NAG Library Chapter Introduction
f03 – Determinants
1 Scope of the Chapter
This chapter is concerned with the calculation of determinants of square matrices.
2 Background to the Problems
The functions in this chapter compute the determinant of a square matrix
. The matrix is assumued to have first been decomposed into triangular factors
using functions from Chapter f07
If is positive definite, then , and the determinant is the product of the squares of the diagonal elements of . Otherwise, the functions in this chapter use the Dolittle form of the decomposition, where has unit elements on its diagonal. The determinant is then the product of the diagonal elements of , 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
is an integer and
For complex valued determinants the real and imaginary parts are scaled separately.
Most of the original functions of the chapter were based on those published in the book edited by Wilkinson and Reinsch (1971)
. We are very grateful to the late Dr J H Wilkinson FRS for his help and interest during the implementation of this chapter of the Library.
3 Recommendations on Choice and Use of Available Functions
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 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 is positive definite, i.e., all eigenvalues of . A real symmetric matrix not known to be positive definite must be treated as a general real matrix.
4 Decision Trees
Note: if at any stage the answer to a question is ‘Don't know’ this should be read as ‘No’.
5 Functionality Index
|Determinants of factorized matrices:|| |
6 Functions Withdrawn or Scheduled for Withdrawal
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