EPODE

Formulation

The step by step numerical algorithm builds an approximate solution by an interactive procedure starting from t=a (if the integration interval is denoted by [a,b]). Note y(t_n) the exact solution at the point t_n of the associated interval division

a=t₀ < ... < t_n < ... < t_M=b,

and y_n, the approximate value computed with the numerical method.

The class of A-methods include all formulae and schemes treated by EpODE, both linear and nonlinear methods. The matrix formula is

Y_n+1=AY_n +hF(t_n , Y_n , h) , n>= 0, Y₀=F(h)

where F is the starting procedure, and A is a matrix which is independently on the solved problem. An A-method consist in

a starting procedure,
the advance formula,
a correct validation Z(t_n,h) of the approximate values Y_n: the global error is the difference Y_n-Z(t_n,h).