EGRR package

The EGRR package implements in Maple 2016 a family of algorithms for transforming a full-rank system to a system having a nonsingular revealing matrix of a desired type.

We consider an arbitrary order linear ordinary system S of differential equations with polynomial coefficients:

An(x) diff(y(x),x$n) + … + A1(x) diff(y(x),x) + A0(x) y(x) = 0,

where

• Ai(x) are matrices whose entries are polynomial of x;
• y(x) is a column vector of unknown functions.

For any full rank system S:

- the algorithm EG constructs an l-embracing system, whose the leading matrix coefficient being invertible, and with the set of solutions containing all the solutions of S;

- the algorithm RR constructs an equivalent system, whose the frontal matrix being invertible;

- the algorithm TriangleEG constructs an l-embracing system, whose the leading matrix coefficient being triangular, and with the set of solutions containing all the solutions of S;

- the algorithm TriangleRR constructs an equivalent system, whose the frontal matrix coefficient being triangular.

The details are to be available in the paper S.A. Abramov, A.A. Ryabenko, and D. E. Khmelnov. Revealing Matrices of Linear Differential Systems of Arbitrary Order. Programming and Computer Software, 2017, Vol.43, No.2, pp. 67-74.

Each procedure of the EGRR package has three input parameters:

• <An | … | A1 | A0> is an explicit matrix of the original system;

• n+1 is the number of blocks of the explicit matrix;

• x is the independent variable of the system.

The values returned are a sequence of two elements:

• res is the explicit matrix of the system obtained as a result of the transformations performed by the algorithm implemented in the procedure;

• full_rank is true if, in the course of the algorithm operation, it was determined that the system has a full rank, and false otherwise.

egrr.mpl - the Maple code of the EGRR package (implemented by D.E.Khmelnov).

egrr.mw - the Maple 2016 session file with examples of using the procedures EG, RR, TriangleEG, TriangleRR.