PREREQUISITE:
MATH 5660:
Numerical Analysis I and MATH 5718: Applied
Linear Algebra
HOURS: MW 4-5:15, CU 641.
INSTRUCTOR:
Prof. Andrew Knyazev
Office: CU (Dravo) 644. Phone: 556-8102.
Office hours: by appointment
WWW: http://www-math.cudenver.edu/~aknyazev/
Email: aknyazev@math.cudenver.edu
TEXTBOOK:
Numerical
Linear Algebra, Lloyd N. Trefethen,
David Bau
$44.00 (varies).
Format: Paperback, 361pp.
ISBN: 0898713617
Publisher: SIAM
Pub. Date: May 1997
Please note that the SIAM offers 30% off list price discounts on books to SIAM members and at present every graduate student at the CU-Denver can get a FREE SIAM membership! Read about Complimentary Graduate Student Memberships and join SIAM today!
SUBJECT:
Computer-based solution of linear equations,
eigenvector and eigenvalue calculations, matrix
error analysis, introduction to iterative methods.
GRADING will be based on COMPUTER PROJECTS.
The course will require knowledge in linear algebra, a basic knowledge of numerical methods, programming experience, and familiarity with the complex plane. A brief review will be presented if necessary. The class will involve MATLAB (and possibly parallel programming) workshops. Projects will be programming assignments using departmental high-performance parallel Beowulf Cluster, supported by the NSF Award DMS MRI 0079719.
This is a high-level graduate class, which is a prerequisite for the MATH 7664: Iterative Methods in Numerical Linear Algebra offered next spring. It will require a significant amount of an independed work and an intellectual effort, in particular, to learn MATLAB and basics of parallel programming, though, help will be provided. It is expected that students solve most of the problems of the textbook, suggested as exersises after every section, as their homework, but solutions will not be collected. Hard problems will be discussed in class.
CONTENTS: The class will follow the outline below, touching on each major topic in a depth that will be determined by the pace of the class.
I Fundamentals
Lecture 1:
Matrix-Vector Multiplication
Lecture 2:
Orthogonal Vectors and Matrices
Lecture 3: Norms
Lecture 4: The
Singular Value Decomposition
Lecture 5: More on
the SVD
II QR Factorization and Least Squares
Lecture 6: Projectors
Lecture 7: QR Factorization
Lecture 8: Gram--Schmidt Orthogonalization
Lecture 9: MATLAB
Lecture 10: Householder Triangularization
Lecture 11: Least Squares Problems
III Conditioning and Stability
Lecture 12: Conditioning and Conditioning Numbers
Lecture 13: Floating Point Arithmetic
Lecture 14: Stability
Lecture 15: More on Stability
Lecture 16: Stability of Householder Transforms
Lecture 17: Stability of Back Substitution
Lecture 18: Conditioning of Least Squares Problems
Lecture 19: Stability of Least Squares Algorithms
IV Systems of Equations
Lecture 20: Gaussian Elimination
Lecture 21: Pivoting
Lecture 22: Stability of Gaussian Elimination
Lecture 23: Cholesky Factorization
V Eigenvalues
Lecture 24: Eigenvalue Problems
Lecture 25: Overview of Eigenvalue Algorithms
Lecture 26: Reduction to Hessenberg or Tridiagonal
Form
Lecture 27: Rayleigh Quotient, Inverse Iteration
Lecture 28: QR Algorithm without Shifts
Lecture 29: QR Algorithm with Shifts
Lecture 30: Other Eigenvalue Algorithms
Lecture 31: Computing the SVD
VI Iterative Methods
Lecture 32: Overview of Iterative Methods
Lecture 33: The Arnoldi Iteration
Lecture 34: How Arnoldi Locates Eigenvalues
Lecture 35: GMRES
Lecture 36: The Lanczos Iteration
Lecture 37: From Lanczos to Gauss Quadrature
Lecture 38: Conjugate Gradients
Lecture 39: Biorthogonalization Methods
Lecture 40: Preconditioning
Past classes: 2001, 1999, 1997.
Other numerical linear algebra books recommended:
Matrix Computations (Johns Hopkins Series in the Mathematical Sciences)
by Gene H. Golub, Charles F. Van Loan
$29.95
Paperback - 694 pages 3rd edition (December 1996)
Johns Hopkins Univ Pr; ISBN: 0801854148
Other Editions: Hardcover
$65.00
The Symmetric Eigenvalue Problem, Beresford Parlett
$48.00
Format: Paperback, 398pp.
ISBN: 0898714028
Publisher: Society for Industrial & Applied Mathematics
Pub. Date: December 1997
Applied
Numerical Linear Algebra, James Demmel
$48.00
Paperback (September 1997)
Society for Industrial & Applied Mathematics; ISBN: 0898713897
MATLAB stuff:
Matlab Guide by Nicholas J. Higham, Desmond J. Higham
Mastering MATLAB 6 by Duane Hanselman, Bruce R. Littlefield.
Matlab Tutorial - MATLAB Primer - A Practical Introduction to Matlab