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# INTRONUMANALYS&COMPUT MATH541

SDSU

GPA 3.64

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This 5 page Study Guide was uploaded by Miss Gladys Lubowitz on Tuesday October 20, 2015. The Study Guide belongs to MATH541 at San Diego State University taught by P.Blomgren in Fall. Since its upload, it has received 25 views. For similar materials see /class/225269/math541-san-diego-state-university in Mathematics (M) at San Diego State University.

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Date Created: 10/20/15

Math 5417 Study Guide 7 o set of topic you should be aware of and questions which you should be able to answer at the end of the course Md studygnidetexv 17 20060928 202858 blomgren Exp t 1 Numerical Approximation Let 1 be an approximation to 1 1 y 0 a b What is the absolute error What is the relative error 2 Fixed Point Iteration Consider the function 91 l 7 2127 the xed point equation 1 917 and the xed point iteration pn1 910n7 a b P0 6 17 ll Does the xed point equation have any solutions ls so7 what are they If we start the xed point iteration at p0 017 is it likely it will converge to any of these solutions Why Why not 3 Taylor Expansion7 Newton s Method Consider the function f1 e 7 cos1 7 1 on the interval 07 ll Write the Taylor polynomial of degree 2 13217 and the remainder term7 R2 Hint You can Taylor expand around any point7 but 10 0 makes life easier e 67 7 cos1 7 sin17 sin1 cos1 Use the remainder term to get an upper bound on the error in the above approximation in the given interva l Now consider 91 ex 7 cos1 7 1 What is the rate of convergence for 9h Hint If you did part a7 this problem is easier than you fearl Write down Newton7s method for 0 for the function For this particular function f7 what do you expect the rate of convergence to be circle one Slower than quadratic Quadratic Better than quadratic Why Write down a scheme which converges faster than the one you wrote down in What is the convergence rate of this scheme 4 Newton s Method Numerical solution of f1 0 Write down Newton7s Method Assume 07 and we have an initial approximation 10 which is close enough77 to 1 What is the convergence rate for Newton7s metho it lff 17 0 iii lff 1 0 How can we improve the convergence rate when 0 Assume f E C21 7 971 In Now view the Newton iteration as a xed point iteration 1n1 9 i What condition on 91 is necessary for convergence of the xed point iteration i What condition on 91 is necessary for convergence of the xed point iteration 39 39l Translate the condition on 9 1 to a condition for the Newton iteration Note this is the expression which quanti es the neighborhood where 10 is close enough77 to H H H 533 7 50 Root Finding For each of the following methods write down the de nition ii What kind of points we need to start the scheme iii How fast does the scheme converge a The Bisection Method i De nition ii Starting Points iii Speed in absolute terms b The Secant Method i De nition ii Starting Points iii Speed compared with the other schemes c Newton7s Method i De nition ii Starting Points iii Speed in absolute terms Multiplicity of zeros Know what it means including impact on the derivatives and how it effects some met 0 s Aitken s A2method Given an already computed linearly convergent sequence p71 how can we manufacture a sequence which converges faster Steffensen s Method In general xedpoint iteration gives linear convergence if it converges that is 7 recall the conditions for convergence of xed point when does it converge faster With the help of Aitken s A2 method we can generate a quadratically converging method which does not require computation of the derivative 7 how Both Newton7s and Steffensen s methods are quadratically con vergent so it seems like Steffensen would be our prime choice why is it not That is what additional restrictions beyond what is needed for Newton is needed for Steffensen s method to converge Polynomials Horner s Method Given the polynomial Pz x4 7 x3 x2 z 7 1 use Horner s method synthetic division to compute P5 and PS De ation with Improvement The method for extracting all real roots of a polynomial Miiller s Method Understand it as a natural extension of the secant method and how it automat ically tells us when we get complex roots The Lagrange Polynomial Understand how the building blocks Lmk work and how they allow us to interpolate any given data set 13 yi n i0 13 Newton s Divided Differences Given the points 2 1011 12 lzn and the function values f f07f17f27 7fn7 Where fi flt1i7 i 07171an a Explain how to ll in the table of Newtonls Divided Differences 7 In particular write down the expression for F3 assuming all entries to the left in the table are known i f lst 2ndl nthl I0 f0 I1 f1 F11 12 f2 F21 F2 13 f3 F31 F32 F33 In fn Fn1 Fn2 Fm b Once the table is full 7 how do we use it Write down the interpolating polynomial of degree n using the appropriate entries from the table 14 Hermite Interpolation Given the points 101112p l l In the function values F f0 f1 f2 fn where i 012Hln and the values of the derivative d d0d1d2l dn where di fzi i 012Hlnl a Explain how to modify the table of Newtonls Divided Differences to compute the Hermite inter polating polynomia i f lst 2ndl nthl 10 f0 11 f1 F11 12 f2 F21 F22 In fn Fn1 Fn2 Fm ii First explain how the table is initialized 239e how we use the values X F and d to get started iii Second how do we compute the rest of the values in the table In particular write down the expression for F53 assuming all entries to the left in the table are known iii Once the table is full 7 how do we use it Write down the interpolating polynomial of degree 2n 1 using the appropriate entries from the table 15 Cubic Spline Interpolation Given the points 2 1011127i 171 and the function values f fmfhfgp i i 7 at those points7 we want to generate a cubic spline7 239e a piecewise third degree polynomial approximation a Why would we want to do this 7 why not just use Newton7s lnterpolatory Divided Difference formula to get an nth degree interpolating polynomial b Write down the interpolant 5k on the subinterval 116 zk1li c Write down the conditions required for the spline 51 to t the data and have two continuous derivatives7 239e 51 6 C2zoznl d How many unknown coefficients do we have to determine e How many equations do we have in c f Suggest additional boundary conditions7 giving enough additional equations to close the system assume we have no additional information about 16 Piecewise Polynomial Approximation Quintic Splines Given the points 2 10 11 12 l l l 1n and the function values f f0 f1 f2 i i at those points we want to generate a quintic spline 239e a piecewise fth degree polynomial approximation a Why would we want to do this 7 why not just use Newton7s Interpolatory Divided Difference formula to get an nth degree interpolating polynomial b Write down the interpolant 5k on the subinterval 116 11644 c Write down the conditions required for the spline 51 to t the data and have four continuous derivatives 239e 51 6 C4101nl d How many unknown coefficients do we have to determine How many equations do we have in c AA Hum VV Suggest additional boundary conditions giving enough additional equations to close the system assume we have no additional information about i H 7 Numerical Integration Simpson s Rule and Composite Simpson s Ruler a We write Simpson7s Rule 7 h5 h h5 fltzgtdz Sfz7a7b WW5 g fa 4fab2 N WW5 Explain how we arrive at this formula using the Lagrange Interpolating Polynomial of degree 2 IGNORE THE ERROR TERM7 AND DO NOT COMPUTE ANY INTEGRALSl b We de ne the Composite Simpson7s Rule by splitting the interval a 12 into smaller subintervals applying Simpson7s Rule on those subintervals and then summing up the results Write down Composite Simpson7s Rule applied to the subintervals 1012 1214 1415 1513 where 1k 177 c Richardson Extrapolation Explain how we can use multiple instances of Composite Simp son7s Rule with pointspacing h h2 h4 to generate a scheme with an error term N 0h5l Warning this question is harder than it looks H 00 l Numerical Integration Gaussian Quadrature We are interested in integrating a function of two variables over the square 711 gtlt 711 239e I fil fil f1y d1dy Build a 9point numerical integration scheme based on 3point Gaussian quadrature in the 1 and ydirectionsl Specify the quadrature points and the summation weightsl H 50 Discrete Least Squares Suppose you are given the data points i 101112p l l 1n and the function values f 1 f1 f2 i fn where 11 gt 0 Vi 012 i n a For some reason you think that h1 a 121 ccos1 is a great model for the data set Find the best t in the least squares sense for this modeli Find the normal equations b By a stroke of luck it turns out that the basis functions 0 1 391 1 392 l1cos1 are orthogonal on the nodes i with respect to summation against the weight function 1 This should help you express the coefficients a b c in a simpler way than in part a i N 0 Discrete Least Squares Approximation Suppose you are given the data points i 10 11 12 l l l 1n and the function values f f0 f1 f2 i i fn where 11 gt 0 Vi 01 2 i i i n a For some reason you think that h1 a b1 c13 is a great model for the data set Find the best t in the least squares sense for this model Find the normal equations b Explain how the normal equations simplify if we have an orthogonal set of basis functions 01 11 21 and we are trying to t the model 91 a 01 b 11 c 21 to the given data

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