# NAG FL Interfacef07fhf (dporfs)

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## 1Purpose

f07fhf returns error bounds for the solution of a real symmetric positive definite system of linear equations with multiple right-hand sides, $AX=B$. It improves the solution by iterative refinement, in order to reduce the backward error as much as possible.

## 2Specification

Fortran Interface
 Subroutine f07fhf ( uplo, n, nrhs, a, lda, af, ldaf, b, ldb, x, ldx, ferr, berr, work, info)
 Integer, Intent (In) :: n, nrhs, lda, ldaf, ldb, ldx Integer, Intent (Out) :: iwork(n), info Real (Kind=nag_wp), Intent (In) :: a(lda,*), af(ldaf,*), b(ldb,*) Real (Kind=nag_wp), Intent (Inout) :: x(ldx,*) Real (Kind=nag_wp), Intent (Out) :: ferr(nrhs), berr(nrhs), work(3*n) Character (1), Intent (In) :: uplo
#include <nag.h>
 void f07fhf_ (const char *uplo, const Integer *n, const Integer *nrhs, const double a[], const Integer *lda, const double af[], const Integer *ldaf, const double b[], const Integer *ldb, double x[], const Integer *ldx, double ferr[], double berr[], double work[], Integer iwork[], Integer *info, const Charlen length_uplo)
The routine may be called by the names f07fhf, nagf_lapacklin_dporfs or its LAPACK name dporfs.

## 3Description

f07fhf returns the backward errors and estimated bounds on the forward errors for the solution of a real symmetric positive definite system of linear equations with multiple right-hand sides $AX=B$. The routine handles each right-hand side vector (stored as a column of the matrix $B$) independently, so we describe the function of f07fhf in terms of a single right-hand side $b$ and solution $x$.
Given a computed solution $x$, the routine computes the component-wise backward error $\beta$. This is the size of the smallest relative perturbation in each element of $A$ and $b$ such that $x$ is the exact solution of a perturbed system
 $(A+δA)x=b+δb |δaij|≤β|aij| and |δbi|≤β|bi| .$
Then the routine estimates a bound for the component-wise forward error in the computed solution, defined by:
 $maxi|xi-x^i|/maxi|xi|$
where $\stackrel{^}{x}$ is the true solution.
For details of the method, see the F07 Chapter Introduction.

## 4References

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

## 5Arguments

1: $\mathbf{uplo}$Character(1) Input
On entry: specifies whether the upper or lower triangular part of $A$ is stored and how $A$ is to be factorized.
${\mathbf{uplo}}=\text{'U'}$
The upper triangular part of $A$ is stored and $A$ is factorized as ${U}^{\mathrm{T}}U$, where $U$ is upper triangular.
${\mathbf{uplo}}=\text{'L'}$
The lower triangular part of $A$ is stored and $A$ is factorized as $L{L}^{\mathrm{T}}$, where $L$ is lower triangular.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
2: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3: $\mathbf{nrhs}$Integer Input
On entry: $r$, the number of right-hand sides.
Constraint: ${\mathbf{nrhs}}\ge 0$.
4: $\mathbf{a}\left({\mathbf{lda}},*\right)$Real (Kind=nag_wp) array Input
Note: the second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the $n×n$ original symmetric positive definite matrix $A$ as supplied to f07fdf.
5: $\mathbf{lda}$Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f07fhf is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
6: $\mathbf{af}\left({\mathbf{ldaf}},*\right)$Real (Kind=nag_wp) array Input
Note: the second dimension of the array af must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the Cholesky factor of $A$, as returned by f07fdf.
7: $\mathbf{ldaf}$Integer Input
On entry: the first dimension of the array af as declared in the (sub)program from which f07fhf is called.
Constraint: ${\mathbf{ldaf}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
8: $\mathbf{b}\left({\mathbf{ldb}},*\right)$Real (Kind=nag_wp) array Input
Note: the second dimension of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
On entry: the $n×r$ right-hand side matrix $B$.
9: $\mathbf{ldb}$Integer Input
On entry: the first dimension of the array b as declared in the (sub)program from which f07fhf is called.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
10: $\mathbf{x}\left({\mathbf{ldx}},*\right)$Real (Kind=nag_wp) array Input/Output
Note: the second dimension of the array x must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
On entry: the $n×r$ solution matrix $X$, as returned by f07fef.
On exit: the improved solution matrix $X$.
11: $\mathbf{ldx}$Integer Input
On entry: the first dimension of the array x as declared in the (sub)program from which f07fhf is called.
Constraint: ${\mathbf{ldx}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
12: $\mathbf{ferr}\left({\mathbf{nrhs}}\right)$Real (Kind=nag_wp) array Output
On exit: ${\mathbf{ferr}}\left(\mathit{j}\right)$ contains an estimated error bound for the $\mathit{j}$th solution vector, that is, the $\mathit{j}$th column of $X$, for $\mathit{j}=1,2,\dots ,r$.
13: $\mathbf{berr}\left({\mathbf{nrhs}}\right)$Real (Kind=nag_wp) array Output
On exit: ${\mathbf{berr}}\left(\mathit{j}\right)$ contains the component-wise backward error bound $\beta$ for the $\mathit{j}$th solution vector, that is, the $\mathit{j}$th column of $X$, for $\mathit{j}=1,2,\dots ,r$.
14: $\mathbf{work}\left(3×{\mathbf{n}}\right)$Real (Kind=nag_wp) array Workspace
15: $\mathbf{iwork}\left({\mathbf{n}}\right)$Integer array Workspace
16: $\mathbf{info}$Integer Output
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

## 6Error 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.

## 7Accuracy

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.

## 8Parallelism and Performance

f07fhf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07fhf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

For each right-hand side, computation of the backward error involves a minimum of $4{n}^{2}$ floating-point operations. Each step of iterative refinement involves an additional $6{n}^{2}$ operations. At most five steps of iterative refinement are performed, but usually only one or two steps are required.
Estimating the forward error involves solving a number of systems of linear equations of the form $Ax=b$; the number is usually $4$ or $5$ and never more than $11$. Each solution involves approximately $2{n}^{2}$ operations.
The complex analogue of this routine is f07fvf.

## 10Example

This example solves the system of equations $AX=B$ using iterative refinement and to compute the forward and backward error bounds, where
 $A= ( 4.16 -3.12 0.56 -0.10 -3.12 5.03 -0.83 1.18 0.56 -0.83 0.76 0.34 -0.10 1.18 0.34 1.18 ) and B= ( 8.70 8.30 -13.35 2.13 1.89 1.61 -4.14 5.00 ) .$
Here $A$ is symmetric positive definite and must first be factorized by f07fdf.

### 10.1Program Text

Program Text (f07fhfe.f90)

### 10.2Program Data

Program Data (f07fhfe.d)

### 10.3Program Results

Program Results (f07fhfe.r)