Consider the following fixed-point iteration algorithm1 = 9(z,)From a graphical solution of z = g(z) you know that z = 2.24 is approximately the fixed point of I = g(z).If you start the fixed-point algorithm with z, = 2.38, how many iterations you need to perform at least in order to reach an error lessthan 10?Use your knowledge about the order of convergence of the fixed-point method to conduct your analysis.!!Answer

Answered: Consider the following fixed-point…

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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Use the Any optimization Algorithm to optimize the following function a. f(x,, x2) = x, -2x, + 4x,² + 7 Given that x1 = [1, 2, 3] & x2 = [3,4] %3D
(Babylonian Method). All numerical answers should be rounded to 6-digit floating-point numbers. (0) If A > 0 is a positive number, then given a real number xo, the sequence (xn) defined recursively by where n > 0 is a natural number, converges to √A, that is, provided that the initial term xo is chosen 'not too badly'. In practice, calculation of terms xn is carried out till this or that stopping criterion is triggered. (i) Let (xn) be the sequence defined recursively by (ii) Generate the terms and x0 = 94 (as it could be guessed from part (0), the sequence (xn) converges to √946). x() = Xn+1= = = (x++) ₁ (n ≥ 0), x1 = x2 = of the sequence (xn) till you find the term x N satisfying the condition x3 = fl(xN) = fl(xN-I (our stopping criterion), where fl(z) denotes the result of rounding of a real number z to a 6-digit floating-point number. Along the way, once the k-th term xk is generated, enter in the corresponding input field the result fl(xk) of rounding of xk to a 6-digit…
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Suppose we want to obtain a root of x-x-10 = 0. We can rewrite it as x = √√x + 10. If we use the fixed-point iteration method on f(x)=√√x + 10 to construct the sequence {x}, with xo = 1.5, to estimate a root of x-x-10 = 0, then the smallest value of n so that x, estimates the said root to three significant digits is Blank 1. Blank 1 Add your answer
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(a) Suggest a value of b so that the iteration formula x = x + b(x² − 3) may converge at a good rate, give that x = √3 is a root.
The Intermediate Value Theorem can be used to approximate a root. The following is an example of binary search in computer science. Suppose you want to approximate /8. You know that it is between 2 and 3. If you consider the function f(x) = x? – 8, then note that f(2) 0. Therefore by the Intermediate Value Theorem, there is a value, 2 < c < 3 such that f(c) = 0. Next choose the midpoint of these two values, 2.5, which is guaranteed to be within 0.5 of the acutal root. f(2.5) will either be less than 0 or greater than 0. You can use the Intermediate Value Theorem again replacing 2.5 with the previous endpoint that has the same sign as 2.5. Continuing this process gives a sequence of approximations xn with x1 = 2.5. How many iterations must you do in order to be within 0.015625 of the root?
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