Rational Function Computing with Poles and Residues

Computer algebra systems (CAS) usually support computation with exact or approximate rational functions stored as ratios of polynomials in “expanded form” with explicit coefficients. We examine the consequences of using a partial-fraction type of form in which all rational functions are expressed as a polynomial plus a sum of terms each of which has a denominator consisting of a monic univariate linear polynomial perhaps to an integer power. We show that some common operations including rational function addition, multiplication, and matrix determinant calculation can be performed many times faster. Polynomial GCD operations, the costliest part of rational additions, are entirely eliminated. The cost of the common case of multiplication is also reduced.