NUMERICAL METH. F/ENGR.(LL)--W/ACCESS
7th Edition
ISBN: 9781260514131
Author: Chapra
Publisher: MCG
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Textbook Question
Chapter 6, Problem 16P
(a) Apply the Newton-Raphson method to the function
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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.
Determine whether the following statement is true or false, and explain why.
Newton’s method converges as long as there is a real root and the function is differentiable.
3. The function f(x) = xe=^(-x) has a unique root x = 0.(a) Compute several iterations of the Newton's method, and conclude that the methoddoes not succeed if x_0 > 1.(b) Draw graphs to illustrate the rst few iteration when x_0 = 0.5 and x_0 = 1.5.
Chapter 6 Solutions
NUMERICAL METH. F/ENGR.(LL)--W/ACCESS
Ch. 6 - 6.1 Use simple fixed-point iteration to locate the...Ch. 6 - 6.2 Determine the highest real root of...Ch. 6 - Use (a) fixed-point iteration and (b) the...Ch. 6 - Determine the real roots of f(x)=1+5.5x4x2+0.5x3:...Ch. 6 - 6.5 Employ the Newton-Raphson method to determine...Ch. 6 - Determine the lowest real root of...Ch. 6 - 6.7 Locate the first positive root of
Where x...Ch. 6 - 6.8 Determine the real root of, with the modified...Ch. 6 - 6.9 Determine the highest real root of:...Ch. 6 - 6.10 Determine the lowest positive root...
Ch. 6 - 6.11 Use the Newton-Raphson method to find the...Ch. 6 - 6.12 Given
Use a root location technique to...Ch. 6 - You must determine the root of the following...Ch. 6 - Use (a) the Newton-Raphson method and (b) the...Ch. 6 - 6.15 The “divide and average” method, an old-time...Ch. 6 - (a) Apply the Newton-Raphson method to the...Ch. 6 - 6.17 The polynomial has a real root between 15...Ch. 6 - Use the secant method on the circle function...Ch. 6 - You are designing a spherical tank (Fig. P6.19) to...Ch. 6 - 6.20 The Manning equation can be written for a...Ch. 6 - 6.21 The function has a double root at. Use (a)...Ch. 6 - 6.22 Determine the roots of the following...Ch. 6 - 6.23 Determine the roots of the simultaneous...Ch. 6 - Repeat Prob. 6.23 except determine the positive...Ch. 6 - A mass balance for a pollutant in a well-mixed...Ch. 6 - Fir Prob. 6.25, the root can be located with...Ch. 6 - 6.27 Develop a user-friendly program for the...Ch. 6 - Develop a user-friendly program for the secant...Ch. 6 - 6.29 Develop a user-friendly program for the...Ch. 6 - 6.30 Develop a user-friendly program for Brent’s...Ch. 6 - 6.31 Develop a user-friendly program for the...Ch. 6 - 6.32 Use the program you developed in Prob. 6.31...
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- Use Newton’s root finding method to determine the zeros for the following functions, from the starting points x0: a) f (x) = sin(x) + x^2 cos(x) − x^2 − x, x_0 = −1, and x_0 = 0 b) f (x) = x tan(x) − x, x_0 = 1, and x_0 = 2 For each starting point case, show all iterations x_1, x_2, ... x_n needed to obtain 3rd decimal precision, |f (x_n)| < 0.001. Root finding iterations may not converge, discuss why. Can you provide typed answer pleasearrow_forwardUse Newton-Raphson method on the function x2 - 10 med startvalue x0 = 3 to find an approximation to √10 with 4 desimal accuracy. Use the intermediate value theorem to show that the desired accuracy have been reached.arrow_forwardThe cubic root of a number N2 can be found by solving x3 – N2 = 0 using modified secant method. Starting with x0 = 0 and taking dx = N1/N2 perform 5 iterations. Assuming the solution obtained by the calculator directly is the exact solution, calculate the true percent error in each iteration. Comment on the convergence of the method. If N1 is zero take it to be 1, if N2 is zero take it to be 11 N1=4 N2=44arrow_forward
- 1. Sketch f(x) = x^3-5.00x^2+1.01x+1.88, showing the roots near+-1 and 5. Write: x = g(x) = (5.00x^2-1.01x-1.88) (x^2) Find the root starting from x0 = 5,4,1,-1. Explain the results. Find a form x = g(x) of f(x) = 0 in problem 1 that yields convergence to the root near x=1.arrow_forwardfind the root value of the f(x) function at the end of the 2nd iteration under the initial condition x0=1. Use the Newton-Raphson method. Take the stopping tolerance 1e-5.arrow_forwardDetermine whether the following statements are true or false? (a) The maximum number of iterations to reach a certain tolerance CAN be predetermined for the bisection method when we use it to solve a single nonlinear equation when starting from an interval that contains a unique root. (b) The number of iterations CANNOT be predetermined for the Newton-Raphson method when we use it to solve a single nonlinear equation. (c) Among the basic formulas for numerical differentiation, the accuracy of the central difference method is MORE ACCURATE than the forward finite difference method and the backward finite difference method. (d)The accuracy of the Euler explicit method for solving ODEs (ordinary differential equations) is always HIGHER than classical Runge-Kuttamethod.arrow_forward
- Solve the initial value problem for x as a function of tarrow_forwardUse (a) Fixed Point Iteration method (b) Newton-Rhapson method and (c) Secant Method to find the solution to the following within error of 10-6. Prepare an Excel file for the finding the root until the error is within 10-6 showing also the graph of the function. sin x - e-x=0, when x = [0,1]arrow_forwardApply Newton’s method twice to approximate 7^(1/3) . Use the initial approximation xsubscript 0=2arrow_forward
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