F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF07AHF (DGERFS)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

F07AHF (DGERFS) returns error bounds for the solution of a real system of linear equations with multiple right-hand sides, $AX=B$ or ${A}^{\mathrm{T}}X=B$. It improves the solution by iterative refinement, in order to reduce the backward error as much as possible.

## 2  Specification

 SUBROUTINE F07AHF ( TRANS, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)
 INTEGER N, NRHS, LDA, LDAF, IPIV(*), LDB, LDX, IWORK(N), INFO REAL (KIND=nag_wp) A(LDA,*), AF(LDAF,*), B(LDB,*), X(LDX,*), FERR(NRHS), BERR(NRHS), WORK(3*N) CHARACTER(1) TRANS
The routine may be called by its LAPACK name dgerfs.

## 3  Description

F07AHF (DGERFS) returns the backward errors and estimated bounds on the forward errors for the solution of a real system of linear equations with multiple right-hand sides $AX=B$ or ${A}^{\mathrm{T}}X=B$. The routine handles each right-hand side vector (stored as a column of the matrix $B$) independently, so we describe the function of F07AHF (DGERFS) 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
 $A+δAx=b+δb δaij≤βaij and δbi≤βbi .$
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.

## 4  References

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

## 5  Parameters

1:     TRANS – CHARACTER(1)Input
On entry: indicates the form of the linear equations for which $X$ is the computed solution.
${\mathbf{TRANS}}=\text{'N'}$
The linear equations are of the form $AX=B$.
${\mathbf{TRANS}}=\text{'T'}$ or $\text{'C'}$
The linear equations are of the form ${A}^{\mathrm{T}}X=B$.
Constraint: ${\mathbf{TRANS}}=\text{'N'}$, $\text{'T'}$ or $\text{'C'}$.
2:     N – INTEGERInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{N}}\ge 0$.
3:     NRHS – INTEGERInput
On entry: $r$, the number of right-hand sides.
Constraint: ${\mathbf{NRHS}}\ge 0$.
4:     A(LDA,$*$) – REAL (KIND=nag_wp) arrayInput
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 $n$ by $n$ original matrix $A$ as supplied to F07ADF (DGETRF).
5:     LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F07AHF (DGERFS) is called.
Constraint: ${\mathbf{LDA}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
6:     AF(LDAF,$*$) – REAL (KIND=nag_wp) arrayInput
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 $LU$ factorization of $A$, as returned by F07ADF (DGETRF).
7:     LDAF – INTEGERInput
On entry: the first dimension of the array AF as declared in the (sub)program from which F07AHF (DGERFS) is called.
Constraint: ${\mathbf{LDAF}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
8:     IPIV($*$) – INTEGER arrayInput
Note: the dimension of the array IPIV must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
On entry: the pivot indices, as returned by F07ADF (DGETRF).
9:     B(LDB,$*$) – REAL (KIND=nag_wp) arrayInput
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 $n$ by $r$ right-hand side matrix $B$.
10:   LDB – INTEGERInput
On entry: the first dimension of the array B as declared in the (sub)program from which F07AHF (DGERFS) is called.
Constraint: ${\mathbf{LDB}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
11:   X(LDX,$*$) – REAL (KIND=nag_wp) arrayInput/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 $n$ by $r$ solution matrix $X$, as returned by F07AEF (DGETRS).
On exit: the improved solution matrix $X$.
12:   LDX – INTEGERInput
On entry: the first dimension of the array X as declared in the (sub)program from which F07AHF (DGERFS) is called.
Constraint: ${\mathbf{LDX}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
13:   FERR(NRHS) – REAL (KIND=nag_wp) arrayOutput
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 $\mathit{j}=1,2,\dots ,r$.
14:   BERR(NRHS) – REAL (KIND=nag_wp) arrayOutput
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 $\mathit{j}=1,2,\dots ,r$.
15:   WORK($3×{\mathbf{N}}$) – REAL (KIND=nag_wp) arrayWorkspace
16:   IWORK(N) – INTEGER arrayWorkspace
17:   INFO – INTEGEROutput
On exit: ${\mathbf{INFO}}=0$ unless the routine detects an error (see Section 6).

## 6  Error Indicators and Warnings

Errors or warnings detected by the routine:
${\mathbf{INFO}}<0$
If ${\mathbf{INFO}}=-i$, the $i$th parameter had an illegal value. An explanatory message is output, and execution of the program is terminated.

## 7  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 $4{n}^{2}$ 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 $Ax=b$ or ${A}^{\mathrm{T}}x=b$; 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 F07AVF (ZGERFS).

## 9  Example

This example solves the system of equations $AX=B$ using iterative refinement and to compute the forward and backward error bounds, where
 $A= 1.80 2.88 2.05 -0.89 5.25 -2.95 -0.95 -3.80 1.58 -2.69 -2.90 -1.04 -1.11 -0.66 -0.59 0.80 and B= 9.52 18.47 24.35 2.25 0.77 -13.28 -6.22 -6.21 .$
Here $A$ is nonsymmetric and must first be factorized by F07ADF (DGETRF).

### 9.1  Program Text

Program Text (f07ahfe.f90)

### 9.2  Program Data

Program Data (f07ahfe.d)

### 9.3  Program Results

Program Results (f07ahfe.r)