CSE 3802 / ECE 3431 Numerical Methods in Scientific Computation Jinbo Bi Department of Computer Science & Engineering http://www.engr.uconn.edu/~jinbo

The Instructor Ph.D in Mathematics Previous professional experience: Siemens Medical Solutions Inc. Department of Defense, Bioanalysis Research interests: biomedical informatics, machine learning, data mining, optimization, mathematical programming, Apply machine learning techniques in biological data, medical image analysis, patient health records analysis Homepage is at http://www.engr.uconn.edu/~jinbo

Class Meetings Lectures are Tuesday and Thursday, 12:30 –1:45 pm No specific lab time, but significant computer time expected. Computers are available in ITEB C25 and C27. Numerical Methods, Lecture 1 Prof. Jinbo Bi CSE, UConn

Class Assignments Homework will be assigned once every week or two and due usually the following week. You may collaborate on the homework, but your submissions should be your own work. Grading: Homework 30% Exam 1 and 2 40% Final Exam 30% Numerical Methods Lecture 1 Prof. Jinbo Bi CSE, UConn

Mathematical Background MATH 2110Q Multivariate Calculus Taylor series MATH 2410Q Introduction to Differential Equations Integration MATH 2210 Linear Algebra Equation systems Numerical Methods, Lecture 1 Prof. Jinbo Bi CSE, UConn

Computer Background Languages to be used: Matlab, C, C++ CSE 1100/1010 programming experience Any OS is acceptable Numerical Methods, Lecture 1 Prof. Jinbo Bi CSE, UConn

Syllabus Go over the course syllabus Course website http://www.engr.uconn.edu/~jinbo/Fall2013_Numerical_Methods.htm Numerical Methods, Lecture 1 Prof. Jinbo Bi CSE, UConn

Today’s Class: Introduction to numerical methods Basic content of course and class expectations Mathematical modeling Numerical Methods, Lecture 1 Prof. Jinbo Bi CSE, UConn

Introduction What are numerical methods? “… techniques by which mathematical problems are formulated so that they can be solved with arithmetic operations.” (Chopra and Canale) What type of mathematical problems? Roots, Integration, Optimization, Curve Fitting, Differential Equations, and Linear Systems Numerical Methods, Lecture 1 Prof. Jinbo Bi CSE, UConn

Introduction How do you solve these difficult mathematical problems? Example: What are the roots of x2-7x+12? Three general non-computer methods Analytical Graphical Manual Let us use some very simple examples to demonstrate the idea the concept. So what we mean numerical methods, numerical solutions. X2-7x+12, we are looking for the roots of this function. You could do three things, these three methods do not require computers. 1 so called analytical solutions 2, graphical way to find solutions 3, you manually calculate the solutions. Numerical Methods, Lecture 1 Prof. Jinbo Bi CSE, UConn

Analytical Solutions This is what you learned in math class Gives exact solutions Example: Roots at 3 and 4 Not always possible for all problems and usually restricted to simple problems with few variables or axes The real world is more complex than the simple problems in math class Analycial solutions are the stuff you learn from your math class, algebra or calculus, differential equations. For the given function, you could factorize the function into the product of these two terms. Based on the examination of the two terms, you will know if you set x=3 or 4, this function will reach value of 0. So you know the roots are 3 and 4. Numerical Methods, Lecture 1 Prof. Jinbo Bi CSE, UConn

Graphical Solution Numerical Methods, Prof. Jinbo Bi CSE, UConn The graphical solution give you a lot intuitive idea about the problem you are trying to solve. You specify some x values, then you calculate the function values of these x, then you draw them as points in this figure, and connect these points Numerical Methods, Lecture 1 Prof. Jinbo Bi CSE, UConn

Manual Solution Using pen and paper, slide rulers, etc. to solve an engineering problem Very time consuming Error-prone Numerical Methods, Lecture 1 Prof. Jinbo Bi CSE, UConn

Numerical Methods What are numerical methods? “… techniques by which mathematical problems are formulated so that they can be solved with arithmetic operations.” (Chopra and Canale) Arithmetic operations map into computer arithmetic instructions Numerical methods allow us to formulate mathematical problems so they can be solved numerically (e.g., by computer) Numerical Methods, Lecture 1 Prof. Jinbo Bi CSE, UConn

Course Overview What is this course about? Using numerical methods to solve mathematical problems that arise in engineering Most of the focus will be on engineering problems Numerical Methods, Lecture 1 Prof. Jinbo Bi CSE, UConn

Basic Materials Introduction Mathematical Problems Programming Mathematical Modeling Error Analysis Mathematical Problems Roots, Linear Systems, Integration, Optimization, Curve Fitting, Differential Equations Numerical Methods, Lecture 1 Prof. Jinbo Bi CSE, UConn

Mathematical Modeling A mathematical model is the formulation of a physical or engineering system in mathematical terms. Empirical Theoretical Mahtemtaicla modeling involves a deep understanding of the engineering problem that you need to solve. You want to fix a car, no matter what fantastic tools you may have, if you don’t understand how the car works, you can not do it. Numerical Methods, Lecture 1 Prof. Jinbo Bi CSE, UConn

Mathematical Modeling A mathematical model is represented as a functional relationship of the form Dependent independent forcing Variable = f variables, parameters, functions Dependent variable: Characteristic that usually reflects the state of the system Independent variables: Dimensions such as time and space along which the systems behavior is being determined Parameters: reflect the system’s properties or composition Forcing functions: external influences acting upon the system In general, a mathematcial model is represented as a funcational relationship of the following form The dependent variable usually reflects the state or a chracteristics of the system; Independenct variables are usually dimensions such as time, space, available resources, the kind of variables on which the system’s state can be determined. Parameters are reflective of the system’s properties, sometimes come from a theory. Numerical Methods, Lecture 1 Prof. Jinbo Bi CSE, UConn

Mathematical Modeling A simple example: In an electrical circuit, I = V/R; The current, I, is dependent on resistance parameter, R, and forcing voltage function, V. Numerical Methods, Lecture 1 Prof. Jinbo Bi CSE, UConn

Example 1 Newton’s 2nd low of Motion states that “the time rate change of momentum of a body is equal to the resulting force acting on it.” The model is formulated as F = m a F=net force acting on the body (N) m=mass of the object (kg) a=its acceleration (m/s2) Numerical Methods, Lecture 1 Prof. Jinbo Bi CSE, UConn

Example 1 What is the velocity of a falling object? First step is to model the system Newton’s second law Total force is gravity and air resistance The gravity is the downward force which is paramterized by gravitational constant, which is approximately 9.8 m/s2 Numerical Methods, Lecture 1 Prof. Jinbo Bi CSE, UConn

Example 1 First order differential equation Analytical solution Numerical Methods, Lecture 1 Prof. Jinbo Bi CSE, UConn

Example 1 m=68.1kg, c=12.5 kg/s Numerical Methods, Lecture 1 Prof. Jinbo Bi CSE, UConn

Example 1 What if we can’t find an analytical solution? How do you get a computer to solve the differential equation? Use numerical methods Numerical Methods, Lecture 1 Prof. Jinbo Bi CSE, UConn

Euler’s Method Use the finite divided difference approximation of the derivative The approximation becomes exact as Δt → 0 Now, we want to re-formulate the problem so it can approximately solved using numerical steps, or arithmetic operations. Numerical Methods, Lecture 1 Prof. Jinbo Bi CSE, UConn

Euler’s Method Using Euler’s method, we can approximate the velocity curve Numerical Methods, Lecture 1 Prof. Jinbo Bi CSE, UConn

Euler’s Method …… Assume Δt=2 Numerical Methods, Lecture 1 Prof. Jinbo Bi CSE, UConn

Euler’s method Numerical Methods, Lecture 1 Prof. Jinbo Bi CSE, UConn

Euler’s Method Avoids solving differential equation Not an exact formula of the function Gets more exact as Δt→0 How do we choose Δt? Dependent on the tolerance of error. How do we estimate the error? Numerical Methods, Lecture 1 Prof. Jinbo Bi CSE, UConn

Overview of the problems Numerical Methods, Lecture 1 Prof. Jinbo Bi CSE, UConn

Overview of the problems Numerical Methods, Lecture 1 Prof. Jinbo Bi CSE, UConn

Next class Programming and Software Read Chapters 1 & 2 Numerical Methods, Lecture 1 Prof. Jinbo Bi CSE, UConn