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Advanced Numerical Linear Algebra - CMSC 760 / AMSC 600, Fall 2009
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Basic Information
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Overview
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The course will survey topics in numerical linear algebra, with emphasis
on solution algorithms for sparse linear systems of equations and numerical
methods for solving eigenvalue problems.
Both theoretical and computational issues will be studied.
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Homework
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Homework #1, Due September 24
(Postscript)
(.pdf)
Matlab input for Problem 4:
Matlab Version 7.3
Other Version 7
Version 6
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Homework #2, Due October 13
(Postscript)
(.pdf)
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Homework #3, Due November 5
(Postscript)
(.pdf)
Matlab input for Problem 5:
Matlab Version 7.3
Other Version 7
Version 6
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Homework #4, Due December 1
(Postscript)
(.pdf)
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Outline of Topics Covered
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- Sparse Direct Methods
- Symmetric positive definite matrices
- Band and profile elimination
- Reordering schemes and general sparse elimination
- Pivoting strategies
- Data structures and mathematical software
- Iterative Methods
- Krylov subspace methods:
the conjugate gradient, MINRES, GMRES methods and generalizations
- Preconditioning:
Incomplete factorization
Other splitting operators
- Multigrid methods for partial differential equations
- Computational Methods for Eigenvalue Problems
- Subspace iteration
- Lanczos and Arnoldi methods
- The implicitly restarted Arnoldi method
- The Jacobi-Davidson method
- Parallel Algorithms
- Parallel sparse LU factorization
- Scalable preconditioning methods
- Multicore architectures
- Sparse data structures
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References
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There will be no assigned text.
Much of the material can be found in the references listed below.
These are largely affordable books and it will be useful to purchase
one or two of them.
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T. A. Davis,
Direct Methods for Sparse Linear Systems,
SIAM Publications, Philadelphia, 2006.
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J. W. Demmel,
Applied Numerical Linear Algebra,
SIAM Publications, Philadelphia, 1997.
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I. S. Duff, A. M. Erisman, and J. K. Reid,
Direct Methods for Sparse Matrices,
Oxford University Press, 1986.
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H. C. Elman, D. J. Silvester and A. J. Wathen,
Fast Solvers and Finite Elements,
Oxford University Press, Oxford, 2006.
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A. George and J. W.-H. Liu
Computer Solution of Large Positive Definite Systems,
Prentice-Hall, New Jersey, 1981.
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G. H. Golub and C. Van Loan,
Matrix Computations,
Third Edition, Johns Hopkins University Press, 1996.
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A. Greenbaum,
Iterative Methods for Solving Linear Systems,
SIAM Publications, Philadelphia, 1997.
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Y. Saad,
Iterative Methods for Sparse Linear Systems,
Second Edition, SIAM Publications, Philadelphia, 2003.
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Y. Saad,
Numerical Methods for Large Eigenvalue Problems,
Manchester University Press, Manchester, 1992.
Available at
http://www-users.cs.umn.edu/~saad/books.html.
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G. W. Stewart,
Matrix Algorithms Volume II: Eigensystems,
SIAM Publications, Philadelphia, 2001.
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D. S. Watkins,
The Matrix Eigenvalue Problem: GR and Krylov Subspace Methods
SIAM Publications, Philadelphia, 2007.
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Grading
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Grades will be determined as follows:
- 5-7 homework assignments: 60%
- Course Project 40%
The homework will consist of both analysis and computational testing.
Computations can be done using any convenient programming tools although
Matlab will be encouraged.
Plagiarism:
You are welcome to discuss assignments in a general way among yourselves,
but you may not use other students' written work or programs.
Use of external references for your work should be cited.
Clear similarities between your work and others will result in a grade
reduction for all parties.
Flagrant violations will be referred to appropriate university authorities.
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