## 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**: w_{i+1} = w_{i} + h*f(t_{i},w_{i})

**Midpoint Method**: w_{i+1} = w_{i} + h*f(t_{i} + h/2,w_{i}+h/2f(t_{i},w_{i})

**Trapezoidal Method**: w_{i+1} = w_{i} + h/2*(f(t_{i},w_{i}) + f(t_{i+1},w_{i+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. tau_{i+1}(h) = 1/h * (w_{i+1} - v_{i+1}); Runge Kutta Fehlberg

Total error; i.e., error_{tot} = SUM(h*tau_{i})

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.

### I will not ask about the following:

memorizing lenghty defintions/theorems.

Stability of the "general problem" (which we discussed in class right after the model problem).

Newton's Method to use Trapezoidal Method

Fast Fourier Transform (FFT)

Definitions or theorems about linear algebra. For example, you don't need to remember the conditions for a norm
to be a norm (if there is a problem on the exam, I would give you the relevant definitions)