Welcome to the class website for Introduction to Numerical Analysis . Computational approaches play an important role in many fields ranging from basic scientific research to the design of financial products. This class will discuss both the mathematical foundations and the practical implementation of modern numerical methods. Examples will also be discussed from applications areas.

Please be sure to read the prerequisites and grading policies for the class.

Topics:

• Differentiation and Integration:
  • Finite Difference Approximation.
  • Numerical Quadrature and Integration.
  • Common Formulas: Simpsons Rule, Newton-Cotes.
  • Gaussian Quadrature.
  • Numerical Accuracy, Order of Accuracy.
  • Composite Integration.
  • Adaptive Methods.
    
    
• Initial-Value Problems for Ordinary Differential Equations
  • Well-posedness of initial-value problems.
  • Euler’s Method of Approximation.
  • Taylor Methods for Higher-order Approximation.
  • Runge-Kutta Methods.
  • Multistep Methods.
  • Convergence Analysis.
  • Order of Accuracy and Stability.
  • Stiff Differential Equations.
    
    
• Solving Linear Systems and Matrix Algebra
  • Linear Equations.
  • Linear Algebra Review.
  • Eigenvalues and Eigenvectors.
  • Direct Methods.
  • Gaussian Elimination.
  • Role of Round-off Errors.
  • Pivoting Methods.
  • Matrix Inversion.
  • LU Factorization.
  • Iterative Methods.
  • Jacobi, Gauss-Siedel, SOR.
  • Conjugate Gradient.

Prerequisites:

Calculus, Linear Algebra, Differential Equations, and experience programming.

Grading:

The grade for the class will be based on the homework assignments (see policy below), midterm exam, and final exam/project as follows:

Homework Assignments 30%
Midterm Exam 30%
Final Exam/Project 40%

Homework Policy:

Assignments will be announced in lecture and posted on the class website. Prompt submission of the homework assignments is required. While no late homework submissions will be accepted, one missed assignment will be allowed without penalty. While you are encouraged to discuss materials with classmates, your submitted homework must be your own work.

Class Announcements:

Supplemental Materials:

• Python 2.7 documentation.
• Python tutorial at Code-Academy.
• Enthought Canopy integrated analysis environment.
• Example Python Code : Neville's Method.

Exams:

A midterm exam will be given in the class on Thursday, February 19th.

Midterm Outline [PDF].

Homework Assignments:

Turn all homeworks into the TA's mailbox John Kaminsky in South Hall 6th Floor by 5pm on the due date. Graded homeworks will be returned in class.

John Kaminsky's TA Office Hours: Wednesday 2pm-3pm in South Hall 6432T (graduate tower) and Wednesday 5pm- 7pm in South Hall Mathlab (ground floor).

Example python code : Neville's Method.

HW1: (Due Tuesday, Jan. 13th) 4.1: 1ab, 2ab, 3, 4ab, 5ad, 8abcd, 10, 13, 21, 22, 24, 25, 26, 28.
HW2: (Due Thursday, Jan. 15th) 4.3: 1abdgh, 2ad, 3, 4, 15, 21af, 24, 26.
HW3: (Due Tuesday, Jan. 20th) 4.7: 1acdeh, 2, 4, 5, 7, 8; 4.4: 1abegh, 3, 8, 11ab, 20, 22, 23.
HW4: (Due Thursday, Jan. 22nd) 5.1: 1acd, 2bc, 3ad, 8ad, 9.
HW5: (Due Tuesday, Feb. 3rd) 5.2: 1bc, 2ad, 3ac, 9abc, 16, 17.
HW6: (Due Tuesday, Feb. 10th) 5.3: 1abd, 2cd, 3, 4, 6ad, 8, 10, 11.
HW7: (Due Thursday, Feb. 19th) 5.4: 1bcd, 2cd, 3ad, 13, 14, 15, 17abd, 25, 27, 28, 29.
HW8: (Due Thursday, March. 3rd) 5.6: 1acd, 2bc, 4, 5, 12; 5.10: 2, 5, 7; 5.11: 1ad, 11.
HW9: (Due Friday, March 13th) 6.1: 1ad, 3a, 5bc, 9; 6.3: 2ab, 7a, 8abc; 6.5: 1ab, 3ab, 9bd; 7.1: 1abc, 4ad; 7.2: 1bf; 7.3:1bc, 3; 7.4: 1ad, 3, 10.

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