SLN 10262, MWF 2:30-3:20, LOW 216
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|Professor Anne Greenbaum
office hours: W 3:30-4:30, Th 10-11
Numerical linear algebra. Review of basic linear algebra and Matlab programming. The singular value decomposition, QR factorization and least squares problems. Conditioning of problems, stability of algorithms. Solving systems of linear equations using Gaussian elimination: timing and accuracy considerations. Solving eigenvalue problems.
The required textbook for this course is "Numerical Linear Algebra" by L.N. Trefethen and D. Bau, SIAM, 1997. We will cover chapters I through V of this book. You also will need access to MATLAB.
Other recommended books (that will not be used in class but are valuable resources) include:
1. "Matrix Computations," by Golub and Van Loan. This is an excellent reference book for practically everything there is to know about numerical linear algebra.
2. "Applied Linear Algebra," by J. Demmel, SIAM, 1997. Goes more deeply into the same topics covered in Trefethen and Bau, especially concerning different variants of computational algorithms.
3. "Numerical Methods: Design, Analysis, and Computer Implementation of Algorithms," by Greenbaum and Chartier, Princeton, 2012. Covers numerical analysis in general but contains chapters on solving linear systems and least squares problems, floating point arithmetic, conditioning and stability, and eigenvalue problems that are along the lines of Trefethen and Bau. Also contains an appendix with a review of basic linear algebra and a chapter on Matlab.
- (1) Review of basic linear algebra and Matlab:
Ways of looking at matrix-vector multiplication. Linear independence and dependence, span of a set of vectors, dimension of a vector space. Orthogonal vectors and matrices. Basic operations with Matlab.
- (2) The singular value decomposition.
What is it and how can it be useful?
- (3) QR factorization and least squares.
- Gram-Schmidt and Householder orthogonalization. Least squares solutions to overdetermined linear systems. Fitting polynomials to data.
- (4) Conditioning of problems, stability of algorithms.
- Floating point arithmetic. What does it mean for a problem to be "ill-conditioned"? Forward and backward error analysis.
- (5) Direct methods for solving systems of linear equations.
- Gaussian elimination. Pivoting. Operation counts and implementation issues for high performance computing. Stability of Gaussian elimination with partial pivoting.
- (6) Eigenvalue problems.
- The power method, inverse iteration, Rayleigh quotient iteration. Reduction to Hessenberg or tridiagonal form. The QR algorithm. Computing the SVD.
Learning objectives and instructor expectations
Schedule and Homework
Follow links in the table below to obtain a copy of the homework in latex (.tex) or Adobe Acrobat (.pdf) format. You may also obtain here solutions to some of the homework and exam problems. An item shown below in plain text is not yet available.
|Homework and Exams||Homework Due Date||Homework Problem Sets||Homework Solutions|
|First day of classes||Wednesday, Sept. 27|
|Homework#1||due Friday, Oct. 6||Homework #1 (hw1)||HW #1 Solutions (hw1_solns.pdf)|
|Homework#2||due Friday, Oct. 13||Homework #2 (hw2)||HW #2 Solutions (hw2_solns.pdf)|
|Homework#3||due Monday, Oct. 23||Homework #3 (hw3)||HW #3 Solutions (hw3_solns.pdf)|
|Homework#4||due Monday, Oct. 30||Homework #4 (hw4)||HW #4 Solutions (hw4_solns.pdf)|
|Midterm||Wednesday, Nov. 1||Midterm Solutions (midterm_solns.pdf)|
|Homework#5||due Monday, Nov. 13||Homework #5 (hw5)||HW #5 Solutions (hw5_solns.pdf)|
|University holiday--Veterans Day||Friday, Nov. 10|
|Homework#6||due Monday, Nov. 20||Homework #6 (hw6)|
|University holiday--Thanksgiving||Thursday-Friday, Nov. 23-24|
|Homework#7||due Friday, Dec. 1||Homework #7 (hw7)|
|Last day of classes||Friday, Dec. 8|
|Final||Tuesday, Dec. 12, 2:30-4:20pm|
There will be weekly homework assignments (usually due on Fridays), a midterm (scheduled for Wed., Nov. 1), and a final. Homework counts 35%, the midterm counts 20% and the final counts for 45% of your grade. You may work together on homework assignments, but each person must write up his/her own answers to the exercises. ONLINE STUDENTS: YOU MUST HAVE A PROCTOR (APPROVED BY EDGE) FOR THE MIDTERM AND FINAL.
Some notes on Stability of Algorithms: stability.pdf
More notes on floating point arithmetic, conditioning of problems, stability of algorithms: numerical.pdf