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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zgetrs (f07as)

## Purpose

nag_lapack_zgetrs (f07as) solves a complex system of linear equations with multiple right-hand sides,
 AX = B ,  ATX = B   or   AHX = B , $AX=B , ATX=B or AHX=B ,$
where A$A$ has been factorized by nag_lapack_zgetrf (f07ar).

## Syntax

[b, info] = f07as(trans, a, ipiv, b, 'n', n, 'nrhs_p', nrhs_p)
[b, info] = nag_lapack_zgetrs(trans, a, ipiv, b, 'n', n, 'nrhs_p', nrhs_p)

## Description

nag_lapack_zgetrs (f07as) is used to solve a complex system of linear equations AX = B$AX=B$, ATX = B${A}^{\mathrm{T}}X=B$ or AHX = B${A}^{\mathrm{H}}X=B$, the function must be preceded by a call to nag_lapack_zgetrf (f07ar) which computes the LU$LU$ factorization of A$A$ as A = PLU$A=PLU$. The solution is computed by forward and backward substitution.
If trans = 'N'${\mathbf{trans}}=\text{'N'}$, the solution is computed by solving PLY = B$PLY=B$ and then UX = Y$UX=Y$.
If trans = 'T'${\mathbf{trans}}=\text{'T'}$, the solution is computed by solving UTY = B${U}^{\mathrm{T}}Y=B$ and then LTPTX = Y${L}^{\mathrm{T}}{P}^{\mathrm{T}}X=Y$.
If trans = 'C'${\mathbf{trans}}=\text{'C'}$, the solution is computed by solving UHY = B${U}^{\mathrm{H}}Y=B$ 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:     trans – string (length ≥ 1)
Indicates the form of the equations.
trans = 'N'${\mathbf{trans}}=\text{'N'}$
AX = B$AX=B$ is solved for X$X$.
trans = 'T'${\mathbf{trans}}=\text{'T'}$
ATX = B${A}^{\mathrm{T}}X=B$ is solved for X$X$.
trans = 'C'${\mathbf{trans}}=\text{'C'}$
AHX = B${A}^{\mathrm{H}}X=B$ is solved for X$X$.
Constraint: trans = 'N'${\mathbf{trans}}=\text{'N'}$, 'T'$\text{'T'}$ or 'C'$\text{'C'}$.
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)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The LU$LU$ factorization of A$A$, as returned by nag_lapack_zgetrf (f07ar).
3:     ipiv( : $:$) – int64int32nag_int array
Note: the dimension of the array ipiv must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The pivot indices, as returned by nag_lapack_zgetrf (f07ar).
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: trans, 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
 |E| ≤ c(n)εP|L||U| , $|E|≤c(n)εP|L||U| ,$
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| / x cond(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)$\mathrm{cond}\left(A\right)$, and cond(AH)$\mathrm{cond}\left({A}^{\mathrm{H}}\right)$ (which is the same as cond(AT)$\mathrm{cond}\left({A}^{\mathrm{T}}\right)$) can be much larger (or smaller) than cond(A)$\mathrm{cond}\left(A\right)$.
Forward and backward error bounds can be computed by calling nag_lapack_zgerfs (f07av), and an estimate for κ(A)${\kappa }_{\infty }\left(A\right)$ can be obtained by calling nag_lapack_zgecon (f07au) with norm = 'I'${\mathbf{norm}}=\text{'I'}$.

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_zgerfs (f07av) to refine the solution and return an error estimate.
The real analogue of this function is nag_lapack_dgetrs (f07ae).

## Example

```function nag_lapack_zgetrs_example
trans = 'N';
a = [ -1.34 + 2.55i,  0.28 + 3.17i,  -6.39 - 2.2i,  0.72 - 0.92i;
-0.17 - 1.41i,  3.31 - 0.15i,  -0.15 + 1.34i,  1.29 + 1.38i;
-3.29 - 2.39i,  -1.91 + 4.42i,  -0.14 - 1.35i,  1.72 + 1.35i;
2.41 + 0.39i,  -0.56 + 1.47i, -0.83 - 0.69i,  -1.96 + 0.67i];
b = [ 26.26 + 51.78i,  31.32 - 6.7i;
6.43 - 8.68i,  15.86 - 1.42i;
-5.75 + 25.31i,  -2.15 + 30.19i;
1.16 + 2.57i,  -2.56 + 7.55i];
% Factorize a
[a, ipiv, info] = nag_lapack_zgetrf(a);

% Compute solution
[bOut, info] = nag_lapack_zgetrs(trans, a, ipiv, b)
```
```

bOut =

1.0000 + 1.0000i  -1.0000 - 2.0000i
2.0000 - 3.0000i   5.0000 + 1.0000i
-4.0000 - 5.0000i  -3.0000 + 4.0000i
0.0000 + 6.0000i   2.0000 - 3.0000i

info =

0

```
```function f07as_example
trans = 'N';
a = [ -1.34 + 2.55i,  0.28 + 3.17i,  -6.39 - 2.2i,  0.72 - 0.92i;
-0.17 - 1.41i,  3.31 - 0.15i,  -0.15 + 1.34i,  1.29 + 1.38i;
-3.29 - 2.39i,  -1.91 + 4.42i,  -0.14 - 1.35i,  1.72 + 1.35i;
2.41 + 0.39i,  -0.56 + 1.47i, -0.83 - 0.69i,  -1.96 + 0.67i];
b = [ 26.26 + 51.78i,  31.32 - 6.7i;
6.43 - 8.68i,  15.86 - 1.42i;
-5.75 + 25.31i,  -2.15 + 30.19i;
1.16 + 2.57i,  -2.56 + 7.55i];
% Factorize a
[a, ipiv, info] = f07ar(a);

% Compute solution
[bOut, info] = f07as(trans, a, ipiv, b)
```
```

bOut =

1.0000 + 1.0000i  -1.0000 - 2.0000i
2.0000 - 3.0000i   5.0000 + 1.0000i
-4.0000 - 5.0000i  -3.0000 + 4.0000i
0.0000 + 6.0000i   2.0000 - 3.0000i

info =

0

```