Math 104A, Summer 2010 - Introduction to Numerical AnalysisMonday, Tuesday, Wednesday & Thursday, 12:30-1:35pm, GIRV 2108

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We will have a ''@@color(red):Midterm exam on Monday, July 12th, 2010@@''. The ''@@color(red):Final exam will be on Wednesday, July 28th, 2010@@''.

Welcome to the Math 104A Wiki!

To navigate this page, simply follow the links on your left. 

# [[Instructor]]: Who I am, and how to contact me.
# [[Teaching Assistant]]: There will be no TA for this course. 
# [[Location]]: Where the class will meet.
# [[Syllabus]]: Information about the course.
# [[Weekly Readings]]: Here I will post the readings for each week, and the lesson objectives. Please take a look at it before the class.
# [[Homework]]: I will post the homework here. 
# [[Lecture Notes]]: These notes created by Prof. Carlos J. García-Cervera  can be useful.
# [[Self-Assessment Questions]]: This will help you study for the midterm and the final exam.
# [[Exams]]: Information about the midterm and final exams.

Once you open a tiddler, you can close it by clicking on its link again. Alternatively you can use the close button over on the right. 

This page is experimental, and hopefully more services will be added as the course progresses, so come back often.

Homework will be assigned on Wednesday, and will be collected at the beginning of the class on the following Wednesday. ''@@color(red): Late homework will not be accepted. If you have a predictable absence, you will need to hand it in earlier@@''. The homework will generally consist of some theoretical questions, and some computational assignments. You will be required to write a program to solve certain problems. The program must be given to me as part of the assignment, together with the output of the program, in the format indicated in the assignment, and an interpretation of the results whenever necessary. You can write the programs either in FORTRAN, C, C++, or Matlab. The book comes with a CD that contains the code for the problems. You may use this code as a guide, but ''@@color(red):you must write your own original code for the assignments. No credit will be given for using the code in the CD@@''.

''Assignments:''
#[[Homework 1|hw1.pdf]]: Due on June 28th, 2010. If you do not have the book, you can download the [[scanned exercises|exercises1.pdf]].
#[[Homework 2|hw2.pdf]]: Due on July 6th, 2010. If you do not have the book, you can download the [[scanned exercises|exercises2.pdf]].
#[[Homework 3|hw3.pdf]]: Due on July 13th, 2010. If you do not have the book, you can download the [[scanned exercises|exercises3.pdf]].
#[[Homework 4|hw4.pdf]]: Due on July 19th, 2010. If you do not have the book, you can download the [[scanned exercises|exercises4.pdf]].
#[[Homework 5|hw5.pdf]]: Due on July 26th, 2010. If you do not have the book, you can download the [[scanned exercises|exercises5.pdf]].

!Francisco de la Hoz Méndez.
South Hall, Room 6709.
Office Phone: (805) 893 5917
Office Fax: (805) 893 2385
Office Hours: Tuesday & Thursday, 11:00am-12:30pm. 
Main Website: http://www.math.ucsb.edu/~delahoz

In this course we will study mostly second order linear ordinary differential equations of the form
ay' '+by'+cy = f(x).

Since the equation is linear, it turns out that the set of solutions forms a finite dimensional vector space, which explains why we study Linear Algebra in this course.

These notes created by Prof. Carlos J. García-Cervera can be useful:

#[[Review Of Calculus|LN/Review of Calculus.pdf]].
#[[Algorithms and Convergence|LN/Algorithms and Convergence.pdf]].
#[[Solutions of Equations in One Variable|LN/Solutions of Equations in one variable.pdf]].
##The code for the Bisection method can be downloaded here: [[bisec.m|bisec.m]], and [[f.m|f.m]].
#[[Polynomial Interpolaton|LN/Polynomial Interpolation.pdf]].
#[[Numerical Differentiation|LN/Numerical Differentiation.pdf]].
#[[Richardson's Extrapolation|LN/Richardson's Extrapolation.pdf]].
#[[Integration - Numerical Quadrature|LN/Integration - Numerical Quadrature.pdf]].
##A Fortran code for Adaptive Quadrature can be downloaded here: [[adapquad.f|adapquad.f]].
#[[Ordinary Differential Equations|LN/Ordinary Differential Equations.pdf]].
#[[Runge-Kutta Methods|LN/Runge-Kutta Methods.pdf]].
#[[Multistep Methods|LN/Multistep Methods.pdf]].
#[[Higher order Equations and Systems|LN/Higher order Equations and Systems.pdf]].

Monday, Tuesday, Wednesday & Thursday, 12:30-1:35pm, GIRV 2108

[[Instructor]]
[[Teaching Assistant]]
[[Location]]
[[Syllabus]]
[[Weekly Readings]]
[[Homework]]
[[Lecture Notes]]
[[Self-Assessment Questions]]
[[Exams]]

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Try to answer the following questions without looking in the book. If you do not feel comfortable with them, please review the corresponding chapters in the book, and them attempt to solve them again.

#[[Self-Assessment Questions 1|SelfAssessment1.pdf]].
#[[Self-Assessment Questions 2|SelfAssessment2.pdf]].

Introduction to Numerical Analysis
Monday, Tuesday, Wednesday & Thursday, 12:30-1:35pm, GIRV 2108

Math 104A, Summer 2010

http://www.math.ucsb.edu/~cgarcia/Courses/Math104A_W09/

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''Textbook:'' //Numerical Analysis//, by Richard L. Burden and J. Douglas Faires, 8th edition.

''Course description:'' This is the first part of a three-quarters introductory course on Numerical Analysis. This quarter we will study numerical methods for the solution of nonlinear algebraic equations, interpolation, extrapolation, numerical differentiation and integration, and numerical solution of ordinary differential equations. Although the emphasis will be in applications, the course will have a strong theoretical component. By the end of the course, the following will be expected:
# Convergence of a numerical algorithm: What it means, and how to determine whether the algorithm converges or not.
# A solid knowledge of Approximation Theory:
## Polynomial Interpolation.
## Numerical Integration: Design and implementation of quadrature rules.
## Numerical Differentiation: Design and implementation of differentiation formulas of arbitrary order of accuracy.
## Estimation of the error in a given approximation.
# Solution of Differential Equations:
## Design and implementation of solvers for Ordinary Differential Equations (~ODEs).
## Being able to obtain the stability region of a multistep method for ~ODEs.
## Estimation of the error and the stability properties of a given method.
# Solid programming skills.

I will not hand out //practice exams// or anything like that. As part of the course, you should develop the necessary skills to study this subject. In order to help you study, I will periodically post [[Self-Assessment|Self-Assessment Questions]] questions. You do not need to hand them in. Those questions will help you know whether you are studying effectively or not, and whether you need to review some parts of the material covered in class.

''Prerequisites:'' Math 5 A, B, and C, or equivalent. Knowledge of a computer language suitable for numerical computing, such as FORTRAN, C, C++, or Matlab.

''Assignments and Grading Policy'': Homework will be assigned on Monday, and will be collected at the beginning of the class on the following Monday. ''@@color(red): Late homework will not be accepted. If you have a predictable absence, you will need to hand it in earlier@@''. The homework will generally consist of some theoretical questions, and some computational assignments. You will be required to write a program to solve certain problems. The program must be given to me as part of the assignment, together with the output of the program, in the format indicated in the assignment, and an interpretation of the results whenever necessary. You can write the programs either in FORTRAN, C, C++, or Matlab. The book comes with a CD that contains the code for the problems. You may use this code as a guide, but ''@@color(red):you must write your own original code for the assignments. No credit will be given for using the code in the CD@@''.

There will be a midterm and a final exam. Your final grade for the course will be given by the formula

Final Grade = 20% Homework + 30% Midterm + 50% Final.

''@@color(red):You must obtain 50% or more in the Final exam@@''.

''Computer Laboratory:'' In order to use the computer laboratory, you must go to Phelps 1523 (between 9am and 5pm) and get a sticker in your ID. You can check the lab schedule [[here|http://calendar.ic.ucsb.edu/]].

For those of you unfamiliar with Matlab, you may want to check this homepage: [[A practical Introduction to Matlab|http://www.math.mtu.edu/~msgocken/intro/intro.html]]. The Matlab Primer is also available in [[pdf|primer.pdf]]. You can find a lot of documentation at the [[MathWorks homepage|http://www.mathworks.com/]], specially in their support page.

You should try and do a search on the ~Internet, since there are a lot of sites dedicated to ~Matlab, and programming in general.

[[Professor Douglas Arnold|http://www.ima.umn.edu/~arnold/]], at the University of Minnesota, has a list of disaters due to bad numerical computing. Check it out [[here|http://www.ima.umn.edu/~arnold/disasters/disasters.html]].

The syllabus and the homework assignments were wrote using a program called ~LaTeX. If you want to learn more about this program, you can find some information and tutorials at the following homepages:

#The ~LaTeX Project.
#Text processing using ~LaTeX.

''Syllabus:'' (Available also in [[pdf|syllabus.pdf]]). During this course we will try to follow the following schedule. However, much like everything said earlier, this is subject to change.

!!!Week 1: Mathematical and Computational Preliminaries.
#Review of Calculus.
#Round-off Errors and Computer Arithmetic.
#Algorithms and Convergence.
#Numerical Software.
#Introduction to programming with Matlab.
!!!Week 2: Solutions of Equations in One Variable.
#The Bisection Method.
#~Fixed-Point iteration.
#Newton's method.
#Error Analysis for Iterative Methods.
#Accelerating Convergence*.
#Zeros of Polynomials and Müller's Method.
!!!Week 3: Interpolation and Polynomial Approximation.
#Interpolation and the Lagrange Polynomial.
#Divided Differences. The lecture notes can be downloaded in postscript or pdf format.
#Hermite Interpolation.
#Cubic Spline Interpolation*.
#Parametric Curves*.
!!!Weeks 4 & 5: Numerical Differentiation and Integration.
#Numerical Differentiation.
#Richardson's Extrapolation.
#Elements of Numerical Integration.
#Composite Numerical Integration.
#Romberg Integration*.
#Adaptive Quadrature Methods*.
#Gaussian Quadrature*.
#Multiple Integrals*.
#Improper Integrals*.
!!!Weeks 5 & 6: ~Initial-Value Problems for Ordinary Differential Equations.
#Elementary Theory of ~Initial-Value Problems.
#Euler's Method.
#~Higher-Order Taylor Methods.
#~Runge-Kutta Methods.
#Error Control and the ~Runge-Kutta-Fehlberg Methods*.
#Multistep Methods.
#Variable ~Step-Size Multistep Methods*.
#Extrapolation Methods*.
#~High-Order Equations and Systems of Differential Equations.
#Stability*.
#Stiff Differential Equations*.

* = If time permits.

There will be no TA for this course.

# Week 1: ''Chapter 1''.
## Objectives for Chapter 1:
### Review of Calculus: Differentiation, Mean Value Theorems, and Taylor's theorem.
### Round-off Errors.
### Convergence of an algorithm, and rate of convergence.
# Week 2: "Chapter 2".
## Objectives for Chapter 2:
### The Bisection Method: A simple strategy to find roots of a scalar algebraic equation. Also, an example of an algorithm, its implementation, and convergence proof.
### Fixed-Point Iteration: A general method for solution, versatile, and that can be used in higher dimensions. One of the main points is the analysis of convergence.
### Newton's method: This is a special Fixed-Point Iteration, which has quadratic convergence. The analysis of the previous section will allow us to understand why and when it converges faster.
### Secant Method: Newton's method requires knowledge of the derivative of the function; the Secant method eliminates this need, while providing a method with superlinear convergence.
### Aitken's method and Steffensen's method for accelerating the convergence of a sequence.
### Müller's method for complex roots of polynomials.
### Obtaining all the roots of a polynomial: Deflation.
# Week 3: "Chapter 3".
## Objectives for Chapter 3:
### Polynomial Interpolation: Approximate a function once its values at certain points are given.
### Existence and uniqueness of the interpolating polynomial.
### Accuracy: Expression for the error in the approximation.
### Practical aspects: Neville's algorithm.
### Construction of the interpolating polynomial using Newton's divided differences.
### Interpretation of the divided differences as derivatives of the function.
### Practical issues: Recursive formula for the divided differences, and Neville's algorithm for the computation of the coefficients.
### Hermite interpolation: How to approximate a function when the values of the function and its derivatives are given at the nodes.