Numerical Methods

Abstract

The course introduces numerical methods according to the type of problemthey tackle. The tutorial will include both theoretical exercises andpractical tasks. The latter will mainly employ on the numerical programming languageMATLAB.Prerequisite is familiarity with basic calculus and linear algebra.

Objective

This course intends to introduce to fundamental numerical methods that form the foundation of numerical simulation in engineering science. Participants should learn about classes of methods, should understand their principles and will be taught how to assess, implement, and apply them. During the course they will become familiar with basic techniques and concepts of numerical analysis. They should be enabled to select and adapt suitable numerical methods for a particular problem.

Content

Computer arithmetic, roundoff errors, elementary error propagation, conditioning of a problem, scalar equations, fix point iteration, computation of zeros of functions, Newton method, secant method, order of convergence, efficiency index, linear systems of equations, Gauss elimination and LU factorization, sparse matrices, band matrices, positive definite matrices, Cholesky factorization, column pivoting for LU factorization, condition of a matrix, error bounds when solving perturbed linear systems of equations, iteration methods for linear systems, convergence, Jacobi or simultaneous iterations, Gauss-Seidel or successive iterations, SOR iterations, nonlinear systems of equations, polynomial interpolation, Neville-Aitken diagram, Newtons interpolation formula, divided differences, interpolation error, cubic splines, solving overdetermined systems of equations, error equations, method of least squares, method of least squares for linear problems, normal equations, Moore-Penrose pseudo inverse, solving the normal equations using QR factorization, initial value problems for systems of ordinary differential equations, local error, error order, one step methods, theta method, Heun method, Runge-Kutta and the Butcher tableau, linear multistep methods, Adam-Bashforth, Adams-Moulton, backward differentiation formulas, stepsize control, stiff differential equations, stability regions, comparison of code for solving initial value problems of ordinary differential equations

Resources