THE SECOND ANNUAL LARGE DENSE LINEAR ALGEBRA SURVEY
Alan Edelman, Dept of Mathematics, University of California,
Berkeley, CA 94720
My calendar file tells me that it is now time for the second
annual large dense linear algebra survey. This year, I hope to
cover two subjects:
1. large dense linear systems (like last year)
2. large dense eigenvalue problems
It is not within my resources to print out and mail letters to
every science department at universities and industries, so I beg
readers to spread my survey by word of mouth, and strongly urge
you and your colleagues to participate. Readers interested in
last year's survey are invited to obtain it by anonymous FTP
from math.berkeley.edu in /pub/edelman/survey1991.
In order to confine the topic of discussion, we do not consider
any matrix that can be parameterized by significantly fewer than n^2
elements to be dense. Thus a Toeplitz matrix or a matrix of the form
A*A' where A is sparse are not considered dense in this context.
I will allow matrices generated for Panel Methods and Moment Methods
to be considered dense.
Part 1: Linear Systems
A. Largest LU or QR factorization
Has anyone solved a system of size bigger than 60,000 using
traditional LINPACK or LAPACK style methods? If so, please
tell me the time it took, why you solved the problem,
how accurate the solution was, and how you know.
Have you tried a condition estimator for your problem?
Did you consider a Krylov space based iterative method for
B. I an interested in the solutions to any dense matrix
bigger than 20,000 for purposes other than Panel Methods
and Moment Methods.
Part II: Eigenvalue Problems
A. I am interested in all eigenvalues problems for dense square matrices
of order at least 5,000. Please carefully describe
where you are in the range of wanting all eigenvalues and all
eigenvectors to merely wanting one eigenvalue. Do your
eigenvalues fall along a curve or cluster or are they scattered
and well separated? Have you evaluated the conditioning of your problem,
and if so, how?
B. Would you like to solve a large dense eigenvalue problem of
order greater than 50,000 if you had the resources? How large
can you foresee your problem getting at this point?
How big is your matrix?
What kind of matrix? (Symmetric, complex, double precision?)
What is the solution method?
What is the time for solution?
On which machine?
How accurate was your solution? (Explain how you know)
What is your confidence in this accuracy?
Could the newly released LAPACK be used for your problem?
(LAPACK replaces LINPACK and EISPACK as the current best linear algebra
software library. Information is available through netlib.)
Please describe your application area.
1. If appropriate please refer to the publication most closely related
to your particular problem. In most cases this will be an article
authored by you or a member of your group.
2. Please suggest an expository article or book that would be
most accessible to a non-specialist trying to understand your