MATH 62263/72263: Numerical Solution of Large Sparse Linear Systems
TIME & PLACE: Tu + Th 2:15 - 3:30 pm, MSB 276.
Y. Saad, Iterative Methods for Sparse Linear Systems, SIAM, Philadelphia, 2003.
as well as journal papers distributed in class. The book is available at
A hard copy can be ordered from SIAM at https://www.siam.org/books/ if you so desire.
Many computational problems that arise in science and engineering give rise to linear
systems of equations with a very large matrix. These problems cannot be solved by using
Gaussian elimination or Cholesky factorization, because these methods require too many
arithmetic floating point operations to be practical. Also, the storage required by the
matrix factors determined by Gaussian elimination or Cholesky factorization can be
It is the purpose of this course to discuss numerical methods suitable for the solution of
large linear systems of equations. These methods are iterative and do not call for the
factorization of the matrix. Instead, they "learn" about the matrix by evaluating matrix-vector
products. In particular, only some of the entries of the matrix have to be stored simultaneously.
Therefore the storage requirement is modest also when the matrix is very large.
The iterative methods are designed to determine an accurate approximation of
the solution in a linear space of fairly low dimension. Generally these linear spaces are
Krylov subspaces. We will describe properties in terms of oblique or
orthogonal projections, and discuss techniques for choosing suitable
subspaces. Most of the course will present and analyze properties of
state-of-the art iterative methods, including the conjugate gradient method,
GMRES, BiCGStab, IDR, and LSQR. The role of preconditioning will be
discussed and so will storage schemes for large sparse matrices. Towards
the end of the course, we will also discuss some techniques for the
computation of a few eigenvalues and associated eigenvectors of a very
large matrix, because several of the techniques for the solution of
large linear systems of equations also, after minor modifications,
can be applied to eigenvalue computation.
Homework will be assigned regularly and collected at the end of each major
section. The homework will include programming and applyzing some of the
methods discussed. Journal papers that complement the textbook will be
distributed. Course assignments will include reading and presenting
state-of-the-art papers on iterative methods.
The programming languages MATLAB and GNU Octave will be used for illustrations,
however, students may use their programming language of choice for their homework.
MATLAB is available on the the Math/CS Network. It is beneficial for students to have
an account on this network. GNU Octave is a public domain language
very similar to MATLAB, and can be used for homework assignments. Instructions
on how to install GNU Octave on your PC are available
STUDENTS WITH DISABILITIES:
Information can be found at
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details. Plagiarism, i.e., presenting someone else's work as your own is
This includes finding the answer of a homework problem in a book or in someone
else's assignment, and copying it. Plagiarism, of course, is unacceptable.
will be provided reasonable accommodations
to ensure their equal access to course content. Futher information can be
found at this web site.