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# NAG Toolbox: nag_linsys_real_posdef_packed_solve (f04be)

## Purpose

nag_linsys_real_posdef_packed_solve (f04be) computes the solution to a real system of linear equations $AX=B$, where $A$ is an $n$ by $n$ symmetric positive definite matrix, stored in packed format, 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

[ap, b, rcond, errbnd, ifail] = f04be(uplo, ap, b, 'n', n, 'nrhs_p', nrhs_p)
[ap, b, rcond, errbnd, ifail] = nag_linsys_real_posdef_packed_solve(uplo, ap, b, 'n', n, 'nrhs_p', nrhs_p)

## Description

The Cholesky factorization is used to factor $A$ as $A={U}^{\mathrm{T}}U$, if ${\mathbf{uplo}}=\text{'U'}$, or $A=L{L}^{\mathrm{T}}$, if ${\mathbf{uplo}}=\text{'L'}$, where $U$ is an upper triangular matrix and $L$ is a lower triangular matrix. 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
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 the matrix $A$ is stored.
If ${\mathbf{uplo}}=\text{'L'}$, the lower triangle of the matrix $A$ is stored.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
2:     $\mathrm{ap}\left(:\right)$ – double array
The dimension of the array ap must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$
The $n$ by $n$ symmetric matrix $A$. The upper or lower triangular part of the symmetric matrix is packed column-wise in a linear array. The $j$th column of $A$ is stored in the array ap as follows:
More precisely,
• if ${\mathbf{uplo}}=\text{'U'}$, the upper triangle of $A$ must be stored with element ${A}_{ij}$ in ${\mathbf{ap}}\left(i+j\left(j-1\right)/2\right)$ for $i\le j$;
• if ${\mathbf{uplo}}=\text{'L'}$, the lower triangle of $A$ must be stored with element ${A}_{ij}$ in ${\mathbf{ap}}\left(i+\left(2n-j\right)\left(j-1\right)/2\right)$ for $i\ge j$.
3:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – 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 $r$ 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:     $\mathrm{ap}\left(:\right)$ – double array
The dimension of the array ap will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$
If ${\mathbf{ifail}}={\mathbf{0}}$ or $\mathbf{n}+{\mathbf{1}}$, the factor $U$ or $L$ from the Cholesky factorization $A={U}^{\mathrm{T}}U$ or $A=L{L}^{\mathrm{T}}$, in the same storage format as $A$.
2:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – 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 $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs_p}}\right)$.
If ${\mathbf{ifail}}={\mathbf{0}}$ or $\mathbf{n}+{\mathbf{1}}$, the $n$ by $r$ solution matrix $X$.
3:     $\mathrm{rcond}$ – 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 ${\mathbf{rcond}}=1/\left({‖A‖}_{1}{‖{A}^{-1}‖}_{1}\right)$.
4:     $\mathrm{errbnd}$ – 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.
5:     $\mathrm{ifail}$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.
${\mathbf{ifail}}=-1$
On entry, uplo not one of 'U' or 'u' or 'L' or 'l'.
${\mathbf{ifail}}=-2$
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=-3$
Constraint: ${\mathbf{nrhs_p}}\ge 0$.
${\mathbf{ifail}}=-6$
Constraint: $\mathit{ldb}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
The integer allocatable memory required is n, and the double allocatable memory required is $3×{\mathbf{n}}$. Allocation failed before the solution could 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. nag_linsys_real_posdef_packed_solve (f04be) uses the approximation ${‖E‖}_{1}=\epsilon {‖A‖}_{1}$ to estimate errbnd. See Section 4.4 of Anderson et al. (1999) for further details.

## Further Comments

The packed storage scheme is illustrated by the following example when $n=4$ and ${\mathbf{uplo}}=\text{'U'}$. Two-dimensional storage of the symmetric matrix $A$:
 $a11 a12 a13 a14 a22 a23 a24 a33 a34 a44 aij = aji$
Packed storage of the upper triangle of $A$:
 $ap= a11, a12, a22, a13, a23, a33, a14, a24, a34, a44$
The total number of floating-point operations required to solve the equations $AX=B$ is proportional to $\left(\frac{1}{3}{n}^{3}+{n}^{2}r\right)$. The condition number estimation typically requires between four and five solves and never more than eleven solves, following the factorization.
In practice the condition number estimator is very reliable, but it can underestimate the true condition number; see Section 15.3 of Higham (2002) for further details.
The complex analogue of nag_linsys_real_posdef_packed_solve (f04be) is nag_linsys_complex_posdef_packed_solve (f04ce).

## Example

This example solves the equations
 $AX=B,$
where $A$ is the symmetric positive definite matrix
 $A= 4.16 -3.12 0.56 -0.10 -3.12 5.03 -0.83 1.18 0.56 -0.83 0.76 0.34 -0.10 1.18 0.34 1.18 and B= 8.70 8.30 -13.35 2.13 1.89 1.61 -4.14 5.00 .$
An estimate of the condition number of $A$ and an approximate error bound for the computed solutions are also printed.
```function f04be_example

fprintf('f04be example results\n\n');

% Solve Ax = b for positive definite A with error bound and condition number
% A is stored in symmetric packed format.
uplo = 'U';
ap = [ 4.16             ...
-3.12  5.03       ...
0.56 -0.83  0.76 ...
-0.10  1.18  0.34 1.18];
b = [  8.7,  8.30;
-13.35, 2.13;
1.89, 1.61;
-4.14, 5.00];

[ap, x, rcond, errbnd, ifail] = f04be(uplo, ap, 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);

```
```f04be example results

Solution
1.0000    4.0000
-1.0000    3.0000
2.0000    2.0000
-3.0000    1.0000

Estimate of condition number
97.3

Estimate of error bound for computed solutions
1.1e-14

```

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