Theory and Methods for One -step Odes

where f is continuous and outputs an Rd vector. approximation on grid functions Let y(t) be the true solution. LetH be the grid dened by step-sizes: h1 , . . . , hN such that 0 < t1 < . . . < tN satises t i = ∑ l=1 h i .1 Let ∣H∣ = maxi h i . 1 is is just the obvious grid where h i = t i − t i−1 . We are parameterizing it by the step-sizes h i instead of times because h is what shows up in the theory. Let yH be the value of y(t) at each of the points in the gridH. We think of this as a set of vectors yk = y(tk), where each yk ∈ Rd . Let ΓH be the space of grid-functions.2 Let z ∈ ΓH, then3 2 You can think of this as the space of d × N matrices where each column is a time-step. 3 e choice of norm inside the max is arbitrary ∥z∥∞ = max i ∥zi∥ .