MA5509/18UNIVERSITY OF KENTFACULTY OF SCIENCESLEVEL 5 EXAMINATIONNUMERICAL METHODSSaturday, 19 May 2018: 2:00pm – 4:00pmThis paper is divided into TWO sections as follows:Section A: Six short questions each marked out of 10.Candidates should attempt ALL SIX questions.Section B: Two longer questions each marked out of 20.Candidates should attempt BOTH questions.Candidates are advised to show their working on theirscripts. Marks might then be allocated for use of a correctmethod, even if the numerical or algebraic result is incorrect.Calculators: Approved calculators are permitted.Stationery: Yellow Answer Booklet.Turn overMA5509/18 2SECTION AThese questions will each be marked out of 10. Candidatesshould attempt ALL SIX questions.1. Given the data(x0, y0) = (0, 0), (x1, y1) = (1, ?1), (x2, y2) = (2, 0), (x3, y3) = (3, 2),construct the Lagrange interpolating polynomial P3(x) that interpolates these points in twodifferent ways: using Newton’s divided differences and using the barycentric formula.[5+5 marks]2. (a) Write the formula for the composite trapezoidal rule with n subintervals to approximatethe value of the integral[2 marks](b) Is the composite trapezoidal rule an open or closed quadrature rule? Justify your answer.[2 marks](c) We apply this rule to approximate the value of the following integral:�If the error of the composite trapezoidal rule iswith h = (b a)/n, then determine how many subintervals we need to take (at least) toapproximate the value of I with an absolute error smaller than 10?4.[6 marks]3. We want to approximate the solution of the initial value problemy(x) = f(x, y(x)), y(x0) = y0using the trapezoidal rule:yn+1 = yn +h2(f(xn, yn) + f(xn+1, yn+1)), n = 0, 1, . . . ,where h = xn+1 xn.(a) Is this method explicit or implicit? Justify your answer.[2 marks](b) Show that this numerical method is of order 2, by studying the local truncation error.3 MA5509/18[8 marks]4. We consider the following function:f(x) = 1 + 2x + sin(x).(a) Show that it has one root in the interval [0, π].[3 marks](b) Using the derivative f0(x), show that this root is unique.[3 marks](c) We apply two different numerical methods to generate sequences {xn}n≥1 and approximatethe root, which is α ≈ 0.335418. The absolute errors ek = |α ? xn| are the following:Iteration Error method 1 Error method 2Based on these observations, would you say that the sequences generated with these methodsconverge or diverge to the rMA5509/18作业代做、代写Java/Python编程语言作业、代做c/c++实验作业、data留学生作业代写 代做oot? If they converge, what is the order in each case?[4 marks]5. We consider the initial value problemy0(x) = f(x, y(x)), y(x0) = y0.(a) Write down the explicit Euler method to solve such problem.[4 marks](b) In the case f(x, y(x)) = 1 + y2(x), take y(0) = 0, h = 0.1 and calculate an approximatevalue for y(0.5) using the explicit Euler method. What is the absolute error with respect tothe value of the exact solution at that point? (Hint: the exact solution is y(x) = tan x).[6 marks]6. (a) Find the coefficients A, B and C that maximise the order in the following centreddifference formula for the second derivative f00(x):D2[f, h] = Af(x + h) + Bf(x) + Cf(x ? h).[6 marks](b) Determine the order of the resulting formula.[4 marks]Turn overMA5509/18 4SECTION BThese questions will each be marked out of 20. Candidatesshould attempt BOTH questions.7. (a) Explain how the Newton–Raphson method is constructed to solve a general root-findingproblem f(x) = 0.[5 marks](b) Let k ≥ 2, we apply the method of Newton-Raphson to the following function:to compute the value of the root p = 21/k. Show that in this example, the Newton–Raphsonmethod is equivalent to the fixed point iteration x = g(x), where�[3 marks](c) If k = 3, use the general theorem about convergence of fixed point iteration to prove thatthere is convergence to the root p = 21/3starting with any initial value in the interval [1, 2].[4 marks](d) What is the order of convergence of the fixed point iteration to p = 21/3 with this functiong(x)?[4 marks](e) Compute the approximations x1, . . . , x4 to p = 21/3, using the fixed point iterationxn+1 = g(xn) and starting from x0 = 2.[4 marks]8. (a) Define the Lagrange interpolating polynomial and the associated interpolation error fora given function f(x) that is known at a set of points x0, . . . , xn in an interval [a, b].[5 marks](b) Construct the Lagrange polynomial L2(x) with nodes x0 = 0, x1 = 1/2 and x2 = 1 forthe function�[3 marks](c) Integrate this Lagrange polynomial between 0 and 1 to get an approximate value Q ofthe integral�(d) Calculate the exact value of the integral I and give the absolute and relative errors whenapproximating I by Q.5 MA5509/18[4 marks](e) If the Gauss–Legendre nodes and weights in [?1, 1] are given bygive the Gauss-Legendre nodes and weights on [0, 1] and calculate an approximation QGL forI using them. What are the absolute and relative errors with respect to I this time?[5 marks]转自：http://www.7daixie.com/2019051813954980.html
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