Set f(x) = 54x6 + 45x5 — 102x4 – 69x³ +35x² + 16x – 4. (b) Use Newton's Method to find all five roots in the interval (correct to eight decimal places). (c) For each root, use the value obtained in part (b) to approximate its exact value |- r| denote the (absolute) error at step i in Part (b). Plot the curve (ei, ei+1). Please use logarithm scale for both the x-axis and y-axis. What can we say about the convergence based on the slope of these curves. (d) For the one with linear convergence, redo part (c) using secant method. =
Set f(x) = 54x6 + 45x5 — 102x4 – 69x³ +35x² + 16x – 4. (b) Use Newton's Method to find all five roots in the interval (correct to eight decimal places). (c) For each root, use the value obtained in part (b) to approximate its exact value |- r| denote the (absolute) error at step i in Part (b). Plot the curve (ei, ei+1). Please use logarithm scale for both the x-axis and y-axis. What can we say about the convergence based on the slope of these curves. (d) For the one with linear convergence, redo part (c) using secant method. =
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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Please show all work and answer the following in Matlab code !!!
For part c, clearly state the root with linear convergence so that part d may be executed.
Please allow the codes to be copied directly into Matlab
This is a part of an ungraded test review so please help!!
![Given the polynomial function:
\[ f(x) = 54x^6 + 45x^5 - 102x^4 - 69x^3 + 35x^2 + 16x - 4. \]
### Tasks:
**(b)** Apply Newton's Method to find all five roots within the interval with an accuracy of eight decimal places.
**(c)** For each root, use the value obtained in part (b) to approximate its exact value \( r \). Define the error as \( e_i = |x_i - r| \), which represents the absolute error at step \( i \) in part (b). Plot the curve \( (e_i, e_{i+1}) \). Utilize a logarithmic scale for both the x-axis and y-axis. Analyze the convergence behavior based on the slope of these curves.
**(d)** For the root with linear convergence observed in part (c), repeat the analysis using the secant method.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdb82eee3-c597-43b7-b98f-cffc18fbca72%2F7513a356-4d01-4b3d-abcc-2473a000ebb4%2Fzmty12e_processed.png&w=3840&q=75)
Transcribed Image Text:Given the polynomial function:
\[ f(x) = 54x^6 + 45x^5 - 102x^4 - 69x^3 + 35x^2 + 16x - 4. \]
### Tasks:
**(b)** Apply Newton's Method to find all five roots within the interval with an accuracy of eight decimal places.
**(c)** For each root, use the value obtained in part (b) to approximate its exact value \( r \). Define the error as \( e_i = |x_i - r| \), which represents the absolute error at step \( i \) in part (b). Plot the curve \( (e_i, e_{i+1}) \). Utilize a logarithmic scale for both the x-axis and y-axis. Analyze the convergence behavior based on the slope of these curves.
**(d)** For the root with linear convergence observed in part (c), repeat the analysis using the secant method.
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Step 1: Analyze the Graph of the function
VIEWStep 2: Part (b) - Approximate all five roots using Newton's method.
VIEWStep 3: Part (c) Study the convergence of the root approximations
VIEWStep 4: (contd.) Part (c) Study the convergence of the root approximations
VIEWStep 5: (contd.) Part (c) Study the convergence of the root approximations
VIEWStep 6: (contd.) part (c) Analyze the graphs
VIEWStep 7: (contd.) part (c) Approximate the roots using secant method
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