... I will be collecting here possible topics
for the midterm as we go ....



Definitions and notions to know
===============================

A Lipschitz function

Def 5.5 = the definition of the well-posed
initial value problem

Algoritms: Taylor methods

Algorithms: Euler, mid-point, modified Euler
(=trapezoid), RKF4 (no need to memorize all the
constants)

Local truncation error for a general one-step
method

Algorithms with varying step --- how to choose the step size?
(RKF 45 as an example)

A general definition of a multistep method

Divided, forward and backward differences, and corresponding
forms of interpolating polynomials.

Adams-Bashforth and Adams-Moulton methods --- a general idea

Def: Consistency, convergence, and stability for one step methods

The root condition

All the types of stability for multistep methods:
0-stable, weakly/strongly stable methods; A-stable
methods.

Stiff ODE's

General definition of a vector norm

Definition of p-norms.
Particular cases: 1-norm, 2-norm, infinity-norm

Definition of the convergence of a sequence of vectors

Defintion of induced (or natural) matrix norm

What is the equivalence of vector norms?



Things to know without a proof
==============================

General Taylor polynomial for a function
of two variables, with the remainder term.

Formula for the binomial coefficients

Thm 5.4 = uniqueness and existence theorem for the
solution of the differential equation
y'=f(t,y).

Thm 5.6 = sufficient condition for IVP to be well-posed.

Thm 5.9, a general idea

Order of RK methods as a function of the number of
funcion evaluations on each step

Thm 5.17


Know how to reduce a higher order ODE to a system
of first order ODE

Solve a difference equation if some of the roots
of the characteristic polynomial are repeated roots
(multiplicity > 1)

Know how to use the test equation to figure out
0-stability or the region of stability of
a multistep method (in particular, that of
the implicit trapezoid and of forward and backward
Euler's methods).


Thm 11.1



Things that require derivations or proofs
=========================================

Derive arbitrary order Taylor method (keep the
full time derivatives)

Derive the second order Taylor method, replace
the full time derivative by the partial derivatives

Derive the midpoint method from the second order
Taylor method  (expand f(t+alpha,y+beta) in
the Taylor polynomial and mutch the coefficients).

Derive a formula for the optimal step size if
two methods are used in parallel

Use an interpolating polynomial of your choice
to derive one of a simplest Adams methods of my
choice (explicit or implicit)

Solve a difference equation if all the roots
of the characteristic polynomial are distinct

Explain in detail the connection between the root
condition and the stability of a difference equation

Corollary 11.2 in connection with Thm 11.1

Obtain a solution of a BVP for a second order ODE
by the linear shooting method

Prove Cauchy-Schwartz inequality (I recommend
the proof I gave in class).

Prove equivalence of the 1, 2, and infinity norms