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Chapter Introduction
NAG Toolbox

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.

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

```

Chapter Contents
Chapter Introduction
NAG Toolbox

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