### Location & Time

Section 1: MWF 2.30 PM-3.20 PM

167 Willard Building

Section 2: MWF 1.25 PM-2.15 PM

365 Willard Building

### Instructor

Kamesh Madduri

*Office hours*:

TR 12 PM-1.30 PM (343E IST)

Appointment (

cal)

### TA

Hongyuan Zhan

*Office hours*:

MWF 10 AM-11 AM (350 IST)

### Important Dates

Jan 11. First class

Feb 08. HW1 due

Feb 17. HW2 due

Feb 24. Exam 1

Mar 23. HW3 due

Apr 01. HW4 due

Apr 04. Exam 2

Apr 13. HW5 due

Apr 22. HW6 due

May 04. Final exam

### 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 February 7.

### Schedule

- Lecture 1, Jan 11
- Course Logistics
- Lecture 2, Jan 13
- Types of error
- Taylor's theorem
- Asymptotic notation for error
- Lecture 3, Jan 15
- Asymptotic notation
- Criteria for assessing numerical methods
- Lecture 4, Jan 20
- Problem conditioning
- Floating-point systems
- Lecture 5, Jan 22
- Floating-point systems
- Rounding rules
- Lecture 6, Jan 25
- IEEE floating-point standards
- Rounding unit
- Lecture 7, Jan 27
- Octave tutorial
- Lecture 8, Jan 29
- Octave tutorial
- Lecture 9, Feb 1
- Error in floating-point arithmetic
- Cancellation error
- Nonlinear equations: Introduction
- Lecture 10, Feb 3
- Interval bisection
- Fixed point iteration
- Lecture 11, Feb 5
- Fixed point theorem
- Order of convergence
- Lecture 12, Feb 8
- Newton's method
- Lecture 13, Feb 10
- Secant iteration
- Java applets
- Lecture 14, Feb 12
- Octave code
- Lecture 15, Feb 15
- Linear independence
- Vector norms
- Lecture 16, Feb 17
- Matrix norms
- Special classes of matrics
- Operation counts
- Lecture 17, Feb 19
- Backward and forward substitution
- Gaussian Elimination
- Lecture 18, Feb 22
- Review for exam 1
- Lecture 19, Feb 24
- Exam 1
- Lecture 20, Feb 26
- Elimination matrices
- LU decomposition
- Lecture 21, Feb 29
- GE with Partial and Complete Pivoting
- Lecture 22, Mar 2
- Gauss-Jordan elimination
- Review: Numerical computations in practice
- Lecture 23, Mar 4
- Exam 1 solutions discussion
- Octave code for linear systems
- Lecture 24, Mar 14
- LU Factorization uniqueness
- Cholesky decomposition
- Lecture 25, Mar 16
- Sparse matrices
- Error and matrix conditioning
- Lecture 26, Mar 18
- Monomial basis
- Lagrange basis
- Lecture 27, Mar 21
- Newton basis
- Divided differences
- Lecture 28, Mar 23
- Interpolation applets
- Error in polynomial interpolation
- Lecture 29, Mar 25
- Chebyshev interpolation
- Interpolation Octave tutorial
- Piecewise polynomial interpolation: Introduction
- Lecture 30, Mar 28
- Piecewise hermite interpolation
- Cubic spline interpolation
- Lecture 31, Apr 1
- Exam 2 review
- Lecture 32, Apr 4
- Exam 2
- Lecture 33, Apr 6
- Numerical Differentiation
- Lecture 34, Apr 11
- Integration: Introduction
- Newton-Cotes quadrature
- Lecture 35, Apr 13
- Clenshaw-Curtis quadrature
- Composite quadrature
- Java applets
- Lecture 36, Apr 15
- Adaptive quadrature
- Octave code
- Exam 2 solutions discussion
- Lecture 37, Apr 18
- IVP ODEs: Introduction
- Lipschitz continuity
- Lecture 38, Apr 20
- ODE solution stability
- Forward Euler method
- Backward Euler method
- Lecture 39, Apr 22
- ODE Error analysis
- Runge-Kutta methods
- Lecture 40, Apr 25
- ODEs Octave code
- Java applets
- Exercises
- Lecture 41, Apr 27
- Final exam review
- Lecture 42, Apr 29
- Final exam review

### Textbook

Uri M. Ascher and Chen Greif,

A First Course in Numerical Methods, SIAM, 2011.

### Prerequisites

MATH 230 or MATH 231; Three credits of programming.

### Class material

Lecture notes and homework assignments will be posted on Canvas.