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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dpbtrs (f07he)

## Purpose

nag_lapack_dpbtrs (f07he) solves a real symmetric positive definite band system of linear equations with multiple right-hand sides,
 $AX=B ,$
where $A$ has been factorized by nag_lapack_dpbtrf (f07hd).

## Syntax

[b, info] = f07he(uplo, kd, ab, b, 'n', n, 'nrhs_p', nrhs_p)
[b, info] = nag_lapack_dpbtrs(uplo, kd, ab, b, 'n', n, 'nrhs_p', nrhs_p)

## Description

nag_lapack_dpbtrs (f07he) is used to solve a real symmetric positive definite band system of linear equations $AX=B$, the function must be preceded by a call to nag_lapack_dpbtrf (f07hd) which computes the Cholesky factorization of $A$. The solution $X$ is computed by forward and backward substitution.
If ${\mathbf{uplo}}=\text{'U'}$, $A={U}^{\mathrm{T}}U$, where $U$ is upper triangular; the solution $X$ is computed by solving ${U}^{\mathrm{T}}Y=B$ and then $UX=Y$.
If ${\mathbf{uplo}}=\text{'L'}$, $A=L{L}^{\mathrm{T}}$, where $L$ is lower triangular; the solution $X$ is computed by solving $LY=B$ and then ${L}^{\mathrm{T}}X=Y$.

## References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{uplo}$ – string (length ≥ 1)
Specifies how $A$ has been factorized.
${\mathbf{uplo}}=\text{'U'}$
$A={U}^{\mathrm{T}}U$, where $U$ is upper triangular.
${\mathbf{uplo}}=\text{'L'}$
$A=L{L}^{\mathrm{T}}$, where $L$ is lower triangular.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
2:     $\mathrm{kd}$int64int32nag_int scalar
${k}_{d}$, the number of superdiagonals or subdiagonals of the matrix $A$.
Constraint: ${\mathbf{kd}}\ge 0$.
3:     $\mathrm{ab}\left(\mathit{ldab},:\right)$ – double array
The first dimension of the array ab must be at least ${\mathbf{kd}}+1$.
The second dimension of the array ab must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The Cholesky factor of $A$, as returned by nag_lapack_dpbtrf (f07hd).
4:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – double 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 second dimension of the array ab.
$n$, 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:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – double 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 ${\mathbf{uplo}}=\text{'U'}$, $\left|E\right|\le c\left(k+1\right)\epsilon \left|{U}^{\mathrm{T}}\right|\left|U\right|$;
• if ${\mathbf{uplo}}=\text{'L'}$, $\left|E\right|\le c\left(k+1\right)\epsilon \left|L\right|\left|{L}^{\mathrm{T}}\right|$,
$c\left(k+1\right)$ is a modest linear function of $k+1$, 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
 $x-x^∞ x∞ ≤ck+1condA,xε$
where $\mathrm{cond}\left(A,x\right)={‖\left|{A}^{-1}\right|\left|A\right|\left|x\right|‖}_{\infty }/{‖x‖}_{\infty }\le \mathrm{cond}\left(A\right)={‖\left|{A}^{-1}\right|\left|A\right|‖}_{\infty }\le {\kappa }_{\infty }\left(A\right)$. Note that $\mathrm{cond}\left(A,x\right)$ can be much smaller than $\mathrm{cond}\left(A\right)$.
Forward and backward error bounds can be computed by calling nag_lapack_dpbrfs (f07hh), and an estimate for ${\kappa }_{\infty }\left(A\right)$ ($\text{}={\kappa }_{1}\left(A\right)$) can be obtained by calling nag_lapack_dpbcon (f07hg).

The total number of floating-point operations is approximately $4nkr$, assuming $n\gg k$.
This function may be followed by a call to nag_lapack_dpbrfs (f07hh) to refine the solution and return an error estimate.
The complex analogue of this function is nag_lapack_zpbtrs (f07hs).

## Example

This example solves the system of equations $AX=B$, where
 $A= 5.49 2.68 0.00 0.00 2.68 5.63 -2.39 0.00 0.00 -2.39 2.60 -2.22 0.00 0.00 -2.22 5.17 and B= 22.09 5.10 9.31 30.81 -5.24 -25.82 11.83 22.90 .$
Here $A$ is symmetric and positive definite, and is treated as a band matrix, which must first be factorized by nag_lapack_dpbtrf (f07hd).
```function f07he_example

fprintf('f07he example results\n\n');

% Symmetric A (one lower/upper off-diagonal) in banded form
uplo = 'Lower';
kd = int64(1);
n  = int64(4);
ab = [5.49,  5.63,  2.60, 5.17;
2.68, -2.39, -2.22, 0.00];

% Factorize
[abf, info] = f07hd( ...
uplo, kd, ab);

% RHS
b  = [22.09,   5.1;
9.31,  30.81;
-5.24, -25.82;
11.83,  22.9];

% Solve Ax = B
[x, info] = f07he( ...
uplo, kd, abf, b);

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

```
```f07he example results

Solution:
5.0000   -2.0000
-2.0000    6.0000
-3.0000   -1.0000
1.0000    4.0000

```