PREREQUISITE:
MATH 5660:
Numerical Analysis I and MATH 5718: Applied
Linear Algebra
HOURS: TR 4:00-5:15 pm CU-Dravo 626
INSTRUCTOR:
Prof. Andrew Knyazev
Office: CU (Dravo) 644. Phone: 556-8102.
Office hours: Wed 3pm - 6pm (or by appointment)
WWW: http://www-math.cudenver.edu/~aknyazev/
Email: aknyazev@math.cudenver.edu
Students are invited to participate in the interactive DISCUSSION
TEXTBOOKS:
Numerical
Linear Algebra, Lloyd N. Trefethen,
David Bau
$37.00
Format: Paperback, 361pp.
ISBN: 0898713617
Publisher: SIAM
Pub. Date: May 1997
Applied
Numerical Linear Algebra, James Demmel
$48.00
Paperback (September 1997)
Society for Industrial & Applied Mathematics; ISBN: 0898713897
SUBJECT:
Solution of linear equations, eigenvector and eigenvalue calculations, matrix
error analysis, iterative methods.
The course will require knowledge in linear algebra, a basic knowledge of numerical methods, some programming experience, and familiarity with the complex plane. A brief review will be presented if necessary. The class will involve MATLAB workshops and some programming assignments in the MATLAB language.
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
GRADING will based on projects.
Other 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
Iterative Solution Methods by Owe Axelsson
$31.95
Paperback (January 1996)
Cambridge Univ Pr (Pap Txt); ISBN: 0521555698