Take-home portion of the final:
1. Interpolation of f(x) on [a,b].
(a) Lagrange interpolate f on [a,b] using n nodes.
(b) Hermite interpolate f on [a,b] using n nodes (a degree
2n-1 algebraic polynimial).
2. Consider the IVP y'=f(t,y) on [a,b] with y(a)=alfa. Solve
numerically, with n steps (or a given h), using the following
methods:
(a) Euler
(b) (Runge-Kutta) Midpoint
(c) Runge-Kutta 4th order
(d) Adams-Bashford two step
note: 2(d) The method is given in the book, so you can just
program it from there. But you also need to derive/justify it,
i.e. compute the coefficients you use for Wi and Wi-1 in the
formula.
Update: demonstrate solutions to 1(a-b) using a concrete
function, for example f(x)=e^x, on [-1,1]. In class I suggested
you use five nodes for Lagrange interpolation and degree 3 (thus, two
nodes, with function and derivative values) for Hermite
interpolation.
Also, demonstrate solutions to problem 2 using the system
y'=1+y/t, t in [1,2], n=10 (or h=0.1).