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There are many numerical methods available for the step-by-step integration of ordinary differential equations. Only few of them, however, take advantage of special properties of the solution that may be known in advance. Examples of such methods are those developed by BROCK and MURRAY [9], and by DENNIS Eg], for exponential type solutions, and a method by URABE and MISE [b~ designed for solutions in whose Taylor expansion the most significant terms are of relatively high order. The present paper is concerned with the case of periodic or oscillatory solutions where the frequency, or some suitable substitute, can be estimated in advance. Our methods will integrate exactly appropriate trigonometric polynomials of given order, just as classical methods integrate exactly algebraic polynomials of given degree. The resulting methods depend on a parameter, v=h~o, where h is the step length and ~o the frequency in question, and they reduce to classical methods if v-~0. Our results have also obvious applications to numerical quadrature. They will, however, not be considered in this paper. Linear functionals of algebraic and trigonometric order .. p)'it is said to be of trigonometric order p, relative to period T, if 2z~ t =Lsln r ~-.t =0 (r=t,2 ..... p).Thus, a functional L is of algebraic order p if it annihilates all algebraic polynomials of degree =

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A collection of subroutinesand examples of their uses, as well as the underlying numerical methods, are described for general:ing orthogonal polynomials relative to arbitrary weight functions. The object of these routines is to produce the coefficients in the three-term recurrence relation satisfied by the orthogonal polynomials.Once these are known, additional data can be generated, such as zeros of orthogonal polynomials and Gauss-type quadrature rules, for which routines are also provided.

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