BisectDemo.m

Contents

Overview

Illustrates the bisection algorithm coded in bisect.m to solve $f(x)=0$ on $[a,b]$. It is important that $f(a)$ and $f(b)$ have opposite signs.

Find the first positive root of the Bessel function $J_0$

We use the built-in MATLAB function besselj to solve

$$ f(x) = J_0(x) = 0 $$

disp('Find the first positive root of the Bessel function J0:')
J0 = @(x) besselj(0,x);
disp('Initial interval is [a,b] = [0,3]')
[x,iter] = bisect(J0,[0,3]);
disp(sprintf('The first positive root of J0 is x = %f.\nIt was computed in %d iterations.\n',x,iter))

Find the first positive root of the Bessel function J0:
Initial interval is [a,b] = [0,3]
x1 = 3.00000000000000
x2 = 3.00000000000000
x3 = 2.62500000000000
x4 = 2.43750000000000
x5 = 2.43750000000000
x6 = 2.43750000000000
x7 = 2.41406250000000
x8 = 2.41406250000000
x9 = 2.40820312500000
x10 = 2.40527343750000
x11 = 2.40527343750000
x12 = 2.40527343750000
x13 = 2.40490722656250
x14 = 2.40490722656250
x15 = 2.40490722656250
x16 = 2.40486145019531
x17 = 2.40483856201172
x18 = 2.40482711791992
x19 = 2.40482711791992
x20 = 2.40482711791992
x21 = 2.40482568740845
x22 = 2.40482568740845
x23 = 2.40482568740845
x24 = 2.40482568740845
x25 = 2.40482559800148
x26 = 2.40482559800148
x27 = 2.40482557564974
x28 = 2.40482556447387
x29 = 2.40482555888593
x30 = 2.40482555888593
x31 = 2.40482555888593
x32 = 2.40482555818744
x33 = 2.40482555783819
x34 = 2.40482555783819
x35 = 2.40482555775088
x36 = 2.40482555770723
x37 = 2.40482555770723
x38 = 2.40482555769631
x39 = 2.40482555769631
x40 = 2.40482555769631
x41 = 2.40482555769631
x42 = 2.40482555769631
x43 = 2.40482555769597
x44 = 2.40482555769580
x45 = 2.40482555769580
x46 = 2.40482555769580
x47 = 2.40482555769578
x48 = 2.40482555769578
x49 = 2.40482555769577
x50 = 2.40482555769577
x51 = 2.40482555769577
x52 = 2.40482555769577
The first positive root of J0 is x = 2.404826.
It was computed in 52 iterations.

Compute $\sqrt{2}$

We solve the equation

$$ f(x) = x^2 - 2 = 0$$

%pause
disp('Compute sqrt(2):')
sqr2 = @(x) x^2-2;
disp('Initial interval is [a,b] = [0,2]')
[x,iter] = bisect(sqr2,[0,2]);
disp(sprintf('sqrt(2) = %f was computed in %d iterations.',x,iter))

Compute sqrt(2):
Initial interval is [a,b] = [0,2]
x1 = 2.00000000000000
x2 = 1.50000000000000
x3 = 1.50000000000000
x4 = 1.50000000000000
x5 = 1.43750000000000
x6 = 1.43750000000000
x7 = 1.42187500000000
x8 = 1.42187500000000
x9 = 1.41796875000000
x10 = 1.41601562500000
x11 = 1.41503906250000
x12 = 1.41455078125000
x13 = 1.41430664062500
x14 = 1.41430664062500
x15 = 1.41424560546875
x16 = 1.41421508789063
x17 = 1.41421508789063
x18 = 1.41421508789063
x19 = 1.41421508789063
x20 = 1.41421508789063
x21 = 1.41421413421631
x22 = 1.41421365737915
x23 = 1.41421365737915
x24 = 1.41421365737915
x25 = 1.41421359777451
x26 = 1.41421356797218
x27 = 1.41421356797218
x28 = 1.41421356797218
x29 = 1.41421356424689
x30 = 1.41421356238425
x31 = 1.41421356238425
x32 = 1.41421356238425
x33 = 1.41421356238425
x34 = 1.41421356238425
x35 = 1.41421356238425
x36 = 1.41421356238425
x37 = 1.41421356238425
x38 = 1.41421356237697
x39 = 1.41421356237333
x40 = 1.41421356237333
x41 = 1.41421356237333
x42 = 1.41421356237333
x43 = 1.41421356237311
x44 = 1.41421356237311
x45 = 1.41421356237311
x46 = 1.41421356237311
x47 = 1.41421356237311
x48 = 1.41421356237310
x49 = 1.41421356237310
x50 = 1.41421356237310
x51 = 1.41421356237310
x52 = 1.41421356237310
x53 = 1.41421356237310
sqrt(2) = 1.414214 was computed in 53 iterations.


© Greg Fasshauer, IIT. Published with MATLAB® 7.11