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

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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 .
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This note was uploaded on 03/22/2011 for the course SS 100 taught by Professor Friedman during the Spring '11 term at NJIT.

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

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