NUMERICAL ANALYSIS I

Compulsory course. 4 credits.

Theory: + Exercises : 4 credits

I. Course description / Overview

This course provides:

- Root finding;

- Linear and nonlinear system of equations

- Interpolation;

- Curve fitting and approximation of functions.

- Eigenvalue problems;

- Initial – value problems and numerical solution boundary – value problems.

II. Prerequisites: Calculus, linear Algebra, Ordinary Differential Equation

III. Contents

Chapter 1: Nonlinear Equations

1.1 The Bisection Method

1.2 The Secant Method

1.3 The Newton – Raphson Method

1.4 Fixed – Point Iteration

1.5 Numerical Solutions of Systems of Nonlinear Equations.

Chapter 2: Systems of Linear Equations

3.1 Gaussian Elimination and Pivoting

3.2 Direct Factorization Methods

3.3 Positive Definite Systems

3.4 Tridiagonal Systems

3.5 Matrix Inversion

3.6 Iterative Methods for Systems of Equations.

Chapter 3: Interpolation & Curve fitting

3.1 The Lagrange Interpolating Polynomial

3.2 The Newton Divided Difference Formulas

3.3 Equal Interval Methods.

3.4 Piecewise Polynomial Approximations: Cubic Splines.

3.5 Least Squares Approximations.

Chapter 4: Numerical Integration and Numerical Differentiation

4.1 First Derivatives & Higher Derivatives.

4.2 Newton – Cotes Formulas

4.3 Chebyshev Integration

4.4 Gaussian Integration

Chapter 5: Eigenvalue and Eigenvectors

5.1 The Power Method

5.2 The Danilevski Method

5.3 The Leverier Method

5.4 The Jacobi Method

5.5 The LR Algorithm and the QR Algorithm.

Chapter 6: Initial – Value problems

6.1 The Picard Method

6.2 The Taylor Series Method

6.3 The Euler Method

6.4 Runge – Kutta Methods

6.5 Multistep Methods

6.6 Systems of ODE.

IV. References

1. Pham Ky Anh – Numerical Analysis, Hanoi, 1996.

2. W. H. Press et al. – Numerical Recips; USA, 1992.

3. John H. Mathews – Numerical Methods; USA, 1987.