Consider the linear differential system

For a rational function matrix  with  the transformation

leads to a new system . One has

The matrices A and  are said to be equivalent.

The super_reduce function computes a matrix  equivalent to A in a special form, called super-irreducible at p. The output is a list of two elements. The first is the super-irreducible matrix; the second gives information about the integer slopes and Newton polynomials of the Newton polygon of the computed system. It is a list of couples, the first element of which is an integer (the slope) and the second a polynomial in u (the Newton polynomial associated with the slope).

The moser_reduce function computes a matrix  equivalent to A which is Moser-irreducible at p. This is an equivalent  matrix of minimal pole order. The output is similar to the super_reduce function, but only the Newton polynomials associated with the two biggest slopes are returned.

If T and invT are passed as arguments, they are assigned to the transformation matrix T and their inverse which transforms A to the super-irreducible (Moser-irreducible) matrix.

These functions are part of the DEtools package, and so they can be used in the form super_reduce(..) and moser_reduce(..) only after executing the command with(DEtools).  However, they can always be accessed through the long form of the command by using DEtools[super_reduce](..) or DEtools[moser_reduce](..).

Pfluegel, E. "An Algorithm For Computing Exponential Solutions of First Order Linear Differential Systems." In Proceedings of ISSAC '97, pp. 164-171. Edited by Wolfgang Kuchlin. New York: ACM Press, 1997.