# NAG Library Routine Document

## 1Purpose

f11mmf computes the reciprocal pivot growth factor of an $LU$ factorization of a real sparse matrix in compressed column (Harwell–Boeing) format.

## 2Specification

Fortran Interface
 Subroutine f11mmf ( n, a, iprm, il, lval, iu, uval, rpg,
 Integer, Intent (In) :: n, icolzp(*), iprm(7*n), il(*), iu(*) Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: a(*), lval(*), uval(*) Real (Kind=nag_wp), Intent (Out) :: rpg
#include <nagmk26.h>
 void f11mmf_ (const Integer *n, const Integer icolzp[], const double a[], const Integer iprm[], const Integer il[], const double lval[], const Integer iu[], const double uval[], double *rpg, Integer *ifail)

## 3Description

f11mmf computes the reciprocal pivot growth factor ${\mathrm{max}}_{j}\left({‖{A}_{j}‖}_{\infty }/{‖{U}_{j}‖}_{\infty }\right)$ from the columns ${A}_{j}$ and ${U}_{j}$ of an $LU$ factorization of the matrix $A$, ${P}_{r}A{P}_{c}=LU$ where ${P}_{r}$ is a row permutation matrix, ${P}_{c}$ is a column permutation matrix, $L$ is unit lower triangular and $U$ is upper triangular as computed by f11mef.

None.

## 5Arguments

1:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
2:     $\mathbf{icolzp}\left(*\right)$ – Integer arrayInput
Note: the dimension of the array icolzp must be at least ${\mathbf{n}}+1$.
On entry: ${\mathbf{icolzp}}\left(i\right)$ contains the index in $A$ of the start of a new column. See Section 2.1.3 in the F11 Chapter Introduction.
3:     $\mathbf{a}\left(*\right)$ – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array a must be at least ${\mathbf{icolzp}}\left({\mathbf{n}}+1\right)-1$, the number of nonzeros of the sparse matrix $A$.
On entry: the array of nonzero values in the sparse matrix $A$.
4:     $\mathbf{iprm}\left(7×{\mathbf{n}}\right)$ – 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.
5:     $\mathbf{il}\left(*\right)$ – 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.
6:     $\mathbf{lval}\left(*\right)$ – 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.
7:     $\mathbf{iu}\left(*\right)$ – 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.
8:     $\mathbf{uval}\left(*\right)$ – 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.
9:     $\mathbf{rpg}$ – Real (Kind=nag_wp)Output
On exit: the reciprocal pivot growth factor ${\mathrm{max}}_{j}\left({‖{A}_{j}‖}_{\infty }/{‖{U}_{j}‖}_{\infty }\right)$. If the reciprocal pivot growth factor is much less than $1$, the stability of the $LU$ factorization may be poor.
10:   $\mathbf{ifail}$ – 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, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=2$
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.

Not applicable.

## 8Parallelism and Performance

f11mmf is not threaded in any implementation.

If the reciprocal pivot growth factor, rpg, is much less than $1$, then the factorization of the matrix $A$ could be poor. This means that using the factorization to obtain solutions to a linear system, forward error bounds and estimates of the condition number could be unreliable. Consider increasing the thresh argument in the call to f11mef.

## 10Example

To compute the reciprocal pivot growth for the factorization of the matrix $A$, where
 $A= 2.00 1.00 0 0 0 0 0 1.00 -1.00 0 4.00 0 1.00 0 1.00 0 0 0 1.00 2.00 0 -2.00 0 0 3.00 .$
In this case, it should be equal to $1.0$.

### 10.1Program Text

Program Text (f11mmfe.f90)

### 10.2Program Data

Program Data (f11mmfe.d)

### 10.3Program Results

Program Results (f11mmfe.r)