## Things you should know for the Final Exam for Math 151B, June 16

The final will last for only 2 hours, 8:30-10:30 on Thursday morning!

### Numerical solution to ODE's (chapter 5; approx. 50% of final)

The form of important methods:
Euler's Method: wi+1 = wi + h*f(ti,wi)
Midpoint Method: wi+1 = wi + h*f(ti + h/2,wi+h/2f(ti,wi)
Trapezoidal Method: wi+1 = wi + h/2*(f(ti,wi) + f(ti+1,wi+1))

Ideas behind deriving Runge-Kutta Methods

Local truncation error; estimate the leading term

How to estimate the error using methods of order n and n+1
i.e. taui+1(h) = 1/h * (wi+1 - vi+1); Runge Kutta Fehlberg

Total error; i.e., errortot = SUM(h*taui)

Ideas behind adaptive timestep selection (do not remember specific formulars)

What is an explicit/implicit method ?

What is a multistep method ? (No tedious derivations in the exam; but something like homework 4, problem 1 is possible).

What is a predictor-corrector scheme ?

Region of stability ? What does this imply for the timestep for the model problem ?

Transform higher order ODE into system of first oder ODE's. Write this in vector/matrix notation.

### Iterative Techniques for Linear Systems (chapter 7, approx. 25%)

Gauss Jacobi, Gauss Seidel, and SOR methods

Sufficient criteria for convergence for Gauss Jacobi, Gauss Seidel

### Approximation Theory (chapter ; approx. 25%)

Least Square fit to a straight line, or a polynomial of degree n.

Least square fit of a continuous function, or a discrete data set.

Normal equations.

Fourier Series.

Fourier coefficients.