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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zpftrs (f07ws)

## Purpose

nag_lapack_zpftrs (f07ws) solves a complex Hermitian positive definite system of linear equations with multiple right-hand sides,
 $AX=B ,$
using the Cholesky factorization computed by nag_lapack_zpftrf (f07wr) stored in Rectangular Full Packed (RFP) format.

## Syntax

[b, info] = f07ws(transr, uplo, ar, b, 'n', n, 'nrhs_p', nrhs_p)
[b, info] = nag_lapack_zpftrs(transr, uplo, ar, b, 'n', n, 'nrhs_p', nrhs_p)

## Description

nag_lapack_zpftrs (f07ws) is used to solve a complex Hermitian positive definite system of linear equations $AX=B$, the function must be preceded by a call to nag_lapack_zpftrf (f07wr) which computes the Cholesky factorization of $A$, stored in RFP format. The RFP storage format is described in Rectangular Full Packed (RFP) Storage in the F07 Chapter Introduction. The solution $X$ is computed by forward and backward substitution.
If ${\mathbf{uplo}}=\text{'U'}$, $A={U}^{\mathrm{H}}U$, where $U$ is upper triangular; the solution $X$ is computed by solving ${U}^{\mathrm{H}}Y=B$ and then $UX=Y$.
If ${\mathbf{uplo}}=\text{'L'}$, $A=L{L}^{\mathrm{H}}$, where $L$ is lower triangular; the solution $X$ is computed by solving $LY=B$ and then ${L}^{\mathrm{H}}X=Y$.

## References

Gustavson F G, Waśniewski J, Dongarra J J and Langou J (2010) Rectangular full packed format for Cholesky's algorithm: factorization, solution, and inversion ACM Trans. Math. Software 37, 2

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{transr}$ – string (length ≥ 1)
Specifies whether the normal RFP representation of $A$ or its conjugate transpose is stored.
${\mathbf{transr}}=\text{'N'}$
The matrix $A$ is stored in normal RFP format.
${\mathbf{transr}}=\text{'C'}$
The conjugate transpose of the RFP representation of the matrix $A$ is stored.
Constraint: ${\mathbf{transr}}=\text{'N'}$ or $\text{'C'}$.
2:     $\mathrm{uplo}$ – string (length ≥ 1)
Specifies how $A$ has been factorized.
${\mathbf{uplo}}=\text{'U'}$
$A={U}^{\mathrm{H}}U$, where $U$ is upper triangular.
${\mathbf{uplo}}=\text{'L'}$
$A=L{L}^{\mathrm{H}}$, where $L$ is lower triangular.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
3:     $\mathrm{ar}\left({\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$ – complex array
The Cholesky factorization of $A$ stored in RFP format, as returned by nag_lapack_zpftrf (f07wr).
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 array ar.
$n$, 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 ${\mathbf{uplo}}=\text{'U'}$, $\left|E\right|\le c\left(n\right)\epsilon \left|{U}^{\mathrm{H}}\right|\left|U\right|$;
• if ${\mathbf{uplo}}=\text{'L'}$, $\left|E\right|\le c\left(n\right)\epsilon \left|L\right|\left|{L}^{\mathrm{H}}\right|$,
$c\left(n\right)$ 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 $\mathrm{cond}\left(A,x\right)={‖\left|{A}^{-1}\right|\left|A\right|\left|x\right|‖}_{\infty }/{‖x‖}_{\infty }\le \mathrm{cond}\left(A\right)={‖\left|{A}^{-1}\right|\left|A\right|‖}_{\infty }\le {\kappa }_{\infty }\left(A\right)$ and ${\kappa }_{\infty }\left(A\right)$ is the condition number when using the $\infty$-norm.
Note that $\mathrm{cond}\left(A,x\right)$ can be much smaller than $\mathrm{cond}\left(A\right)$.

The total number of real floating-point operations is approximately $8{n}^{2}r$.
The real analogue of this function is nag_lapack_dpftrs (f07we).

## Example

This example solves the system of equations $AX=B$, where
 $A= 3.23+0.00i 1.51-1.92i 1.90+0.84i 0.42+2.50i 1.51+1.92i 3.58+0.00i -0.23+1.11i -1.18+1.37i 1.90-0.84i -0.23-1.11i 4.09+0.00i 2.33-0.14i 0.42-2.50i -1.18-1.37i 2.33+0.14i 4.29+0.00i$
and
 $B= 3.93-06.14i 1.48+06.58i 6.17+09.42i 4.65-04.75i -7.17-21.83i -4.91+02.29i 1.99-14.38i 7.64-10.79i .$
Here $A$ is Hermitian positive definite, stored in RFP format, and must first be factorized by nag_lapack_zpftrf (f07wr).
```function f07ws_example

fprintf('f07ws example results\n\n');

% Symmetric matrix in RFP format
transr = 'n';
uplo   = 'l';
ar = [ 4.09 + 0.00i  2.33 - 0.14i;
3.23 + 0.00i  4.29 + 0.00i;
1.51 + 1.92i  3.58 + 0.00i;
1.90 - 0.84i -0.23 - 1.11i;
0.42 - 2.50i -1.18 - 1.37i];
n  = int64(4);
n2 = (n*(n+1))/2;
ar  = reshape(ar,[n2,1]);

% RHS
b = [ 3.93 -  6.14i,  1.48 +  6.58i;
6.17 +  9.42i,  4.65 -  4.75i;
-7.17 - 21.83i, -4.91 +  2.29i;
1.99 - 14.38i,  7.64 - 10.79i];

% Factorize a
[ar, info] = f07wr(transr, uplo, n, ar);

if info == 0
% Compute solution
[b, info] = f07ws( ...
transr, uplo, ar, b);
fprintf('\n');
ncols  = int64(80);
indent = int64(0);
form   = 'f7.4';
title  = 'Solutions';

[ifail] = x04db( ...
'g', ' ', b, 'bracket', form, title, ...
'int', 'int', ncols, indent);
else
fprintf('\na is not positive definite.\n');
end

```
```f07ws example results

Solutions
1                 2
1  ( 1.0000,-1.0000) (-1.0000, 2.0000)
2  (-0.0000, 3.0000) ( 3.0000,-4.0000)
3  (-4.0000,-5.0000) (-2.0000, 3.0000)
4  ( 2.0000, 1.0000) ( 4.0000,-5.0000)
```