Numerical differentiation: Consider a smooth function f(r). The central difference approxima- tion to the first derivative for small h> 0 is Def (1) while f'(x) = D.f(x) + Ch² for some constant C that depends on f". Discarding the error term we have f'(x) Def(x). Similarly we define the forward difference approximation to the first derivative for small h as f(x+h)-f(x) Dif(2)= and the backward difference approximation D₂ (1) with truncation error of O(h). There is a general numerical procedure to determine the order of accuracy of the approximation (i.e. the exponent p in the error formula ChP): We consider a problem with known solution. In our case we consider a given function f(r) and values h = h₁, h₂, h with hi+1
Numerical differentiation: Consider a smooth function f(r). The central difference approxima- tion to the first derivative for small h> 0 is Def (1) while f'(x) = D.f(x) + Ch² for some constant C that depends on f". Discarding the error term we have f'(x) Def(x). Similarly we define the forward difference approximation to the first derivative for small h as f(x+h)-f(x) Dif(2)= and the backward difference approximation D₂ (1) with truncation error of O(h). There is a general numerical procedure to determine the order of accuracy of the approximation (i.e. the exponent p in the error formula ChP): We consider a problem with known solution. In our case we consider a given function f(r) and values h = h₁, h₂, h with hi+1
Computer Networking: A Top-Down Approach (7th Edition)
7th Edition
ISBN:9780133594140
Author:James Kurose, Keith Ross
Publisher:James Kurose, Keith Ross
Chapter1: Computer Networks And The Internet
Section: Chapter Questions
Problem R1RQ: What is the difference between a host and an end system? List several different types of end...
Related questions
Question
Could you please do question 2 from the image? Python coding is required in the answer.
![Numerical differentiation: Consider a smooth function f(r). The central difference approxima-
tion to the first derivative for small h> 0 is
Def(r):
while f'(x) = D.f(x) + Ch² for some constant C that depends on f". Discarding the error term
we have f'(r) Def(x).
Similarly we define the forward difference approximation to the first derivative for small h as
Djf(x)
and the backward difference approximation
1.
2.
f(r+h)-f(r-h)
2h
Dof(x)=
with truncation error of O(h). There is a general numerical procedure to determine the order of
accuracy of the approximation (i.e. the exponent p in the error formula Chp): We consider a problem
with known solution. In our case we consider a given function f(r) and values h = h₁, h₂,... hn
with hi+1 <h. We compute the quantity Df(z) and the exact value f'(x) and then we compute
the error E₁ = f'(x) - Df(r)] for each h,. This error must be E; = Ch. Taking the values E;
and E+1 we can approximate the order p as follows: We compute the fraction
E₁/E₁+1 = (hi/hi+1)P
and then using logarithms and solving for p we get the approximation
log(E₁/E+1)
log(h₂/hi+1)
p=
f(x+h)-f(x)
(x + h) -
f(x)-f(x-h)
h
Consider the function f(x)=sin(z). Approximate the derivative f'(x) with central,
forward and backward differences for values h = 0.2, 0.1, 0.05, 0.01, 0.005, 0.001, 0.0005,
0.0001 and plot the graph of the error f'(x) - Df() for the corresponding values of h.
Compute the values of p using the above method and make a table with the values.
Comment on the results.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fba18de34-fc06-47a6-b1ea-c54726b84874%2Fc40593b2-c288-47f5-b24d-0483dfc2c0d7%2Fzjy749_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Numerical differentiation: Consider a smooth function f(r). The central difference approxima-
tion to the first derivative for small h> 0 is
Def(r):
while f'(x) = D.f(x) + Ch² for some constant C that depends on f". Discarding the error term
we have f'(r) Def(x).
Similarly we define the forward difference approximation to the first derivative for small h as
Djf(x)
and the backward difference approximation
1.
2.
f(r+h)-f(r-h)
2h
Dof(x)=
with truncation error of O(h). There is a general numerical procedure to determine the order of
accuracy of the approximation (i.e. the exponent p in the error formula Chp): We consider a problem
with known solution. In our case we consider a given function f(r) and values h = h₁, h₂,... hn
with hi+1 <h. We compute the quantity Df(z) and the exact value f'(x) and then we compute
the error E₁ = f'(x) - Df(r)] for each h,. This error must be E; = Ch. Taking the values E;
and E+1 we can approximate the order p as follows: We compute the fraction
E₁/E₁+1 = (hi/hi+1)P
and then using logarithms and solving for p we get the approximation
log(E₁/E+1)
log(h₂/hi+1)
p=
f(x+h)-f(x)
(x + h) -
f(x)-f(x-h)
h
Consider the function f(x)=sin(z). Approximate the derivative f'(x) with central,
forward and backward differences for values h = 0.2, 0.1, 0.05, 0.01, 0.005, 0.001, 0.0005,
0.0001 and plot the graph of the error f'(x) - Df() for the corresponding values of h.
Compute the values of p using the above method and make a table with the values.
Comment on the results.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 1 images
Recommended textbooks for you
Computer Networking: A Top-Down Approach (7th Edi…
Computer Engineering
ISBN:
9780133594140
Author:
James Kurose, Keith Ross
Publisher:
PEARSON
Computer Organization and Design MIPS Edition, Fi…
Computer Engineering
ISBN:
9780124077263
Author:
David A. Patterson, John L. Hennessy
Publisher:
Elsevier Science
Network+ Guide to Networks (MindTap Course List)
Computer Engineering
ISBN:
9781337569330
Author:
Jill West, Tamara Dean, Jean Andrews
Publisher:
Cengage Learning
Computer Networking: A Top-Down Approach (7th Edi…
Computer Engineering
ISBN:
9780133594140
Author:
James Kurose, Keith Ross
Publisher:
PEARSON
Computer Organization and Design MIPS Edition, Fi…
Computer Engineering
ISBN:
9780124077263
Author:
David A. Patterson, John L. Hennessy
Publisher:
Elsevier Science
Network+ Guide to Networks (MindTap Course List)
Computer Engineering
ISBN:
9781337569330
Author:
Jill West, Tamara Dean, Jean Andrews
Publisher:
Cengage Learning
Concepts of Database Management
Computer Engineering
ISBN:
9781337093422
Author:
Joy L. Starks, Philip J. Pratt, Mary Z. Last
Publisher:
Cengage Learning
Prelude to Programming
Computer Engineering
ISBN:
9780133750423
Author:
VENIT, Stewart
Publisher:
Pearson Education
Sc Business Data Communications and Networking, T…
Computer Engineering
ISBN:
9781119368830
Author:
FITZGERALD
Publisher:
WILEY