Video Lectures for Numerical Analysis 4315

Course Information and Syllabus

Go to The Start of 2011 Numerical Analysis Lectures then scroll down to the current lecture video.
Go to The Start of 2009 Numerical Analysis Lectures then scroll down to the current lecture video.

Additional Resources

1-Resources for Matlab
a- check the videos that come up on the Matlab itself (on top of command windows once you open Matlab).
b- check videos by Charles F. Van Loan and K-Y Daisy Fan from Cornell University from their text Insight Through Computing. (You need to register on the site.)
c- check videos at MathWorks.

2-Online Texts and Notes
a- Numerical Analysis by Detruck Wilf Notes on ODE and Linear Algebra
b- Notes on Numerical Analysis by Catherine Powell
c- check notes at MathWorks.

3- Interactive Apps.
a- Function analyzer. Evaluation, graphing, differentiation, integration, solving, Taylor series generation. (Fractional powers of negative numbers should be handled carefully here.)
b- Multiple Function Graph Explorer 1.
c- Multiple Function Graph Explorer 2.
d- Wolfram Natural Language Portal.
e- Wolfram Integrator.

4- Calculators
a- InstaCalc with some spreadsheet capbility.
b- Online Calculator.
c- Calc98 a downloadable calculator.
d- Power Calculator a downloadable calculator.

5- Advanced Software
a- Scilab a numerical software package similar to matlab.
Some syntax differences with matlab:
- Matlab: function end -vs- SciLab: function endfunction
- Matlab: [first, second, third] -vs- Scilab: [third, second, first]




Lectures for Spring 2009
Intro to Matlab+ Bisection + Newton Method. Description of basic functionality of Matlab. How to write a function. Input, output. First program: bisection method. Error handling. fprintf and g formating. Basic descritpion of Newton Method.
Newton's Method Introduction to solving f(x)=0 by Newton's Method
Full error analysis for Newton's Method. Shows e_(k+1)= C e_k ^2 where C = f"/2f' at the root, uses Taylor Series.
Introduction to Secant method.

Introduction to Parograming in Matlab. Updating formula for secant method , basic Matlab, programing a basic newton algorithm.
Assignment 1 1st program: find averages, newton method with exit criteria maxiter, xtol, ytol, loops, conditionals, logical operators, advanced project: generalized averages.
Assignment 2 Error analysis for newton method, bisection algorithm, advanced project: generalized averages
Assignment 3 Error analysis for quadratures
Assignment 4 Richardson Extrapolation
Assignment 5 Romberg Algorithm
Assignment 6 Newton Basins
Introduction to Ordinary Differential EQuations
Assignment 7 Introduction to ODEs, Analysis of derivative
Epsilon, error of difference quotient in approximating derivative, error of representation and truncation, Euler method
Assignment 8 Euler Method, vectorization in Matlab
Higher order ODE solvers based on Taylor Series and higher order derivatives
Review of ODE, linear and separable cases, finite difference approximation of second derivative
Higher order and explocit Runge-Kutta methods Brief explanation of Mid-point, Heun, 2/3, and RK4 methods
Assignment 11 Explicit Runge-Kutta Methods Euler, Mid-point, Huen-2, 2/3, RK4 methods explained. No proof.
Design of Higher Order Runge-Kutta Methods, Part 1 Derivation of Taylor Series formula for functions of 2 or more variables.
Design of Higher Order Runge-Kutta Methods, Part 2a Derivation of 2nd order methods.
Design of Higher Order Runge-Kutta Methods, Part 2b Derivation of 2nd order methods.
Numerical Differential Equations 3. Third order RK methods: Nystrom, Nearly Optimal, Classical, Heun. Parameter array for a method. (There is some audio problem toward the end of this file.)




Lectures for Spring 2011
Introduction to NUmerical Analysis symmetric averages, first program in matlab, Newton Method.






Older lectures from 2006-08:
Intro to Matlab+ Bisection + Newton Method. Description of basic functionality of Matlab. How to write a function. Input, output. First program: bisection method. Error handling. fprintf and g formating. Basic descritpion of Newton Method.
Secant Method. Secant method for finding roots of f(x)=0.
False Position Method. False position method for finding roots of f(x)=0.
Simplified Error Analysis for Bisection and Newton Methods. Error analysis for bisection method. Simplified error analysis for newton method. Experimental determination of order of newton method. Introduction to catastrophic cancellation.
Simplified Error Analysis for Secant Method. Application of Fibonacci sequence in eror analysis of secant method. Experimental determination of the order.
Numerical Integration 1. Left-end point rule, right end-point rule, trapezoid rule, mid-point rule, Experimental determination of the order. Introduction to Simpson's rule.
Numerical Integration 2. How to create a quadrature (numerical integration) rule from scratch. We make a rule using polynomial interpolation. We approximate the function, on each panel, by a parabola and integrate the parabola.
Numerical Integration 3. Introduction to Gaussian Quadrature rules. Comparison agianst Newton-type rules. Two-point GQR is derived and three-point GQR is presented.
Numerical Differential Equations 1. Intrduction to differential equations. Direction field. Euler Method. Sample problem.
Numerical Differential Equations 2. Higher order methods. Taylor Series. Taylor methods. Runge-Kutta methods (Mid, Euler-cauchy, Heun).
Numerical Differential Equations 3. Third order RK methods: Nystrom, Nearly Optimal, Classical, Heun. Parameter array for a method. (There is some audio problem toward the end of this file.)
Numerical Differential Equations 4. Informal theoretical derivation of coefficients for second order Runge-Kutta methods. An application of Taylor series theorem and multi-varibale chain rule. Mid-point, Euler Cauchy, and Huen methods are derived as examples of a one-parameter family of methods.
Corrections:
On the first page I write K_2 = h f ( t + ah, y + b K_2)
it should be k_2 = h f ( t + a h, y + b K_1)

video 2. Review of Derivatives. derivative is limit of a difference quotient. Linearization formula for a function. Taylor series formula for a function of one variable. Taylor series formula for a function of two variables.

video 3. Review of Chain Rule. Chain rule for functions of several variables. Application to ODE.

video 4. Numerical Solution of ODEs 1. Euler method, Taylor methods, second order Taylor method, second order Ruge-Kutta methods (Midpoint, modified Euler,and Huen methods)

video 5. Numerical Solution of ODEs 2. Derivation of coefficients for second order Runge-Kutta Methods (Midpoint, modified Euler,and Huen methods)

video 6. Stability of Numerical Methods for ODEs. Simple description of stability, stability in the context of Euler method applied to the model problem, y'=lambda*y, lambda <0, brief description of stability for systems of differential equations