Numerical Methods - Weeblynumericalanalysis.weebly.com/uploads/1/3/8/6/13867400/fixed_point.pdf · Numerical Methods Finding Roots. Fixed Point Iteration Rewrite f(x) = 0to x =g(x)
Post on 17-Mar-2018
225 Views
Preview:
Transcript
Numerical Methods
Finding Roots
Fixed Point Iteration
�Rewrite f(x) = 0 to x = g(x)�To solve x = g(x), we iteratively calculate
�xi+1 = g(xi)
�The problem is how to choose g(x) so as �The problem is how to choose g(x) so as to ensure convergence
y = f(x)
4
6
8
10
12f(
x)
y = g(x) and y = x
1
2
3
4
5
y-6
-4
-2
0
2
-3 -2 -1 0 1 2 3
x
f(x)
-4
-3
-2
-1
0
-3 -2 -1 0 1 2 3
x
y
Fixed Point IterationThe The equation f(x) = 0, where f(x) = x3 − 7x + 3, may be re-
arranged to give x = (x3 + 3)/7.
3
4
y = (x3 + 3)/7
Intersection of the graphs of y = x and y = (x3 + 3)/7 represent roots of the original equation x3 − 7x + 3 = 0.
-4
-3
-2
-1
0
1
2
3
-5 -4 -3 -2 -1 0 1 2 3 4 5
x
y
y = x
The rearrangement x = (x3 + 3)/7 leads to the iteration
To find the middle root α, let initial approximation x0 = 2.
Fixed Point Iteration
...,3,2,1,0,7
33
1 =+
=+ nx
x nn
57143.17
32
7
3 330
1 =+=+
=x
x
98292.0357143.13 33
1 =+=+
=x
x
The iteration slowly converges to give α = 0.441 (to 3 s.f.)
etc.etc.
98292.07
357143.1
7
312 =+=
+=
xx
56423.07
398292.0
7
3 332
3 =+=+
=x
x
45423.07
356423.0
7
3 333
4 =+=+
=x
x
Fixed-point Iteration Example(1)
Fixed-point Iteration Example(2)
Convergence of Fixed Point Iteration
� |g’(x)| < 1: (a), (b)� |g’(x)| > 1: (c), (d)
Example
�Find a root of x3 - x2 - 1 = 0 with x0 = 1.5
x - 1 - x-2 = 0 x3 = 1 + x2 x2(x - 1) = 1
g(x) 1 + x-2 (1 + x2)1/3 (x - 1)-1/2
g’(x) -2x-3 2x / [3(1+x2)2/3] -1 / [2(x-1)3/2]
Comment|g’(x)| < 1 forx > 1.3
|g’(x)| < 1 for1 ≤ x ≤ 2
|g’(x)| > 1 forx < 1.6
The rearrangement x = (x3 + 3)/7 leads to the iteration
For x0 = 2 the iteration will converge on the middle root α, since g’(α) < 1.
1.5
2
Fixed Point Iteration
n x0 21 1.571432 0.98292
...,3,2,1,0,7
33
1 =+
=+ nx
x nn
0
0.5
1
0 0.5 1 1.5 2x
y
2 0.982923 0.564234 0.454235 0.441966 0.44097 0.440828 0.44081
α = 0.441 (to 3 s.f.)
α x0x2 x1x3
y = (x3 + 3)/7
y = x
top related