# Math 254

**Numerical Methods**

**Credit Hours: 3(3 + 0) **

**Prerequisite**: M-107 or M-202 or M-244 + CompSci-101 or CompSci-206 or CompSci-207

**TEXTBOOK**: **Introduction to Numerical Analysis using MATLAB**

**AUTHOR**: **Rizwan Butt **

**REFERENCE TEXTBOOK**: **Numerical Analysis (Seventh Edition)**

**AUTHORS**: Richard L. Burden and J. Douglas Faires

**Note: ****The Contents of the course will be covered by the following sections**:

**CHAPTER 2: 2.1,2.2,2.4,2.5,2.6,2.7,2.8,2.10.**

**CHAPTER 3: 3.1,3.2,3.3,3.4,3.6,3.7.**

**CHAPTER 4: 4.1,4.2,4.3. **

**CHAPTER 5: 5,1,5.2,5.3,5.5,5.6.**

**CHAPTER 6: 6.1,6.2,6.3.**

**Theorems with Proofs: 2.2,3.28,5.2,5.3,5.4.**

**Note: The proofs of Linear Lagrange formula+unique Lagrange polynomial+Trapezoida rule for two-points + Simpsons rule for three-point + dierentiation [two-point + three point (Forward+Central+Backward)formulas]can be ask.**

**Theorems and Lemmas without Proofs:**

**Theorems 2.1,3.1,3.2,3.7,3.8,3.14,3.20,3.21,3.22,4.1,4.2,4.3,4.4,5.1,5.5,6.1. Lemmas 2.1,2.2,2.3,2.4.**

**Note: Before we start Chapter 2, we must discuss Computer Representation of Num ber, Error(in details), the Taylors polynomial. These topic can be found in the and fourth chapters of the recommended book(sections:1.2,1.3,1.4,and 4.1).**

**Note: About the 10 marks we do the following:**

**2 Quizzes + computer assignments + Attendance (4 + 3 + 3) marks**

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**The students should study the following topics:**

**Chapter 2: Solution of Nonlinear Equations**

**The bisection method: How to apply it and to compute an error bound for the approxi****mate solutions derived by the method.**

**The Newtons method: How to apply it and the analysis of its error.**

**The secant method: How to apply it.**

**The fixed point iterative method: How to formulate the function g(x) which will satises the conditions of the Theorem 2.2, then apply the iterative scheme and the analysis of the error.**

**The rate of convergence of the iterative methods including the Newtons method.**

**The multiple roots: How to dene it and discuss the conditions under which the root is said to be simple or multiple. Here some attention should be given for the rate of the convergence of Newtons method for both the simple and the multiple roots. Also, th following modied Newtons methods should be discussed:**

**where m is the multiplicity of the multiple roots, and the other one is**

**The Newtons method for the nonlinear systems (only for two nonlinear equations)**

**Chapter 3 Systems of Linear Equations**

**How to apply the Gaussian elimination method (without pivoting) (algorithmic approach) and also, discuss the partial pivoting. Give examples showing that the system has innit number of solutions or no solution at all (singular matrix).**

**How to apply LU factorization [only lii = 1 (in lecture class), and uii = 1 (in tutorial class)]. How to apply the iterative methods (Jacobi and Gauss-Seidel) to solve a linear system. The analysis of the error related to these methods (condition for convergence, diagonally dominant matrix ...). Also, how to compute an error bound for both methods. Error in solving linear systems. Residual vector, condition number of a given matrix... etc. How to compute an upper bound for both absolute and relative errors. Here we must give the denitions of the vector and matrix norms (l norm only).**

**Chapter 4: Approximating Functions**

**How to construct the Lagrange polynomial which approximate a function f(x) at an (n + 1) distinct numbers(not the uniqueness).**

**How to apply the divided dierences to construct the Newtons polynomial (without d cussing the forward or backward cases).**

**Error in polynomial interpolation: How to compute an error bound for any**

*x*value in the interval*[ab]*and a given*x*=*x [ab].*Interpolation using spline functions(Linear Spline Only).
**Chapter 5 Differentiation and Integration**

**How to derive the rst and second order nite dierence formulas for approximating th**

**rst and second derivatives of the function f(x) at a point x0 using Lagrange and Taylor polynomials (note that for the rst derivative we study (Forward+Central+Backward and for the second derivative we study only the central dierence formula). How to apply these formulas including the estimation of an error bound (also, discuss the eect of erro in function values).**

**How to derive the Trapezoidal and the Simpson rules, how to apply them and to compute the error bounds (for both single and composite formulas).**

**Chapter 6: Ordinary Dierential Equations**

**How to use the Eulers method, the Taylor method of order N and the Runge Kutt method of order two (only modied Eulers method) and order four for solving rs order initial-value problems in ordinary dierential equations. Also, discuss the loca truncation errors for Euler and Taylor methods.**