Applied Numerical Methods For Engineering & Science Students

Engineering, Science, Finance, Economics

What you'll learn

Develop an understanding of numerical Root Finding techniques.

Develop an understanding of Linear and Nonlinear Equation Solvers.

Develop an understanding of Numerical Integration and Differentiation.

Develop an understanding of Curve Fitting.

Develop an understanding of errors inherent in numerical methods and the use of Taylor Series Expansions.

Improved computer programming skills.

Requirements

Knowledge of basic calculus, a programming language, and access to a computer.

Description

Motivation:Many, if not most, mathematical formulations resulting from the application of physical laws in science and engineering are not amenable to analytic solutions.  This leads to a numerical approach which generally involves the formulation of problems so that they may be solved using what is typically a large number of arithmetic operations, ideally suited for programming on a computer.  Numerical methods are also widely employed in fields of finance and economics.This Course:This is a first course in applied numerical methods for engineering and science university students, or those interested in a refresher course in numerical methods.  It may also be of interest to students with interests in the fields of finance and economics.  Course content is aimed toward students at the sophomore or junior level who have a basic knowledge of calculus and computer programming.  Topics covered include errors and Taylor series expansions, root finding, solution of systems of linear and nonlinear equations, numerical differentiation and integration, and curve fitting.  The subject of numerical solutions to ordinary and partial differential equations is covered in a separate course as is the subject of optimization.  To receive the most benefit from this course, students should have completed a basic calculus class and have access to a computer with a programming language installed.  Several programs are made available for download with Fortran being used for the algorithms, while Octave is used for graphics.  All PowerPoint slides used in the lectures are also available for download.

Overview

Section 1: Introduction

Lecture 1 Introduction

Section 2: Errors and Taylor Series Expansions

Lecture 2 Numerical Errors

Lecture 3 Taylor Series Expansions

Section 3: Root Finding

Lecture 4 Introduction and Example Problem

Lecture 5 Interval Halving

Lecture 6 Interval Halving Error Analysis

Lecture 7 Linear Interpolation

Lecture 8 Fixed Point Iteration

Lecture 9 Fixed Point Example

Lecture 10 Newton's Method

Lecture 11 Newton's Method Example

Lecture 12 Secant Method

Lecture 13 Nonlinear System of Equations

Section 4: Simultaneous Equation Solvers

Lecture 14 Introduction

Lecture 15 Matrix Notation and Operations

Lecture 16 Basic Elimination Methods

Lecture 17 Partial Pivoting

Lecture 18 LU Decomposition

Lecture 19 Matrix Inverse

Lecture 20 Matrix Norms and Condition Number

Lecture 21 Iterative Solvers

Lecture 22 Iterative Convergence

Lecture 23 Spectral Radius Example

Lecture 24 SOR/Spectral Radius Example

Section 5: Numerical Differentiation and Integration

Lecture 25 Finite Difference Approximations I

Lecture 26 Finite DIfference Approximations II

Lecture 27 Finite Difference Example

Lecture 28 Numerical Integration: Trapezoidal Rule

Lecture 29 Trapezoidal Rule Error

Lecture 30 Simpson's Rules

Lecture 31 Numerical Integration Example

Lecture 32 Richardson Extrapolation

Section 6: Curve Fitting

Lecture 33 Least Squares Fit Introduction

Lecture 34 Linear Least Squares Fit

Lecture 35 Quantification of Least Squares Fit Errors

Lecture 36 Quadratic Least Squares Fit and Example

Lecture 37 Exponential Least Squares Fits

Lecture 38 Lagrange Polynomials

Lecture 39 Linear Spline Interpolation

Lecture 40 Quadratic and Cubic Splines with Example

Advanced high school and college-level students in mathematics, engineering, and the sciences.