Math 1080:      Numerical Linear Algebra      Spring 2016

Instructor: David Swigon

Office: Thackeray 511, 412-624-4689, swigon@pitt.edu

Lectures:  MWF 10:00-10:50am, Thackeray 524

Office Hours: M 11:00am-12:30pm, W 12:30-2:00pm, Thackeray 511, or by appointment.

Grader: Sarah Khankan, Thackeray 615, sak169@pitt.edu

Course Web Page: (check frequently for changes and updates) http://www.math.pitt.edu/~swigon/math1080.html

Course Description

This course gives an introduction to the basic areas of numerical linear algebra. It will cover the development and analysis of algorithms that are used in the solution of linear algebraic equations, algebraic eigenvalue problems and linear least-square minimization problems. 

Prerequisites

Basic knowledge of matrix theory and linear algebra (one of MATH 0250, MATH 0280, or MATH 1180) plus knowledge of computer programming (one of CS 0002, CS 0007, CS 0401, CS 0132) is expected.

Textbook

L. N. Trefethen & D. Bau III, Numerical Linear Algebra, SIAM, 1997, ISBN 0898713617

This is a course in applied mathematics and hence emphasis will be placed on practical usage of methods and algorithms.  Other texts on numerical linear algebra you can use for review of the theory or for enhancement include:

D. Poole , Linear Algebra; a Modern Introduction; G. Strang, Linear Algebra and Its Applications; C. G. Cullen, An Introduction to Numerical Linear Algebra; C. F. van Loan, Introduction to Scientific Computing

Grading Scheme

Homework assignments: 30%

Two midterm exams: 20% + 20%

Cumulative final exam: 30%

Programming

Many assignments will require computer programming. You are expected to be proficient in at least one computer language, such as Matlab, Fortran, C, Basic, JAVA, etc. to such an extent that you can write a program, debug it, run it, and print out the output.  I recommend Matlab as it makes manipulation of matrices easy and requires the least coding overhead. Feel free to use your personal computer or any of the University computing labs to work on your assignments.

Matlab resources

Matlab Primer of Professor Sigmon of the University of Florida:  http://www.math.pitt.edu/~swigon/Matlab/primer.pdf

Matlab documentation: http://www.mathworks.com/access/helpdesk/help/techdoc/matlab.html

Disability Resource Services

If you have a disability for which you are or may be requesting an accommodation, you are encouraged to contact both your instructor and Disability Resources and Services, 140 William Pitt Union, 412-648-7890 or 412-383-7355 (TTY) as early as possible in the term. DRS will verify your disability and determine reasonable accommodations for this course.

Academic Integrity

Cheating/plagiarism will not be tolerated. Students suspected of violating the University of Pittsburgh Policy on Academic Integrity will incur a minimum sanction of a zero score for the quiz, exam or paper in question. Additional sanctions may be imposed, depending on the severity of the infraction.  On homework, you may work with other students or use library resources, but each student must write up his or her solutions independently. Copying solutions from other students will be considered cheating, and handled accordingly.

Syllabus

Date

Reading

Topics

Homework

Jan 6-8

I.1-2

Matrix Multiplication, Fundamental Theorem

HW 1

Due Jan 15

solutions

Jan 11-15

I.3

II.6

II.7

Orthogonality, Norms

Projectors

QR factorization

HW 2

Due Jan 22

solutions

Jan 18

 

No Class

 

Jan 20-22

II.8

Gram-Schmidt Orthogonalization

HW 3

Due Monday, Feb 1

solutions

Jan 25-29

II.10

 

II.11

Householder Triangularization,

Householder QR factorization

Applications of QR factorization

Feb 3

Review (Midterm Exam I Review Topics)

Feb 5

Midterm Exam I (Covers Sections I and  II)

Feb 8-12

III.12

III.13

III.14

Conditioning

Floating point arithmetic

Stability

HW 4

Due Feb 19

solutions

Feb 15-19

III.15

III.16

IV.20

Accuracy

Stability of Householder triangularization

Gaussian Elimination

HW 5

Due Monday, Feb 29

solutions

Feb 22-26

IV.20

IV.21

Gaussian Elimination (cont’d)

Pivoting

Feb 29 - Mar 4

IV.22

IV.23

Stability of Gaussian Elimination

Cholesky Factorization

HW 6

Due Mar 16

solution

Mar 7-11

 

Spring Break – No Classes

Mar 14

IV.23

Cholesky Factorization (cont’d)

Mar 16

Review (Midterm Exam II Review Topics)

Mar 18

Midterm Exam II (Covers Sections III and IV)

Mar 21-25

V.24

V.25

V.27

Eigenvalue Problems

Overview of Eigenvalue Algorithms

Power Iteration, Rayleigh Quotient, Inverse iteration

HW 7

Due Apr 1

solution

Mar 28 - Apr 1

V.27

V.26

Rayleigh Quotient, Power Iteration, Inverse iteration

Reduction to Hessenberg Form

HW 8

Due Apr 8

solutions

Apr 4-8

V.28

V.29

QR Algorithm

QR Algorithm with shifts

HW 9

Due Apr 15

solutions

Apr 11-15

I.4

I.5

V.31

Singular Value Decomposition

More on the SVD

Computing the SVD

HW 10

Due Apr 22

Apr 18-20

 

Applications of SVD (image compression)

 

Apr 22

Review (Final Exam Review Topics)

 

Apr 25

FINAL EXAM (2-3:50pm, Thack 524)