## Final Exam: 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), 2.13-2.15, 1.17;
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

1. Basic knowledge of Matlab: input and output, data structure, operations, vectors and matrices, programming, graphics, etc.
2. 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.
3. 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.
4. 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.
5. 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.
6. Derivation of some basic numerical differentiation formulas; Matlab codes.
7. 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.
8. Rectangular, trapezoidal, and Simpson's rule for numerical integration, their derivation and degrees of exactness; related Matlab coding; general ideas of Newton-Cotes formulas.
9. Gaussian quadrature: concepts, derivation, and basic formulas (up to n = 2); use Gaussian quadrature to approximate definite integrals; related Matlab codes.
10. Gauss-Chebyshev formulas and Filon's since and cosine formulas.
11. 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.
12. Concept of explicit and implicit methods; predictor-corrector methods; Matlab programming.
13. 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.
14. Concept of Lagrange interpolation; Lagrange formula and Newton formula; Neville's and Aitken's algorithms; all related Matlab commands and coding.
15. Natural cubic splines: concept, calculations, Matlab commands and coding.
16. Least-squares fitting: concept, calculations, Matlab commands and coding.