Analytical and Computational Methods in Engineering ECE6203 (1st Year MSc Program)

Chapter 1: Finite precision arithmetic and numerical errors

Background, bits, bytes, words and types

floating point arithmetic

sources of flop errors (representation, rounding, truncation, propagation, catastrophic)

zero test (absolute/machine zero tolerance)

convergence tests

NaNs and Inf

Chapter 2: Solution of nonlinear algebraic equations

Bisection, False Position, Secant, and Newton methods Roots of polynomials

Chapter 3: Solution of linear system of equations

Direct methods: Gauss elimination, LU factorization Iterative methods: Jacobi iteration, Gauss-Seidel iteration

Chapter 4: Solution of nonlinear system of equations

Fixed point iteration Newton’s method

Chapter 5: Numerical approximation and interpolation

Least squares approximation

Polynomial approximation and interpolation Splines

Band limitted approximation

Chapter 6: Higher Dimensions, Irregular Data, Extrapolation and Approximation

Interpolation in two or more dimensions.

Data sampled on regular lattices.

Bilinear and trilinear interpolation.

Interpolating irregularly sampled data in one dimension. Interpolation of data over a triangular mesh: barycentric coordinates. Extrapolation: Motivation, e.g. forecasting.

Applying interpolation schemes beyond input data range. Need for a functional form.

Suitable applications for extrapolation and accuracy. Approximation, suitable kernels (e.g. B-spline)

Chapter 7: Numerical differentiation

Forward, backward, central differencing Higher order derivatives

Chapter 8: Numerical integration

Trapezoidal rule Simpson’s rule Gaussian quadrature

Chapter 9: Unconstrained Optimization Algorithms

Real-life motivation,

residual, objective function, stationary points (vs. roots),

Local and global minima.

Example: least squares formulation.

Gradient-based algorithms: steepest descent, Gauss-Newton,

Levenberg-Marquardt. Analysis (scalability,

convergence behaviors)

Chapter 10: Statistical methods

Random sampling, sample estimation Hypotheses testing

Statistical quality control

Monte Carlo methods

————REFERENCE BOOKs————————

Joel H. Ferziger, “Numerical Methods for Engineering Application”, John Wiley & Sons, Inc.

Thomas J.R., “The Finite Element Method: Linear Static and Dynamic Finite Element Analysis”.

C. T. Kelley, “Iterative Methods for Optimization”. SIAM, 1999.

R. Fletcher, “Practical Methods of Optimization”, 2nd ed., John Wiley & Sons.

D. Logan, “Applied Partial Differential Equations”, Springer-Verlag., 2004

A. Quarteroni,, and A. Valli, “Numerical Approximation of Partial Differential Equations”, Springer, 2008

U. Asher, “Numerical Methods for Evolutionary Differential Equations”.

Alastair, “Wood,Introduction to Numerical Analysis”, Addison Wesley, 1999.

J. H. Mathews, K. D. Fink, “ Numerical Methods Using MATLAB”, 3rd Ed., Prentice Hall, 1999.

R. E. Walpole, R. H. Myers, S. L. Myers, “Probability and Statistics for Engineers and Scientists”, 6th Ed., Prentice Hall, 1998.

W. Mendenhall, T. Sincich, “Statistics for Engineering and the Sciences”, 4th Ed., Prentice Hall, 1995.

Laurene V. Fausett, “Applied Numerical Analysis Using MATLAB”, Prentice-Hall, NJ, 1999.