Solvef(x) =x2−xcos(x) +14−sin2(x)4= 0,withx0=π2.(1) Does Newton’s method converge quadratically to the rootr=r1∈[0,1]? If not, explain why?(2) Find the multiplicity of the rootr=r1off(x).(3) Write out the Modified Newton’s Method such that we havequadratical convergence. ### Educational Content for Advanced Calculus#### Problem StatementConsider the function:\[ f(x) = x^2 - x \cos(x) + \frac{1}{4} - \frac{\sin^2(x)}{4} = 0 \]We begin with an initial approximation:\[ x_0 = \frac{\pi}{2} \]#### Questions1. **Convergence of Newton's Method** Does Newton’s method converge quadratically to the root \( r = r_1 \in [0, 1] \)? If not, explain why.2. **Multiplicity of the Root** Find the multiplicity of the root \( r = r_1 \) of \( f(x) \).3. **Modified Newton’s Method** Write out the Modified Newton’s Method such that we have quadratic convergence.#### ExplanationFor educational purposes, we explore:- **Quadratic Convergence**: Quadratic convergence occurs when the error in the numerical approximation decreases quadratically as iterations proceed.- **Multiplicity of Roots**: This refers to the number of times a particular root is repeated. It affects the speed of convergence in methods like Newton’s.- **Modified Newton’s Method**: When a root has a multiplicity greater than one, the standard Newton's method might not converge quadratically. Adjustments are needed in the iterative formula to ensure rapid convergence.#### ConclusionUnderstanding these concepts is crucial for students studying numerical methods and their applications.

Answered: Image /qna-images/answer/24324a38-0cdd-4c81-bcf0-1f5a0380c973.jpg

1 sin (r) f(x) = x² = x cos(x)+ 0, with xo 2 4 (1) Does Newton's method converge quadratically to the root r Ti E (0, 1]? If not, explain why? (2) Find the multiplicity of the root r = r1 of f(x). (3) Write out the Modified Newton's Method such that we have quadratical convergence.

1 sin (r) f(x) = x² = x cos(x)+ 0, with xo 2 4 (1) Does Newton's method converge quadratically to the root r Ti E (0, 1]? If not, explain why? (2) Find the multiplicity of the root r = r1 of f(x). (3) Write out the Modified Newton's Method such that we have quadratical convergence.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

Q3) Find the root of the equation F(x)=x² - 3(sinx)² by using Newton Raphson method with initial value of Xo-6. Take for i= 1.2.3
Q4: Find the real root of (x) = x' - 7x² +8x - 3; x, = 5 Newton Raphson method.
the root of the function f(x)= (x power 3) +x -1 obtained after first iteration on application of newton - raphson secheme using initial guess of x0=1 is
Q1) Use the Newton-Raphson Method to estimate the root of f(x) = x − 2sin (x), employing an initial guess of xo 2, DO ONLY THREE ITERATIONS.
Q2] : solve xZx + yZ, – 4Z = 0 Q3] Find the roots of the equation by: (the error <0.001) f(x) = x* – 3x² + x + 1
Apply Newton’s Method using the given initial guess, and explain why the method fails. y = 2x3 − 6x2 + 6x − 1, x1 = 1
Q4: Solve f(x) = x² + 4x² - 10 using the Newton-Raphson method for a root in [1, 2]
Solve for x using secant method using four iterations. Assume xo=1 and x1=1.5 f(x) — 3х* — х + 1 Show all the steps and calculations, including the rules.
Use Newton's Method to approximate the zeros for f(x), and be sure to continue the process until two successive iterations differ by no more than 0.001, which is your delta-x value. You should find three (3) answers between x = - 4 and x = 4. Please round your answers to four (4) decimal places. f(x) = 2x³ - 8x² + 3x +5