clarke@csri.toronto.edu (Jim Clarke) (01/18/88)
(SF = Sandford Fleming Building, 10 King's College Road)
(GB = Galbraith Building, 35 St. George Street)
(WB = Wallberg Building, 184 College Street)
SUMMARY:
NUMERICAL ANALYSIS SEMINAR, Tuesday, January 26, 10 am, WB242 -- John Butcher:
"Singly-Implicit Methods"
A.I. SEMINAR, Tuesday, January 26, 2 pm, SF1105 -- Robin Cohen:
"Implementing a model for understanding goal-oriented discourse"
NUMERICAL ANALYSIS SEMINAR, Friday, January 29, 3 pm, GB220 -- James Demmel:
"The Probability that a Numerical Analysis Problem is Difficult"
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NUMERICAL ANALYSIS SEMINAR, Tuesday, January 26, 10 am, WB242
Professor John Butcher
The University of Auckland
"Singly-Implicit Methods"
For an implicit multistage method, such as a Runge-Kutta method, for which
$A$ denotes the matrix showing the dependence of the internal stage values
on the derivatives at these stage values, the implementation costs depend
heavily on the cost of solving a linear system based on a matrix of the
form $I-hAJ$. In the case of an $s$-stage method and an $N$ dimensional
problem, the linear algebra costs go up like $s sup 3 N sup 3$. To avoid
the factor $s sup 3$, diagonally-implicit methods are often proposed but
these have some limitations. A more general approach is considered here,
in which $A$ is constrained only by the requirement of having a 1-point
spectrum. Advantages and disadvantages of this generalization will be dis-
cussed and prospects are assessed for the construction of efficient algo-
rithms based on singly-implicit methods.
A.I. SEMINAR, Tuesday, January 26, 2 pm, SF1105
Professor Robin Cohen
University of Waterloo
"Implementing a model for understanding goal-oriented discourse"
NUMERICAL ANALYSIS SEMINAR, Friday, January 29, 3 pm, GB220
Dr. James Demmel
Courant Institute
"The Probability that a Numerical Analysis Problem is Difficult"
Numerous problems in numerical analysis, including matrix inversion, eigen-
value calculations and polynomial zero finding, share the following pro-
perty: the difficulty of solving a given problem is large when the distance
from that problem to the nearest "ill-posed" one is small. For example,
the closer a matrix is to the set of noninvertible matrices, the larger its
condition number with respect to inversion. We show that the sets of ill-
posed problems for matrix inversion, eigenproblems, and polynomial zero
finding all have a common algebraic and geometric structure which lets us
compute the probability distribution of the distance from a "random" prob-
lem to the set. From this probability distribution we derive, for example,
the distribution of the condition number of a random matrix. We examine the
relevance of this theory to the analysis and construction of numerical
algorithms destined to be run in finite precision arithmetic.
--
Jim Clarke -- Dept. of Computer Science, Univ. of Toronto, Canada M5S 1A4
(416) 978-4058
{allegra,cornell,decvax,linus,utzoo}!utcsri!clarke