CS 2262: Numerical Methods

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CS 2262 Numerical Methods

• Schedule TTh 310-430
• Room Turead 0229
• Instructor Rahul Shah
• Office 285 Coates
• Phone [phone]
• Office Hours Wed 230-430, Th 11-12
• Email rahul_at_csc.lsu.edu
• Web http//www.csc.lsu.edu/rahul/2262
• Text Atkinson and Han, Elementary Numerical
  Analysis, 3rd edition, John Wiley Sons, Inc.,
  2004
• Grader ?

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Grading

• Midterm 30
• Final 30
• Homework/Project 2510 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

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Prerequisites and Background

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

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

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

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Overview

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

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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

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Interpolation

• Given a set of points (x1, y1) , (x2, y2), ,
  (xn, yn)
• Find a polynomial which passes through them
• Find a line which fits these the best
• Find a smooth curve which passes through them

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Matrices

• Given a set of n linear equations in variables
  x1, x2, x3, , xn
• Find the values of xi s
• Find best values of xi s
• Linear programming , Optimization
• Find Eigenvalues of the matrix
• Google
• Differential Equations

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Differential Equations

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

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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.

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Modeling contd
Theoretical Science
Computational Science
Experimental Science
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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(t0) N0
• 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 0.2975.

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Population Data
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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.

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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

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Google contd..

• Say there a n webpages
• Construct a n x n probability matrix
• With Ai,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

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Foundations Calculus

• Intermediate Value Theorem
• Mean Value Theorem
• Extended Mean Value Theorem
• Integral Mean Value Theorem

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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.

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IVT
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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)

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MVT
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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!

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