CS 2262: Numerical Methods Schedule: TTh 3:10-4:30 Room: Turead 0229 Instructor: Rahul Shah Office: 285 Coates Phone: Office Hours: Wed 2:30-4:30, Th Web: Text: Atkinson and Han, Elementary Numerical Analysis, 3rd edition, John Wiley & Sons, Inc., 2004 Grader: ?

Grading Midterm : 30 % Final : 30% Homework/Project: = 35% Class Participation : 5% Relative – on the curve Homeworks will involve writing Matlab programs …about 4 to 5 of them Mini project will involve solving real-life problem using Matlab and/or C

Prerequisites and Background Math 1552 and CS 1251 or 1351 or 2290 The course will involve concepts from –Calculus –Linear Algebra Programming –Matlab –C

Applications of this course Numerical Methods Data Mining Algorithms/Optimization Solving large scale systems Sales data Search Engines Network problems Fluid dynamics Stock Market

Course Contents Foundations: Calculus, Computer Architecture, Matlab Taylor Series Root Finding Polynomial Interpolation Numerical Integration/Differentiation Linear Equations/Matrices Differential Equations

Overview Taylor Series –Evaluationg functions like sin x, e x etc –Processors only have support for additions and multiplications –Errors involved, number of iterations needed

Overview: Root finding Reverse process of evaluating the function Given function f, find the value of x such that f(x) = 0 Methods for general functions Methods for polynomials Rate of convergence

Interpolation Given a set of points (x 1, y 1 ), (x 2, y 2 ), …, (x n, y n ) –Find a polynomial which passes through them –Find a line which fits these the best –Find a smooth curve which passes through them

Matrices Given a set of n linear equations in variables x 1, x 2, x 3, …, x n –Find the values of x i s –Find best values of x i s Linear programming, Optimization Find Eigenvalues of the matrix –Google –Differential Equations

Differential Equations Modelling/Simulations of Engineering systems Population Modeling Financial Models, Stocks/Options pricing

Motivation1: Modelling Traditionally, engineering and science had a two-sided approach to understanding a subject: the theoretical and the experimental. More recently, a third approach has become equally important: the computational. Traditionally we would build an understanding by building theoretical mathematical models, and we would solve these for special cases. For example, we would study the flow of an incompressible irrotational fluid past a sphere, obtaining some idea of the nature of fluid flow. But more practical situations could seldom be handled by direct means, because the needed equations were too difficult to solve. Thus we also used the experimental approach to obtain better information about the flow of practical fluids. The theory would suggest ideas to be tried in the laboratory, and the experiemental results would often suggest directions for a further development of theory.

Modeling contd Theoretical Science Experimental Science Computational Science

Modeling: Population This is the simplest model for population growth. Let N(t) denote the number of individuals in a population (rabbits, people, bacteria, etc). Then we model its growth by –N’(t) = cN(t), t≥ 0, N(t 0 ) = N 0 The constant c is the growth constant, and it usually must be determined empirically. Over short periods of time, this is often an accurate model for population growth. For example, it accurately models the growth of US population over the period of 1790 to 1860, with c =

Population Data

Predator-Prey Let F(t) denote the number of foxes at time t; and let R(t) denote the number of rabbits at time t. A simple model for these populations is called the Lotka-Volterra predator-prey model: –dR/dt = a [1 − bF(t)] R(t) –dF/dt = c [−1 + gR(t)] F(t) –with a, b, c, g positive constants. If one looks carefully at this, then one can see how it is built from the logistic equation. In some cases, this is a very useful model and agrees with physical experiments. Of course, we can substitute other interpretations, replacing foxes and rabbits with other predator and prey. The model will fail, however, when there are other populations that affect the first two populations in a significant way.

Motivation2: Google Term frequency, location, meaning based search engines : Altavista, Lycos etc Spamming Google used social concepts to reduce effect of spamming A webpage is good if many good webpages link to it So how to find goodness score

Google contd.. Say there a n webpages Construct a n x n probability matrix –With A[i,j] = likelihood that a user will jump to page j from i Find dominant eigenvalue of this matrix Corresponding eigenvector gives the goodness scores How to solve the problem on such a large scale, which method to use, how many iterations, etc

Foundations: Calculus Intermediate Value Theorem Mean Value Theorem Extended Mean Value Theorem Integral Mean Value Theorem

Intermediate Value Theorem Let f(x) be a continuous function in interval a ≤ x ≤ b, –Let M = max f(x) in the interval [a,b] –Let m = min f(x) in [a,b] Then, for any value v such that m ≤ v ≤ M –There is at least one point c such that f(c) = v.

IVT

Mean Value Theroem Let f(x) be continuous and differentiable on [a,b] Then there is at least one point c in (a,b) –Such that f(b) – f(a) = f’(c) (b-a)

MVT

Extension Let f(x) be continuous and n-times differentiable on [a,b] –Then there is c such that f(b) = f(a) + f’(c)(b-a) –There is d such that f(b) = f(a) + f’(a) (b-a) + f’’(d) (b-a) 2 /2 –….. –There is t in [a,b], such that f(b) = f(a) + f’(a)(b- a) + f’’(a) (b-a) 2 /2 …+ f (n-1) (a) (b-a) n-1 /(n-1)! + f ( n) (t) (b-a) n /n!