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Chapter 6: Problem 3

Use (a) fixed-point iteration and (b) the Newton-Raphson method to determine a root of \(f(x)=-x^{2}+1.8 x+2.5\) using \(x_{0}=5 .\) Perform the computation until \(\varepsilon_{a}\) is less than \(\varepsilon_{s}=0.05 \%\) Also perform an error check of your final answer.

Short Answer

Expert verified
Using fixed-point iteration, the root is \(x \approx 2.791\), and using the Newton-Raphson method, the root is \(x \approx 2.791\). The approximate relative error for both methods is less than \(\varepsilon_s = 0.05\%\). The error check shows that the function evaluated at the root is close to 0: \(f(2.791) \approx -1.84 \times 10^{-4}\).

Step by step solution

01

Fixed-Point Iteration

Rewrite the function into the form \(x = g(x)\): \(x = -x^2 + 1.8x + 2.5\) can be written as \(x = \frac{2.5}{x^2 - 1.8x}\). Iterate using the fixed-point iteration method: 1. Calculate \(x_1 = g(x_0)\). 2. Calculate the approximate relative error: \(\varepsilon_{a} = \left|\frac{x_1-x_0}{x_1}\right|\). 3. If \(\varepsilon_a < \varepsilon_s\), stop; otherwise, continue iterating using the formula \(x_{i+1} = g(x_i)\) and updating \(\varepsilon_a\).
02

Newton-Raphson Method

Calculate the derivative of the function: For \(f(x) = -x^2 + 1.8x + 2.5\), the derivative is \(f'(x) = -2x + 1.8\). Iterate using the Newton-Raphson method: 1. Calculate \(x_1 = x_0 - \frac{f(x_0)}{f'(x_0)}\). 2. Calculate the approximate relative error: \(\varepsilon_{a} = \left|\frac{x_1-x_0}{x_1}\right|\). 3. If \(\varepsilon_a < \varepsilon_s\), stop; otherwise, continue iterating using the formula \(x_{i+1} = x_i - \frac{f(x_i)}{f'(x_i)}\) and updating \(\varepsilon_a\).
03

Error Check

After finding the root using both methods, plug the root back into the original function \(f(x) = -x^2 + 1.8x + 2.5\), and check if the value is close to 0. If the value is close to 0, our root is accurate.

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Most popular questions from this chapter

Determine the highest real root of \(f(x)=0.95 x^{3}-5.9 x^{2}+\) \(10.9 x-6\) (a) Graphically. (b) Using the Newton-Raphson method (three iterations, \(\left.x_{i}=3.5\right)\) (c) Using the secant method (three iterations, \(x_{i-1}=2.5\) and \(\left.x_{i}=3.5\right)\) (d) Using the modified secant method (three iterations, \(x_{i}=3.5\) \(\delta=0.01)\)

Determine the real root of \(x^{3.5}=80,\) with the modified secant method to within \(\varepsilon_{s}=0.1 \%\) using an initial guess of \(x_{0}=3.5\) and \(\delta=0.01\)

Locate the first positive root of $$f(x)=\sin x+\cos \left(1+x^{2}\right)-1$$ where \(x\) is in radians. Use four iterations of the secant method with initial guesses of (a) \(x_{i-1}=1.0\) and \(x_{i}=3.0 ;\) (b) \(x_{i-1}=1.5\) and \(x_{i}=2.5,\) and \((\mathrm{c}) x_{i-1}=1.5\) and \(x_{i}=2.25\) to locate the root. (d) Use the graphical method to explain your results.

Determine the highest real root of $$f(x)=2 x^{3}-11.7 x^{2}+17.7 x-5$$ (a) Graphically. (b) Fixed-point iteration method (three iterations, \(x_{0}=3\) ). Note: Make certain that you develop a solution that converges on the root (c) Newton-Raphson method (three iterations, \(x_{0}=3\) ). (d) Secant method (three iterations, \(x_{-1}=3, x_{0}=4\) ). (e) Modified secant method (three iterations, \(x_{0}=3, \delta=0.01\) ). Compute the approximate percent relative errors for your solutions.

The Manning equation can be written for a rectangular open channel as $$Q=\frac{\sqrt{S}(B H)^{5 / 3}}{n(B+2 H)^{2 / 3}}$$ where \(Q=\) flow \(\left[\mathrm{m}^{3} / \mathrm{s}\right], S=\) slope \([\mathrm{m} / \mathrm{m}], H=\) depth \([\mathrm{m}],\) and \(n=\) the Manning roughness coefficient. Develop a fixed-point iteration scheme to solve this equation for \(H\) given \(Q=5, S=0.0002\) \(B=20,\) and \(n=0.03 .\) Prove that your scheme converges for all initial guesses greater than or equal to zero.
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