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

The functions in this chapter compute the determinant of a square matrix . The matrix is first decomposed into triangular factors

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 Crout 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

where  is an integer and

Most of the functions of the chapter are 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 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.

General Purpose functions,

    including the decomposition into triangular factors:

        complex matrix: f03ahc (nag_complex_lu)

        real matrix: f03afc (nag_real_lu)

        real symmetric positive-definite matrix: f03aec (nag_real_cholesky)

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

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