ENEE 241/MATH 242
Numerical Techniques in Engineering
8:00-10:00, Friday, 5/17, ENG 1202
Neither rescheduled nor make-up 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 close-book, close-note, no-calculator exam.
It will be cumulative, and will cover the following sections
of the textbook and related topics presented in lectures:
Chapter 1. 1.1-1.10, 1.12, 1.13, 1.5-1.7;
Chapter 2. 2.1-2.3, 2.5-2.11, 2.12 (exclude under-determined systesm),
Chapter 3. 3.1-3.5, 3.7, 3.10, 3.14;
Chapter 4. 4.1-4.7, 4.9, 4.10;
Chapter 5. 5.1-5.6, 5.10, 5.14;
Chapter 7. 7.1-7.3, 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 Gauss-Seidel iterative methods;
all related Matlab commands.
- Householder matrices; QR decomposition;
concept of pseudo-inverse matrices;
over-determined 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 Newton-Cotes formulas.
Gaussian quadrature: concepts, derivation, and basic formulas (up to
n = 2); use Gaussian quadrature to approximate definite integrals;
related Matlab codes.
Gauss-Chebyshev formulas and Filon's since and cosine formulas.
Initial-value problems of ODEs;
Euler's method and related stability issue; the trapezoidal method;
basic second-order and forth-order Runge-Kutta methods; Adams methods: ideas
and basic formulas; all related Matlab coding; Matlab ode45.
- Concept of explicit and implicit methods; predictor-corrector methods;
Write a high-order ODE into a system of first-order ODEs;
solving a system of first-order 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.
- Least-squares fitting: concept, calculations, Matlab commands and coding.