Week 9

Week 9 - 604 Karp, R. M., and V. Ramachandran. 1990....

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
604 IV. Branches of Mathematics Karp, R. M., and V. Ramachandran. 1990. Parallel algorithms for shared-memory machines. In Handbook of Theoret- ical Computer Science , volume A, Algorithms and Com- plexity , edited by J. van Leeuwen. Cambridge, MA: MIT Press/Elsevier. Kearns, M. J., and U. V. Vazirani. 1994. An Introduction to Computational Learning Theory . Cambridge, MA: MIT Press. Kitaev, A., A. Shen, and M. Vyalyi. 2002. Classical and Quan- tum Computation . Providence, RI: American Mathematical Society. Kushilevitz, E., and N. Nisan. 1996. Communication Com- plexity . Cambridge: Cambridge University Press. Ron, D. 2001. Property testing (a tutorial). In Handbook on Randomized Computing , volume II. Dordrecht: Kluwer. Shaltiel, R. 2002. Recent developments in explicit construc- tions of extractors. Bulletin of the European Association for Theoretical Computer Science 77:67–95. Sipser, M. 1997. Introduction to the Theory of Computation . Boston, MA: PWS. Strassen, V. 1990: Algebraic complexity theory. In Hand- book of Theoretical Computer Science , volume A, Algo- rithms and Complexity , edited by J. van Leeuwen. Cam- bridge, MA: MIT Press/Elsevier. IV.21 Numerical Analysis Lloyd N. Trefethen 1 The Need for Numerical Computation Everyone knows that when scientists and engineers need numerical answers to mathematical problems, they turn to computers. Nevertheless, there is a wide- spread misconception about this process. The power of numbers has been extraordinary. It is often noted that the scientiFc revolution was set in motion when Galileo and others made it a principle that everything must be measured. Numerical measure- ments led to physical laws expressed mathematically, and, in the remarkable cycle whose fruits are all around us, Fner measurements led to reFned laws, which in turn led to better technology and still Fner measure- ments. The day has long since passed when an advance in the physical sciences could be achieved, or a signiF- cant engineering product developed, without numerical mathematics. Computers certainly play a part in this story, yet there is a misunderstanding about what their role is. Many people imagine that scientists and mathemati- cians generate formulas, and then, by inserting num- bers into these formulas, computers grind out the nec- essary results. The reality is nothing like this. What really goes on is a far more interesting process of exe- cution of algorithms . In most cases the job could not be done even in principle by formulas, for most mathe- matical problems cannot be solved by a Fnite sequence of elementary operations. What happens instead is that fast algorithms quickly converge to “approximate” answers that are accurate to three or ten digits of pre- cision, or a hundred. ±or a scientiFc or engineering application, such an answer may be as good as exact. We can illustrate the complexities of exact versus approximate solutions by an elementary example. Sup- pose we have one polynomial of degree 4, p(z) = c 0 + c 1 z + c 2 z 2 + c 3 z 3 + c 4 z 4 , and another of degree 5, q(z) = d 0 + d 1 z + d 2 z 2 + d 3 z 3 + d 4 z 4 + d 5 z 5 .
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 12

Week 9 - 604 Karp, R. M., and V. Ramachandran. 1990....

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online