Question: For the following functions, does the fixed point iteration for finding the fixed point in $\left [ 0,1 \right ]$ converge for all first points $ p_{0} \in \left [ 0,1 \right ]$?
Justify your answer.
a.$ g(x) = e^{\frac{-x}{2}}$
b.$ g(x) = 3x - 1$
Let me attempt for part a first...
Hey! :o
We have the function $f(x)=x^5-\frac{5}{16}$.
I have approximated the root of that function using three steps of Newton's method with initla value $x_0=\frac{1}{2}$ :
\begin{align*}x_1&=x_0-\frac{f(x_0)}{f'(x_0)}\approx \frac{7}{5} \\ x_2&=x_1-\frac{f(x_1)}{f'(x_1)} \approx...
first of all i simply don't want to give up learning numerical methods ...
i am trying to follow fixed point iteration method from this link ...
http://www.ce.utexas.edu/prof/mckinney/ce311k/Overheads/14-NonlinearEquations_1_FixedPoint.pdf fixed point iteration can be used to solve...
Here I do not perceive the a sequence generated by fixed-point iteration. First would you like to explain this. How can it be that if lim n->∞ pn=P, then lim n-> ∞ Pn+1 ?
Source: Numerical Methods Using Matlab by Kurtis D. Fink and John Matthews.
Homework Statement
For which of them will the corresponding fixed point iteration xk+1 = g(xk) be locally convergent to the solution xbar in [0, 1]? (The condition to check is whether |g'(xbar)| < 1.)
A) 1/x2 -1
B)...
C)...
compute xbar to within absolute error 10-4.
Homework Equations
3. The...
Hi everyone, I has been learning numerical method recently, i am very wonder how fixed point iteration method and Newton raphson method works (a more insight explanation rather than mathematical proof ) thanks!
Homework Statement
Apply fixed point iteration to find the solution of each equation to eight correct decimal places
x3=2x+2
The Attempt at a Solution
I have tried to rewrite the equation for in every possible way to solve for one x and pluggin in my guess( have tried...
If we want to solve $$f(x)=0$$ we can re-write the equation as
$$g(x)=x$$ and use the fixed point method, i.e, $$x_{n+1}=g(x_n)$$ starting with a guess $$x_0.$$ I was wondering if something similar can be done with
$$\Lambda(x,y)=h(x,y).$$
Hi, I'm new to posting questions on forums, so I apologise if the problem is poorly described.
My problem is solving a simulation of the state of a mineral processing froth flotation plant. In the form x@i+1 = f(x@i), f represents the flotation plant. f is a computationally intensive solution...
Homework Statement
Consider the system
x = \frac{1}{\sqrt{2}} * \sqrt{1+(x+y)^2} - 2/3
y = x = \frac{1}{\sqrt{2}} * \sqrt{1+(x-y)^2} - 2/3
Find a region D in the x,y-plane for which a fixed point iteration
xn+1 = \frac{1}{\sqrt{2}} * \sqrt{1+(x_n + y_n)^2} - 2/3
yn+1 =...
Let p>0 and x = \sqrt{p+\sqrt{p+\sqrt{p+ \cdots }}} , where all the square roots are positive. Design a fixed point iteration x_{n+1} = F (x_{n}) with some F which has x as a fixed point. We prove that the fixed point iteration converges for all choices of initial guesses greater than -p+1/4...
Homework Statement
Fixed Point Iteration MATLAB program
Homework Equations
To test for convergence: abs(g'(x))<1
The Attempt at a Solution
Hi all, I am trying to write a Fixed Point Iteration program but when I enter in the command line it kept giving me an error message. Can you...
Homework Statement
The question wants me to first estimate the roots by drawing the graph and then by using a 'suitable' fixed point method to determine the first 4 positive roots.
Homework Equations
0=x-tan (x)
I rearranged to get x=arctan (x) so that the series x_n will converge...
Fixed Point iteration using matlab, what's wrong with my code??
Homework Statement
We are suppose to use MatLab to make a program using the fixed point iteration to find the root of an equation.
I just can't figure out what I'm doing wrong here...
I'm pretty sure a while loop is the...
Homework Statement
Most functions can be rearranged in several ways to give x = g(x) with which to begin the fixed-point
iteration method. For f(x) = e^x − 2x^2 , one g(x) is x = +- sqrt(e^x/2)
a) Using the convergence criteria, show that this converges to the root near 1.5 if the positive...
Hi
I wrote a numerical analysis midterm earlier this week and there was one question I couldn't figure out. I was wondering if anyone had some insight.
What I've been told and what I've read in many many places is that
f(x) will converge to a fixed point on an interval I if
1. f(x) is...
I just want to know why in the world this works?
I am speaking about the simple iteration of taking a function, f(x), setting it to 0, f(x) = 0, solving for x in the function and setting it equal to g(x)...and then iterating.
For example the function :f(x) = x^2 +2x - 1
Setting it to 0 and...
Consider the fixed point iteration formula:
*x_(n+1) = (2/3)[(x_n)^3 - 1] - 3(x_n)^2 + 4x_n = g(x)
*Note: "_" precedes a subscript and "^" precedes a superscript
(a) Find an interval in which every starting point x_0 will definitely converge to alpha = 1.
(b) Show that the order of the...