### Location & Time

TR 8.00 AM-9.15 AM

169 Willard Building

### Instructor

Kamesh Madduri

*Office hours*:

TR 9.30 AM-11.00 AM (343E IST)

Appointment (

cal)

### Important Dates

Jan 08. First class

Jan 17. HW1 out

Jan 29. HW1 due

Feb 05. HW2 out

Feb 12. HW2 due

Feb 14. Midterm exam 1

Mar 12. HW3 out

Mar 21. HW3 due

Mar 26. HW4 out

Apr 02. HW4 due

Apr 02. Make-up midterm 1

Apr 04. Midterm exam 2

Apr 16. HW5 out

Apr 25. HW5 due

May 02. Final exam

### Overview

This class introduces students to key topics in numerical analysis, with emphasis on the computational mathematics behind numerical methods and on proving theorems. CMPSC/MATH 451 is a related course that has a significant overlap with topics studied in CMPSC/MATH 455 and CMPSC/MATH 456. Students may obtain credit for either this class or CMPSC/MATH 451, but not both.

### Topics

Floating Point Arithmetic, Approximations in Scientific Computing, Non-linear equations in one variable, Linear Systems: Direct Methods, Polynomial Interpolation, Piecewise Polynomial Interpolation, Numerical Differentiation, Numerical Integration.

### Syllabus

Last updated April 18.

### Schedule

- Lecture 1, Jan 8
- Course Logistics
- Introduction to Scientific Computing

- Lecture 2, Jan 10
- Error types
- Taylor series
- Big-O notation for error analysis

- Lecture 3, Jan 15
- Conditioning of a problem
- Assessing Algorithms
- Floating-point systems

- Lecture 4, Jan 17
- Floating-point systems, properties
- Roundoff error
- Octave introduction

- Lecture 5, Jan 22
- Rounding unit
- Sources of roundoff error
- Decimal to binary conversion

- Lecture 6, Jan 24
- IEEE floating-point system
- Avoiding cancellation and roundoff error
- Introduction to nonlinear methods

- Lecture 7, Jan 29
- Interval bisection
- Fixed point theorem
- Fixed point iteration

- Lecture 8, Jan 31
- Speed of convergence
- Newton iteration
- Secant iteration

- Lecture 9, Feb 5
- Nonlinear equations Octave tutorial
- Multiple roots
- Function minimization

- Lecture 10, Feb 7
- Matrices review
- Vector and matrix norms

- Lecture 11, Feb 12
- Midterm 1 review

- Lecture 12, Feb 19
- Special classes of matrices
- Operation counts
- Backward substitution

- Lecture 13, Feb 21
- Forward substitution
- Gaussian Elimination

- Lecture 14, Feb 26
- Gaussian Elimination algorithm, operation count
- Partial pivoting

- Lecture 15, Feb 28
- Complete pivoting
- Gauss-Jordan elimination
- Uniqueness of LU decomposition
- Cholesky factorization

- Lecture 16, Mar 12
- Midterm 1 solutions, discussion

- Lecture 17, Mar 14
- Cholesky factorization example
- Banded matrices
- Error, residual, conditioning
- Linear systems: Octave tutorial

- Lecture 18, Mar 19
- Linear systems review
- Polynomial Interpolation: Introduction
- Monomial basis

- Lecture 19, Mar 21
- Lagrange basis
- Newton basis

- Lecture 20, Mar 26
- Divided differences
- Polynomial interpolation error
- Chebyshev interpolation

- Lecture 21, Mar 28
- Octave interlation tutorial
- Piecewise polynomial interpolation introduction

- Lecture 22, Apr 9
- Midterm 2 solutions discussion
- Hermite cubic interpolation
- Cubic splines

- Lecture 23, Apr 11
- Finite difference formulas
- Richardson Extrapolation

- Lecture 24, Apr 16
- Numerical Quadrature introduction
- Newton-Cotes quadrature rules

- Lecture 25, Apr 18
- Precision of a quadrature rule
- Deriving error with a quadrature formula
- Clenshaw-Curtis quadrature
- Composite quadrature

- Lecture 26, Apr 23
- Octave tutorial on numerical integration
- Gaussian quadrature
- Adaptive quadrature

- Lecture 27, Apr 25
- Final exam 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.