(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."