Heronverfahren, Näherung mit Rechtecken
Author: Prof. Dr. DörteHaftendorn
Date: 26 Mai 2013

 \sqrt{ r } 

Heronverfahren mit Rechtecken

Ha 11 und 13

a_{n+1}= \frac{ 1 }{ 2}(a_n +  \frac{ r  }{a_n } ) 

Rechteck 2

Rechteck 1

Rechteck 3

Rechteck 4

Schritt 1

Schritt 2

Schritt 3

Schritt 4

Liste anzeigen

r = 14 (slider from 0 to 16, step 1)
A = Point(xAxis)
a = OrthogonalLine(A, xAxis)
f(x) = r / x
B = Intersect(f, a, (14.002311350036457, 0.9998349308211568))
b = OrthogonalLine(B, yAxis)
C = (-2, -2)
D = Intersect(xAxis, yAxis)
c = Line(D, C)
E = Intersect(c, b)
F = Midpoint(B, E)
d = OrthogonalLine(F, xAxis)
G = Intersect(f, d, (7.501073140428807, 1.866399612149319))
e = OrthogonalLine(G, yAxis)
H = Intersect(c, e)
I = Midpoint(H, G)
g = OrthogonalLine(I, xAxis)
J = Intersect(f, g, (4.683736376289063, 2.989066607350826))
h_1 = OrthogonalLine(J, yAxis)
K = Intersect(c, h_1)
L = Midpoint(K, J)
i = OrthogonalLine(L, e)
M = Intersect(f, i, (3.8364014918199443, 3.6492530903898075))
j = Segment(A, B)
k = Segment(E, B)
l = Segment(F, G)
m = Segment(H, G)
n = Segment(I, J)
p = Segment(J, K)
q = Segment(L, M)
Text1: f + ""
N = Intersect(e, yAxis)
O = Intersect(d, xAxis)
P = Intersect(xAxis, yAxis)
Vieleck1, n_1, d_1, o, g_1 = Polygon(N, D, O, G)
s = true
Q = Intersect(h_1, yAxis)
R = Intersect(g, xAxis)
S = Intersect(b, yAxis)
Vieleck2, d_2, a_1, b_1, s_1 = Polygon(D, A, B, S)
t = true
Vieleck3, d_3, r_1, j_1, q_1 = Polygon(D, R, J, Q)
u = true
c_1 = OrthogonalLine(M, yAxis)
T = Intersect(c_1, yAxis)
U = Intersect(i, xAxis)
Vieleck4, d_4, u_1, m_1, t_1 = Polygon(D, U, M, T)
v = true
w = true
z = true
e_1 = true
f_1 = true
wr: sqrt(r)
Text3: "=" + wr + ""
mi: (m_1 + u_1) / 2
Text7: "" + (LaTeX[u_1]) + " ≤ Wurzel(r) ≤ " + (LaTeX[m_1]) + "."
h(x) = 1 / 2 * (x + f(x))
liste: {x(A), h(x(A)), h(h(x(A))), h(h(h(x(A)))), h(h(h(h(x(A))))), h(h(h(h(h(x(A)))))), h(h(h(h(h(h(x(A)))))))}
Text8: "" + liste + ""
i_1 = true