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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_linsys_complex_posdef_tridiag_solve (f04cg)

## Purpose

nag_linsys_complex_posdef_tridiag_solve (f04cg) computes the solution to a complex system of linear equations $AX=B$, where $A$ is an $n$ by $n$ Hermitian positive definite tridiagonal matrix and $X$ and $B$ are $n$ by $r$ matrices. An estimate of the condition number of $A$ and an error bound for the computed solution are also returned.

## Syntax

[d, e, b, rcond, errbnd, ifail] = f04cg(d, e, b, 'n', n, 'nrhs_p', nrhs_p)
[d, e, b, rcond, errbnd, ifail] = nag_linsys_complex_posdef_tridiag_solve(d, e, b, 'n', n, 'nrhs_p', nrhs_p)

## Description

$A$ is factorized as $A=LD{L}^{\mathrm{H}}$, where $L$ is a unit lower bidiagonal matrix and $D$ is a real diagonal matrix, and the factored form of $A$ is then used to solve the system of equations.

## 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
Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{d}\left(:\right)$ – double array
The dimension of the array d must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
Must contain the $n$ diagonal elements of the tridiagonal matrix $A$.
2:     $\mathrm{e}\left(:\right)$ – complex array
The dimension of the array e must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$
Must contain the $\left(n-1\right)$ subdiagonal elements of the tridiagonal matrix $A$.
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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs_p}}\right)$.
The $n$ by $r$ matrix of right-hand sides $B$.

### Optional Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
Default: the first dimension of the array b.
The number of linear equations $n$, 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.
The number of right-hand sides $r$, i.e., the number of columns of the matrix $B$.
Constraint: ${\mathbf{nrhs_p}}\ge 0$.

### Output Parameters

1:     $\mathrm{d}\left(:\right)$ – double array
The dimension of the array d will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
If ${\mathbf{ifail}}={\mathbf{0}}$ or $\mathbf{n}+{\mathbf{1}}$, d stores the $n$ diagonal elements of the diagonal matrix $D$ from the $LD{L}^{\mathrm{H}}$ factorization of $A$.
2:     $\mathrm{e}\left(:\right)$ – complex array
The dimension of the array e will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$
If ${\mathbf{ifail}}={\mathbf{0}}$ or $\mathbf{n}+{\mathbf{1}}$, e stores the $\left(n-1\right)$ subdiagonal elements of the unit lower bidiagonal matrix $L$ from the $LD{L}^{\mathrm{H}}$ factorization of $A$. (e can also be regarded as the conjugate of the superdiagonal of the unit upper bidiagonal factor $U$ from the ${U}^{\mathrm{H}}DU$ factorization of $A$.)
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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs_p}}\right)$.
If ${\mathbf{ifail}}={\mathbf{0}}$ or $\mathbf{n}+{\mathbf{1}}$, the $n$ by $r$ solution matrix $X$.
4:     $\mathrm{rcond}$ – double scalar
If ${\mathbf{ifail}}={\mathbf{0}}$ or $\mathbf{n}+{\mathbf{1}}$, an estimate of the reciprocal of the condition number of the matrix $A$, computed as ${\mathbf{rcond}}=1/\left({‖A‖}_{1},{‖{A}^{-1}‖}_{1}\right)$.
5:     $\mathrm{errbnd}$ – double scalar
If ${\mathbf{ifail}}={\mathbf{0}}$ or $\mathbf{n}+{\mathbf{1}}$, an estimate of the forward error bound for a computed solution $\stackrel{^}{x}$, such that ${‖\stackrel{^}{x}-x‖}_{1}/{‖x‖}_{1}\le {\mathbf{errbnd}}$, where $\stackrel{^}{x}$ is a column of the computed solution returned in the array b and $x$ is the corresponding column of the exact solution $X$. If rcond is less than machine precision, then errbnd is returned as unity.
6:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

${\mathbf{ifail}}>0 \text{and} {\mathbf{ifail}}\le {\mathbf{n}}$
The principal minor of order $_$ of the matrix $A$ is not positive definite. The factorization has not been completed and the solution could not be computed.
W  ${\mathbf{ifail}}={\mathbf{n}}+1$
A solution has been computed, but rcond is less than machine precision so that the matrix $A$ is numerically singular.
${\mathbf{ifail}}=-1$
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=-2$
Constraint: ${\mathbf{nrhs_p}}\ge 0$.
${\mathbf{ifail}}=-6$
Constraint: $\mathit{ldb}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
The double allocatable memory required is n. In this case the factorization and the solution $X$ have been computed, but rcond and errbnd have not been 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. nag_linsys_complex_posdef_tridiag_solve (f04cg) uses the approximation ${‖E‖}_{1}=\epsilon {‖A‖}_{1}$ to estimate errbnd. See Section 4.4 of Anderson et al. (1999) for further details.

The total number of floating-point operations required to solve the equations $AX=B$ is proportional to $nr$. The condition number estimation requires $\mathit{O}\left(n\right)$ floating-point operations.
See Section 15.3 of Higham (2002) for further details on computing the condition number of tridiagonal matrices.
The real analogue of nag_linsys_complex_posdef_tridiag_solve (f04cg) is nag_linsys_real_posdef_tridiag_solve (f04bg).

## Example

This example solves the equations
 $AX=B,$
where $A$ is the Hermitian positive definite tridiagonal matrix
 $A= 16.0i+00.0 16.0+16.0i 0.0i+0.0 0.0i+0.0 16.0-16.0i 41.0i+00.0 18.0-9.0i 0.0i+0.0 0.0i+00.0 18.0+09.0i 46.0i+0.0 1.0-4.0i 0.0i+00.0 0.0i+00.0 1.0+4.0i 21.0i+0.0$
and
 $B= 64.0+16.0i -16.0-32.0i 93.0+62.0i 61.0-66.0i 78.0-80.0i 71.0-74.0i 14.0-27.0i 35.0+15.0i .$
An estimate of the condition number of $A$ and an approximate error bound for the computed solutions are also printed.
```function f04cg_example

fprintf('f04cg example results\n\n');

% Solve complex Ax = b for Hermitian tridiagonal A
% with error bound and condition number.
d = [ 16         41         46        21];
e = [ 16 + 16i   18 -  9i    1 - 4i     ];
b = [ 64 + 16i, -16 - 32i;
93 + 62i,  61 - 66i;
78 - 80i,  71 - 74i;
14 - 27i,  35 + 15i];

[d, e, x, rcond, errbnd, ifail] = ...
f04cg(d, e, b);

disp('Solution');
disp(x);
disp('Estimate of condition number');
fprintf('%10.1f\n\n',1/rcond);
disp('Estimate of error bound for computed solutions');
fprintf('%10.1e\n\n',errbnd);

```
```f04cg example results

Solution
2.0000 + 1.0000i  -3.0000 - 2.0000i
1.0000 + 1.0000i   1.0000 + 1.0000i
1.0000 - 2.0000i   1.0000 - 2.0000i
1.0000 - 1.0000i   2.0000 + 1.0000i

Estimate of condition number
9206.6

Estimate of error bound for computed solutions
1.0e-12

```