# NAG Library Chapter Introduction

## 1Scope of the Chapter

This chapter is concerned with the calculation of determinants of square matrices.

## 2Background to the Problems

The routines in this chapter compute the determinant of a square matrix $A$. The matrix is assumed to have first been decomposed into triangular factors
 $A=LU ,$
using routines from Chapter F07.
If $A$ is positive definite, then $U={L}^{\mathrm{T}}$, and the determinant is the product of the squares of the diagonal elements of $L$. Otherwise, the routines in this chapter use the Dolittle form of the $LU$ decomposition, where $L$ has unit elements on its diagonal. The determinant is then the product of the diagonal elements of $U$, taking account of possible sign changes due to row interchanges.
To avoid overflow or underflow in the computation of the determinant, some scaling is associated with each multiplication in the product of the relevant diagonal elements. The final value is represented by
 $det⁡A=d×2id$
where $id$ is an integer and
 $116≤d<1 .$
For complex valued determinants the real and imaginary parts are scaled separately.

## 3Recommendations on Choice and Use of Available Routines

It is extremely wasteful of computer time and storage to use an inappropriate routine, for example to use a routine requiring a complex matrix when $A$ is real. Most programmers will know whether their matrix is real or complex, but may be less certain whether or not a real symmetric matrix $A$ is positive definite, i.e., all eigenvalues of $A>0$. A real symmetric matrix $A$ not known to be positive definite must be treated as a general real matrix. In all other cases either the band routine or the general routines must be used.
The routines in this chapter are general purpose routines. These give the value of the determinant in its scaled form, $d$ and $id$, given the triangular decomposition of the matrix from a suitable routine from Chapter F07.

## 4Decision Trees

### Tree 1

 Is $A$ a real matrix? Is $A$ a symmetric positive definite matrix? Is $A$ a band matrix? f07hdf and f03bhf yes yes yes no no no f07fdf and f03bff f07adf and f03baf f07arf and f03bnf

## 5Functionality Index

 Determinants of factorized matrices,
 complex matrix f03bnf
 real matrix f03baf
 real symmetric band positive definite matrix f03bhf
 real symmetric positive definite matrix f03bff

None.

## 7Routines Withdrawn or Scheduled for Withdrawal

The following lists all those routines that have been withdrawn since Mark 19 of the Library or are scheduled for withdrawal at one of the next two marks.
 WithdrawnRoutine Mark ofWithdrawal Replacement Routine(s) f03aaf 25 f07adf (dgetrf) and f03baf f03abf 25 f07fdf (dpotrf) and f03bff f03acf 25 f07hdf (dpbtrf) and f03bhf f03adf 25 f07arf (zgetrf) and f03bnf f03aef 25 f07fdf (dpotrf) and f03bff f03aff 25 f07adf (dgetrf) and f03baf
Fox L (1964) An Introduction to Numerical Linear Algebra Oxford University Press
Wilkinson J H and Reinsch C (1971) Handbook for Automatic Computation II, Linear Algebra Springer–Verlag
© The Numerical Algorithms Group Ltd, Oxford, UK. 2017