NAG Library Chapter Introduction

F03 (det)


Scope of the Chapter

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

Background 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=LT, 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
where id is an integer and
116d<1 .  
For complex valued determinants the real and imaginary parts are scaled separately.

Recommendations 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.

Decision Trees

Tree 1

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

Functionality Index

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

Auxiliary Routines Associated with Library Routine Arguments


Routines Withdrawn or Scheduled for Withdrawal

The following lists all those routines that have been withdrawn since Mark 19 of the Library or are scheduled for withdrawal at one of the next two marks.
Mark of

Replacement Routine(s)
f03aaf25f07adf (dgetrf) and f03baf
f03abf25f07fdf (dpotrf) and f03bff
f03acf25f07hdf (dpbtrf) and f03bhf
f03adf25f07arf (zgetrf) and f03bnf
f03aef25f07fdf (dpotrf) and f03bff
f03aff25f07adf (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