The following results were obtained in a certain experiment:1.502.503.5044.50f(z)2303454365275007738649551070a) Use Romberg's algorithm to the order of O(h°) (evel 3) to obtain anapproximation of a? f(x) d22df(r)b) Use the above table to determine5 with accuracyof O(h?).atx =dr

The values of fx given are as follows: x fx 1 230 1.5 345 2 436 2.5 527 3 500 3.5 773…

Answered: The following results were obtained in…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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The auxiliary equation of y"" – 4y" – 4y' + 16y = 0 is r³ – 4r² – 4r + 16 = 0. The auxiliary equation has a root in the interval [-0.75, 3.75]. Let f(r) = r³ - 4r² - 4r + 16 so that the roots " of the auxiliary quation can be determined by solving f(r) = 0. Then, applying four iterations of the Bisection Method to f(r), using the 55
18.10 Use Newton's interpolating polynomial to determine y at x = 8 to the best possible accuracy. Compute the finite divided differences as in Fig. 18.5 and order your points to attain optimal accuracy and convergence. 1 2 5.5 11 13 16 18 y 0.5 3.134 5.3 9.9 10.2 9.35 7.2 6.2
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Use the adaptive quadrature algorithm to evaluate 2 L²₁ dt -1 sin(√|t|)* State how you arrive at your result i.e. what tolerances do you choose and what degree of accuracy can you argue that they provide you? Why do you not encounter divide-by-zero difficulties at t = 0?
D5
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МCBS МАT 255 Note: For Problems 3 and 4, you may use the following two facts about the Euler Phi-function: a) If integers m and n are coprime, then o(mn) = ¢(m)o(n). b) If p is a prime number and a is a positive integer, then o(p") = pª-1(p – 1) 1. In a class of 16 students, there are 6 male students and 10 female students. In how many ways can you select a group of 5 students from this class such that 3 of the 5 students are males and the other two are females? 2. Use the inclusion-exclusion principle to find the number of solutions to the equation x + y + z = 19, where x, y, and z are positive integers. [Hint: You know how to find the number of non-negative solutions to the equation above.] 3. Let ø: {1, 2, 3, ...} → {1, 2, 3, ...} denote the Euler phi-function. If ø injective? Is it surjective? Justify your answers. 4. Define a relation - on the set N of natural numbers as follows: For natural numbers n and m, n ~ m if and only if ø(n) = ¢(m), where o denotes the Euler phi-function.…
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The following results were obtained in a certain experiment: 1.50 2.50 3.50 4 4.50 f(z) 773 955 230 345 436 527 500 864 1070 a) Use Romberg's algorithm to the order of O(h°) (evel 3) to obtain an approximation of rf(x)dz df(z) b) Use the above table to determine 5 with accuracy of atx =