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FIXED POINT ITERATION The idea of the xed point iteration methods is to rst reformulate a equation to an equivalent xed point problem: f(x) = 0 x = g(x) and then to use the iteration: with an initial guess x 0 chosen, compute a sequence x n+1 = g(x n); n 0 in the hope that x n! . There are in nite many ways to introduce an equivalent xed point

Algorithm - Fixed Point Iteration Scheme Given an equation f(x) = 0 Convert f(x) = 0 into the form x = g(x) Let the initial guess be x0 Do xi+1= g(xi) while (none of the convergence criterion C1 or C2 is met) C1. Fixing apriori the total number of iterations N. C2. |(whereiis the iteration number) less than some tolerance

many with fixed points at : Example: ≔ − ≔ +3 … If has fixed point at , then = − ( ) has a zero at 2 . Why study fixed-point iteration? 3 1. Sometimes easier to analyze 2. Analyzing fixed-point problem can help us find good root-finding methods

1. Algorithm & Example-1 f(x)= ...

to a xed point iteration x n+1 = g(x n) by doing algebra on f(x) = 0. Newton’s method is also a xed point iteration of the form x n+1 = g(x n), where g(x n) = x n f(xn) f0(xn). But we didn’t get this xed point iteration by algebra like the 5 in the example, we got it …

Iteration, Fixed points Paul Seidel 18.01 Lecture Notes, Fall 2011 Take a function f(x). De nition. A xed point is a point x such that f(x) = x : Graphically, these are exactly those points where the graph of f, whose equation is y = f(x), crosses the diagonal, whose equation is y = x. You can often solve for them exactly: Example.

Sample Problem. Functional (Fixed-Point) Iteration. Fixed-Point Algorithm. To find the fixed point of g in an interval [a,b], given the equation.

The resulting iteration method may or may not converge, though. Page 2. Example. We begin with an example. Consider solving the two equations.

Example 1 (From BFB p60) Consider the equation f(x) = x3 +4x2 10. By the Interme-diate Value Theorem there exists a root in the interval [1;2]. There are many ways to change the equation f(x) = 0 to a xed point iteration of the form x= g(x). Here are 5 such examples (which you can verify are equivalent to f(x) = 0 yourself with algebra): a) x= g

Example 1 (From BFB p60) Consider the equation f(x) = x3 +4x2 −10. By the Interme- diate Value Theorem there exists a root in the interval [1,2]. There are ...

The rationale behind that definition is the fact that this sequence will become stationary after some index k, if (and only if) x(k) is a fixed point of Φ.

Example. To determine the xed points of the function f(x) = x3, we solve x3 = x)x3 x = 0)x(x2 1) = 0 so the xed points are x = 0 and x = 1, x = +1. Example. The function f(x) = x2 + x+ 1 has no xed points. Example. The function f(x) = x 2 + 1 x has a xed point at x = p 2. Example. The function f(x) = cos(x) has a xed point, as one can see by looking at the graph.

Examples Example 1. Consider the equation x = 1 + 0:5 sinx: Here g(x) = 1 + 0:5 sinx: Note that 0:5 g(x) 1:5 for any x 2R. Also, g(x) is a continuous function. Applying the existence lemma, we conclude that the equation x = 1 + 0:5 sinx has a solution in [a;b] with a 0:5 and b 1:5. Example 2. Similarly, the equation x = 3 + 2 sinx

Examples[edit] ; x n + 1 = cos ⁡ x n ; f ( x ) = cos ⁡ x ; x 0 . ; | x n − x | ≤ q n 1 − q | x 1 − x 0 | = C q n ; q = 0.85 ...

• A number is a fixed point for a given function if = • Root finding =0 is related to fixed-point iteration = –Given a root-finding problem =0, there are many with fixed points at : Example: ≔ − ≔ +3 … If has fixed point at , then = − ( ) has

• A first simple and useful example is the Babylonian method for computing the square root of a>0, which consists in taking , i.e. the mean value of x and a/x, to approach the limit (from whatever starting point ). This is a special case of Newton's method quoted below.• The fixed-point iteration converges to the unique fixed point of the function fo…

Fixed point Iteration:The transcendental equation f(x) = 0can be converted algebraically into the form x = g(x)and then using the iterative scheme with the recursive relation xi+1= g(xi), i = 0, 1, 2, . . ., with some initial guessx0 is called the fixed point iterative scheme. Algorithm - Fixed Point Iteration Scheme

However, it is important to ensure that the conversion yields a function g for which fixed-point iteration will converge. Example We use fixed-point iteration ...