Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int

Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dsprfs (f07ph)

## Purpose

nag_lapack_dsprfs (f07ph) returns error bounds for the solution of a real symmetric indefinite system of linear equations with multiple right-hand sides, AX = B$AX=B$, using packed storage. It improves the solution by iterative refinement, in order to reduce the backward error as much as possible.

## Syntax

[x, ferr, berr, info] = f07ph(uplo, ap, afp, ipiv, b, x, 'n', n, 'nrhs_p', nrhs_p)
[x, ferr, berr, info] = nag_lapack_dsprfs(uplo, ap, afp, ipiv, b, x, 'n', n, 'nrhs_p', nrhs_p)

## Description

nag_lapack_dsprfs (f07ph) returns the backward errors and estimated bounds on the forward errors for the solution of a real symmetric indefinite system of linear equations with multiple right-hand sides AX = B$AX=B$, using packed storage. The function handles each right-hand side vector (stored as a column of the matrix B$B$) independently, so we describe the function of nag_lapack_dsprfs (f07ph) in terms of a single right-hand side b$b$ and solution x$x$.
Given a computed solution x$x$, the function computes the component-wise backward error β$\beta$. This is the size of the smallest relative perturbation in each element of A$A$ and b$b$ such that x$x$ is the exact solution of a perturbed system
 (A + δA)x = b + δb |δaij| ≤ β|aij|   and   |δbi| ≤ β|bi| .
$(A+δA)x=b+δb |δaij|≤β|aij| and |δbi|≤β|bi| .$
Then the function estimates a bound for the component-wise forward error in the computed solution, defined by:
 max |xi − x̂i| / max |xi| i i
$maxi|xi-x^i|/maxi|xi|$
where $\stackrel{^}{x}$ is the true solution.
For details of the method, see the F07 Chapter Introduction.

## References

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

## Parameters

### Compulsory Input Parameters

1:     uplo – string (length ≥ 1)
Specifies whether the upper or lower triangular part of A$A$ is stored and how A$A$ is to be factorized.
uplo = 'U'${\mathbf{uplo}}=\text{'U'}$
The upper triangular part of A$A$ is stored and A$A$ is factorized as PUDUTPT$PUD{U}^{\mathrm{T}}{P}^{\mathrm{T}}$, where U$U$ is upper triangular.
uplo = 'L'${\mathbf{uplo}}=\text{'L'}$
The lower triangular part of A$A$ is stored and A$A$ is factorized as PLDLTPT$PLD{L}^{\mathrm{T}}{P}^{\mathrm{T}}$, where L$L$ is lower triangular.
Constraint: uplo = 'U'${\mathbf{uplo}}=\text{'U'}$ or 'L'$\text{'L'}$.
2:     ap( : $:$) – double array
Note: the dimension of the array ap must be at least max (1,n × (n + 1) / 2)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$.
The n$n$ by n$n$ original symmetric matrix A$A$ as supplied to nag_lapack_dsptrf (f07pd).
3:     afp( : $:$) – double array
Note: the dimension of the array afp must be at least max (1,n × (n + 1) / 2)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$.
The factorization of A$A$ stored in packed form, as returned by nag_lapack_dsptrf (f07pd).
4:     ipiv( : $:$) – int64int32nag_int array
Note: the dimension of the array ipiv must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
Details of the interchanges and the block structure of D$D$, as returned by nag_lapack_dsptrf (f07pd).
5:     b(ldb, : $:$) – double array
The first 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 second dimension of the array must be at least max (1,nrhs)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$
The n$n$ by r$r$ right-hand side matrix B$B$.
6:     x(ldx, : $:$) – double array
The first dimension of the array x 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,nrhs)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$
The n$n$ by r$r$ solution matrix X$X$, as returned by nag_lapack_dsptrs (f07pe).

### Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The first dimension of the arrays b, x The dimension of the array ipiv.
n$n$, the order of the matrix A$A$.
Constraint: n0${\mathbf{n}}\ge 0$.
2:     nrhs_p – int64int32nag_int scalar
Default: The second dimension of the arrays b, x. (An error is raised if these dimensions are not equal.)
r$r$, the number of right-hand sides.
Constraint: nrhs0${\mathbf{nrhs}}\ge 0$.

### Input Parameters Omitted from the MATLAB Interface

ldb ldx work iwork

### Output Parameters

1:     x(ldx, : $:$) – double array
The first dimension of the array x 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,nrhs)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$
ldxmax (1,n)$\mathit{ldx}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The improved solution matrix X$X$.
2:     ferr(nrhs_p) – double array
ferr(j)${\mathbf{ferr}}\left(\mathit{j}\right)$ contains an estimated error bound for the j$\mathit{j}$th solution vector, that is, the j$\mathit{j}$th column of X$X$, for j = 1,2,,r$\mathit{j}=1,2,\dots ,r$.
3:     berr(nrhs_p) – double array
berr(j)${\mathbf{berr}}\left(\mathit{j}\right)$ contains the component-wise backward error bound β$\beta$ for the j$\mathit{j}$th solution vector, that is, the j$\mathit{j}$th column of X$X$, for j = 1,2,,r$\mathit{j}=1,2,\dots ,r$.
4:     info – int64int32nag_int scalar
info = 0${\mathbf{info}}=0$ unless the function detects an error (see Section [Error Indicators and Warnings]).

## Error Indicators and Warnings

info = i${\mathbf{info}}=-i$
If info = i${\mathbf{info}}=-i$, parameter i$i$ had an illegal value on entry. The parameters are numbered as follows:
1: uplo, 2: n, 3: nrhs_p, 4: ap, 5: afp, 6: ipiv, 7: b, 8: ldb, 9: x, 10: ldx, 11: ferr, 12: berr, 13: work, 14: iwork, 15: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.

## Accuracy

The bounds returned in ferr are not rigorous, because they are estimated, not computed exactly; but in practice they almost always overestimate the actual error.

For each right-hand side, computation of the backward error involves a minimum of 4n2$4{n}^{2}$ floating point operations. Each step of iterative refinement involves an additional 6n2$6{n}^{2}$ operations. At most five steps of iterative refinement are performed, but usually only 1$1$ or 2$2$ steps are required.
Estimating the forward error involves solving a number of systems of linear equations of the form Ax = b$Ax=b$; the number is usually 4$4$ or 5$5$ and never more than 11$11$. Each solution involves approximately 2n2$2{n}^{2}$ operations.
The complex analogues of this function are nag_lapack_zhprfs (f07pv) for Hermitian matrices and nag_lapack_zsprfs (f07qv) for symmetric matrices.

## Example

```function nag_lapack_dsprfs_example
uplo = 'L';
ap = [2.07;
3.87;
4.2;
-1.15;
-0.21;
1.87;
0.63;
1.15;
2.06;
-1.81];
afp = [2.07;
4.2;
0.2230413840558341;
0.6536583767489105;
1.15;
0.8115010321439103;
-0.5959697237786296;
-2.59067708640519;
0.3030846795506181;
0.4073851981348882];
ipiv = [int64(-3);-3;3;4];
b = [-9.5, 27.85;
-8.38, 9.9;
-6.07, 19.25;
-0.96, 3.93];
x = [-4, 1;
-1, 4;
2, 3;
5, 2];
[xOut, ferr, berr, info] = nag_lapack_dsprfs(uplo, ap, afp, ipiv, b, x)
```
```

xOut =

-4     1
-1     4
2     3
5     2

ferr =

1.0e-13 *

0.2307
0.3196

berr =

1.0e-16 *

0.5738
0.2552

info =

0

```
```function f07ph_example
uplo = 'L';
ap = [2.07;
3.87;
4.2;
-1.15;
-0.21;
1.87;
0.63;
1.15;
2.06;
-1.81];
afp = [2.07;
4.2;
0.2230413840558341;
0.6536583767489105;
1.15;
0.8115010321439103;
-0.5959697237786296;
-2.59067708640519;
0.3030846795506181;
0.4073851981348882];
ipiv = [int64(-3);-3;3;4];
b = [-9.5, 27.85;
-8.38, 9.9;
-6.07, 19.25;
-0.96, 3.93];
x = [-4, 1;
-1, 4;
2, 3;
5, 2];
[xOut, ferr, berr, info] = f07ph(uplo, ap, afp, ipiv, b, x)
```
```

xOut =

-4     1
-1     4
2     3
5     2

ferr =

1.0e-13 *

0.2307
0.3196

berr =

1.0e-16 *

0.5738
0.2552

info =

0

```