# What is Fixed point iteration: Definition and 19 Discussions

In numerical analysis, fixed-point iteration is a method of computing fixed points of a function.
More specifically, given a function

f

{\displaystyle f}
defined on the real numbers with real values and given a point

x

0

{\displaystyle x_{0}}
in the domain of

f

{\displaystyle f}
, the fixed point iteration is

x

n
+
1

=
f
(

x

n

)
,

n
=
0
,
1
,
2
,

{\displaystyle x_{n+1}=f(x_{n}),\,n=0,1,2,\dots }
which gives rise to the sequence

x

0

,

x

1

,

x

2

,

{\displaystyle x_{0},x_{1},x_{2},\dots }
which is hoped to converge to a point

x

{\displaystyle x}
. If

f

{\displaystyle f}
is continuous, then one can prove that the obtained

x

{\displaystyle x}
is a fixed point of

f

{\displaystyle f}
, i.e.,

f
(
x
)
=
x
.

{\displaystyle f(x)=x.\,}
More generally, the function

f

{\displaystyle f}
can be defined on any metric space with values in that same space.

View More On Wikipedia.org
1. ### MHB Fixed point iteration convergence

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...
2. ### Solve a nonlinear equation using fixed-point iteration in MATLAB

My attempt is attached below. When I tried to solve it , nothing comes up. However, there are no errors !
3. ### MHB Fixed point iteration: g is a contraction mapping

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...
4. ### A few more questions about fixed point iteration ....?

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...
5. ### How Does a Fixed Point Theorem Explain Convergence in Iterative Methods?

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.
6. ### Fixed point iteration, locally convergent

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...
7. ### Newton Raphson method and Fixed Point Iteration method ?

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!
8. ### Fixed Point Iteration for Solving Equations

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...
9. ### How to generalize the fixed point iteration

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 $$\Lambda(x,y)=h(x,y).$$
10. ### Convergence and stability in multivariate fixed point iteration

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...
11. ### Fixed Point Iteration Convergence

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 =...
12. ### How Does Fixed Point Iteration Converge with Nested Square Roots?

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...
13. ### MATLAB: with My Fixed Point Iteration Program

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...
14. ### Fixed point iteration to find the roots of 0=x-tan(x)

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...
15. ### Fixed Point iteration using matlab, whats wrong with my code?

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...
16. ### Convergence criteria for fixed point iteration

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...
17. ### Fixed Point Iteration Requirements

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...
18. ### Simple Fixed Point Iteration for Root-solving

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...
19. ### Numerical Analysis: Fixed Point Iteration

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...