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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_linsys_real_posdef_solve_1rhs (f04as)

## Purpose

nag_linsys_real_posdef_solve_1rhs (f04as) calculates the accurate solution of a set of real symmetric positive definite linear equations with a single right-hand side, Ax = b$Ax=b$, using a Cholesky factorization and iterative refinement.

## Syntax

[a, c, ifail] = f04as(a, b, 'n', n)
[a, c, ifail] = nag_linsys_real_posdef_solve_1rhs(a, b, 'n', n)

## Description

Given a set of real linear equations Ax = b$Ax=b$, where A$A$ is a symmetric positive definite matrix, nag_linsys_real_posdef_solve_1rhs (f04as) first computes a Cholesky factorization of A$A$ as A = LLT$A=L{L}^{\mathrm{T}}$ where L$L$ is lower triangular. An approximation to x$x$ is found by forward and backward substitution. The residual vector r = bAx$r=b-Ax$ is then calculated using additional precision and a correction d$d$ to x$x$ is found by solving LLTd = r$L{L}^{\mathrm{T}}d=r$. x$x$ is then replaced by x + d$x+d$, and this iterative refinement of the solution is repeated until machine accuracy is obtained.

## References

Wilkinson J H and Reinsch C (1971) Handbook for Automatic Computation II, Linear Algebra Springer–Verlag

## Parameters

### Compulsory Input Parameters

1:     a(lda, : $:$) – double array
The first dimension of the array a must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The upper triangle of the n$n$ by n$n$ positive definite symmetric matrix A$A$. The elements of the array below the diagonal need not be set.
2:     b(max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$) – double array
Note: the dimension of the array b must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The right-hand side vector b$b$.

### Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The first dimension of the array a The second dimension of the arrays a, b.
n$n$, the order of the matrix A$A$.
Constraint: n0${\mathbf{n}}\ge 0$.

lda wk1 wk2

### Output Parameters

1:     a(lda, : $:$) – double array
The first dimension of the array a will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
ldamax (1,n)$\mathit{lda}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The elements of the array below the diagonal are overwritten; the upper triangle of a${\mathbf{a}}$ is unchanged.
2:     c(max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$) – double array
The solution vector x$x$.
3:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

## Error Indicators and Warnings

Errors or warnings detected by the function:
ifail = 1${\mathbf{ifail}}=1$
The matrix A$A$ is not positive definite, possibly due to rounding errors.
ifail = 2${\mathbf{ifail}}=2$
Iterative refinement fails to improve the solution, i.e., the matrix A$A$ is too ill-conditioned.
ifail = 3${\mathbf{ifail}}=3$
 On entry, n < 0${\mathbf{n}}<0$, or lda < max (1,n)$\mathit{lda}<\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.

## Accuracy

The computed solutions should be correct to full machine accuracy. For a detailed error analysis see page 39 of Wilkinson and Reinsch (1971).

The time taken by nag_linsys_real_posdef_solve_1rhs (f04as) is approximately proportional to n3${n}^{3}$.
The function must not be called with the same name for parameters b and c.

## Example

```function nag_linsys_real_posdef_solve_1rhs_example
a = [5, 7, 6, 5;
7, 10, 8, 7;
6, 8, 10, 9;
5, 7, 9, 10];
b = [23;
32;
33;
31];
[aOut, c, ifail] = nag_linsys_real_posdef_solve_1rhs(a, b)
```
```

aOut =

5.0000    7.0000    6.0000    5.0000
3.1305   10.0000    8.0000    7.0000
2.6833   -0.8944   10.0000    9.0000
2.2361         0    2.1213   10.0000

c =

1
1
1
1

ifail =

0

```
```function f04as_example
a = [5, 7, 6, 5;
7, 10, 8, 7;
6, 8, 10, 9;
5, 7, 9, 10];
b = [23;
32;
33;
31];
[aOut, c, ifail] = f04as(a, b)
```
```

aOut =

5.0000    7.0000    6.0000    5.0000
3.1305   10.0000    8.0000    7.0000
2.6833   -0.8944   10.0000    9.0000
2.2361         0    2.1213   10.0000

c =

1
1
1
1

ifail =

0

```