Merge branch 'main' of https://gitea.fhgr.ch/zimmersandro/cds-404

src/FHGR_CDS_Nume_E_PYNU_SVD_Bildkompression.py

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+# Python
+# AUEM
+# 2026-05-06
+# Begin
+# --------------------------------------------------------------------------------------
+# Python initialisieren:
+from PIL import Image, ImageOps
+import matplotlib.pyplot as pl
+import numpy as np
+
+# Parameter:
+N = 10
+sc_r = 100
+pr = 3
+fig = 1
+# Import:
+img = Image.open("./Bild.JPG")
+img_sw = ImageOps.grayscale(img)
+A = np.asarray(img_sw)
+# Berechnungen;
+[U, S, Vt] = np.linalg.svd(A)
+U_red = U[:, :N]
+S_red = S[:N]
+Vt_red = Vt[:N, :]
+A_red = U_red @ np.diag(S_red) @ Vt_red
+G = A.size
+G_red = U_red.size + S_red.size + Vt_red.size
+r = G_red / G
+# Anzeige der Bilder:
+fh = pl.figure(fig)
+pl.imshow(A, cmap="gray")
+fig = fig + 1
+fh = pl.figure(fig)
+pl.imshow(A_red, cmap="gray")
+# Ausgabe:
+print("--------------------------------------------------")
+print(__file__)
+print("--------------------------------------------------")
+print(f"Groesse original:        G = {G}")
+print(f"Groesse komprimiert: G_red = {G_red}")
+print(f"Datenkompression:        r = {r*sc_r:#.{pr}g}%")
+print("--------------------------------------------------")
+# --------------------------------------------------------------------------------------
+# End

src/klausurvorbereitung/frobenius_norm.py

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+"""
+Frobenius-Norm ||A||F
+
+A Matrix:
+
+[4 9]
+[2 7]
+"""
+
+import numpy as np
+import sympy as sp
+
+# Numpy
+A = np.array([[4, 9], [2, 7]])
+n = np.linalg.norm(A, ord="fro")
+print(n.round(4))
+
+# Sympy
+A = sp.Matrix([[4, 9], [2, 7]])
+n = A.norm(ord="fro")
+print(n)
+
+"""
+Ausgabe:
+
+12.2474
+5*sqrt(6)
+"""

src/klausurvorbereitung/integration_simpson.py

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+"""
+Simpson-Regel
+
+Integration der funktion:
+    2
+I = | (x + sin(x))^(2/3)) dx
+    0
+"""
+
+import numpy as np
+import scipy.integrate as ig
+
+# Parameter
+x_0 = 0
+x_E = 2
+n = 10
+N = 201
+pr = 6
+
+f = lambda x: np.sqrt(x + np.sin(x))
+
+# Berechnung
+for k in range(0, n):
+    x_data = np.linspace(x_0, x_E, N)
+    y_data = f(x_data)
+    I = ig.simpson(y=y_data, x=x_data)
+    print(f"I = {I:#.16g} | N = {N:g}")
+    N *= 2
+print(f"I = {I:#.{pr}g}")
+
+"""
+Ausgabe:
+
+I = 2.490070783046884 | N = 201
+I = 2.490260152603558 | N = 402
+I = 2.490326897146905 | N = 804
+I = 2.490350466364332 | N = 1608
+I = 2.490358796310514 | N = 3216
+I = 2.490361741357393 | N = 6432
+I = 2.490362782708113 | N = 12864
+I = 2.490363150933651 | N = 25728
+I = 2.490363281138151 | N = 51456
+I = 2.490363327177379 | N = 102912
+I = 2.49036
+"""

src/klausurvorbereitung/integration_trapez.py

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+"""
+Trapez-Regel
+
+Integration der funktion:
+    2
+I = | (x + sin(x))^(2/3)) dx
+    0
+"""
+
+import numpy as np
+import scipy.integrate as ig
+
+# Parameter
+x_0 = 0
+x_E = 2
+n = 10
+N = 201
+pr = 6
+
+f = lambda x: np.sqrt(x + np.sin(x))
+
+# Berechnung
+for k in range(0, n):
+    x_data = np.linspace(x_0, x_E, N)
+    y_data = f(x_data)
+    I = ig.trapezoid(y=y_data, x=x_data)
+    print(f"I = {I:#.16g} | N = {N:g}")
+    N *= 2
+print(f"I = {I:#.{pr}g}")
+
+"""
+Ausgabe:
+
+I = 2.490070783046884 | N = 201
+I = 2.490260152603558 | N = 402
+I = 2.490326897146905 | N = 804
+I = 2.490350466364332 | N = 1608
+I = 2.490358796310514 | N = 3216
+I = 2.490361741357393 | N = 6432
+I = 2.490362782708113 | N = 12864
+I = 2.490363150933651 | N = 25728
+I = 2.490363281138151 | N = 51456
+I = 2.490363327177379 | N = 102912
+I = 2.49036
+"""

src/klausurvorbereitung/lr_zerlegung.py

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+"""
+LR-Zerlegung
+
+Matrix A:
+
+[3, 2]
+[1, 4]
+
+"""
+
+import numpy as np
+import scipy as sc
+
+# Parameter
+A = np.array([[3, 2], [1, 4]])
+pr = 3
+
+[P, L, R] = sc.linalg.lu(A)
+
+with np.printoptions(precision=pr):
+    print(f"P = \n{P}\n\nL = \n{L}\n\nR = \n{R}")
+
+"""
+Ausgabe:
+
+P = 
+[[1. 0.]
+ [0. 1.]]
+
+L = 
+[[1.    0.   ]
+ [0.333 1.   ]]
+
+R = 
+[[3.    2.   ]
+ [0.    3.333]]
+"""

src/klausurvorbereitung/maclaurin_entwicklungen.py

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+#%%
+"""
+Maclaurin Entwicklungen
+"""
+
+import IPython.display as dp
+import sympy as sp
+
+# Konfig
+sp.init_printing()
+x = sp.symbols("x")
+
+# Parameter
+n = 2
+F = sp.sqrt(1 + x)
+
+# Berechnungen
+T = sp.series(F, x, 0, n+2)
+
+# Ausgabe
+dp.display(F)
+dp.display(T)
+
+# Parameter
+n = 2
+F = sp.log(sp.sqrt(sp.cos(x)))
+
+# Berechnungen
+T = sp.series(F, x, 0, n+3)
+
+# Ausgabe
+dp.display(F)
+dp.display(T)
+
+# Parameter
+n = 2
+F = 1 / (1 + 2 * sp.sin(x))
+
+# Berechnungen
+T = sp.series(F, x, 0, n+1)
+
+# Ausgabe
+dp.display(F)
+dp.display(T)
+# %%

src/klausurvorbereitung/qr_zerlegung.py

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+"""
+QR-Zerlegung
+
+A Matrix:
+
+[3 1]
+[6 9]
+"""
+
+import numpy as np
+import scipy as sc
+
+# Parameter
+A = np.array([[3, 1], [6, 9]])
+pr = 3
+
+# Berechnung
+[Q, R] = sc.linalg.qr(A)
+
+# Ausgabe
+with np.printoptions(precision=pr):
+    print(f"Q = \n{Q}\n\n R = \n{R}")
+
+"""
+Ausgabe:
+
+Q = 
+[[-0.447 -0.894]
+ [-0.894  0.447]]
+
+ R = 
+[[-6.708 -8.497]
+ [ 0.     3.13 ]]
+"""

src/klausurvorbereitung/regression_polyfit.py

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+# %%
+# Polyfit tutorial mit Plot
+# Python initialisieren
+import matplotlib.pyplot as plt
+import numpy as np
+
+# Parameter
+x_0 = 1
+x_E = 7.0
+y_a = -2
+y_b = 3
+dg = 1
+pr = 3
+lw = 3
+fig = 1
+tc_x = np.r_[x_0 : x_E + 0.5 : 0.5]
+tc_y = np.r_[y_a : y_b + 0.5 : 0.5]
+
+# Daten
+x_data = np.r_[x_0 : x_E + 1]
+y_data = np.array([2.5, 2.2, 1.5, 1.0, 0.6, -0.3, -1.4])
+
+# Berechnungen
+p = np.polyfit(x_data, y_data, dg)
+g_data = np.polyval(p, x_data)
+
+# Ausgabe
+print(60 * "*")
+print("__file__")
+print(60 * "*")
+print(f"Steigung:           m = {p[0]:#.{pr}g}")
+print(f"y-Achsenabschnitt:  q = {p[1]:#.{pr}g}")
+
+# Plot
+fh = plt.figure(fig)
+plt.plot(x_data, g_data, linewidth=lw)
+plt.plot(x_data, y_data, "o", linewidth=lw)
+plt.xlabel(r"$x$")
+plt.ylabel(r"$y$")
+plt.xticks(tc_x)
+plt.yticks(tc_y)
+plt.grid(visible=True)
+plt.axis("image")

src/klausurvorbereitung/spektral_norm.py

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+"""
+Spektral-Norm ||A||2
+
+A Matrix:
+
+[4 9]
+[2 7]
+"""
+
+import numpy as np
+import sympy as sp
+
+# Numpy
+A = np.array([[4, 9], [2, 7]])
+n = np.linalg.norm(A, ord=2)
+print(n.round(4))
+
+# Sympy
+A = sp.Matrix([[4, 9], [2, 7]])
+n = A.norm(ord=2)
+print(n)
+
+"""
+Ausgabe:
+
+12.2201
+sqrt(5*sqrt(221) + 75)
+"""

src/klausurvorbereitung/svd_zerlegung.py

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+"""
+Singulärwertzerlegung SVD
+
+A Matrix:
+
+[4  6]
+[3 -8]
+"""
+
+import numpy as np
+import scipy as sc
+
+# Parameter
+A = np.array([[4.0, 6.0], [3.0, -8.0]])
+pr = 3
+
+# Berechnung
+[U, S, Vt] = sc.linalg.svd(A)
+
+# Ausgabe
+with np.printoptions(precision=pr):
+    print(f"U = \n{U}\n\nS = \n{S}\n\nV = \n{Vt.T}")
+
+"""
+Ausgabe:
+
+U = 
+[[-0.6 -0.8]
+ [ 0.8 -0.6]]
+
+S = 
+[10.  5.]
+
+V = 
+[[-0. -1.]
+ [-1. -0.]]
+"""
+
+# S = Vektor mit den Singulärwerten (o1, o2)
+# Nicht die ganze Matrix:
+# [10 0]
+# [0  5]