Numerical Analysis
MATH 4271/6271
Spring 2012
D. P. Dwiggins, PhD
35 lectures, 2 tests, 4 review days, final exam
Text: An Introduction to Numerical Analysis, K. E.
Atkinson (Univ. Iowa)
second edition (1989), Wiley & Sons, ISBN 0-471-62489-6, QA.297.A84
Chapter One. Error: Sources, Propagation, and Analysis
1.1. Preliminaries
1.2. Floating-Point Numbers
1.3. Definition and Sources of Error
1.4. Propagation of Errors
1.5. Error Analysis
1.6. Stability Theory
Chapter Two. Finding Roots of Equations and Nonlinear Systems
2.1. The Bisection Method
2.2.
2.3. The Secant Method
2.4. Muller’s Method
2.5. One-Point Iteration: General Theory
2.6. Aitken’s Method
2.7. Multiple Roots
2.8. Brent’s Algorithm
2.9. Roots of Polynomials
2.10. Systems of Nonlinear Equations
2.11.
2.12. Unconstrained Optimization
Chapter Three. Interpolation Theory
3.1. Polynomial Interpolation (Splines)
3.2. Divided Differences
3.3. Finite Difference Tables
3.4. Forward Differences, Error Propagation
3.5. Interpolation Error Analysis
3.6. Hermite Interpolation
3.7. Spline Theory
3.8. Trigonometric Splines
Chapter Four. Functional Analysis: Approximation Theory
4.1. From
4.2. The Minimax Problem
4.3. The Least Squares Problem
4.4. Orthogonal Polynomials
4.5. L2 Approximation of Functions
4.6. Minimax Approximations
4.7. Near-Minimax Approximation
Chapter Five. Numerical Integration
5.1. What was covered in Calc II
5.2. Spline Integration (Newton-Cotes Method)
5.3. Gaussian Quadrature
5.4. Error Analysis
5.5. Automatic (?) Numerical Integration
5.6. Improper Integrals
5.7. Numerical Differentiation
Chapter Six. Numerical Solutions for Differential Equations
6.1. Existence, Uniqueness, and Stability
6.2. Euler’s Method
6.3. Multistep Methods
6.4. Midpoint Method
6.5. Trapezoidal Method
6.6-11 = other topics (6.10 = Runge-Kutta, 6.11 = BVP’s)
Chapter Seven. Linear Algebra
7.1. Linear Spaces and Systems of Equations
7.2. Eigenvalues and Eigenvectors
7.3. Vector Norms and Matrix Norms
7.4. Convergence and Perturbation
Chapter Eight. Matrix Methods
8.1. Gaussian Elimination
8.2. Pivoting and Scaling
8.3. Other ways to improve Gaussian Elimination
8.4. Error Analysis
8.5. Residual Method
8.6. Iteration Methods
8.7. Error Propagation
8.8-9 = other topics
Chapter Nine. Numerical Analysis of Eigenvalues
9.1. Error and Stability
9.2-4 = special cases (symmetric tridiagonal systems)
9.5-7 = other stuff (QR Method)
Professor Atkinson designed his textbook for a two-semester course, so we will not be able to cover every section.
My proposal is to have a mid-February test on Chapters One, Two, Three (error analysis, iteration methods, and interpolation theory), a second test near the end of March on Chapters Seven and Eight (linear algebra and matrix methods), with the final exam also covering selected topics from Chapters Four (functional approximation theory), Five (numerical integration), and Six (numerical solutions for differential equations).
In addition to homework assignments (exercises from the textbook), students might expect selected topics to lead to mid-term and final projects, such as comparing various methods for solving a given differential equation or a large system of equations.
Nonlinear Analysis
Tentative Schedule of Lectures and Exams
Spring 2012
Friday Jan 13 = Course Introduction and Overview
Weds Jan 18 = §§ 1.1-2 = Representation of Numbers as Numerals
Friday Jan 20 = §§ 1.3-4 = Error: Definition, Sources, and Propagation
Mon Jan 23 = §§ 1.5-6 = Introduction to Error Analysis
Weds Jan 25 = §§ 2.1-2 = Finding Roots of Nonlinear Equations
Friday Jan 27 = §§ 2.3-5 = Iteration Methods: General Theory
Mon Jan 30 = §§ 2.6-8 = Comparing Newton’s Method with other methods
Weds Feb 1 = §§ 2.9-11 = other topics
Friday Feb 3 = §§ 3.1-2 = Interpolation Theory
Mon Feb 6 = §§ 3.3-4 = Finite Difference Tables
Weds Feb 8 = §§ 3.5-6 = Error Analysis
Friday Feb 10 = §§ 3.7-8 = Spline Theory
Mon Feb 13 = Review Chapters One and Two
Weds Feb 15 = Review Chapters Two and Three
Friday Feb 17 = Test # 1
Mon Feb 20 = §§ 7.1-2 = Review of Linear Algebra
Weds Feb 22 = § 7.3 = Norms and Linear Spaces
Friday Feb 24 = § 7.4 = Convergence and Perturbation
Mon Feb 27 = §§ 8.1-2 = Gaussian Elimination
Weds Feb 29 = §§ 8.3-4 = Error Analysis
Friday March 1 =
March 2-11 = Spring Break
Mon Mar 12 = § 8.4 = Condition Number of a Matrix
Weds Mar 14 = § 8.5 = Residual Methods
Friday Mar 16 = § 8.6 = Other Iteration Methods
Mon Mar 19 = §§ 8.7-9 = other topics
Weds Mar 21 = Chapter Nine = Numerical Eigenvalues
Friday Mar 23 = Review
Mar 26 = Test # 2
Weds Mar 28 = § 4.1 =
Functional Approximation: From
Friday Mar 30 = §§ 4.2-4 = Inner Product Spaces and Orthogonal Polynomials
Mon Apr 2 = §§ 4.4-6 = Least Squares Approximation
Weds Apr 4 = § 5.1 = Numerical Integration
Friday Apr 6 = § 5.2 = Splines and the Newton-Cotes Method
Mon Apr 9 = § 5.3 = Gaussian Quadrature
Weds Apr 11 = § 5.4 = Error Analysis
Friday Apr 13 = §§ 6.1-2 = Differential Equations
Mon Apr 16 = §§ 6.2-3 = Euler’s Method
Weds Apr 18 = § 6.10 = Runge-Kutta Methods
Friday Apr 20 = §§ 6.4-5 = Other Methods
Mon Apr 23 = Review
Weds Apr 25 = Review
Friday Apr 27 = Final Exam
Two tests, 100 points each
Two homework scores, 100 points each
Final Exam (covers Chapters One through Eight), 100 points
Semester Average based on 500 points
Lowest of the test/homework scores can be replaced with final exam score
20% of test/exam scores will come from projects based on selected topics
Instructor: D. P. Dwiggins
Office: Room 368, Dunn Hall
678.4174, ddwiggns@memphis.edu