1. In the bisection method, assuming all convergence conditions are satisfied, what is the least number of required iterations if the tolerated interval size is 10-6 units and if the size of [a, b] is 4 units?
1. In the bisection method, assuming all convergence conditions are satisfied, what is the least number of required iterations if the tolerated interval size is 10-6 units and if the size of [a, b] is 4 units?
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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![1. In the bisection method, assuming all convergence conditions are satisfied, what is the least number
of required iterations if the tolerated interval size is 10-6 units and if the size of [a, b] is 4 units?
2. Přove the Bolzano's Theorem for the case f(a) > 0 and ƒ(b) < 0. ·
3. Determine a root of x4 + 2.02.x3 – 14.2913.x² – 15.4645z + 30.4932. State your answer in 8 places
after the decimal point. Attach your code or Excel file.
(a) Using bisection method with tolerance |f|< 10-6
(b) Using bisection method with tolerance interval size < 10-6.
(c) Using the Regula Falsi with tolerance 10-6. r'as-
(d) Using the fixed-point iteration with tolerance 10-7.
4. Consider a bracketing method where the candidate point is computed as p; = a; * [a; + b;] where
a; ~ U (0, 1), i.e., a random locator using the Uniform Distribution. Assuming that f is continuous
on [ao, bo] and f (ao)f(bo) < 0, do you think the algorithm will converge? Why or why not?
5. In the previous question, affirm your answer by implementing the algorithm to the problem given in
number 3.
In Python, you may first import the numpy library and use the command numpy.random.uniform(0,
1) to generate a uniformly distributed variable. In Excel, you may use the rand() command.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6fba2ab3-f434-4e63-b8c8-f99564734b14%2F71b759bc-5815-445d-881c-c5cc3fa54279%2Fakp4jh4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. In the bisection method, assuming all convergence conditions are satisfied, what is the least number
of required iterations if the tolerated interval size is 10-6 units and if the size of [a, b] is 4 units?
2. Přove the Bolzano's Theorem for the case f(a) > 0 and ƒ(b) < 0. ·
3. Determine a root of x4 + 2.02.x3 – 14.2913.x² – 15.4645z + 30.4932. State your answer in 8 places
after the decimal point. Attach your code or Excel file.
(a) Using bisection method with tolerance |f|< 10-6
(b) Using bisection method with tolerance interval size < 10-6.
(c) Using the Regula Falsi with tolerance 10-6. r'as-
(d) Using the fixed-point iteration with tolerance 10-7.
4. Consider a bracketing method where the candidate point is computed as p; = a; * [a; + b;] where
a; ~ U (0, 1), i.e., a random locator using the Uniform Distribution. Assuming that f is continuous
on [ao, bo] and f (ao)f(bo) < 0, do you think the algorithm will converge? Why or why not?
5. In the previous question, affirm your answer by implementing the algorithm to the problem given in
number 3.
In Python, you may first import the numpy library and use the command numpy.random.uniform(0,
1) to generate a uniformly distributed variable. In Excel, you may use the rand() command.
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