ENEE 241/MATH 242
Numerical Techniques in Engineering
Spring, 2002
Final Exam:
8:0010:00, Friday, 5/17, ENG 1202
Neither rescheduled nor makeup exams will be allowed unless a
written verification of a valid excuse (such as hospitalization, family
emergency, religious observance, court appearance, etc.) is provided and
approved by the instructor.
The final exam will be a closebook, closenote, nocalculator exam.
It will be cumulative, and will cover the following sections
of the textbook and related topics presented in lectures:

Chapter 1. 1.11.10, 1.12, 1.13, 1.51.7;

Chapter 2. 2.12.3, 2.52.11, 2.12 (exclude underdetermined systesm),
2.132.15, 1.17;

Chapter 3. 3.13.5, 3.7, 3.10, 3.14;

Chapter 4. 4.14.7, 4.9, 4.10;

Chapter 5. 5.15.6, 5.10, 5.14;

Chapter 7. 7.17.3, 7.5.
Final Exam Review Outline
 Basic knowledge of Matlab: input and output, data structure,
operations, vectors and matrices, programming, graphics, etc.
 Basic linear algebra: vectors and matrices, and their operations;
special matrices; definition and basic properties of symmetric
positive definite matrices; definition and basic properties of orthogonal
matrices; eigenvalues and eigenvectors.
 Mathematical results on Ax = b;
elementary row reduction; RREF; Gaussian elimination; diagonal and
triangular systems; LU decomposition; Cholesky decomposition;
sparse matrices; Jacobi and GaussSeidel iterative methods;
all related Matlab commands.
 Householder matrices; QR decomposition;
concept of pseudoinverse matrices;
overdetermined systems of linear equations using Matlab;
use Matlab to find eigenvalues and eigenvectors; SVD
(singular value decomposition): concept and Matlab command.

The bisection method and its number of steps; Newton's method for single equation
and systems of two equations with two unknowns; fixed point iteration and its
convergence (basic results); related Matlab coding.

Derivation of some basic numerical differentiation formulas; Matlab codes.

The concept of degree of exactness for numerical integration formulas;
the method of undetermined coefficients for deriving numerical
integration formulas; basic and composite numerical integration formulas.

Rectangular, trapezoidal, and Simpson's rule for numerical integration,
their derivation and degrees of exactness; related Matlab coding; general
ideas of NewtonCotes formulas.

Gaussian quadrature: concepts, derivation, and basic formulas (up to
n = 2); use Gaussian quadrature to approximate definite integrals;
related Matlab codes.

GaussChebyshev formulas and Filon's since and cosine formulas.

Initialvalue problems of ODEs;
Euler's method and related stability issue; the trapezoidal method;
basic secondorder and forthorder RungeKutta methods; Adams methods: ideas
and basic formulas; all related Matlab coding; Matlab ode45.
 Concept of explicit and implicit methods; predictorcorrector methods;
Matlab programming.

Write a highorder ODE into a system of firstorder ODEs;
solving a system of firstorder ODEs by Euler's method; Matlab code.
 Concept of Lagrange interpolation; Lagrange formula and Newton formula;
Neville's and Aitken's algorithms; all related Matlab commands and coding.
 Natural cubic splines: concept, calculations, Matlab commands and coding.
 Leastsquares fitting: concept, calculations, Matlab commands and coding.