677466558 - 1.394217865916694i,  0 + 0i];
b = [ -12.42 + 68.42i,  54.3 - 56.56i;
-9.93 + 0.88i,  18.32 + 4.76i;
-27.3 - 0.01i,  -4.4 + 9.97i;
5.31 + 23.63i,  9.43 + 1.41i];
x = [ -1 + 8i,  5 - 6i;
2 - 3i,  2 + 3i;
-4 - 5i,  -8 + 4i;
7 + 6i,  -1 - 7i];
[xOut, ferr, berr, info] = f07hv(uplo, kd, ab, afb, b, x)
```
```

xOut =

-1.0000 + 8.0000i   5.0000 - 6.0000i
2.0000 - 3.0000i   2.0000 + 3.0000i
-4.0000 - 5.0000i  -8.0000 + 4.0000i
7.0000 + 6.0000i  -1.0000 - 7.0000i

ferr =

1.0e-13 *

0.3448
0.3219

berr =

1.0e-16 *

0.4425
0.6017

info =

0

```

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