MTH3051 Introduction to Computational Mathematics - User Web ...

School of Mathematical Sciences
Monash University
Fixed point iterations x n+1 = e −xn
Iteration n Old guess x n New guess x n+1 x n+1 − e −x n+1
0 0.500000000000 -1.065e-01
1 0.500000000000 0.606530659713 6.129e-02
2 0.606530659713 0.545239211893 -3.446e-02
3 0.545239211893 0.579703094878 1.964e-02
4 0.579703094878 0.560064627939 -1.111e-02
5 0.560064627939 0.571172148977 6.309e-03
6 0.571172148977 0.564862946980 -3.575e-03
7 0.564862946980 0.568438047570 2.029e-03
8 0.568438047570 0.566409452747 -1.150e-03
9 0.566409452747 0.567559634262 6.524e-04
10 0.567559634262 0.566907212935 -3.700e-04
It appears to be converging, the last column seems to be getting smaller with each
iteration, but the its seems to be a slow convergence. Here are the results at every 10-th
iteration.
Fixed point iterations x n+1 = e −xn
Iteration n Old guess x n New guess x n+1 x n+1 − e −x n+1
1 0.500000000000 0.606530659713 6.129e-02
11 0.566907212935 0.567277195971 2.098e-04
21 0.567142477551 0.567143751417 7.225e-07
31 0.567143287611 0.567143291997 2.487e-09
41 0.567143290400 0.567143290415 8.564e-12
51 0.567143290410 0.567143290410 2.931e-14
61 0.567143290410 0.567143290410 1.110e-16
Clearly the algorithm worked, we have a solution, but it took over 60 iterations. For
functions as simple as this, 61 iterations does not take too much time to compute. So we
might feel that this is a good result. Not quite. If this happened to be part of a much
larger computation, where we needed to compute the root many thousands of times (for
example) then we have good reason to want to improve on 60 iterations per root. As we
shall see in later lectures there are algorithms that can converge in (usually) less than
about 5 iterations. This is a significant improvement.
You might ask how well does this algorithm work for other choices of the initial guess
x 1 ? It converges for a wide range of values! By direct trial and error you can verify
(you’ll need a program!) that it converges (at least) for any initial guess in the range
−5 < x 1 < +5. Again this is encouraging. Flush with confidence you might try rewriting
the original equation 0 = x − e −x as x = − log(x) and thus create the sequence
x n+1 = − log(x n ) n = 1, 2, 3, · · ·
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