CMPSC/MATH 451: Numerical Computations

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