Numerical Methods Contents - SAM

Termination criterion for contractive fixed point iteration, c.f. (3.2.3), with contraction factor 0 ≤ L < 1:
3.3.1 Bisection
∥
∥x (k+m) − x (k)∥ ∥
△-ineq.
k+m−1 ∑
∥
≤ ∥x (j+1) − x (j)∥ k+m−1
∥
∑ ∥ ≤ L j−k ∥x (k+1) − x (k)∥ ∥ j=k
j=k
= 1 − Lm
∥
∥x (k+1) − x (k)∥ ∥ 1 − L m ∥ ≤
1 − L
1 − L Lk−l ∥x (l+1) − x (l)∥ ∥ .
hence for m → ∞, with x ∗ := lim
k→∞ x(k) :
Idea: use ordering of real numbers & intermediate value theorem
F(x)
Input: a,b ∈ I such that F(a)F(b) < 0
(different signs !)
⇒
∃ x∗ ∈] min{a,b}, max{a,b}[:
x ∗
F(x ∗ ) = 0 .
a
b
x
∥
∥x ∗ − x (k)∥ ∥ L k−l
∥
≤ ∥x (l+1) − x (l)∥ ∥ . (3.2.5)
1 − L
Algorithm 3.3.1 (Bisection method).
Fig. 34
Ôº¾¿ ¿º¾
Ôº¾ ¿º¿
Set l = 0 in (3.2.5)
Set l = k − 1 in (3.2.5)
a priori termination criterion
a posteriori termination criterion
∥
∥x ∗ − x (k)∥ ∥ L k
∥
≤ ∥x (1) − x (0)∥ ∥ ∥
(3.2.6) ∥x ∗ − x (k)∥ ∥ L
∥
≤ ∥x (k) − x (k−1)∥ ∥ 1 − L
1 − L
(3.2.7)
3.3 Zero Finding
△
MATLAB-CODE: bisection method
function x = bisect( F ,a,b,tol)
% Searching zero by bisection
if (a>b), t=a; a=b; b=t; end;
fa = F(a); fb = F(b);
if (fa*fb>0)
error(’f(a), f(b) same sign’); end;
if (fa > 0), v=-1; else v = 1; end
x = 0.5*(b+a);
while((b-a > tol) & ((a