Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zsysv (f07nn)

## Purpose

nag_lapack_zsysv (f07nn) computes the solution to a complex system of linear equations
 $AX=B ,$
where $A$ is an $n$ by $n$ symmetric matrix and $X$ and $B$ are $n$ by $r$ matrices.

## Syntax

[a, ipiv, b, info] = f07nn(uplo, a, b, 'n', n, 'nrhs_p', nrhs_p)
[a, ipiv, b, info] = nag_lapack_zsysv(uplo, a, b, 'n', n, 'nrhs_p', nrhs_p)

## Description

nag_lapack_zsysv (f07nn) uses the diagonal pivoting method to factor $A$ as $A=UD{U}^{\mathrm{T}}$ if ${\mathbf{uplo}}=\text{'U'}$ or $A=LD{L}^{\mathrm{T}}$ if ${\mathbf{uplo}}=\text{'L'}$, where $U$ (or $L$) is a product of permutation and unit upper (lower) triangular matrices, and $D$ is symmetric and block diagonal with $1$ by $1$ and $2$ by $2$ diagonal blocks. The factored form of $A$ is then used to solve the system of equations $AX=B$.

## References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{uplo}$ – string (length ≥ 1)
If ${\mathbf{uplo}}=\text{'U'}$, the upper triangle of $A$ is stored.
If ${\mathbf{uplo}}=\text{'L'}$, the lower triangle of $A$ is stored.
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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The $n$ by $n$ symmetric matrix $A$.
• If ${\mathbf{uplo}}=\text{'U'}$, the upper triangular part of $a$ must be stored and the elements of the array below the diagonal are not referenced.
• If ${\mathbf{uplo}}=\text{'L'}$, the lower triangular part of $a$ must be stored and the elements of the array above the diagonal are not referenced.
3:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – 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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs_p}}\right)$.
Note: to solve the equations $Ax=b$, where $b$ is a single right-hand side, b may be supplied as a one-dimensional array with length $\mathit{ldb}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
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 array a.
$n$, the number of linear equations, i.e., 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, i.e., the number of columns of the matrix $B$.
Constraint: ${\mathbf{nrhs_p}}\ge 0$.

### Output Parameters

1:     $\mathrm{a}\left(\mathit{lda},:\right)$ – complex array
The first dimension of the array a will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array a will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
If ${\mathbf{info}}={\mathbf{0}}$, the block diagonal matrix $D$ and the multipliers used to obtain the factor $U$ or $L$ from the factorization $A=UD{U}^{\mathrm{T}}$ or $A=LD{L}^{\mathrm{T}}$ as computed by nag_lapack_zsytrf (f07nr).
2:     $\mathrm{ipiv}\left(:\right)$int64int32nag_int array
The dimension of the array ipiv will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
Details of the interchanges and the block structure of $D$. More precisely,
• if ${\mathbf{ipiv}}\left(i\right)=k>0$, ${d}_{ii}$ is a $1$ by $1$ pivot block and the $i$th row and column of $A$ were interchanged with the $k$th row and column;
• if ${\mathbf{uplo}}=\text{'U'}$ and ${\mathbf{ipiv}}\left(i-1\right)={\mathbf{ipiv}}\left(i\right)=-l<0$, $\left(\begin{array}{cc}{d}_{i-1,i-1}& {\stackrel{-}{d}}_{i,i-1}\\ {\stackrel{-}{d}}_{i,i-1}& {d}_{ii}\end{array}\right)$ is a $2$ by $2$ pivot block and the $\left(i-1\right)$th row and column of $A$ were interchanged with the $l$th row and column;
• if ${\mathbf{uplo}}=\text{'L'}$ and ${\mathbf{ipiv}}\left(i\right)={\mathbf{ipiv}}\left(i+1\right)=-m<0$, $\left(\begin{array}{cc}{d}_{ii}& {d}_{i+1,i}\\ {d}_{i+1,i}& {d}_{i+1,i+1}\end{array}\right)$ is a $2$ by $2$ pivot block and the $\left(i+1\right)$th row and column of $A$ were interchanged with the $m$th row and column.
3:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – 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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs_p}}\right)$.
Note: to solve the equations $Ax=b$, where $b$ is a single right-hand side, b may be supplied as a one-dimensional array with length $\mathit{ldb}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
If ${\mathbf{info}}={\mathbf{0}}$, the $n$ by $r$ solution matrix $X$.
4:     $\mathrm{info}$int64int32nag_int scalar
${\mathbf{info}}=0$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_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.
W  ${\mathbf{info}}>0$
Element $_$ of the diagonal is exactly zero. The factorization has been completed, but the block diagonal matrix $D$ is exactly singular, so the solution could not be computed.

## Accuracy

The computed solution for a single right-hand side, $\stackrel{^}{x}$, satisfies an equation of the form
 $A+E x^=b ,$
where
 $E1 = Oε A1$
and $\epsilon$ is the machine precision. An approximate error bound for the computed solution is given by
 $x^-x1 x1 ≤ κA E1 A1 ,$
where $\kappa \left(A\right)={‖{A}^{-1}‖}_{1}{‖A‖}_{1}$, the condition number of $A$ with respect to the solution of the linear equations. See Section 4.4 of Anderson et al. (1999) and Chapter 11 of Higham (2002) for further details.
nag_lapack_zsysvx (f07np) is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, nag_linsys_complex_symm_solve (f04dh) solves $Ax=b$ and returns a forward error bound and condition estimate. nag_linsys_complex_symm_solve (f04dh) calls nag_lapack_zsysv (f07nn) to solve the equations.

The total number of floating-point operations is approximately $\frac{4}{3}{n}^{3}+8{n}^{2}r$, where $r$ is the number of right-hand sides.
The real analogue of this function is nag_lapack_dsysv (f07ma). The complex Hermitian analogue of this function is nag_lapack_zhesv (f07mn).

## Example

This example solves the equations
 $Ax=b ,$
where $A$ is the complex symmetric matrix
 $A = -0.56+0.12i -1.54-2.86i 5.32-1.59i 3.80+0.92i -1.54-2.86i -2.83-0.03i -3.52+0.58i -7.86-2.96i 5.32-1.59i -3.52+0.58i 8.86+1.81i 5.14-0.64i 3.80+0.92i -7.86-2.96i 5.14-0.64i -0.39-0.71i$
and
 $b = -6.43+19.24i -0.49-01.47i -48.18+66.00i -55.64+41.22i .$
Details of the factorization of $A$ are also output.
```function f07nn_example

fprintf('f07nn example results\n\n');

% Symmetric indefinite matrix A (Upper triangular part stored)
uplo = 'Upper';
a = [-0.56 + 0.12i, -1.54 - 2.86i,  5.32 - 1.59i,  3.80 + 0.92i;
0    + 0i,    -2.83 - 0.03i, -3.52 + 0.58i, -7.86 - 2.96i;
0    + 0i,     0    + 0i,     8.86 + 1.81i,  5.14 - 0.64i;
0    + 0i,     0    + 0i,     0    + 0i,    -0.39 - 0.71i];

% RHS
b = [ -6.43 + 19.24i;
-0.49 -  1.47i;
-48.18 + 66.00i;
-55.64 + 41.22i];

% Solve
[af, ipiv, x, info] = f07nn( ...
uplo, a, b);

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

[ifail] = x04da( ...
uplo, 'Non-unit', af, 'Details of factorization');

fprintf('\nPivot indices\n   ');
fprintf('%11d', ipiv);
fprintf('\n');

```
```f07nn example results

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

Details of factorization
1          2          3          4
1     -2.0954    -0.1071    -0.4823     0.4426
-2.2011    -0.3157     0.0150     0.1936

2                 4.4079    -0.6078     0.5279
5.3991     0.2811    -0.3715

3                           -2.8300    -7.8600
-0.0300    -2.9600

4                                      -0.3900
-0.7100

Pivot indices
1          2         -2         -2
```

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

© The Numerical Algorithms Group Ltd, Oxford, UK. 2009–2015