Numerik, Statistik und Datenanalyse

p = [1 -3 2];                 % x^2 - 3x + 2
roots(p)'
polyval(p, 3)
polyder(p)
polyint(p)
conv([1 1], [1 -1])           % (x+1)(x-1)

x = 0:5;
y = [1.1 2.9 5.2 7.1 8.8 11.2];
koef = polyfit(x, y, 1)       % Gerade y = a*x + b
y_fit = polyval(koef, x);
fehler = sqrt(mean((y - y_fit).^2))
round(polyval(koef, 10) * 100) / 100

ans =

   2   1

ans = 2
ans =

   2  -3

ans =

   0.3333  -1.5000   2.0000        0

ans =

   1   0  -1

koef =

   2.0029   1.0429

fehler = 0.1499
ans = 21.070

x = [0 1 2 3];
y = [0 1 4 9];
interp1(x, y, 1.5)
interp1(x, y, [0.5 2.5], 'linear')
interp1(x, y, 1.5, 'spline')

f = @(x) sin(x);
integral(f, 0, pi)
trapz(linspace(0, pi, 1001), sin(linspace(0, pi, 1001)))
quad(@(x) exp(-x.^2), -5, 5)
sqrt(pi)

fzero(@(x) cos(x) - x, [0 1])
x_min = fminbnd(@(x) (x - 2).^2 + 3, 0, 5)
[x_opt, wert] = fminsearch(@(v) (v(1) - 1)^2 + (v(2) + 2)^2, [0 0]);
round(x_opt * 1000) / 1000
round(wert * 1e6) / 1e6

ans = 2.5000
ans =

   0.5000   6.5000

ans = 2.2500
ans = 2.0000
ans = 2.0000
ans = 1.7725
ans = 1.7725
ans = 0.7391
x_min = 2
ans =

   1  -2

ans = 0

[t, y] = ode45(@(t, y) -2 * y, [0 2], 1);
exakt = exp(-2 * t);
fehler = max(abs(y - exakt))
fehler < 1e-3
fprintf('y(2) = %.4f (exakt %.4f)\n', y(end), exp(-4));

% Eigenes Euler-Verfahren
h = 0.01;
ys = 1;
for k = 1:200
    ys = ys + h * (-2 * ys);
end
fprintf('Euler: %.4f\n', ys);

% Räuber-Beute-System (Lotka-Volterra)
lv = @(t, z) [z(1) * (1 - 0.5 * z(2)); z(2) * (-0.75 + 0.25 * z(1))];
[t, z] = ode45(lv, [0 10], [2; 1]);
fprintf('Schritte > 20: %d, max Beute %.2f\n', numel(t) > 20, max(z(:, 1)));

fehler = 2.2228e-06
ans = 1
y(2) = 0.0183 (exakt 0.0183)
Euler: 0.0176
Schritte > 20: 1, max Beute 6.11

A = [4 -2; 1 1];
[V, D] = eig(A);
diag(D)'
A * V(:, 1) - D(1, 1) * V(:, 1) < 1e-10
norm(A)
cond(A)
[Q, R] = qr(A);
round(Q * R)
[L, U, P] = lu(A);
round(P' * L * U)
pinv([1 2; 2 4])
null([1 1; 1 1])'
svd([3 0; 0 4])'
expm(zeros(2))
b = [2; 3];
x = A \ b
A * x

ans =

   3   2

ans =

  1
  1

ans = 4.4966
ans = 3.3699
ans =

   4  -2
   1   1

ans =

   4  -2
   1   1

ans =

   0.040000   0.080000
   0.080000   0.160000

ans =

   0.7071  -0.7071

ans =

   4   3

ans =

Diagonal Matrix

   1   0
   0   1

x =

   1.3333
   1.6667

ans =

   2
   3

rand('seed', 1);
randn('seed', 1);
x = [2 4 4 4 5 5 7 9];
mean(x), median(x), mode(x), std(x), var(x)
range(x)
prctile(x, [25 50 75])
histc(x, 0:2:10)
corr([1 2 3 4]', [2 4 5 9]')
corrcoef([1 2 3 4], [2 4 6 8])
cov([1 2 3 4], [1 2 3 4])
cumsum(x) ./ (1:numel(x))
numel(randperm(10))
all(randi(6, 1, 100) >= 1)
zufall_in_bereich = all(rand(1, 100) < 1)
1 / sqrt(2*pi)                      % Dichte der Standardnormalverteilung bei 0
0.5 * (1 + erf(1.96 / sqrt(2)))     % Verteilungsfunktion bei 1.96
nchoosek(5, 2)
factorial(5)
primes(30)
isprime(97)
gcd(48, 18)
lcm(4, 6)

ans = 5
ans = 4.5000
ans = 4
ans = 2.1381
ans = 4.5714
ans = 7
ans =

   4.0000   4.5000   6.0000

ans =

   0   1   5   1   1   0

ans = 0.9648
ans =

   1   1
   1   1

ans = 1.6667
ans =

   2.0000   3.0000   3.3333   3.5000   3.8000   4.0000   4.4286   5.0000

ans = 10
ans = 1
zufall_in_bereich = 1
ans = 0.3989
ans = 0.9750
ans = 10
ans = 120
ans =

    2    3    5    7   11   13   17   19   23   29

ans = 1
ans = 6
ans = 12