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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zpttrs (f07js)

## Purpose

nag_lapack_zpttrs (f07js) 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, using the $LD{L}^{\mathrm{H}}$ factorization returned by nag_lapack_zpttrf (f07jr).

## Syntax

[b, info] = f07js(uplo, d, e, b, 'n', n, 'nrhs_p', nrhs_p)
[b, info] = nag_lapack_zpttrs(uplo, d, e, b, 'n', n, 'nrhs_p', nrhs_p)

## Description

nag_lapack_zpttrs (f07js) should be preceded by a call to nag_lapack_zpttrf (f07jr), which computes a modified Cholesky factorization of the matrix $A$ as
 $A=LDLH ,$
where $L$ is a unit lower bidiagonal matrix and $D$ is a diagonal matrix, with positive diagonal elements. nag_lapack_zpttrs (f07js) then utilizes the factorization to solve the required equations. Note that the factorization may also be regarded as having the form ${U}^{\mathrm{H}}DU$, where $U$ is a unit upper bidiagonal matrix.

## 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

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{uplo}$ – string (length ≥ 1)
Specifies the form of the factorization as follows:
${\mathbf{uplo}}=\text{'U'}$
$A={U}^{\mathrm{H}}DU$.
${\mathbf{uplo}}=\text{'L'}$
$A=LD{L}^{\mathrm{H}}$.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
2:     $\mathrm{d}\left(:\right)$ – 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 diagonal matrix $D$ from the $LD{L}^{\mathrm{H}}$ or ${U}^{\mathrm{H}}DU$ factorization of $A$.
3:     $\mathrm{e}\left(:\right)$ – complex array
The dimension of the array e must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$
If ${\mathbf{uplo}}=\text{'U'}$, e must contain the $\left(n-1\right)$ superdiagonal elements of the unit upper bidiagonal matrix $U$ from the ${U}^{\mathrm{H}}DU$ factorization of $A$.
If ${\mathbf{uplo}}=\text{'L'}$, e must contain the $\left(n-1\right)$ subdiagonal elements of the unit lower bidiagonal matrix $L$ from the $LD{L}^{\mathrm{H}}$ factorization of $A$.
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 $r$ matrix of right-hand sides $B$.

### Optional Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
Default: the first dimension of the array b and the dimension of the array d.
$n$, 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, i.e., the number of columns of the matrix $B$.
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

The computed solution for a single right-hand side, $\stackrel{^}{x}$, satisfies an equation of the form
 $A+E x^=b ,$
where
 $E1 =OεA1$
and $\epsilon$ is the machine precision. An approximate error bound for the computed solution is given by
 $x^ - x 1 x1 ≤ κA E1 A1 ,$
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) for further details.
Following the use of this function nag_lapack_zptcon (f07ju) can be used to estimate the condition number of $A$ and nag_lapack_zptrfs (f07jv) can be used to obtain approximate error bounds.

The total number of floating-point operations required to solve the equations $AX=B$ is proportional to $nr$.
The real analogue of this function is nag_lapack_dpttrs (f07je).

## 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
 $B = 64.0+16.0i -16.0-32.0i 93.0+62.0i 61.0-66.0i 78.0-80.0i 71.0-74.0i 14.0-27.0i 35.0+15.0i .$
```function f07js_example

fprintf('f07js example results\n\n');

% Hermitian tridiagonal A stored as two diagonals
d = [ 16            41          46            21];
e = [ 16 + 16i      18 - 9i      1 - 4i         ];

% Factorize
[df, ef, info] = f07jr( ...
d, e);

% RHS
b = [ 64 + 16i,  -16 - 32i;
93 + 62i,   61 - 66i;
78 - 80i,   71 - 74i;
14 - 27i,   35 + 15i];

% Solve
uplo = 'L';
[x, info] = f07js( ...
uplo, df, ef, b);

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

```
```f07js example results

Solution(s)
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

```