# Exponential fitting of matricial multistep methods for ordinary differential equations

## Abstract

We study a class of explicit or implicit multistep integration formulas for solving NXN systems of ordinary differential equations. The coefficients of these formulas are diagonal matrices of order N, depending on a diagonal matrix of parameters Q of the same order. By definition, the formulas considered here are exact with respect to y =-Dy + 4>(x, y) provided Q-hD, h is the integration step, and <p belongs to a certain class of polynomials in the independent variable x. For arbitrary step number k > 1, the coefficients of the formulas are given explicitly as functions of Q. The present formulas are generalizations of the Adams methods (Q = 0) and of the backward differentiation formulas (Q). For arbitrary Q they are fitted exponentially at Q in a matricial sense. The implicit formulas are unconditionally fixed-ft stable. We give two different algorithmic implementations of the methods in question. The first is based on implicit formulas alone and utilizes the Newton Raphson method; it is well suited for stiff problems. The second implementation is a predictor-corrector approach. An error analysis is carried out for arbitrarily large Q. Finally, results of numerical test calculations are presented. © 1974, American Mathematical Society.