f03 Chapter Contents
f03 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_real_lu (f03afc)

## 1  Purpose

nag_real_lu (f03afc) computes an $LU$ factorization of a real matrix, with partial pivoting, and evaluates the determinant.

## 2  Specification

 #include #include
 void nag_real_lu (Integer n, double a[], Integer tda, Integer pivot[], double *detf, Integer *dete, NagError *fail)

## 3  Description

nag_real_lu (f03afc) computes an $LU$ factorization of a real matrix $A$ with partial pivoting: $PA=LU$, where $P$ is a permutation matrix, $L$ is lower triangular and $U$ is unit upper triangular. The determinant of $A$ is the product of the diagonal elements of $L$ with the correct sign determined by the row interchanges.

## 4  References

Wilkinson J H and Reinsch C (1971) Handbook for Automatic Computation II, Linear Algebra Springer–Verlag

## 5  Arguments

1:     nIntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 1$.
2:     a[${\mathbf{n}}×{\mathbf{tda}}$]doubleInput/Output
Note: the $\left(i,j\right)$th element of the matrix $A$ is stored in ${\mathbf{a}}\left[\left(i-1\right)×{\mathbf{tda}}+j-1\right]$.
On entry: the $n$ by $n$ matrix $A$.
On exit: $A$ is overwritten by the lower triangular matrix $L$ and the off-diagonal elements of the upper triangular matrix $U$. The unit diagonal elements of $U$ are not stored.
3:     tdaIntegerInput
On entry: the stride separating matrix column elements in the array a.
Constraint: ${\mathbf{tda}}\ge {\mathbf{n}}$.
4:     pivot[n]IntegerOutput
On exit: ${\mathbf{pivot}}\left[i-1\right]$ gives the row index of the $i$th pivot.
5:     detfdouble *Output
6:     deteInteger *Output
On exit: the determinant of $A$ is given by ${\mathbf{detf}}×{2.0}^{{\mathbf{dete}}}$. It is given in this form to avoid overflow or underflow.
7:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_2_INT_ARG_LT
On entry, ${\mathbf{tda}}=〈\mathit{\text{value}}〉$ while ${\mathbf{n}}=〈\mathit{\text{value}}〉$. The arguments must satisfy ${\mathbf{tda}}\ge {\mathbf{n}}$.
NE_ALLOC_FAIL
Dynamic memory allocation failed.
NE_INT_ARG_LT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 1$.
NE_SINGULAR
The matrix $A$ is singular, possibly due to rounding errors. The factorization could not be completed. detf and dete are set to zero.

## 7  Accuracy

The accuracy of the determinant depends on the conditioning of the original matrix. For a detailed error analysis, see Wilkinson and Reinsch (1971).

The time taken by nag_real_lu (f03afc) is approximately proportional to ${n}^{3}$.

## 9  Example

To compute the $LU$ factorization with partial pivoting, and calculate the determinant, of the real matrix
 $33 16 72 -24 -10 -57 -8 -4 -17 .$

### 9.1  Program Text

Program Text (f03afce.c)

### 9.2  Program Data

Program Data (f03afce.d)

### 9.3  Program Results

Program Results (f03afce.r)