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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zhetrs (f07ms)

## Purpose

nag_lapack_zhetrs (f07ms) solves a complex Hermitian indefinite system of linear equations with multiple right-hand sides,
 $AX=B ,$
where $A$ has been factorized by nag_lapack_zhetrf (f07mr).

## Syntax

[b, info] = f07ms(uplo, a, ipiv, b, 'n', n, 'nrhs_p', nrhs_p)
[b, info] = nag_lapack_zhetrs(uplo, a, ipiv, b, 'n', n, 'nrhs_p', nrhs_p)

## Description

nag_lapack_zhetrs (f07ms) is used to solve a complex Hermitian indefinite system of linear equations $AX=B$, this function must be preceded by a call to nag_lapack_zhetrf (f07mr) which computes the Bunch–Kaufman factorization of $A$.
If ${\mathbf{uplo}}=\text{'U'}$, $A=PUD{U}^{\mathrm{H}}{P}^{\mathrm{T}}$, where $P$ is a permutation matrix, $U$ is an upper triangular matrix and $D$ is an Hermitian block diagonal matrix with $1$ by $1$ and $2$ by $2$ blocks; the solution $X$ is computed by solving $PUDY=B$ and then ${U}^{\mathrm{H}}{P}^{\mathrm{T}}X=Y$.
If ${\mathbf{uplo}}=\text{'L'}$, $A=PLD{L}^{\mathrm{H}}{P}^{\mathrm{T}}$, where $L$ is a lower triangular matrix; the solution $X$ is computed by solving $PLDY=B$ and then ${L}^{\mathrm{H}}{P}^{\mathrm{T}}X=Y$.

## References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{uplo}$ – string (length ≥ 1)
Specifies how $A$ has been factorized.
${\mathbf{uplo}}=\text{'U'}$
$A=PUD{U}^{\mathrm{H}}{P}^{\mathrm{T}}$, where $U$ is upper triangular.
${\mathbf{uplo}}=\text{'L'}$
$A=PLD{L}^{\mathrm{H}}{P}^{\mathrm{T}}$, where $L$ is lower triangular.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
2:     $\mathrm{a}\left(\mathit{lda},:\right)$ – complex array
The first dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
Details of the factorization of $A$, as returned by nag_lapack_zhetrf (f07mr).
3:     $\mathrm{ipiv}\left(:\right)$int64int32nag_int array
The dimension of the array ipiv must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
Details of the interchanges and the block structure of $D$, as returned by nag_lapack_zhetrf (f07mr).
4:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – complex array
The first dimension of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs_p}}\right)$.
The $n$ by $r$ right-hand side matrix $B$.

### Optional Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
Default: the first dimension of the arrays a, b and the second dimension of the arrays a, ipiv.
$n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
2:     $\mathrm{nrhs_p}$int64int32nag_int scalar
Default: the second dimension of the array b.
$r$, the number of right-hand sides.
Constraint: ${\mathbf{nrhs_p}}\ge 0$.

### Output Parameters

1:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – complex array
The first dimension of the array b will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array b will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs_p}}\right)$.
The $n$ by $r$ solution matrix $X$.
2:     $\mathrm{info}$int64int32nag_int scalar
${\mathbf{info}}=0$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

## Accuracy

For each right-hand side vector $b$, the computed solution $x$ is the exact solution of a perturbed system of equations $\left(A+E\right)x=b$, where
• if ${\mathbf{uplo}}=\text{'U'}$, $\left|E\right|\le c\left(n\right)\epsilon P\left|U\right|\left|D\right|\left|{U}^{\mathrm{H}}\right|{P}^{\mathrm{T}}$;
• if ${\mathbf{uplo}}=\text{'L'}$, $\left|E\right|\le c\left(n\right)\epsilon P\left|L\right|\left|D\right|\left|{L}^{\mathrm{H}}\right|{P}^{\mathrm{T}}$,
$c\left(n\right)$ is a modest linear function of $n$, and $\epsilon$ is the machine precision
If $\stackrel{^}{x}$ is the true solution, then the computed solution $x$ satisfies a forward error bound of the form
 $x-x^∞ x∞ ≤cncondA,xε$
where $\mathrm{cond}\left(A,x\right)={‖\left|{A}^{-1}\right|\left|A\right|\left|x\right|‖}_{\infty }/{‖x‖}_{\infty }\le \mathrm{cond}\left(A\right)={‖\left|{A}^{-1}\right|\left|A\right|‖}_{\infty }\le {\kappa }_{\infty }\left(A\right)$.
Note that $\mathrm{cond}\left(A,x\right)$ can be much smaller than $\mathrm{cond}\left(A\right)$.
Forward and backward error bounds can be computed by calling nag_lapack_zherfs (f07mv), and an estimate for ${\kappa }_{\infty }\left(A\right)$ ($\text{}={\kappa }_{1}\left(A\right)$) can be obtained by calling nag_lapack_zhecon (f07mu).

The total number of real floating-point operations is approximately $8{n}^{2}r$.
This function may be followed by a call to nag_lapack_zherfs (f07mv) to refine the solution and return an error estimate.
The real analogue of this function is nag_lapack_dsytrs (f07me).

## Example

This example solves the system of equations $AX=B$, where
 $A= -1.36+0.00i 1.58+0.90i 2.21-0.21i 3.91+1.50i 1.58-0.90i -8.87+0.00i -1.84-0.03i -1.78+1.18i 2.21+0.21i -1.84+0.03i -4.63+0.00i 0.11+0.11i 3.91-1.50i -1.78-1.18i 0.11-0.11i -1.84+0.00i$
and
 $B= 7.79+05.48i -35.39+18.01i -0.77-16.05i 4.23-70.02i -9.58+03.88i -24.79-08.40i 2.98-10.18i 28.68-39.89i .$
Here $A$ is Hermitian indefinite and must first be factorized by nag_lapack_zhetrf (f07mr).
```function f07ms_example

fprintf('f07ms example results\n\n');

% Hermitian indefinite matrix A (Lower triangular part stored)
uplo = 'L';
a = [-1.36 + 0i,     0    + 0i,     0    + 0i,      0    + 0i;
1.58 - 0.90i, -8.87 + 0i,     0    + 0i,      0    + 0i;
2.21 + 0.21i, -1.84 + 0.03i, -4.63 + 0i,      0    + 0i;
3.91 - 1.50i, -1.78 - 1.18i,  0.11 - 0.11i,  -1.84 + 0i];

% Factorize
[af, ipiv, info] = f07mr( ...
uplo, a);

% RHS
b = [ 7.79 +  5.48i, -35.39 + 18.01i;
-0.77 - 16.05i,   4.23 - 70.02i;
-9.58 +  3.88i, -24.79 -  8.40i;
2.98 - 10.18i,  28.68 - 39.89i];

% Solve
[x, info] = f07ms( ...
uplo, af, ipiv, b);

disp('Solution(s)');
disp(x);

```
```f07ms example results

Solution(s)
1.0000 - 1.0000i   3.0000 - 4.0000i
-1.0000 + 2.0000i  -1.0000 + 5.0000i
3.0000 - 2.0000i   7.0000 - 2.0000i
2.0000 + 1.0000i  -8.0000 + 6.0000i

```