# NAG Library Routine Document

## 1Purpose

f11mff solves a real sparse system of linear equations with multiple right-hand sides given an $LU$ factorization of the sparse matrix computed by f11mef.

## 2Specification

Fortran Interface
 Subroutine f11mff ( n, iprm, il, lval, iu, uval, nrhs, b, ldb,
 Integer, Intent (In) :: n, iprm(7*n), il(*), iu(*), nrhs, ldb Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: lval(*), uval(*) Real (Kind=nag_wp), Intent (Inout) :: b(ldb,*) Character (1), Intent (In) :: trans
#include <nagmk26.h>
 void f11mff_ (const char *trans, const Integer *n, const Integer iprm[], const Integer il[], const double lval[], const Integer iu[], const double uval[], const Integer *nrhs, double b[], const Integer *ldb, Integer *ifail, const Charlen length_trans)

## 3Description

f11mff solves a real system of linear equations with multiple right-hand sides $AX=B$ or ${A}^{\mathrm{T}}X=B$, according to the value of the argument trans, where the matrix factorization ${P}_{r}A{P}_{c}=LU$ corresponds to an $LU$ decomposition of a sparse matrix stored in compressed column (Harwell–Boeing) format, as computed by f11mef.
In the above decomposition $L$ is a lower triangular sparse matrix with unit diagonal elements and $U$ is an upper triangular sparse matrix; ${P}_{r}$ and ${P}_{c}$ are permutation matrices.

None.

## 5Arguments

1:     $\mathbf{trans}$ – Character(1)Input
On entry: specifies whether $AX=B$ or ${A}^{\mathrm{T}}X=B$ is solved.
${\mathbf{trans}}=\text{'N'}$
$AX=B$ is solved.
${\mathbf{trans}}=\text{'T'}$
${A}^{\mathrm{T}}X=B$ is solved.
Constraint: ${\mathbf{trans}}=\text{'N'}$ or $\text{'T'}$.
2:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3:     $\mathbf{iprm}\left(7×{\mathbf{n}}\right)$ – Integer arrayInput
On entry: the column permutation which defines ${P}_{c}$, the row permutation which defines ${P}_{r}$, plus associated data structures as computed by f11mef.
4:     $\mathbf{il}\left(*\right)$ – Integer arrayInput
Note: the dimension of the array il must be at least as large as the dimension of the array of the same name in f11mef.
On entry: records the sparsity pattern of matrix $L$ as computed by f11mef.
5:     $\mathbf{lval}\left(*\right)$ – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array lval must be at least as large as the dimension of the array of the same name in f11mef.
On entry: records the nonzero values of matrix $L$ and some nonzero values of matrix $U$ as computed by f11mef.
6:     $\mathbf{iu}\left(*\right)$ – Integer arrayInput
Note: the dimension of the array iu must be at least as large as the dimension of the array of the same name in f11mef.
On entry: records the sparsity pattern of matrix $U$ as computed by f11mef.
7:     $\mathbf{uval}\left(*\right)$ – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array uval must be at least as large as the dimension of the array of the same name in f11mef.
On entry: records some nonzero values of matrix $U$ as computed by f11mef.
8:     $\mathbf{nrhs}$ – IntegerInput
On entry: $\mathit{nrhs}$, the number of right-hand sides in $B$.
Constraint: ${\mathbf{nrhs}}\ge 0$.
9:     $\mathbf{b}\left({\mathbf{ldb}},*\right)$ – Real (Kind=nag_wp) arrayInput/Output
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 ${\mathbf{n}}$ by ${\mathbf{nrhs}}$ right-hand side matrix $B$.
On exit: the ${\mathbf{n}}$ by ${\mathbf{nrhs}}$ solution matrix $X$.
10:   $\mathbf{ldb}$ – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f11mff is called.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
11:   $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, . If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value  is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value  is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{ldb}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{nrhs}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nrhs}}\ge 0$.
On entry, ${\mathbf{trans}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{trans}}=\text{'N'}$ or $\text{'T'}$.
${\mathbf{ifail}}=2$
Incorrect row permutations in array iprm.
${\mathbf{ifail}}=3$
Incorrect column permutations in array iprm.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

For each right-hand side vector $b$, the computed solution $x$ is the exact solution of a perturbed system of equations $\left(A+E\right)x=b$, where
 $E≤cnεLU,$
$c\left(n\right)$ is a modest linear function of $n$, and $\epsilon$ is the machine precision, when partial pivoting is used.
If $\stackrel{^}{x}$ is the true solution, then the computed solution $x$ satisfies a forward error bound of the form
 $x-x^∞ x∞ ≤cncondA,xε$
where $\mathrm{cond}\left(A,x\right)={‖\left|{A}^{-1}\right|\left|A\right|\left|x\right|‖}_{\infty }/{‖x‖}_{\infty }\le \mathrm{cond}\left(A\right)={‖\left|{A}^{-1}\right|\left|A\right|‖}_{\infty }\le {\kappa }_{\infty }\left(A\right)$. Note that $\mathrm{cond}\left(A,x\right)$ can be much smaller than $\mathrm{cond}\left(A\right)$, and $\mathrm{cond}\left({A}^{\mathrm{T}}\right)$ can be much larger (or smaller) than $\mathrm{cond}\left(A\right)$.
Forward and backward error bounds can be computed by calling f11mhf, and an estimate for ${\kappa }_{\infty }\left(A\right)$ can be obtained by calling f11mgf.

## 8Parallelism and Performance

f11mff is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f11mff 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.

f11mff may be followed by a call to f11mhf to refine the solution and return an error estimate.

## 10Example

This example solves the system of equations $AX=B$, where
 $A= 2.00 1.00 0 0 0 0 0 1.00 -1.00 0 4.00 0 1.00 0 1.00 0 0 0 1.00 2.00 0 -2.00 0 0 3.00 and B= 1.56 3.12 -0.25 -0.50 3.60 7.20 1.33 2.66 0.52 1.04 .$
Here $A$ is nonsymmetric and must first be factorized by f11mef.

### 10.1Program Text

Program Text (f11mffe.f90)

### 10.2Program Data

Program Data (f11mffe.d)

### 10.3Program Results

Program Results (f11mffe.r)