From 3664961747f2928f90ceab2e5787b0a9ad158047 Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Fri, 3 Oct 2025 16:16:46 +0200 Subject: [PATCH] [NumCS] Add more notes and tricks They came up while solving the utter mess and clusterf of exercises of this subject and are trying to make future us's live easier --- semester3/numcs/numcs-summary.pdf | Bin 499633 -> 503210 bytes .../03_lagrange-and-barzycentric-formula.tex | 63 +++++++++++++++++- .../04_chebyshev-interpolation.tex | 2 + 3 files changed, 64 insertions(+), 1 deletion(-) diff --git a/semester3/numcs/numcs-summary.pdf b/semester3/numcs/numcs-summary.pdf index 9661011cff5fe904eac2ebf6437ff8aab392c62d..3dd26fbaf6e49290f183c64ad4b42da7843bcb15 100644 GIT binary patch delta 404695 zcmV)QK(xQHzaFaF9tb5+L`E$!E;q3VKna)eQU?ssyYb(JJ~B(NOWezbBC)E{V=H zb11T$fI?G6;QN<^(mN7@dJ>=k62cb_LLd>O9R?^0A}Pdw7%^GKT_mOz$vH=fWJF}S zkG12(mE)8ED1cSDMPSSDfj954!E%Q!J%P)cMM#t;R7;dFL5vI$s$qEAzD?8ndZ@-* zQg%)H&AF<;NoofdE{PM#@pf#Qbzj@)m_=LYQgf1X?7CY 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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 @@ -5,6 +5,7 @@ \newsection \subsection{Lagrange- und Baryzentrische Interpolationsformeln} % Session: Gemäss TA sehr gut beschrieben im alten Script +\label{sec:barycentric-interpolation} \begin{definition}[]{Lagrange Polynome} Für Knoten (auch gennannt Stützstellen) $x_0, x_1, \ldots, x_n \in \R$ definieren wir die Lagrange-Polynome: @@ -42,8 +43,34 @@ Falls $j = i$ im Produkt, so überspringt $j$ diese Zahl. \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. -Mit der \bi{baryzentrischen Interpolationsformel} wird zuerst ein Pre-Computing auf Teilen der Lagrange-Polynome durchgeführt, was dann dazu führt, dass die Laufzeit auf $\tco{n^2}$ sinkt ($\tco{n}$ für die Auswertung der Formel und $\tco{n^2}$ für die Berechnung der $\lambda_k$): +Mit der \bi{baryzentrischen Interpolationsformel} wird zuerst ein Pre-Computing auf Teilen der Lagrange-Polynome durchgeführt, was dann dazu führt, dass die Laufzeit auf $\tco{n^2}$ sinkt ($\tco{n}$ für die Auswertung der Formel und $\tco{n^2}$ für die Berechnung der $\lambda_k$). +Man berechnet die baryzentrischen Gewichte $\lambda_k$ folgendermassen: +\rmvspace +\begin{align*} + \lambda_k = \prod_{j \neq k} \frac{1}{x_k - x_j} +\end{align*} +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 +\end{code} + +Mit dem können wir dann ein Polynom mit der baryzentrischen Interpolationsformel interpolieren: \setcounter{numberingConfig}{0} \begin{formula}[]{Baryzentrische Interpolationsformel} \vspace{-1.5pc} @@ -60,6 +87,40 @@ Mit anderen $\lambda_k$ eröffnet die baryzentrische Formel einen Weg zur Verall Eine weitere Anwendung der Formel ist als Ausganspunkt für die Spektralmethode für Differenzialgleichungen. +\begin{code}{python} + def interp_barycentric( + data_point_x: np.ndarray, + data_point_y: np.ndarray, + barweight: np.ndarray, + x: np.ndarray + ): + """Compute an Interpolation polynomial p(x) using the barycentric interpolation formula + + Args: + data_point_x: The data points' x-coordinate from which to interpolate (Stützstellen) + data_point_y: The data points' y-coordinates (Stützwerte) + barweight: Barycentric weights + x: The argument of the polynomial (the x in p(x)) + + Returns: + The Interpolation polynomial evaluated at each x + """ + 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 + + return p_x +\end{code} + % ──────────────────────────────────────────────────────────────────── \newpage diff --git a/semester3/numcs/parts/01_interpolation/00_polynomial/04_chebyshev-interpolation.tex b/semester3/numcs/parts/01_interpolation/00_polynomial/04_chebyshev-interpolation.tex index c6f7aa2..3816618 100644 --- a/semester3/numcs/parts/01_interpolation/00_polynomial/04_chebyshev-interpolation.tex +++ b/semester3/numcs/parts/01_interpolation/00_polynomial/04_chebyshev-interpolation.tex @@ -53,6 +53,8 @@ Das $n$-te Chebyshev-Polynom ist ein Polynom von Grad $n$ und für $x \in [-1, 1 \fancydef{Chebyshev-Abszissen} Die $(n - 1)$ Chebyshev-Abszissen $x_0, \ldots, x_{n - 2}$ im Intervall $[-1, 1]$ sind die Extrema des Chebyshev-Polynoms $T_n(x)$ und zeitgleich die Nullstellen von $U_{n - 1}(x)$. Je nach Kontext nimmt man noch die Grenzen des Intervalls ($1$ und $-1$) hinzu und hat dann $(n + 1)$ Abszissen. +Die Baryzentrischen Gewichte sind dann viel einfacher zu berechnen: $\lambda_k = (-1)^k$ (siehe Bemerkung unterhalb der Baryzentrischen Interpolationsformel, Kapitel \ref{sec:barycentric-interpolation}) + \fancyremark{Chebyshev-Abszissen für beliebiges Intervall} Für $I = [a, b]$ sind die Chebyshev-Abszissen: \rmvspace \begin{align*}