octave-3.0.3:3> x0 = [1; 2; 3]
x0 =

   1
   2
   3

octave-3.0.3:4> size(x0)
ans =

   3   1

octave-3.0.3:5> M = @(alpha) [ cos(alpha), -sin(alpha); sin(alpha), cos(alpha) ]
M =

@(alpha) [cos(alpha), -sin(alpha); sin(alpha), cos(alpha)]

octave-3.0.3:6> M(pi)
ans =

  -1.0000e+00  -1.2246e-16
   1.2246e-16  -1.0000e+00

octave-3.0.3:7> pi
ans =  3.1416
octave-3.0.3:8> format long e
octave-3.0.3:9> pi
ans =  3.14159265358979e+00
octave-3.0.3:10> M(pi)
ans =

  -1.00000000000000e+00  -1.22464679914735e-16
   1.22464679914735e-16  -1.00000000000000e+00

octave-3.0.3:11> f = @(x) M(pi) * x
f =

@(x) M (pi) * x

octave-3.0.3:12> x0 = [ 1; 0 ]
x0 =

   1.00000000000000e+00
   0.00000000000000e+00

octave-3.0.3:13> format short
octave-3.0.3:14> x0
x0 =

   1
   0

octave-3.0.3:15> X = euler(f, x0, 1, 2*pi)
X =

   1.00000   0.10240   0.01049   0.00107   0.00011   0.00001   0.00000   0.00000
   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000   0.00000

octave-3.0.3:16> plot (X(1,:), X(2,:))
octave-3.0.3:17> 
octave-3.0.3:17> X = euler(f, x0, 1e-3, 2*pi);
octave-3.0.3:18> plot (X(1,:), X(2,:))
octave-3.0.3:19> f(x0)
ans =

  -1.0000e+00
   1.2246e-16

octave-3.0.3:20> x0
x0 =

   1
   0

octave-3.0.3:21> M(pi)
ans =

  -1.0000e+00  -1.2246e-16
   1.2246e-16  -1.0000e+00

octave-3.0.3:22> M = @(alpha) [ cos(alpha), -sin(alpha); sin(alpha), cos(alpha) ]
M =

@(alpha) [cos(alpha), -sin(alpha); sin(alpha), cos(alpha)]

octave-3.0.3:23> M(pi)
ans =

  -1.0000e+00  -1.2246e-16
   1.2246e-16  -1.0000e+00

octave-3.0.3:24> f = @(x) M(pi/2) * x
f =

@(x) M (pi / 2) * x

octave-3.0.3:25> X = euler(f, x0, 1e-3, 2*pi);
octave-3.0.3:26> plot (X(1,:), X(2,:))
octave-3.0.3:27> axis equal
octave-3.0.3:28> 
octave-3.0.3:28> 
octave-3.0.3:28> X = euler(f, x0, 1e-2, 2*pi);
octave-3.0.3:29> 
octave-3.0.3:29> plot (X(1,:), X(2,:))
octave-3.0.3:30> X = euler(f, x0, 1e-1, 2*pi);
octave-3.0.3:31> plot (X(1,:), X(2,:))
octave-3.0.3:32> X = euler(f, x0, 1, 2*pi);
octave-3.0.3:33> plot (X(1,:), X(2,:))
octave-3.0.3:34> X = euler(f, x0, 1, 10*pi);
octave-3.0.3:35> plot (X(1,:), X(2,:))
octave-3.0.3:36> X = euler(f, x0, 1e-10, 10*pi);
octave-3.0.3:36> X = euler(f, x0, 1e-5, 10*pi);
octave-3.0.3:37> tic
octave-3.0.3:38> X = euler(f, x0, 1e-2, 10*pi);
octave-3.0.3:39> toc
Elapsed time is 9.9034 seconds.
octave-3.0.3:40> tic
octave-3.0.3:41> toc
Elapsed time is 0.59786 seconds.
octave-3.0.3:42> tic
octave-3.0.3:43> toc
Elapsed time is 4.1441 seconds.