Initial

uebung_01/exercise_1.py

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+import numpy as np
+
+for n in [3,5,10,100,1000]:
+    A = np.zeros((n,n))
+    x = np.zeros((n,1))
+    y_c = np.zeros((n,1))
+
+    for i in range(0,n):
+        x[i] = i+1
+        y_c[i] = (i+1) * n;
+
+        for j in range(0,n):
+            A[i,j] =(i+1) / (j+1)
+
+    y = A@x
+    
+    if n == 3 or n == 5:
+        print(x)
+        print(A)
+        print(y)
+
+    print("Fuer n =",n,":",np.linalg.norm(y - y_c))

uebung_01/exercise_2a.py

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+import numpy as np
+import matplotlib.pyplot as plt
+
+# x = np.linspace(0, 2 * np.pi, 100)
+x_1 = np.arange(0, 2 * np.pi, 1)
+x_0_5 = np.arange(0, 2 * np.pi, 0.5)
+x_0_1 = np.arange(0, 2 * np.pi, 0.1)
+
+y_s_1 = np.sin(x_1)
+y_s_0_5 = np.sin(x_0_5)
+y_s_0_1 = np.sin(x_0_1)
+y_c = np.cos(x_1)
+
+plt.plot(x_1,y_s_1,"purple", label="sin(x) with stepsize 1")
+plt.plot(x_0_5,y_s_0_5,"pink", label="sin(x) with stepsize 0.5")
+plt.plot(x_0_1,y_s_0_1,"red", label="sin(x) with stepsize 0.1")
+plt.plot(x_1,y_c,"green", label="cos(x)")
+
+plt.title("A2.a) Sine and Cosine")
+plt.xlabel("x")
+plt.ylabel("y")
+plt.legend()
+
+plt.show()   

uebung_01/exercise_2b.py

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+import numpy as np
+import matplotlib.pyplot as plt
+
+plt.rcParams['text.usetex'] = True
+
+# x = np.linspace(0, 2 * np.pi, 100)
+x = np.arange(-2, 2, 0.1)
+
+f = np.exp(-x**2)
+g = np.sin(x**2)
+h = np.sin(1 / (x**3 + 9))
+
+plt.plot(x, f, "purple", label=r"$f(x) = \exp(-x^2)$")
+plt.plot(x, g, "red", label=r"$g(x) = \sin(x^2)$")
+plt.plot(x, h, "green", label=r"$h(x) = \frac{1}{x^3 + 9}$")
+
+plt.title("A2.b)")
+plt.xlabel("x")
+plt.ylabel("y")
+plt.legend()
+
+plt.show()   

uebung_01/exercise_3.py

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+import numpy as np
+import matplotlib.pyplot as plt
+
+def runge(x):
+    return 1 / (1 + 25 * x**2)
+
+def divided_differences(x, y):
+    n = len(y)
+    a = np.zeros((n,n))
+    
+    a[:,0] = y
+    
+    for i in range(1,n):
+        for j in range(1,i+1):
+            a[i,j] = (a[i,j-1] - a[i-1,j-1]) / (x[i] - x[i-j]) # (links daneben - links darüber) / (x[zeile] - x[zeile - spalte])
+    
+    return a
+    
+def newton_interpolation(a, data, x):
+    n = len(a)
+    
+    p = np.zeros(len(x))
+    
+    for i in range(1,n+1):
+        p = a[n-i] + (x - data[n-i]) * p # Horner Schema; '-' and '*' are overloaded for numpy arrays: so at least one argument 'data' or 'x' has to be numpy array 
+    
+    return p
+
+####################################################################################################################################################################
+n = 12
+
+x = np.linspace(-1, 1, 200)
+x_e = np.linspace(-1, 1, n) # equidistant grid points
+
+# Chebyshev grid points
+x_c = np.zeros(n)
+for i in range(0,n):
+    x_c[i] = np.cos((2 * i + 1) * np.pi / (2 * n))
+    
+f = runge(x)
+y_e = runge(x_e) # values for grid points for interpolation with equidistant grid points
+y_c = runge(x_c) # values for grid points for interpolation with Chebyshev grid points
+
+# Interpolation with equidistant grid points and evaluation of interpolated values at x
+a_e = np.diag(divided_differences(x_e, y_e))
+p_e = newton_interpolation(a_e, x_e, x)
+
+# Interpolation with Chebyshev grid points and evaluation of interpolated values at x
+a_c = np.diag(divided_differences(x_c, y_c))
+p_c = newton_interpolation(a_c, x_c, x)
+       
+       
+# Plotting of Runge and the two interpolated polynomials
+plt.rcParams['text.usetex'] = True
+
+plt.plot(x, f, label=r"$f(x) = \frac{1}{1 + 25 x^2}$")
+plt.plot(x, p_e, label=r"$p_n(x)$ equidistant")
+plt.plot(x, p_c, label=r"$p_n(x)$ Chebyshev")
+plt.plot([], [], ' ', label="$n = {}$".format(n))
+
+plt.title("A3)")
+plt.xlabel("x")
+plt.ylabel("y")
+plt.legend()
+
+plt.show()
+
+# braendel@math.tu-freiberg.de