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In this Installment, we describe the bareiss method for finding the determinant of an integer matrix where all the arithmetic is done in integer mode. Of course, it will also work for any kind of matrix even if not using integers, but there are better ways for doing that. The Bareiss method is useful when you really need an exact answer. For example, suppose you need to know if the determinants of two very large matrices are equal. You could use integer arithmetic and this method. Note that the question is one of equality rather than approximate equality. And if the elements of the matrices themselves are very large, you could combine this with the method in a previous Computing Prescription^{1} on the Chinese Remainder theorem. (More about this connection later.) As another example, suppose you needed to decide which side of a plane a point is on-this is the same as finding the sign of a certain determinant. It's a very delicate and unstable operation, and so you might need a determinant algorithm that is useful for exact computation. Computing a larger determinant exactly also arises in computing the Pfaffian, which is a matrix operation whose square is the determinant. Sometimes the matrices are huge-here, the Bareiss method and the Chinese remainder theorem would both be useful.

Citation

IEEE Computing in Science and Engineering

Volume

2

Issue

No. 5

Pub Type

Journals

Keywords

determinants, exact arithmetic, integer arithmetic, linear algebra

Citation

Beichl, I.
and Sullivan, F.
(2000),
Determining the Determinant, IEEE Computing in Science and Engineering
(Accessed June 3, 2023)