NUMERICAL COMPUTING CSCI 4800
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This 15 page Class Notes was uploaded by Santos Fadel on Monday October 19, 2015. The Class Notes belongs to CSCI 4800 at Rensselaer Polytechnic Institute taught by Staff in Fall. Since its upload, it has received 69 views. For similar materials see /class/224866/csci-4800-rensselaer-polytechnic-institute in ComputerScienence at Rensselaer Polytechnic Institute.
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Date Created: 10/19/15
MATHCSCI 4800 Introduction to Numerical Computing What is numerical computing 0 Numerical Computing is concerned with the design and analysis of algorithms for solving mathematical problems arising in applications 0 We model continuous quantities using discrete approximations 0 Most problems are so hard that we cannot find an exact solution Instead we follow an infinite iterative process In practice we only go up to N finite iterations Analysis of Algorithms 0 Once we have an algorithm we consider the effects of the approximation and we study the rate of convergence of it 0 Sometimes we must improve the speed by modifying this algorithm 0 Good algorithms need to be accurate despite all approximations Computational Simulation 0 Computational simulation is the representation and emulation of a physical system using a computer 0 Computational Simulation in general include Mathematical Model Algorithm that solves the model numericallyEl Implementation Run Simulation Visual representation of results ie graphs Interpret and validate results Strategies for solving computational problems 0 Replace the problem With a simpler one 2 Slmple pendulum d 0 9 7 7 sm 0 0 dt2 1 0 Replace With line equationdgi 6 0 it I 0 Replace complicate functions With simple functions 5 8111 7 m a 5 Effects of Approximations 0 Once we pose the problem we already have many errors Modeling we omit and simplify some effects like viscosity in uids or energy dissipation Empirical Measurements our measuring equipment is limited by nature and affected by noise Example Fundamental constants such as C speed of light and G Newton s gravitational constants We do our calculations with collected data and hence by default it contains and error Example averaging of temperature In this course we cannot control any of the above error We will work with them as needed Effects of Approximations 0 These are errors we can control Truncation or discretization some feature of the mathematical model are omitted or simplified EX to truncate an infinite series Rounding representation of real numbers and arithmetic operations Example rounding 313 or e Types of errors 0 Absolute error Approx value True value 0 Relative error Absolute error True value The relative error is used to obtain a percentage of accuracy 0 In general we do not know our True value instead we use bounds to approximate these errors We normally tend to be quite pessimistic about our approximations Truncation and Rounding Error 0 Truncation Error Is the difference between true result the result that would be produced by a given algorithm using exact arithmetic 0 Rounding Error Is the result given by using exact arithmetic and the result produced by using the same algorithm using finite precision rounded arithmetic 0 There are trade offs between them Analysis of Error Example 0 Consider the differentiable function f we will approximate the derivative ofit by using fm z fir h H117 h f72 h M web f 0 Since we assume f is nicely behave we have that f must be bounded By Taylor series we have that We also assume that x and fxh have a round off error M h 2 Hence we have that error is given by EM h 0739 By using Calculus we have that there is a minimum point at which we have an error Whenever h 2 EM Forward and Backward Error 0 Forward error Deals with absolute error of the output 9 f 1 A y me m Forward Error y Backward error Deals with absolute error 0f 11mm Backward Error Sensitivity and Conditioning 0 Sensitivity given a small perturbation of an input how big is the change of the output 0 We describe this qualitatively using th condition number Condition Number y 5370 1 Phrase in terms of Forward and BackW rd Error lRealiveForwardErrorl ConditionNumber x lRelaileBackwardErrorl Computer Arithmetic 0 Computers have store a finite set of real numbers 0 Thus they have a smallest positive number and a largest number When we try to access a smaller number we say there is an underflow and when we try to access a larger number we say there is an over ow Floating point numbers In a computer number are represented by a oating point number system Present day machine use a binary representation of numbers Real numbers represented exactly are machine numbers Since not all number are represented they must be approximated by a nearby number For example 13 has to be approximated because the base 2 representation of this number is a repeated decimal This of course introduces an error Next Time 0 Introduction to linear algebra in numerical computing