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eine manuelle implementation des Bisektionsverfahren. -Der Code ist jedoch (at the time of writing) nicht ausführbar aufgrund von \texttt{IndentationErrors} - Das Bisektionsverfahren konvergiert linear und kann nur für Funktionen verwenden, bei welchen die Nullstellen auf beiden Seiten jeweils ungleiche Vorzeichen haben. Für jeden Iterationsschritt ermitteln wir die Mitte des Intervalls und berechnen die Funktionswerte an den Rändern, wie auch dem Mittelpunkt. Dann ersetzen wir den Rand des Intervalls, dessen Funktionswert dasselbe Vorzeichen hat, wie der Funktionswert des Mittelpunkts. + +\innumpy Lässt sich das Intervallhalbierungsverfahren (Bisektion) leicht direkt implementieren: + +\begin{code}{python} +def bisection_method(f, a: float, b: float, tol=1e-12, maxIter=100): + """ Bisection method on f using initial bounds a,b """ + if a > b: a, b = b, a + fa = f(a); fb = f(b) + if fa*fb > 0: raise ValueError("f(a) & f(b) must have different signs for bisection") + + sgn_fb = 1 + if fa > 0: sgn_fb = -1 + x = 0.5 * (a + b) + + iter = 1 + while (b-a > tol and a 0: b = x + else: a = x + + x = 0.5 * (a + b) + iter += 1 + + return x, iter +\end{code} \ No newline at end of file diff --git a/semester3/numcs/parts/03_zeros/04_newton-one-d.tex b/semester3/numcs/parts/03_zeros/04_newton-one-d.tex index 7d47de5..208c701 100644 --- a/semester3/numcs/parts/03_zeros/04_newton-one-d.tex +++ b/semester3/numcs/parts/03_zeros/04_newton-one-d.tex @@ -15,3 +15,17 @@ Beim Newtonverfahren verwendet man für jeden Iterationsschritt die lineare Funk \drmvspace falls $F(x^*) = 0$ und $F^(x^*) \neq 0$ + +\newpage + +\innumpy Ist das Newton-Verfahren mit sehr wenig code implementierbar: + +\begin{code}{python} +def newton_method(f, df, x: float, tol=1e-12, maxIter=100): + """ Use Newton's method to find zeros, requires derivative of f """ + iter = 0 + while (np.abs(df(x)) > tol and iter < maxIter): + x -= f(x)/df(x); iter += 1 + + return x, iter +\end{code} diff --git a/semester3/numcs/parts/03_zeros/05_sectant-method.tex b/semester3/numcs/parts/03_zeros/05_sectant-method.tex index 29ad1e6..38b95e9 100644 --- a/semester3/numcs/parts/03_zeros/05_sectant-method.tex +++ b/semester3/numcs/parts/03_zeros/05_sectant-method.tex @@ -6,3 +6,20 @@ Dann ist ein Schritt: \begin{align*} \tilde{F}(x) = F(x^{(k)}) + q^{(k)} (x - x^{(k)}) \Longrightarrow x^{(k + 1)} := x^{(k)} - \frac{F(x^{(k)})}{q^{(k)}}, \smallhspace \text{ falls } q^{(k)} \neq 0 \end{align*} + +\innumpy Das Sekanten-Verfahren lässt sich im Prinzip ähnlich implementieren wie Newton: + +\begin{code}{python} +def secant_method(f, x0: float, x1: float, tol=1e-12, maxIter=100): + """ Use secant method, which approximates the derivative """ + f0 = f(x0) + for i in range(maxIter): + fn = f(x1) + secant = fn * (x1-x0) / (fn - f0) # Approximate derivative via secant + x0 = x1; x1 -= secant + + if np.abs(secant) < tol: return x1, i + else: f0 = fn + + return None, maxIter +\end{code}