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a/semester3/numcs/parts/01_interpolation/00_polynomial/03_lagrange-and-barzycentric-formula.tex b/semester3/numcs/parts/01_interpolation/00_polynomial/03_lagrange-and-barzycentric-formula.tex index a49edf0..55c354f 100644 --- a/semester3/numcs/parts/01_interpolation/00_polynomial/03_lagrange-and-barzycentric-formula.tex +++ b/semester3/numcs/parts/01_interpolation/00_polynomial/03_lagrange-and-barzycentric-formula.tex @@ -9,37 +9,37 @@ \begin{definition}[]{Lagrange Polynome} Für Knoten (auch gennannt Stützstellen) $x_0, x_1, \ldots, x_n \in \R$ definieren wir die Lagrange-Polynome für $n = \text{Anzahl Stützstellen}$, also haben wir $n - 1$ Brüche, da wir eine Iteration überspringen, weil bei dieser $j = i$ ist: - \begin{align*} - l_i(x) = \prod_{j = 0 \atop j \neq i}^n \frac{x - x_j}{x_i - x_j} - \end{align*} + \begin{align*} + l_i(x) = \prod_{\elementstack{j = 0}{j \neq i}}^n \frac{x - x_j}{x_i - x_j} + \end{align*} \end{definition} Falls $j = i$ im Produkt, so überspringt $j$ diese Zahl. \inlineex Seien $x_0, x_1, x_2$ die Stützstellen für die Lagrange-Polynome (mit $n = 2$): \begin{align*} - l_0(x) & = \frac{x - x_1}{x_0 - x_1} \cdot \frac{x - x_2}{x_0 - x_2} & - l_1(x) & = \frac{x - x_0}{x_1 - x_0} \cdot \frac{x - x_2}{x_1 - x_2} & - l_2(x) & = \frac{x - x_0}{x_2 - x_0} \cdot \frac{x - x_1}{x_2 - x_1} + l_0(x) & = \frac{x - x_1}{x_0 - x_1} \cdot \frac{x - x_2}{x_0 - x_2} & + l_1(x) & = \frac{x - x_0}{x_1 - x_0} \cdot \frac{x - x_2}{x_1 - x_2} & + l_2(x) & = \frac{x - x_0}{x_2 - x_0} \cdot \frac{x - x_1}{x_2 - x_1} \end{align*} \begin{theorem}[]{Lagrange-Interpolationsformel} - Die Lagrange-Polynome $l_i$ zu den Stützstellen $(x_0, y_0), \ldots, (x_n, y_n)$ bilden eine Basis der Polynome $\mathcal{P}_n$ und es gilt: - \begin{align*} - p(x) = \sum_{i = 0}^{n} y_i l_i(x) \text{ mit } l_i(x) = \prod_{j \neq i} \frac{x - x_j}{x_i - x_j} - \end{align*} + Die Lagrange-Polynome $l_i$ zu den Stützstellen $(x_0, y_0), \ldots, (x_n, y_n)$ bilden eine Basis der Polynome $\mathcal{P}_n$ und es gilt: + \begin{align*} + p(x) = \sum_{i = 0}^{n} y_i l_i(x) \text{ mit } l_i(x) = \prod_{j \neq i} \frac{x - x_j}{x_i - x_j} + \end{align*} \end{theorem} \fancyremark{Eigenschaften der Lagrange-Polynome} \rmvspace \begin{multicols}{2} - \begin{enumerate} - \item $l_i(x_j) = 0 \smallhspace \forall j \neq i$ - \item $l_i(x_i) = 1 \smallhspace \forall i$ - \item $\deg(l_i) = n \smallhspace \forall i$ - \item $\sum_{k = 0}^{n} l_k(x) = 1 \text{ und } \sum_{k = 0}^{n} l_k^{(m)}(x) = 0 \text{ für } m > 0$ - \end{enumerate} + \begin{enumerate} + \item $l_i(x_j) = 0 \smallhspace \forall j \neq i$ + \item $l_i(x_i) = 1 \smallhspace \forall i$ + \item $\deg(l_i) = n \smallhspace \forall i$ + \item $\sum_{k = 0}^{n} l_k(x) = 1 \text{ und } \sum_{k = 0}^{n} l_k^{(m)}(x) = 0 \text{ für } m > 0$ + \end{enumerate} \end{multicols} Da eine Implementation, welche direkt auf den Lagrange-Polynomen basiert, eine Laufzeit von $\tco{n^3}$ hätte, suchte man nach einer besseren Methode. @@ -52,44 +52,44 @@ Man berechnet die baryzentrischen Gewichte $\lambda_k$ folgendermassen: oder das ganze mithilfe von Numpy: \begin{code}{python} def barycentric_weights(x: np.ndarray) -> np.ndarray: - n = len(x) - # Initialize to zeros - barweight = np.ones(n) - for k in range(n): - # Vectorized differences between $x_k$ and all $x$s - differences = x[k] - x - # Remove the $k$-th element (and handle edge cases for $k = 0$ and $k = n - 1$) - if k < n - 1 and k > 0: - diff_processed = np.concatenate((differences[:k], differences[(k + 1) :])) - barweight[k] = 1 / np.prod(diff_processed) - elif k == 0: - barweight[k] = 1 / np.prod(differences[1:]) - else: - barweight[k] = 1 / np.prod(differences[:k]) - return barweight + n = len(x) + # Initialize to zeros + barweight = np.ones(n) + for k in range(n): + # Vectorized differences between $x_k$ and all $x$s + differences = x[k] - x + # Remove the $k$-th element (and handle edge cases for $k = 0$ and $k = n - 1$) + if k < n - 1 and k > 0: + diff_processed = np.concatenate((differences[:k], differences[(k + 1) :])) + barweight[k] = 1 / np.prod(diff_processed) + elif k == 0: + barweight[k] = 1 / np.prod(differences[1:]) + else: + barweight[k] = 1 / np.prod(differences[:k]) + return barweight \end{code} Gleiche funktion, etwas kürzer: \begin{code}{python} def barycentric_weights(x: np.ndarray) -> np.ndarray: - n = len(x) - w = np.ones(n) # = barweight - # Compute the (non-inverted) product, avoiding case (x[i] - x[i]) = 0 - for i in range(0, n, 1): - if (i-1 > 0): w[0:(i-1)] *= (x[0:(i-1)] - x[i]) - if (i+1 < n): w[i+1:n] *= (x[i+1:n] - x[i]) - # Invert all at once - return 1/w + n = len(x) + w = np.ones(n) # = barweight + # Compute the (non-inverted) product, avoiding case (x[i] - x[i]) = 0 + for i in range(0, n, 1): + if (i-1 > 0): w[0:(i-1)] *= (x[0:(i-1)] - x[i]) + if (i+1 < n): w[i+1:n] *= (x[i+1:n] - x[i]) + # Invert all at once + return 1/w \end{code} Mit dem können wir dann ein Polynom mit der baryzentrischen Interpolationsformel interpolieren: \numberingOff \begin{formula}[]{Baryzentrische Interpolationsformel} - \vspace{-1.5pc} - \begin{align*} - p(x) = \frac{\displaystyle \sum_{k = 0}^{n} \frac{\lambda_k}{x - x_k} y_k}{\displaystyle \sum_{k = 0}^{n} \frac{\lambda_k}{x - x_k}} - \end{align*} + \vspace{-1.5pc} + \begin{align*} + p(x) = \frac{\displaystyle \sum_{k = 0}^{n} \frac{\lambda_k}{x - x_k} y_k}{\displaystyle \sum_{k = 0}^{n} \frac{\lambda_k}{x - x_k}} + \end{align*} \end{formula} \numberingOn @@ -102,25 +102,25 @@ Eine weitere Anwendung der Formel ist als Ausganspunkt für die Spektralmethode \begin{code}{python} def interp_barycentric( - data_point_x: np.ndarray, - data_point_y: np.ndarray, - barweight: np.ndarray, - x: np.ndarray + data_point_x: np.ndarray, + data_point_y: np.ndarray, + barweight: np.ndarray, + x: np.ndarray ): - p_x = np.zeros_like(x) - n = data_point_x.shape[0] + p_x = np.zeros_like(x) + n = data_point_x.shape[0] - for i in range(x.shape[0]): - # Separate sums to divide in the end - upper_sum = 0 - lower_sum = 0 - for k in range(n): - frac = barweight[k] / (x[i] - data_point_x[k]) - upper_sum += frac * data_point_y[k] - lower_sum += frac - p_x[i] = upper_sum / lower_sum + for i in range(x.shape[0]): + # Separate sums to divide in the end + upper_sum = 0 + lower_sum = 0 + for k in range(n): + frac = barweight[k] / (x[i] - data_point_x[k]) + upper_sum += frac * data_point_y[k] + lower_sum += frac + p_x[i] = upper_sum / lower_sum - return p_x + return p_x \end{code} @@ -130,14 +130,14 @@ Eine weitere Anwendung der Formel ist als Ausganspunkt für die Spektralmethode Falls an den Stützstellen $x_i$ durch beispielsweise ungenaue Messungen unpräzise Werte $\tilde{y_i}$ haben, so entsteht logischerweise auch ein unpräzises Polynom $\tilde{p}(x)$. Verglichen in der Lagrange-Basis zum korrekten Interpolationspolynom $p(x)$ ergibt sich folgender Fehler: \begin{align*} - |p(x) - \tilde{p}(x)| = \left| \sum_{i = 0}^{n} (y_i - \tilde{y_i}) l_i(x) \right| \leq \max_{i = 0, \ldots, n} |y_i - \tilde{y_i}| \cdot \sum_{i = 0}^{n} |l_i(x)| + |p(x) - \tilde{p}(x)| = \left| \sum_{i = 0}^{n} (y_i - \tilde{y_i}) l_i(x) \right| \leq \max_{i = 0, \ldots, n} |y_i - \tilde{y_i}| \cdot \sum_{i = 0}^{n} |l_i(x)| \end{align*} \fancydef{Lebesgue-Konstante} Zu den Stützstellen $x_0, \ldots, x_n$ im Intervall $[a, b]$ ist sie definiert durch \rmvspace \begin{align*} - \Lambda_n = \max_{x \in [a, b]} \sum_{i = 0}^{n} |l_i(x)| + \Lambda_n = \max_{x \in [a, b]} \sum_{i = 0}^{n} |l_i(x)| \end{align*} @@ -145,11 +145,11 @@ Verglichen in der Lagrange-Basis zum korrekten Interpolationspolynom $p(x)$ ergi \fancytheorem{Auswirkung von Messfehlern} Es gilt (wenn $\Lambda_n$ die beste Lebesgue-Konstante für die Ungleichung ist): \rmvspace \begin{align*} - \max_{x \in [a, b]} |p(x) - \tilde{p}(x)| \leq \Lambda_n \max_{i = 0, \ldots, n} |y_i - \tilde{y_i}| + \max_{x \in [a, b]} |p(x) - \tilde{p}(x)| \leq \Lambda_n \max_{i = 0, \ldots, n} |y_i - \tilde{y_i}| \end{align*} \begin{theorem}[]{Fehler} - Sei $f : [a, b] \rightarrow \R$ und $p$ das Interpolationspolynom zu $f$. Seien $x_0, \ldots, x_n$ die Stützstellen, dann gilt: + Sei $f : [a, b] \rightarrow \R$ und $p$ das Interpolationspolynom zu $f$. Seien $x_0, \ldots, x_n$ die Stützstellen, dann gilt: \rmvspace \begin{align*} ||f(x) - p(x)||_{\infty} = \max_{x \in [a, b]}|f(x) - p(x)| \leq (1 + \Lambda_n) \min_{q \in \mathcal{P}_n} \max_{x \in [a, b]} |f(x) - q(x)| diff --git a/semester3/numcs/parts/02_quadrature/00_introduction.tex b/semester3/numcs/parts/02_quadrature/00_introduction.tex index c7f4d6f..c8aae13 100644 --- a/semester3/numcs/parts/02_quadrature/00_introduction.tex +++ b/semester3/numcs/parts/02_quadrature/00_introduction.tex @@ -49,7 +49,7 @@ Falls wir $c_k(x) = x^k$ haben (was oft der Fall ist, je nach Funktion aber kön Für die Knoten $x_0, x_1, \ldots, x_n \in \R$ definieren wir die Polynome \rmvspace \begin{align*} - l_i(x) = \prod_{j = 0 \atop j \neq i}^n \frac{x - x_j}{x_i - x_j} + l_i(x) = \prod_{\substack{j = 0\\ j \neq i}}^n \frac{x - x_j}{x_i - x_j} \end{align*} \rmvspace @@ -129,7 +129,7 @@ Für dieses Referenzintervall können wir die Gewichte $\hat{w}_j$ und die Knote \rmvspace \begin{align*} \int_{a}^{b} f(t) \dx t \approx \frac{1}{2}(b - a) \sum_{j = 1}^{n} \hat{w}_j \hat{f}(\hat{c}_j) = \sum_{j = 1}^{n} w_j f(c_j) - & & \text{ mit } {c_j = \frac{1}{2} (1 - \hat{c}_j) a + \frac{1}{2}(1 + \hat{c}_j) b \atop{w_j = \frac{1}{2}(b - a)\hat{w}_j}} + & & \text{ mit } \elementstack{c_j = \frac{1}{2} (1 - \hat{c}_j) a + \frac{1}{2}(1 + \hat{c}_j) b}{w_j = \frac{1}{2}(b - a)\hat{w}_j} \end{align*} \rmvspace\rmvspace