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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zsytrs (f07ns)

## Purpose

nag_lapack_zsytrs (f07ns) solves a complex symmetric system of linear equations with multiple right-hand sides,
 $AX=B ,$
where $A$ has been factorized by nag_lapack_zsytrf (f07nr).

## Syntax

[b, info] = f07ns(uplo, a, ipiv, b, 'n', n, 'nrhs_p', nrhs_p)
[b, info] = nag_lapack_zsytrs(uplo, a, ipiv, b, 'n', n, 'nrhs_p', nrhs_p)

## Description

nag_lapack_zsytrs (f07ns) is used to solve a complex symmetric system of linear equations $AX=B$, this function must be preceded by a call to nag_lapack_zsytrf (f07nr) which computes the Bunch–Kaufman factorization of $A$.
If ${\mathbf{uplo}}=\text{'U'}$, $A=PUD{U}^{\mathrm{T}}{P}^{\mathrm{T}}$, where $P$ is a permutation matrix, $U$ is an upper triangular matrix and $D$ is a symmetric block diagonal matrix with $1$ by $1$ and $2$ by $2$ blocks; the solution $X$ is computed by solving $PUDY=B$ and then ${U}^{\mathrm{T}}{P}^{\mathrm{T}}X=Y$.
If ${\mathbf{uplo}}=\text{'L'}$, $A=PLD{L}^{\mathrm{T}}{P}^{\mathrm{T}}$, where $L$ is a lower triangular matrix; the solution $X$ is computed by solving $PLDY=B$ and then ${L}^{\mathrm{T}}{P}^{\mathrm{T}}X=Y$.

## 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.
${\mathbf{uplo}}=\text{'U'}$
$A=PUD{U}^{\mathrm{T}}{P}^{\mathrm{T}}$, where $U$ is upper triangular.
${\mathbf{uplo}}=\text{'L'}$
$A=PLD{L}^{\mathrm{T}}{P}^{\mathrm{T}}$, 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_zsytrf (f07nr).
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_zsytrf (f07nr).
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 ${\mathbf{uplo}}=\text{'U'}$, $\left|E\right|\le c\left(n\right)\epsilon P\left|U\right|\left|D\right|\left|{U}^{\mathrm{T}}\right|{P}^{\mathrm{T}}$;
• if ${\mathbf{uplo}}=\text{'L'}$, $\left|E\right|\le c\left(n\right)\epsilon P\left|L\right|\left|D\right|\left|{L}^{\mathrm{T}}\right|{P}^{\mathrm{T}}$,
$c\left(n\right)$ 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_zsyrfs (f07nv), and an estimate for ${\kappa }_{\infty }\left(A\right)$ ($\text{}={\kappa }_{1}\left(A\right)$) can be obtained by calling nag_lapack_zsycon (f07nu).

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_zsyrfs (f07nv) 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
 $A= -0.39-0.71i 5.14-0.64i -7.86-2.96i 3.80+0.92i 5.14-0.64i 8.86+1.81i -3.52+0.58i 5.32-1.59i -7.86-2.96i -3.52+0.58i -2.83-0.03i -1.54-2.86i 3.80+0.92i 5.32-1.59i -1.54-2.86i -0.56+0.12i$
and
 $B= -55.64+41.22i -19.09-35.97i -48.18+66.00i -12.08-27.02i -0.49-01.47i 6.95+20.49i -6.43+19.24i -4.59-35.53i .$
Here $A$ is symmetric and must first be factorized by nag_lapack_zsytrf (f07nr).
```function f07ns_example

fprintf('f07ns example results\n\n');

% Complex symmetrix matrix A, lower triangle stored.
uplo = 'L';
a = [-0.39 - 0.71i,  0    + 0i,     0    + 0i,     0    + 0i;
5.14 - 0.64i,  8.86 + 1.81i,  0    + 0i,     0    + 0i;
-7.86 - 2.96i, -3.52 + 0.58i, -2.83 - 0.03i,  0    + 0i;
3.80 + 0.92i,  5.32 - 1.59i, -1.54 - 2.86i, -0.56 + 0.12i];

%Factorize A
[af, ipiv, info] = f07nr( ...
uplo, a);

% RHS
b = [ -55.64 + 41.22i, -19.09 - 35.97i;
-48.18 + 66.00i, -12.08 - 27.02i;
-0.49 -  1.47i,   6.95 + 20.49i;
-6.43 + 19.24i,  -4.59 - 35.53i];

% Solve Ax=b
[x, info] = f07ns( ...
uplo, af, ipiv, b);

disp('Solution:');
disp(x);

```
```f07ns example results

Solution:
1.0000 - 1.0000i  -2.0000 - 1.0000i
-2.0000 + 5.0000i   1.0000 - 3.0000i
3.0000 - 2.0000i   3.0000 + 2.0000i
-4.0000 + 3.0000i  -1.0000 + 1.0000i

```