Consider the recursion formula:

 

with n = 0,…, 1000, h= 1/1000, and a = -0.5. Create a list u. Store in its first three elements e 0 , e ha, and e 2ha 1. . These represent the starting values u₀, u₁, and u₂ in the given formula. Build up the complete list from the recursion formula.

2. Construct a second list, td, in which you store the values nh, with n = 0, …, 1000. Plot td versus u (refer section Basic plotting in Chapter 6, Plotting, for more information). Make a second plot in which you plot the difference, that is, |e^at, – u_n|, where t_n represents the values inside the vector td. Set axis labels and a title. The recursion is a multistep formula to solve the differential equation u' = au with the initial value u(0) = u₀ = 1. u_n approximates u(nh) = e^anhu0.

U=U+ha