"); grf.document.close(); } function dontshow(){ grf.window.close(); } function cfc(){ cf=window.open('','','toolbar=0,width=300,height=350,top=100,left=100'); cf.document.open(); cf.document.writeln("Condition for Convergence-Proof"); cf.document.writeln("By the mean value theorem of differential calculus there is a t between x and s such that
g(x)-g(s) = g'(t)(x-s)

Since g(s) = s and x1 = g(x0), x2 = g(x1), ..., we obtain
|xn-s| = |g(xn-1)-g(s)| = |g'(t)||xn-1-s|
                                  <a |xn-1-s|
                                  <a2 |xn-2-s|
                               ...<an |x0-s|.
Since a < 1, we have an --> 0; hence |xn-s| --> 0
as n --> infinity.

We mention that g is sometimes called a contraction since we have |g(x)-g(v)|< a |x-v|. Furthermore, a gives information on the speed of convergence. For instance, if a = 0.5, since 0.57~0.008, we gain at least 2 digits more accuracy in 7 steps. "); cf.document.close(); } function cfccls(){ cf.window.close(); } function hlts(){ hg=window.open('','','toolbar=0,width=240,height=170'); hg.document.open(); hg.document.writeln("fixed-point highlights"); hg.document.writeln(" "); hg.document.close(); } function hltsclose(){ hg.window.close(); }


FIXED POINT ITERATION METHOD

Fixed point :  A point, say, s is called a fixed point if it satisfies the equation x = g(x). 

Fixed point Iteration : The transcendental equation f(x) = 0 can 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 guess x0  is called the fixed point iterative scheme.


 

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. By testing the condition  | xi+1 - g(xi) | (where i is the iteration number) less than some tolerance limit, say epsilon, fixed apriori.

    •  

     
     
     

    Numerical Example :

    Find a root of  x4-x-10 = 0                                [ Graph] 
    Consider g1(x) = 10 / (x3-1) and the fixed point iterative scheme xi+1=10 / (xi3 -1),      i = 0, 1, 2, . . .let the initial guess x0 be 2.0
    i
    0
    1
    2
    3
    4
    5
    6
    7
    8
    xi
    2
    1.429
    5.214
    0.071
    -10.004
    -9.978E-3
    -10
    -9.99E-3
    -10
    So the iterative process with g1 gone into an infinite loop without converging. 

    Consider another function g2(x) = (x + 10)1/4   and the fixed point iterative scheme 
    xi+1= (xi + 10)1/4,     i = 0, 1, 2, . . .

    let the initial guess x0 be 1.0, 2.0 and 4.0
    i
    0
    1
    2
    3
    4
    5
    6
    xi
    1.0
    1.82116
    1.85424
    1.85553
    1.85558
    1.85558
    		  
    xi
    2.0
    1.861
    1.8558
    1.85559
    1.85558
    1.85558
    		  
    xi
    4.0
    1.93434
    1.85866
    1.8557
    1.85559
    1.85558
    1.85558
    That is for g2  the iterative process is converging to 1.85558 with any initial guess. 

    Consider g3(x) =(x+10)1/2/x  and the fixed point iterative scheme 

    xi+1=( xi+10)1/2 /xi,     i = 0, 1, 2, . . .
    let the initial guess x0 be 1.8, 
    i
    0
    1
    2
    3
    4
    5
    6
    . . .
    98
    xi
    1.8
    1.9084
    1.80825
    1.90035
    1.81529
    1,89355
    1.82129 
    . . .
    1.8555
    That is for g3 with any initial guess the iterative process is converging but very slowly to 

    Geometric interpretation of convergence with g1, g2 and g3 


     

    gx1.jpg gx2.jpg gx3.jpg
    Fig g1
    Fig g2
    Fig g3
    The graphs Figures Fig g1, Fig g2 and Fig g3 demonstrates the Fixed point Iterative Scheme with g1, g2 and g3 respectively for some initial approximations. It's clear from the 
     
  • Fig g1, the iterative process does not converge for any initial approximation.
  • Fig g2, the iterative process converges very quickly to the root which is the intersection point of y = x and y = g2(x) as shown in the figure.
  • Fig g3, the iterative process converges but very slowly.

    •  

    Example 2 :The equation  x4 +  x = Î, where Î is a small number , has a root which is close to Î. Computation of this root is done by the expression x = Î - Î 4 + 4Î7  Then find an iterative formula of the form xn+1 = g(xn ), if we start with x0 = 0 for the computation then show that we get the expression given above as a solution. Also find the error in the approximation in the interval [0, 0.2] .

    Proof

    Given  x4 +  x = Î

    x(x3 +  1) = Î

    x = Î/(1 +  x3)    or      xi = Î/(1 + xi 3)         i = 0, 1, 2, . . .

    x0 = 0

    x1 = Î

    x2 = Î/ (1 + Îi3) = Î(1 + Îi3 )-1
         = Î(1 - Î3 + Î6 + . . .)
         = Î - Î4 + Î7 + . . .

    x3 = Î/( 1 + (Î - Î 4 + Î7)3) = Î[1 + ( Î - Î4 + Î7)-3] = Î - Î4 + 4Î 7

    Now taking x = Î - Î4 + 4Î7

    error = x 4 + x - Î
            = (Î - Î 4 + 4Î7)4 + (Î - Î 4 + 4Î7) - Î
            = 22Î10 + higher order power of Î

    Condition for Convergence :

    If g(x) and g'(x) are continuous on an interval J about their root s of the equation x = g(x), and if |g'(x)|<1 for all x in the interval J then the fixed point iterative process xi+1=g( xi), i = 0, 1, 2, . . ., will converge to the root x = s for any  initial approximation x0 belongs to the interval J .

    Worked out problems
     Exapmple 1  Find a root of cos(x) - x * exp(x) = 0  Solution
     Exapmple 2  Find a root of x4-x-10 = 0  Solution
     Exapmple 3  Find a root of x-exp(-x) = 0  Solution
     Exapmple 4  Find a root of exp(-x) * (x2-5x+2) + 1= 0  Solution
     Exapmple 5  Find a root of x-sin(x)-(1/2)= 0  Solution
     Exapmple 6  Find a root of exp(-x) = 3log(x)  Solution
    Problems to workout

     
    Work out with the Fixed Point Iteration method here 


    Note :Few examples of how to enter equations are given below . . . (i) exp[-x]*(x^2+5x+2)+1  (ii) x^4-x-10  (iii) x-sin[x]-(1/2) 
    (iv) exp[(-x+2-1-2+1)]*(x^2+5x+2)+1  (v) (x+10) ^ (1/4) 
    Solution of Transcendental Equations | Solution of Linear System of Algebraic Equations | Interpolation & Curve Fitting
    Numerical Differentiation & Integration | Numerical Solution of Ordinary Differential Equations
    Numerical Solution of Partial Differential Equations