Course Details

1. Discuss the need for numerical methods and how they can be successfully applied to many important scientific problems that cannot be solved exactly within a reasonable amount of time.
2. Identify limitations and compromises inherent in numerical computation.
3. Examine the influence of the nature of the problem to be solved, understand the properties of floating-point arithmetic, the architecture of available computers and determine the effect of round off errors or loss of significance.
4. Analyze various methods for solving non-linear equations, including bracketing, bisection, Newton-Raphson, secant and iterative methods. Evaluate their appropriateness for different examples, and assess their robustness and accuracy, as well as their rate of convergence.
5. Acquire a basic knowledge of numerical approximation techniques (Taylor-series Method or Newton-Cotes Rules) for mathematical expressions, such as derivatives and definite integrals, and learn how, why, and when these techniques can be expected to work. Evaluate various methods (Euler, Midpoint and Multistep method) for solving first-order Ordinary Differential Equation.
6. Compare a number of different polynomial interpolation techniques for Curve Fitting (Monomial Basis, Newton’s Divided-Difference and Langrange Interpolating Polynomials) and illustrate their applicability.
7. Write simple programs for the proposed numerical algorithms in Matlab or other programming environments.

Tools for Scientific Computation: Mathematical background from Calculus, Floating point representation and arithmetic; rounding errors and its consequences. Approximation of functions and derivatives; measuring and controlling errors.

Solving non-linear equations: Iterative methods; bracketing and bisection method; Newton-Raphson & secant method; convergence rates and criteria.

Curve Fitting: Polynomial Interpolation with Monomial Basis; Newton’s Divided-Difference Interpolating Polynomials; Langrange Interpolating Polynomials; Spline Interpolation; Least Squares Regression Approximation

Numerical Integration and Differentiation: Newton-Cotes Rules  (Trapezoidal and Simpson’s Rules); Taylor-series Method; Richardson’s Extrapolation.

Solving first-order Ordinary Differential Equations: Euler and Midpoint Method; Multistep Method.

• J.H. Mathews, K.D. Fink. Numerical Methods using MATLAB, (Fourth Edition), Pearson Prentice Hall, 2004.

• S.C. Chapra and R.P. Canale, Numerical Methods for Engineers, 6th Edition, McGraw Hill, 2010.

For the delivery of the class material, power point presentations are primarily used, along with the whiteboard. The lecture notes, consisting of slides presented in class, and additional material, are made available to the students through the course website and are complementary of the course textbooks. The theoretical part of each lecture is accompanied with detailed solved examples on which emphasis is given in the class. The solutions to these exercises, as well as specimen solutions for all tests and assignments, are discussed with students.