Question

Choosing convergent iterations: find a constant k for which...

Answer 1

Find a constant k for which
\(kx _{r+1}+x _{r+1}=kx _{r}+f(x _{r})\)
is a better form than \(x _{r+1}=f(x _{r})\) to use to find the root of the equation x=f(x), near \(x _{0}\), Find this root correct to 4sf for the equation below:
\(F(x) = 2-5\ln x, \) \(x _{0}=1\)

Answer 2

what

Answer 3

@coolor

Answer 4

f(x) and F(x) are the same thing?

Answer 5

@Javk
last line it is f(x) right??

Answer 6

@javk
Have you done fixed point iterations in numerical analysis?

First, it would be beneficial for you to understand what a fixed point iteration is. The following link may help you, if you are not already familiar with the subject.

https://mat.iitm.ac.in/home/sryedida/public_html/caimna/transcendental/iteration%20methods/fixed-point/iteration.html

Notice that that theorem at the end of the article \(assumes\) that a fixed point exists in the interval J.

The following theorems will ensure that there is a unique fixed point in the interval [a,b] AND any value of p0 \(\in\) [a,b] WILL converge to the unique fixed point, which is the case you have posted.

Start with that, and try to work on the example problems if fixed point iteration is new to you.

Here are two theorems that might help you:
Theorem 1:
If a\(\in C\{a,b\}\) and g(x) \(\in\{a,b\}\) then g has a fixed point in \(\{a,b\}\). Moreover, if g'(x) exists on \(\{a,b\}\) and a positive constant k<1 exists with |g'(x)|\(\le\)k for all x\(\in\{a,b\}\), then the fixed point in \(\{a,b\}\) is unique.

Theorem 2:
Let g \(\in\{a,b\}\) be such that g(x) \(\in\{a,b\}\) for all x \(\in\{a,b\}\). Suppose, in addition that g' exists on \(\{a,b\}\) and that a constant 0<k<1 exists with |g'(x)| \(\le\) k, for all x\(\in\{a,b\}\). Then, for any number \(p_0\) \(\in\{a,b\}\), the sequence defined by \(p_n=g(p_{n-1})\) for n\(\ge\)1converges to the unique fixed point p \(\in\{a,b\}\).

(Ref. Numerical Analysis, Burden and Faire, 8th ed. or higher).