NAG Toolbox Chapter Introduction

F03 — determinants

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

The functions in this chapter compute the determinant of a square matrix $A$. The matrix is assumed to have first been decomposed into triangular factors using functions 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 functions 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

For complex valued determinants the real and imaginary parts are scaled separately.

It is extremely wasteful of computer time and storage to use an inappropriate function, for example to use a function 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 function or the general functions must be used.

The functions in this chapter are general purpose functions. These give the value of the determinant in its scaled form, $d$ and $id$, given the triangular decomposition of the matrix from a suitable function from Chapter F07.

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