Using the following methods, determine the real roots of ƒ(x)=2x³-11.1x² −26.75x+70.6 1. a) Rearrange the original function [with f(x) = 0] to find three iterative functions for x. Call them g₁(x), g2(x), and g(x). Use successive substitution (also known as fixed-point iteration) to solve for the roots in Excel: b) Use initial guesses of x= 1 for each of the iterative functions. How many iterations does it take each function to converge to an error of less than 0.0001? Do they all converge? What are the values of the roots found? c) Use initial guesses of x = 3 for each of the iterative functions given. How many iterations does it take each function to converge to an error of less than 0.0001? Do they all converge? What are the values of the roots found?
Using the following methods, determine the real roots of ƒ(x)=2x³-11.1x² −26.75x+70.6 1. a) Rearrange the original function [with f(x) = 0] to find three iterative functions for x. Call them g₁(x), g2(x), and g(x). Use successive substitution (also known as fixed-point iteration) to solve for the roots in Excel: b) Use initial guesses of x= 1 for each of the iterative functions. How many iterations does it take each function to converge to an error of less than 0.0001? Do they all converge? What are the values of the roots found? c) Use initial guesses of x = 3 for each of the iterative functions given. How many iterations does it take each function to converge to an error of less than 0.0001? Do they all converge? What are the values of the roots found?
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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![Using the following methods, determine the real roots of
f(x)=2x³-11.1x² -26.75x+70.6
1. a) Rearrange the original function [with f(x) = 0] to find three iterative functions for x. Call
them g₁(x), g₂(x), and g(x).
Use successive substitution (also known as fixed-point iteration) to solve for the roots in
Excel:
b) Use initial guesses of x = 1 for each of the iterative functions. How many iterations does it
take each function to converge to an error of less than 0.0001? Do they all converge? What
are the values of the roots found?
c) Use initial guesses of x = 3 for each of the iterative functions given. How many iterations
does it take each function to converge to an error of less than 0.0001? Do they all
converge? What are the values of the roots found?
2. Use Newton's Method to solve for the roots of the original equation given in Problem 1. Write
a MATLAB.mlx program to implement Newton's Method for finding roots using user inputted
initial guesses of:
a)
x = 1
b)
x = 3
For both initial guesses, use a convergence tolerance of: f(x₁)| <0.0001.
Use fprintf statement(s) to output the root and iteration number for each iteration until the
tolerance is met.
Extra Credit: Run your code with initial guesses of x = 4 and x = 5. Turn in your results.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc9b629bd-7b7c-4b52-85cc-46a95271ddc8%2F6248d531-3a38-412d-9716-63ee424f294e%2F77vbe2h_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Using the following methods, determine the real roots of
f(x)=2x³-11.1x² -26.75x+70.6
1. a) Rearrange the original function [with f(x) = 0] to find three iterative functions for x. Call
them g₁(x), g₂(x), and g(x).
Use successive substitution (also known as fixed-point iteration) to solve for the roots in
Excel:
b) Use initial guesses of x = 1 for each of the iterative functions. How many iterations does it
take each function to converge to an error of less than 0.0001? Do they all converge? What
are the values of the roots found?
c) Use initial guesses of x = 3 for each of the iterative functions given. How many iterations
does it take each function to converge to an error of less than 0.0001? Do they all
converge? What are the values of the roots found?
2. Use Newton's Method to solve for the roots of the original equation given in Problem 1. Write
a MATLAB.mlx program to implement Newton's Method for finding roots using user inputted
initial guesses of:
a)
x = 1
b)
x = 3
For both initial guesses, use a convergence tolerance of: f(x₁)| <0.0001.
Use fprintf statement(s) to output the root and iteration number for each iteration until the
tolerance is met.
Extra Credit: Run your code with initial guesses of x = 4 and x = 5. Turn in your results.
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