### Location & Time

Section 1: MWF 11.15AM-12.05PM

Section 2: MWF 12.20PM-01.10PM

110 Walker Building

### Instructor

Kamesh Madduri

*Office hours*:

MWF 1.30 PM-2.30 PM (343E IST)

Appointment (

cal)

### TA

Yu-Hsuan Kuo

*Office hours*:

TR 4.00 PM-5.00 PM (339 IST)

### Important Dates

Aug 26. First class

Sep 16. HW1 due

Sep 30. HW2 due

Oct 02. Exam 1

Oct 28. HW3 due

Nov 04. HW4 due

Nov 04. Exam 2

Dec 06. HW5 due

Dec 09. HW6 due

Dec 13. Exam 3

### Overview

This class introduces students to key ideas behind numerical computation, with
emphasis on the implementation of common numerical methods. CMPSC/MATH 455 is a
related course that covers a subset of the topics that we will study in this class.
Students may obtain credit for either this class or CMPSC/MATH 455, but not both.

### Topics

Floating Point Arithmetic, Approximations in Scientific Computing, Nonlinear equations in one variable, Linear Systems: Direct Methods, Polynomial Interpolation, Piecewise Polynomial Interpolation, Numerical Differentiation, Numerical Integration, Initial Value Problems for ODEs.

### Syllabus

Last updated October 28.

### Schedule

- Lecture 1, Aug 26
- Course Logistics
- Introduction to Scientific Computing
- Lecture 2, Aug 28
- Error types
- Taylor series
- Asymptotic notation
- Lecture 3, Aug 30
- Error bounds
- Criteria for assessing numerical methods
- Lecture 4, September 4
- Problem conditioning
- Floating-point systems
- Lecture 5, September 6
- IEEE floating-point standards
- Rounding rules
- Lecture 6, September 9
- Octave tutorial
- Rounding unit
- Avoiding roundoff error
- Lecture 7, September 11
- Floating-point arithmetic
- Cancellation error
- Lecture 8, September 13
- Nonlinear equations: Introduction
- Iterative methods
- Interval bisection
- Lecture 9, September 16
- Fixed-point iteration
- Order of convergence
- Lecture 10, September 18
- Rate of convergence
- Newton's method
- Secant iteration
- Lecture 11, September 20
- Function minimization
- Nonlinear equations: Octave programs
- Lecture 12, September 23
- Linear systems: Introduction
- Linear independence
- Lecture 13, September 25
- Vector and matrix norms
- Lecture 14, September 27
- Special classes of matrices
- Orthogonal matrices
- Lecture 15, September 30
- Operation counts
- Forward and backward substitution
- Lecture 16, October 2
- Exam 1 review
- Exam 1 (evening)
- Lecture 17, October 4
- Gaussian elimination
- Lecture 18, October 7
- Gaussian elimination (continued)
- LU decomposition
- Lecture 19, October 9
- Gaussian Elimination with pivoting
- Lecture 20, October 11
- Exam 1 solutions discussion
- Lecture 21, October 14
- Cholesky factorization
- Lecture 22, October 18
- Gauss-Jordan Elimination
- Octave programs
- Lecture 23, October 21
- Octave programs
- Sparse matrices
- Relative Error and Matrix Conditioning
- Lecture 24, October 23
- Polynomial Interpolation: Introduction
- Monomial basis
- Lagrange basis
- Lecture 25, October 25
- Newton basis
- Divided differences
- Lecture 26, October 28
- Error in polynomial interpolation
- Lecture 27, October 30
- Interpolation with derivative values
- Piecewise polynomial Interpolation: Introduction
- Lecture 28, November 1
- Hermite cubic interpolation
- Cubic splines
- Lecture 29, November 4
- Exam 2 review
- Exam 2 (evening)
- Lecture 30, November 6
- Piecewise polynomial interpolation exercises
- Lecture 31, November 8
- Exam 2 solutions discussion
- Numerical differentiation: Introduction
- Lecture 32, November 11
- Finite difference schemes for computing derivative
- Richardson extrapolation
- Lecture 33, November 13
- Numerical Integration: Introduction
- Newton-Cotes quadrature
- Lecture 34, November 15
- Deriving quadrature rules
- Error in quadrature rules
- Lecture 35, November 18
- Clenshaw-Curtis quadrature
- Gaussian quadrature
- Composite quadrature
- Lecture 36, November 22
- Adaptive quadrature
- Octave examples
- Lecture 37, December 2
- Romberg Integration
- Multidimensional Integration
- Lecture 38, December 4
- ODEs: Introduction
- Stability, Global error
- Lecture 39, December 6
- Lipschitz continuity
- Euler's methods
- Lecture 40, December 9
- Runge-Kutta methods
- Lecture 41, December 11
- ODE exercises
- Lecture 42, December 13
- Exam 3 review

### Textbook

Uri M. Ascher and Chen Greif,

A First Course in Numerical Methods, SIAM, 2011.

### Prerequisites

MATH 220; MATH 230 or MATH 231; Three credits of programming (CMPSC 201 or CMPSC 202 or CMPSC 121).

### Class material

Presentations, lecture notes, and homework assignments will be posted on Angel.