In this problem you will need to use complex number arithmetic. Assume that the equation f(x) = 0 has a unique solution in an interval [a, b], and that ƒ € C³[a, b]. (a) Let h> 0 be a small number. Using appropriate Taylor expansion show that f'(ro) = Im(f(xo +ih))/h+h²f"" (xo)/6+0(hª), (b) Where i is the imaginary unit such that i² = −1. Since f'(ro) Im(f(xo + ih))/h, consider the following modification of Newton's method with complex step.
In this problem you will need to use complex number arithmetic. Assume that the equation f(x) = 0 has a unique solution in an interval [a, b], and that ƒ € C³[a, b]. (a) Let h> 0 be a small number. Using appropriate Taylor expansion show that f'(ro) = Im(f(xo +ih))/h+h²f"" (xo)/6+0(hª), (b) Where i is the imaginary unit such that i² = −1. Since f'(ro) Im(f(xo + ih))/h, consider the following modification of Newton's method with complex step.
Computer Networking: A Top-Down Approach (7th Edition)
7th Edition
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Author:James Kurose, Keith Ross
Publisher:James Kurose, Keith Ross
Chapter1: Computer Networks And The Internet
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Could you please do the Python coding with the correct indentations and answer for question 5a, 5b, 5c? The questions are in the image.
![5.
In this problem you will need to use complex number arithmetic. Assume that the
equation f(x) = 0 has a unique solution in an interval [a, b], and that fe C³ [a, b].
(a)
Let h> 0 be a small number. Using appropriate Taylor expansion show that
f'(ro) = Im(f(xo +ih))/h + h²f" (xo)/6+0(hª),
(b)
Where i is the imaginary unit such that i2= -1.
Since f'(ro) Im(f(xo + ih))/h, consider the following modification of Newton's
method with complex
step.
f(xk)
Im(f(xk+ih))
Tk+1=kh-
k = 0, 1, 2,...
with given ro. Modify your code with Newton's method of problem 4 so as the default value
of the derivative df to be the new approximation Im(f(ro+ih))/h.
Assume that
|æk+1=x*|
kx|kx* |r
where 2* is such that f(r*) = 0. This means that the convergence rate of the new method is
r. Estimate the convergence rate r using the equation r(e/2 + 1) = 0 and ro = 2.5. Study
the influence of the parameter h in the convergence rate.
lim
= C > 0,](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fba18de34-fc06-47a6-b1ea-c54726b84874%2Ffd625d6c-091d-46f5-8b26-da4621af152e%2F81uze7h_processed.jpeg&w=3840&q=75)
Transcribed Image Text:5.
In this problem you will need to use complex number arithmetic. Assume that the
equation f(x) = 0 has a unique solution in an interval [a, b], and that fe C³ [a, b].
(a)
Let h> 0 be a small number. Using appropriate Taylor expansion show that
f'(ro) = Im(f(xo +ih))/h + h²f" (xo)/6+0(hª),
(b)
Where i is the imaginary unit such that i2= -1.
Since f'(ro) Im(f(xo + ih))/h, consider the following modification of Newton's
method with complex
step.
f(xk)
Im(f(xk+ih))
Tk+1=kh-
k = 0, 1, 2,...
with given ro. Modify your code with Newton's method of problem 4 so as the default value
of the derivative df to be the new approximation Im(f(ro+ih))/h.
Assume that
|æk+1=x*|
kx|kx* |r
where 2* is such that f(r*) = 0. This means that the convergence rate of the new method is
r. Estimate the convergence rate r using the equation r(e/2 + 1) = 0 and ro = 2.5. Study
the influence of the parameter h in the convergence rate.
lim
= C > 0,
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Could you please clarify if the this is the Python coding with the correct indentations and answer for question 5a, 5b, 5c? The questions are in the image.
![5.
In this problem you will need to use complex number arithmetic. Assume that the
equation f(x) = 0 has a unique solution in an interval [a, b], and that fe C³ [a, b].
(a)
Let h> 0 be a small number. Using appropriate Taylor expansion show that
f'(ro) = Im(f(xo +ih))/h + h²f" (xo)/6+0(hª),
(b)
Where i is the imaginary unit such that i2= -1.
Since f'(ro) Im(f(xo + ih))/h, consider the following modification of Newton's
method with complex
step.
f(xk)
Im(f(xk+ih))
Tk+1=kh-
k = 0, 1, 2,...
with given ro. Modify your code with Newton's method of problem 4 so as the default value
of the derivative df to be the new approximation Im(f(ro+ih))/h.
Assume that
|æk+1=x*|
kx|kx* |r
where 2* is such that f(r*) = 0. This means that the convergence rate of the new method is
r. Estimate the convergence rate r using the equation r(e/2 + 1) = 0 and ro = 2.5. Study
the influence of the parameter h in the convergence rate.
lim
= C > 0,](https://content.bartleby.com/qna-images/question/ba18de34-fc06-47a6-b1ea-c54726b84874/36ce01ad-8b9b-4668-a78d-93afa25fe819/jrh8e7_processed.jpeg)
Transcribed Image Text:5.
In this problem you will need to use complex number arithmetic. Assume that the
equation f(x) = 0 has a unique solution in an interval [a, b], and that fe C³ [a, b].
(a)
Let h> 0 be a small number. Using appropriate Taylor expansion show that
f'(ro) = Im(f(xo +ih))/h + h²f" (xo)/6+0(hª),
(b)
Where i is the imaginary unit such that i2= -1.
Since f'(ro) Im(f(xo + ih))/h, consider the following modification of Newton's
method with complex
step.
f(xk)
Im(f(xk+ih))
Tk+1=kh-
k = 0, 1, 2,...
with given ro. Modify your code with Newton's method of problem 4 so as the default value
of the derivative df to be the new approximation Im(f(ro+ih))/h.
Assume that
|æk+1=x*|
kx|kx* |r
where 2* is such that f(r*) = 0. This means that the convergence rate of the new method is
r. Estimate the convergence rate r using the equation r(e/2 + 1) = 0 and ro = 2.5. Study
the influence of the parameter h in the convergence rate.
lim
= C > 0,
Solution
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