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

## References

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

## Parameters

### Compulsory Input Parameters

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

### Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The first dimension of the arrays a, b The second dimension of the arrays a, ipiv.
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 array b.
r$r$, the number of right-hand sides.
Constraint: nrhs0${\mathbf{nrhs}}\ge 0$.

lda ldb

### Output Parameters

1:     b(ldb, : $:$) – complex array
The first dimension of the array b 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)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$
ldbmax (1,n)$\mathit{ldb}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The n$n$ by r$r$ solution matrix X$X$.
2:     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: nrhs_p, 4: a, 5: lda, 6: ipiv, 7: b, 8: ldb, 9: 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

For each right-hand side vector b$b$, the computed solution x$x$ is the exact solution of a perturbed system of equations (A + E)x = b$\left(A+E\right)x=b$, where
• if uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, |E|c(n)εP|U||D||UH|PT$|E|\le c\left(n\right)\epsilon P|U||D||{U}^{\mathrm{H}}|{P}^{\mathrm{T}}$;
• if uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, |E|c(n)εP|L||D||LH|PT$|E|\le c\left(n\right)\epsilon P|L||D||{L}^{\mathrm{H}}|{P}^{\mathrm{T}}$,
c(n)$c\left(n\right)$ is a modest linear function of n$n$, and ε$\epsilon$ is the machine precision
If $\stackrel{^}{x}$ is the true solution, then the computed solution x$x$ satisfies a forward error bound of the form
 (‖x − x̂‖∞)/(‖x‖∞) ≤ c(n)cond(A,x)ε $‖x-x^‖∞ ‖x‖∞ ≤c(n)cond(A,x)ε$
where cond(A,x) = |A1||A||x| / xcond(A) = |A1||A|κ(A)$\mathrm{cond}\left(A,x\right)={‖|{A}^{-1}||A||x|‖}_{\infty }/{‖x‖}_{\infty }\le \mathrm{cond}\left(A\right)={‖|{A}^{-1}||A|‖}_{\infty }\le {\kappa }_{\infty }\left(A\right)$.
Note that cond(A,x)$\mathrm{cond}\left(A,x\right)$ can be much smaller than cond(A)$\mathrm{cond}\left(A\right)$.
Forward and backward error bounds can be computed by calling nag_lapack_zherfs (f07mv), and an estimate for κ(A)${\kappa }_{\infty }\left(A\right)$ ( = κ1(A)$\text{}={\kappa }_{1}\left(A\right)$) can be obtained by calling nag_lapack_zhecon (f07mu).

The total number of real floating point operations is approximately 8n2r$8{n}^{2}r$.
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

```function nag_lapack_zhetrs_example
uplo = 'L';
a = [complex(-1.36),  0 + 0i,  0 + 0i,  0 + 0i;
3.91 - 1.5i,  -1.84 + 0i,  0 + 0i,  0 + 0i;
0.3100287981271241 + 0.04333020743962702i, ...
0.5637050486508776 + 0.2850349501519716i,  -5.417624387291579 + 0i,  0 + 0i;
-0.1518120207240102 + 0.3742958425613706i, ...
0.339658279960361 + 0.03031451811355637i,  0.2997244646075835 + 0.1578268372785777i,  -7.102809895801842 + 0i];
ipiv = [int64(-4);-4;3;4];
b = [ 7.79 + 5.48i,  -35.39 + 18.01i;
-0.77 - 16.05i,  4.23 - 70.02i;
-9.58 + 3.88i,  -24.79 - 8.4i;
2.98 - 10.18i,  28.68 - 39.89i];
[bOut, info] = nag_lapack_zhetrs(uplo, a, ipiv, b)
```
```

bOut =

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

info =

0

```
```function f07ms_example
uplo = 'L';
a = [complex(-1.36),  0 + 0i,  0 + 0i,  0 + 0i;
3.91 - 1.5i,  -1.84 + 0i,  0 + 0i,  0 + 0i;
0.3100287981271241 + 0.04333020743962702i, ...
0.5637050486508776 + 0.2850349501519716i,  -5.417624387291579 + 0i,  0 + 0i;
-0.1518120207240102 + 0.3742958425613706i, ...
0.339658279960361 + 0.03031451811355637i,  ...
0.2997244646075835 + 0.1578268372785777i,  -7.102809895801842 + 0i];
ipiv = [int64(-4);-4;3;4];
b = [ 7.79 + 5.48i,  -35.39 + 18.01i;
-0.77 - 16.05i,  4.23 - 70.02i;
-9.58 + 3.88i,  -24.79 - 8.4i;
2.98 - 10.18i,  28.68 - 39.89i];
[bOut, info] = f07ms(uplo, a, ipiv, b)
```
```

bOut =

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

info =

0

```