(c) Given f: R → R, suppose that f is differentiable and |f' (x)| < 1 for all x E R. Show that the sequence generated by the fixed point iteration method applied to f converges to a fixed point of f for any initial Xo ER.
(c) Given f: R → R, suppose that f is differentiable and |f' (x)| < 1 for all x E R. Show that the sequence generated by the fixed point iteration method applied to f converges to a fixed point of f for any initial Xo ER.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Please please do only number c
![(a) Given f (x)
determine the minimum number of iterations required to have an accuracy of
10-9 in the root. If no root exists on the interval, then write “No root exists."
COS x
xe" on [0, 1], with reference to the Bisection Method,
||
(b) Given f (x)
f on [0, 1], accurate to 3 decimal digits. If no root exists on the interval, then
= cos x – xe", use the Bisection Method to approximate a root of
write "No root exists."
(c) Given f: R → R, suppose that f is differentiable and |f' (x)| < 1 for all x E R.
Show that the sequence generated by the fixed point iteration method applied
to f converges to a fixed point of f for any initial xo E R.
(d) Apply the fixed point iteration to sin x – x = -1 four times, with xo = ;. Use
an accuracy of 6 decimal digits throughout.
(e) Determine if the fixed point iteration applied in the previous question will con-
verge.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F18affe90-918d-4daa-82c6-8a455017a2f0%2F6e29f67d-e578-4c7f-9793-a323056c7ed7%2Flbmjmkh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(a) Given f (x)
determine the minimum number of iterations required to have an accuracy of
10-9 in the root. If no root exists on the interval, then write “No root exists."
COS x
xe" on [0, 1], with reference to the Bisection Method,
||
(b) Given f (x)
f on [0, 1], accurate to 3 decimal digits. If no root exists on the interval, then
= cos x – xe", use the Bisection Method to approximate a root of
write "No root exists."
(c) Given f: R → R, suppose that f is differentiable and |f' (x)| < 1 for all x E R.
Show that the sequence generated by the fixed point iteration method applied
to f converges to a fixed point of f for any initial xo E R.
(d) Apply the fixed point iteration to sin x – x = -1 four times, with xo = ;. Use
an accuracy of 6 decimal digits throughout.
(e) Determine if the fixed point iteration applied in the previous question will con-
verge.
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