This chapter is concerned with the calculation of determinants of square matrices.
The routines in this chapter compute the determinant of a square matrix
A. The matrix is assumued to have first been decomposed into triangular factors
using routines from
Chapter F07.
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
d2 is an integer and
For complex valued determinants the real and imaginary parts are scaled separately.
Most of the original routines 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.
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,
d1 and
d2, given the triangular decomposition of the matrix from a suitable routine from
Chapter F07.
None.
Wilkinson J H and Reinsch C (1971)
Handbook for Automatic Computation II, Linear Algebra Springer–Verlag