Numerical Methods Questions


(Fall 2000 - Demmel & Kahan):
"Q1:  The word  "stability"  is used in conjunction with 
O.D.E.  initial value problems,  and also in conjunction with
numerical methods for solving them.  Use examples to
illustrate the difference in meanings,  mentioning also the
extent to which the order  and the stability of a numerical
method may limit each other.

Q2: ``Shooting methods'' characterizes a family of ways to
solve two-point boundary-value problems.  Describe these
methods and point out how the concept of stability influences
the outcomes.

Q3: In a conversation, an engineer says that
``roundoff can hit me in three ways: cancellation, accumulation
of large numbers of tiny errors, and dividing by a tiny number.''
Is this right? If not, what would you say to the person to 
explain the effects of roundoff?

Q4: Define ``foolproof algorithm.'' Is your definition the
same for a mathematician, a numerical analyst, and a practicing
engineer? Give examples of algorithms may be foolproof
for one person but not another, as well as examples of problems
for which foolproof algorithms remain to be found, and what
the current best (but non-foolproof) options are."

September 2000