A hybrid approach for the integration of a rational function

Abstract

A hybrid integration algorithm obtaining an indefinite integral of a rational function (say q/r, q and r are polynomials) with floating-point but real coefficients is proposed. The algorithm consists of four steps and is based on combinations of symbolic and numeric computations (hybrid computation). The first step is a hybrid preprocessing stage. An integrand is decomposed into rational and logarithmic parts by using an approximate Horowitz' method which allows floating-point coefficients. Here, we replace the Euclidean GCD algorithm with an approximate-GCD algorithm which was proposed by Sasaki and Noda recently. It is easy to integrate the rational part. The logarithmic part is integrated numerically in the second step. Zeros of a denominator of it are computed by the numerical Durand-Kerner method which computes all zeros of a polynomial equation simultaneously. The integrand is then decomposed into partial fractions in the third step. Coefficients of partial fractions are determined by residue theory. Finally, in the fourth step, partial fractions are transformed into the resulting indefinite integral by using well-known rules of integrals. The hybrid algorithm proposed here gives both indefinite integrals and accurate values of definite integrals. Numerical errors in the hybrid algorithm depend only on errors in the second step. The algorithm evaluates some problems where numerical methods are inefficient or incapable, or a pure symbolic method is theoretically insufficient.