Math 104B, Spring 2009 - Introduction to Numerical AnalysisMonday, Wednesday & Friday, 9:00-9:50am, South Hall 6635

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We will have a ''@@color(red):Midterm exam on Friday, May 8th, 2009@@''. The ''@@color(red):Final exam will be on Wednesday, June 10th, 2009, from 8-11am@@''.

The final exam consists of two parts: One practical, take-home exam, and an in-class, theoretical part. The take-home exam can be found [[here|FinalSpring2009_TakeHome.pdf]].

Welcome to the Math 104B 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 one 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]]: I will post the notes from class here.
# [[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 April 8th, 2009. 
#[[Homework 2|hw2.pdf]]: Due on April 15th, 2009. 
#[[Homework 3|hw3.pdf]]: Due on April 22nd, 2009. 
#[[Homework 4|hw4.pdf]]: Due on April 29th, 2009. 
#[[Homework 5|hw5.pdf]]: Due on May 6th, 2009.
#[[Homework 6|hw6.pdf]]: Due on May 20th, 2009.
#[[Homework 7|hw7.pdf]]: Due on May 27th, 2009.
#[[Homework 8|hw8.pdf]]: Due on June 3rd, 2009.

!Carlos J. ~García-Cervera.
South Hall, Room 6707.
Office Phone: (805) 893 3681
Office Fax: (805) 893 2385
Office Hours: Monday and Wednesday, from 10-11:30am.
Main Website: http://www.math.ucsb.edu/~cgarcia

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 are the notes from class:

#[[Direct Methods for solving Linear Systems|LN/Direct Methods for solving Linear Systems.pdf]].
#[[Iterative Techniques in Matrix Algebra.pdf|LN/Iterative Techniques in Matrix Algebra.pdf]].
#[[The Conjugate Gradient Method|LN/The Conjugate Gradient Method.pdf]].
#[[Approximation Theory|LN/Approximation Theory.pdf]].
#[[Trigonometric Approximation|LN/Trigonometric Approximation.pdf]].
#[[Approximation Eigenvalues|LN/Approximation Eigenvalues.pdf]]

Monday, Wednesday & Friday, 9:00-9:50am, South Hall [[6635|http://math.ucsb.edu/images/6thfloor/6635.gif]].

[[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 1|SelfAssessment1.pdf]].

Introduction to Numerical Analysis
Monday, Wednesday & Friday, 9:00-9:50am, South Hall 6635

Math 104B, Spring 2009

http://www.math.ucsb.edu/~cgarcia/Courses/Math104B_S09/

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

''Course description:'' This is the second part of a three-quarters introductory course on Numerical Analysis. The focus this quarter will be the numerical solution of linear systems of equations, and eigenvalue problems. We will also study families of orthogonal polynomials and their approximation properties. 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:
# Solid knowledge on direct and iterative methods for solving linear systems of equations: Design and implementation of solvers, computational complexity and efficiency.
# A solid knowledge of Approximation Theory:
## Least squares approximations.
## Orthogonal polynomials.
## Fast Fourier Transform.
# Approximation of eigenvalues:
## Power Method and Subspace Iteration.
## The QR algorithm.
# 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; Math 104A. 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 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@@''.

There will be a midterm and a final exam. The Final exam will be split into a take-home, computationally oriented exam, and an in-class, more theoretical 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]].

I wrote the syllabus and the homework assignments 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.

!!!Weeks 1 & 2: Direct Methods for Solving Linear Systems.
#Linear Systems of Equations.
#Pivoting Strategies.
#Linear Algebra and Matrix Inversion.
#The Determinant of a Matrix.
#Matrix Factorization.
#Special Types of Matrices.
!!!Weeks 3 & 4: Iterative Techniques in Matrix Algebra.
#Norms of Vectors and Matrices.
#Eigenvalues and Eigenvectors.
#Iterative Techniques for Solving Linear Systems.
#Error bounds and Iterative Refinement.
#The Conjugate Gradient Method.
!!!Weeks 5, 6, & 7: Approximation Theory.
#Discrete Least Squares Approximation.
#Orthogonal Polynomials and Least Squares Approximation.
#Chebyshev Polynomials and Economization of Power Series.
#Rational Function Approximation.
#Trigonometric Polynomial Approximation.
#Fast Fourier Transforms.
!!!Weeks 8, 9 & 10: Approximating Eigenvalues.
#Linear Algebra and Eigenvalues.
#The Power Method.
#Computing multiple eigenvalues: Subspace Iteration*.
#Householder's Method.
#The QR Algorithm.

* = If time permits.

!!Pat Plunkett
South Hall, Room 6432X.
Office Hours: Wednesdays from 2-4pm.
Computer Laboratory: Mondays from 2-3:50pm, Miramar lab (Phelps 1526) except for Monday 4/13: They will be held in the Mesa lab next door (Phelps 1525).

# Week 1: ''Sections 6.1, 6.2, 6.3, 6.4, and 6.5''.
## Objectives for Chapter 6:
### Solving linear systems of equations: Gauss elimination.
### Pivoting.
### Matrix Factorization.
### Computational complexity: How computationally expensive are our algorithms? How do they scale with the number of equations?
# Week 2: ''Sections 6.6, 6.7''.
## Objectives:
### To understand how direct methods can be simplified for special types of matrices, such as tridiagonal matrices.
### To learn new forms of factorization for special matrices, in particular, the Choleskii and the LDL^t factorization for positive definite matrices.
### To develop good programming skills.
# Week 3: ''Sections 7.1, 7.2''.
## Objectives:
### To learn the concept of norm for vectors and matrices. A norm allows us to define convergence in a general setting.
### Review of eigenvalues and eigenvectors.
### To learn the relation between the norm of a matrix, and its eigenvalues (in particular, the spectral radius of the matrix).
### Convergence of a sequence of matrices, and in particular, of A^n. 
# Week 4: ''Sections 7.3, 7.4''.
## Objectives:
### The classical iterative methods for linear systems of equations: Jacobi, ~Gauss-Seidel, and SOR(w).
### Convergence of iterative methods for linear systems, and the relation to fixed point theory studied in 104A.
### Computational complexity of Jacobi and ~Gauss-Seidel for the model problem.
### Error bounds, and the condition number of a matrix.
# Week 5: ''Section 7.5''.
## Objectives:
### The conjugate gradient method: Relation to variational problems.
### Convergence of the conjugate gradient, and preconditioning.
# Week 6: ''Sections 8.1, 8.2, and 8.3''.
## Objectives:
### Least square approximation: Continuous and discrete case.
### Orthogonal polynomials and Gram-Schmidt orthogonalization procedure.
### Chebyshev polynomials.