Expocon.mpl

A Maple package for the generation of order conditions for the construction of exponential integrators.

Expocon.mpl was presented at the CASC 2019 workshop (=>Slides of the Talk), see also

[1] H. Hofstätter, W. Auzinger, O. Koch, An Algorithm for Computing Coefficients of Words in Expressions Involving Exponentials and its Application to the Construction of Exponential Integrators, Proceedings of CASC 2019, Lecture Notes in Computer Science 11661, pp. 197-214.

[2] H. Hofstätter, Order conditions for exponential integrators.

It provides the Maple function wcoeff for computing coefficients of words in expressions involving exponentials.

Example

with(Physics):
read "/path/to/Expocon.mpl":

Setup(noncommutativeprefix = {A, B}):
X := exp((1/2)*B)*exp(A)*exp((1/2)*B)-exp(A+B):
W := [[A], [B], [A, A], [A, B], [B, A], [B, B], 
     [A, A, A], [A, A, B], [A, B, A], [A, B, B], 
     [B, A, A], [B, A, B], [B, B, A], [B, B, B]]:
seq(wcoeff(w, X), w in W);

Setup(noncommutativeprefix = {A, B});
C := Commutator;
X := exp(b*B)*exp(a*A)*exp(c*B+d*C(B, C(A, B)))*exp(a*A)*exp(b*B)-exp(A+B);
W := lyndon_words([A, B], [1, 3]);

eqs := [seq(simplify(wcoeff(w, X)), w in W)];

sol := solve(eqs);

W5 := lyndon_words([A, B], [5]);

B5 := lyndon_basis([A, B], [5]);

T5 := Matrix([seq([seq(wcoeff(w, b), b in B5)], w in W5)]);

c_w := [seq(wcoeff(w, subs(sol, X)), w in W5)];

c_b := evalm(`&*`(LinearAlgebra[MatrixInverse](T5), c_w));