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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

```