Scientific Computation ECS 231
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This 3 page Class Notes was uploaded by Ashleigh Dare on Tuesday September 8, 2015. The Class Notes belongs to ECS 231 at University of California - Davis taught by Zhaojun Bai in Fall. Since its upload, it has received 99 views. For similar materials see /class/187717/ecs-231-university-of-california-davis in Engineering Computer Science at University of California - Davis.
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Date Created: 09/08/15
ECS231 Handout 1 Introduction March 31 2009 1 to 03 Scienti c computing and computational science Scienti c computing numerical computing is about the design and analysis of algorithms and engineering software for solving mathematical problems in nite precision arithmetic Computational science involves innovative and essential use of high perfor mance computation andor the development of computational technologies to advance knowledge or capabilities in scienti c and engineering disciplines A necessary element in computational science is a strong close tie to an appli cation discipline Research in computation is inherently multidisciplinary and includes for example environmental modeling simulation of complex physical systems that generate energy semiconductor design modeling DNA sequences and protein structure and the simulation and analysis of ow through geologic structures Ref DOE s computational science graduate fellowship program Algorithms as a technology computational simulation as the third pillar of science General strategy to replace a di icultproblem with an easier one that has the same solution or at least a closely related solution For example solve Am b If we can write A LU where L and U are lower and upper triangular matrices respectively then it is equivalent to solve Ly b for y and Um y for x can be easily computed by forward and back substitution We will discuss this in more detail when we study the solution of linear system of equations Approximation and error not mistake are the facts of life Sources of errors measurement and data uncertainty modeling truncation discretization rounding in nite precision arithmetic It is important to note that no mistake were made For example consider 1 R a R 90 a 1 We have an inexact input 2 and approximate function fconstructued by some algorithm then as 7 M Ma 7 1 ME 7 1 computational errors propagated data errors total error trunctionrounding conditioning x data error 5 a T 00 Absolute error and relative error Let E be an approximation of z Then the absolute error is de ned by abserrz lie ml and the relatiue error assume that z is a nonzero number is de ned by relerrz lpl By the de nition of the relative error E z1 p Relative error is indepen dent of scaling Rule of Thumb if the relative error is approximately 10 d then x and E agree to about at signi cant digits and conversely Forward error analysis and backward error analysis Suppose that an approximation 3 to y fz is computed How should we measure the quality of y Ideally we would like to have the forward error relerry ly 7 WM tiny However we can also ask for what set of data have we actually solved our problem77 That is for what Ax do we have 27 m Am lAml or min M if there are many such Ax is called backward error Two main motiviations for using backward error analysis 0 interprets errors as being equivalent to perturbations in the data 0 reduces the question of bounding or estimating the forward error to perturbation theory for which many problems is well understood and only has to be developed once for the given problem and not for each method An algorithm for computing y fz is called backward stable if for any x it produces a computed ywith a small backward error that is y fz Ax for some small Ax Conditioning of problems the relationship between forward and backward errors for a problem is governed by the conditioning of the problem that is the sensitivity of the solution to perturbation in the data Example compute y Let the computed results in terms of backward error 3 fz Ax Then the absolute error is if y m Am 7 M mm 0 on Correspondingly the relative error is given by 3 9 mic95 A95 OA239 y 1 7 where f We mg The quantity afz is called the condition number of f at m It measures approximately how much the relative backward error in z is magni ed by evaluating of f at m Rule of Thumb forward errorl condition number x lbackward error The computed solution to an ill conditioned ie large condition number problem can have a large forward error even for small backward error 9 Mathematical software engineering 7 desirable qualities 0 reliability 0 robustness 0 accuracy ef ciency speed maintainability structure modules documentation easy to modify portability o usability easy of use 0 applicability functionality Tradeoffs among the different desirable qualities are common Different prior ity for different users 10 Three programming paradigms for exploiting other experts7 software 0 Traditional software libraries and packages For examples BLAS LAPACK 0 Scienti c computing environment a much easier to use environment but at the cost of some performance For examples Matlab Mathematica o Templates for assembling complicated algorithms out of simpler building blocks For examples Templates for the solution of linear systems Numerical Recipes Netlib wwwmetliborg is a valuable resource for free numerical software to solve all kinds of computational problems GAMS httpgamsnistgov is a cross index and repository of mathe matical and statistical software components of use in computational science and engineering
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