Fixed point iteration method for root finding

1. Fixed Point Iteration
• Also known as one-point iteration or
successive substitution
• To find the root for f(x) = 0, we rearrange f(x) = 0
so that there is an x on one side of the equation.
x
x
g
x
f 

 )
(
0
)
(
• If we can solve g(x) = x, we solve f(x) = 0.
• We solve g(x) = x by computing
until xi+1 converges to xi.
given
with
)
( 0
1 x
x
g
x i
i 


2. Fixed Point Iteration – Example
0
3
2
)
( 2



 x
x
x
f
2
3
)
(
2
3
3
2
0
3
2
2
1
2
2
2
i
i
i
x
x
g
x
x
x
x
x
x
x














Reason: When x converges, i.e. xi+1  xi
0
3
2
2
3
2
3
2
2
2
1










i
i
i
i
i
i
x
x
x
x
x
x

3. Example
Find root of f(x) = e-x - x = 0.
(Answer: α= 0.56714329)
i
x
i e
x 
 
1
put
We
εt (%)
εa (%)
xi
i
100.0
0
0
76.3
100.0
1.000000
1
35.1
171.8
0.367879
2
22.1
46.9
0.692201
3
11.8
38.3
0.500473
4
6.89
17.4
0.606244
5
3.83
11.2
0.545396
6
2.20
5.90
0.579612
7
1.24
3.48
0.560115
8
0.705
1.93
0.571143
9
0.399
1.11
0.564879
10

4. Two Curve Graphical
Method
The point, x, where
the two curves,
f1(x) = x and
f2(x) = g(x),
intersect is the solution
to f(x) = 0.

5. Fixed Point Iteration
0
3
2
)
( 2



 x
x
x
f
2
3
)
(
2
3
3
2
0
3
2
2
2
2
2












x
x
g
x
x
x
x
x
x
2
3
)
(
2
3
0
3
)
2
(
0
3
2
2













x
x
g
x
x
x
x
x
x
3
2
)
(
3
2
3
2
0
3
2
2
2












x
x
g
x
x
x
x
x
x
• There are infinite ways to construct g(x) from f(x).
For example,
So, which one is better?
(ans: x = 3 or -1)
Case a: Case b: Case c:

6. 3
2
a
Case
1 

 i
i x
x
1. x0 = 4
2. x1 = 3.31662
3. x1 = 3.10375
4. x1 = 3.03439
5. x1 = 3.01144
6. x1 = 3.00381
2
3
b
Case
1



i
i
x
x
2
3
c
Case
2
1



i
i
x
x
1. x0 = 4
2. x1 = 1.5
3. x1 = -6
4. x1 = -0.375
5. x1 = -1.263158
6. x1 = -0.919355
7. -1.02762
8. -0.990876
9. -1.00305
1. x0 = 4
2. x1 = 6.5
3. x1 = 19.625
4. x1 = 191.070
Converge!
Converge, but slower
Diverge!

7. How to choose g(x)?
• Can we know which function g(x) would
converge to solution before we do the
computation?

8. Convergence of Fixed Point Iteration
By definition
)
2
(
)
1
(
1
1 
 



i
i
i
i
x
x




Fixed point
iteration
)
4
(
)
(
and
)
3
(
)
(
1 i
i x
g
x
g





)
6
(
)
(
)
(
)
5
(
in
)
2
(
Sub
)
5
(
)
(
)
(
)
4
(
)
3
(
1
1
i
i
i
i
x
g
g
x
g
g
x















9. Convergence of Fixed Point Iteration
According to the derivative mean-value theorem, if a g(x)
and g'(x) are continuous over an interval xi ≤ x ≤ α, there
exists a value x = ξ within the interval such that
)
7
(
)
(
)
(
)
(
'
i
i
x
x
g
g
x
g





• Therefore, if |g'(x)| < 1, the error decreases with each
iteration. If |g'(x)| > 1, the error increase.
• If the derivative is positive, the iterative solution will be
monotonic.
• If the derivative is negative, the errors will oscillate.
)
(
)
(
have
we
(6),
and
(1)
From 1 i
i
i
i x
g
g
and
x 


  



i
i
i
i
x
g
x
g 



)
(
'
)
(
'
)
7
(
Thus 1
1



 


10. (a) |g'(x)| < 1, g'(x) is +ve
 converge, monotonic
(b) |g'(x)| < 1, g'(x) is -ve
 converge, oscillate
(c) |g'(x)| > 1, g'(x) is +ve
 diverge, monotonic
(d) |g'(x)| > 1, g'(x) is -ve
 diverge, oscillate