Power Series Utilities

_matrix_inv_newton_iteration(M, Minv)

Performs a single step of Newton iteration for inverting M with Minv being a partial result

ps_diff(ps)

Input:

• ps - (absolute capped) univariate power series

Output:

• the derivative of ps

ps_integrate(ps)

Input:

• ps - (absolute capped) univariate power series

Output:

• the integral of ps without constant term

ps_matrix_homlinear_de(A, Y0, prec)

Input:

• A - a square matrix with entries in a univariate power series ring
• Y0 - a square invertible matrix over the base field

Output:

• matrix Y such that Y' = AY up to precision of A - 1 and Y(0) = Y0

ps_matrix_inv(M, prec)

Input:

• M - a square matrix with entries in a univariate power series ring it is assumed that M(0) is invertible and all entries having the same precision
• prec - an integer, precision, if -1 then defaults to precision of M

Output:

• the inverse of M computed up to prec

ps_matrix_linear_de(A, B, Y0, prec)

Input:

• A, B - square matrices with entries in a univariate power series ring
• Y0 - a matrix over the base field with the rows number the same as A

Output:

• matrix Y such that Y' = AY + B up to precision of A - 1 and Y(0) = Y0

ps_matrix_log(M)

Input:

• M - a square matrix with entries in a univariate power series ring it is assumed that M(0) is the identity

Output:

• the natural log of M

ps_ode_solution(equations, ic, inputs, prec)

Input:

• equations - a system of the form $A(x, u, mu)x' - B(x, u, mu) = 0$, where $A$ is a generically nonsingular square matrix. Assumption: $A$ is nonzero at zero
• ic - initial conditions for $x$'s (dictionary)
• inputs - power series for inputs represented as arrays (dictionary)
• prec - precision of the solution

Output:

• power series solution of the system