(i) Rewrite the equation (3.1) in the form 1 3! 5! 7! suitable for application of the Fixed-Point Iteration method, and then show that the iteration function 73 g(x) = 3! 1 5! 7! takes the interval [a, b) = [0.5,0.5861] into itself and there is a constant kE (0, 1) such that lg'(x)| < k for all z € [a, b) = [0.5, 0.5861]. This will imply that given any po € [a, b), the sequence (pn) generated by the Fixed-Point Iteration method converges to the fixed point p of the function g(x) in [a, b), and hence to the minimal positive solution of the equation (3.1).
(i) Rewrite the equation (3.1) in the form 1 3! 5! 7! suitable for application of the Fixed-Point Iteration method, and then show that the iteration function 73 g(x) = 3! 1 5! 7! takes the interval [a, b) = [0.5,0.5861] into itself and there is a constant kE (0, 1) such that lg'(x)| < k for all z € [a, b) = [0.5, 0.5861]. This will imply that given any po € [a, b), the sequence (pn) generated by the Fixed-Point Iteration method converges to the fixed point p of the function g(x) in [a, b), and hence to the minimal positive solution of the equation (3.1).
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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Numerical Analysis & it’s applications.
Q3 Part (i)
The equation (3.1) is written in the beginning of the question which is
x-x^3/3! +.........
![3. (Fired-Point Iteration Method;
Consider the equation
73
1
(3.1)
%3D
3!
5!
7!
2
Since we have the seventh Taylor polynomial of the sine function centered at zero in the left-hand side of the equation, it is reasonable to expect
that the minimal positive solution of the equation can be a good enough approximation of 1/6.
(i) Rewrite the equation (3.1) in the form
73
3!
5!
+
7!
suitable for application of the Fixed-Point Iteration method, and then show that the iteration function
77
1
g(x) =
3!
5!
7!
2
takes the interval [a, b] = [0.5,0.5861] into itself and there is a constant k e (0, 1) such that
|g'(x)| < k
for all a € [a, b) = [0.5, 0.5861]. This will imply that given any po € [a, b], the sequence (pn) generated by the Fixed-Point Iteration method
converges to the fixed point p of the function g(x) in [a, b], and hence to the minimal positive solution of the equation (3.1).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdc92fb0c-7643-4ce5-8ef2-0330ed27f016%2F8cef8b9d-8757-4b27-9525-e74d15fd1bc5%2Fd8m9exo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3. (Fired-Point Iteration Method;
Consider the equation
73
1
(3.1)
%3D
3!
5!
7!
2
Since we have the seventh Taylor polynomial of the sine function centered at zero in the left-hand side of the equation, it is reasonable to expect
that the minimal positive solution of the equation can be a good enough approximation of 1/6.
(i) Rewrite the equation (3.1) in the form
73
3!
5!
+
7!
suitable for application of the Fixed-Point Iteration method, and then show that the iteration function
77
1
g(x) =
3!
5!
7!
2
takes the interval [a, b] = [0.5,0.5861] into itself and there is a constant k e (0, 1) such that
|g'(x)| < k
for all a € [a, b) = [0.5, 0.5861]. This will imply that given any po € [a, b], the sequence (pn) generated by the Fixed-Point Iteration method
converges to the fixed point p of the function g(x) in [a, b], and hence to the minimal positive solution of the equation (3.1).
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