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zI=Aux+O=u7HDZuU#6Y+i=_2ff)IjMaKAr3X@@%z7A4$luWn=&iDm$Wf#C2R;V_x>amwM(x+ zut$F`m^$Q{FqII#=iT2~S5;%&*V-zou}s=|vdieiZlnY8>FCR~ALOOF?Z1mZ?3D_H zs0!h9L{ zL85iHY-C96w7z`XuPLzj)!^7mYTZ)2<7*z+bEfLE^zDF{hgw-cYci78>b>}18UJ7tgW6k$|!fk4{ zWB3V2m%o_N7JwgbEq}49Zx(MlEjpfpZUBP9zK4GR{U$`NpYVSG;N0_ 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 +def weights_barycentric(x: np.ndarray): + """ All x should be pairwise distinct, else zero-divison error. """ + n = x.size + w = np.zeros(n) + for i in range(n): + w[i] = 1/( np.prod(x[i] - x[0:i]) * np.prod(x[i] - x[i+1:n]) ) + return w \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 -\end{code} Mit dem können wir dann ein Polynom mit der baryzentrischen Interpolationsformel interpolieren: \numberingOff @@ -101,26 +79,20 @@ 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 - ): - 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 +def eval_barycentric(w: np.ndarray, data: np.ndarray, y: np.ndarray, x: np.ndarray): + """ Sequentially calculate the barycentric formula """ + n = x.size + tmp = np.ones(n) + for i in range(n): tmp[i] = eval_barycentric_scalar(w, data, y, x[i]) + return tmp + + +def eval_barycentric_scalar(w: np.ndarray, data: np.ndarray, y: np.ndarray, x): + """ Barycentric interpolation formula for a single value x """ + n = data.size; + bottom = np.sum( w / (x - data) ) + top = np.sum( w / (x - data) * y) + return top / bottom \end{code} 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 133d13c..8b949f5 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 @@ -139,20 +139,15 @@ Auf der nächsten Seite findet sich eine saubere, effiziente Implementation des \newpage \begin{code}{python} - def clenshaw(coeffs: np.ndarray, x: np.ndarray): - n = len(coeffs) - 1 - # initialise temporary variables - d_prev_prev, d_prev, d_curr = ( - np.zeros_like(x), - np.zeros_like(x), - np.zeros_like(x), - ) - - for k in range(n, -1, -1): # backward recursion - d_curr = coeffs[k] + 2 * x * d_prev - d_prev_prev - d_prev_prev, d_prev = d_prev, d_curr - - return d_prev - x * d_prev_prev +def clenshaw(c: np.ndarray, x: np.ndarray): + """ Clenshaw algorithm to evaluate polynomial using chebyshev coeffs """ + n = c.size; m = x.size + d = np.zeros((m, 3)) # Save vectors [curr, prev1, prev2] as matrix + + for i in range(n-1, -1, -1): + d[:, 2] = d[:, 1]; d[:, 1] = d[:, 0] + d[:, 0] = c[i] + (2*x)*d[:, 1] - d[:, 2] + return d[:, 0] - x*d[:, 1] \end{code} diff --git a/semester3/numcs/parts/01_interpolation/01_trigonometric/03_interpolation/01_zero-padding.tex b/semester3/numcs/parts/01_interpolation/01_trigonometric/03_interpolation/01_zero-padding.tex index 10cdc27..9bf2c40 100644 --- a/semester3/numcs/parts/01_interpolation/01_trigonometric/03_interpolation/01_zero-padding.tex +++ b/semester3/numcs/parts/01_interpolation/01_trigonometric/03_interpolation/01_zero-padding.tex @@ -8,21 +8,49 @@ Ein trigonometrisches Polynom $p_{N - 1}(t)$ kann effizient an den äquidistante Dazu muss das Polynom $p_{N - 1} \in \mathcal{T}_N \subseteq \mathcal{T}_M$ in der trigonometrischen Basis $\mathcal{T}_M$ neugeschrieben werden, in dem man \bi{Zero-Padding} verwendet, also Nullen im Koeffizientenvektor an den Stellen höheren Frequenzen einfügt. -\TODO Insert cleaned up code from Page 95 (part of exercises) - -Die folgende Funktion wird im Script \texttt{evaliptrig} genannt. +\innumpy Dieses Verfahren lässt sich in Python leicht via \verb|slices| umsetzen: \rmvspace \begin{code}{python} - def evaluate_trigonometric_interpolation_polynomial(y: np.ndarray, N: int): - n = len(y) - if (n % 2) == 0: - c = np.fft.ifft(y) # Fourier coefficients - a = np.zeros(N, dtype=complex) +def zero_pad(v: np.ndarray, N: int): + """Apply zero-padding to size N to a vector v """ + n = v.size + if (N < n): raise ValueError(f"ERROR: Zeropadding for N smaller than vector length: {N} < {n}") + + u = np.zeros(N, dtype=complex) + u[:n//2] = v[:n//2] + u[N-n//2:] = v[n//2:] + return u - # Zero padding - a[: n // 2] = c[: n // 2] - a[N - n // 2 :] = c[n // 2 :] - return np.fft.fft(a) - else: - raise ValueError("odd length") + +def eval_trig_poly(y: np.ndarray, N: int): + """ Evaluate trig poly generated using y on N points """ + n = y.size + if (n % 2 != 0): raise ValueError(f"ERROR: y must be of even length, len(y)={n}") + + coeffs = np.fft.fft(y) * 1/n + coeffs = zero_pad(coeffs, N) + return np.fft.ifft(coeffs) * N \end{code} + +\subsubsection{Ableitungen mit Zero-Padding} +Mit dem Trick aus Bemerkung 3.2.13 lassen sich auch direkt die Ableitungen berechnen. + +\innumpy Geht dies direkt durch leichte Modifikation der obigen Funktionen: + +\begin{code}{python} +def eval_trig_poly_d1(y: np.ndarray, N: int): + """ Evaluates first der. of trig poly generated using y on N points """ + n = y.size + if (n % 2 != 0): raise ValueError(f"ERROR: y must be of even length, len(y)={n}") + + coeffs = np.fft.fft(y) * 1/n + + for i in range(0, n//2): + coeffs[i] *= (2.0j * np.pi * i) + for i in range(n//2, n): + coeffs[i] *= (2.0j * np.pi * (i - n)) + + coeffs = zero_pad(coeffs, N) + return np.fft.ifft(coeffs) * N +\end{code} + diff --git a/semester3/numcs/parts/01_interpolation/02_piece-wise/00_intro.tex b/semester3/numcs/parts/01_interpolation/02_piece-wise/00_intro.tex index cbfb430..0f6d827 100644 --- a/semester3/numcs/parts/01_interpolation/02_piece-wise/00_intro.tex +++ b/semester3/numcs/parts/01_interpolation/02_piece-wise/00_intro.tex @@ -39,3 +39,28 @@ also gilt: \end{align*} Dies ist eine lokale Interpolation und $s_j$ ist $0$ ausser im definierten Intervall. Die Idee des Variablenwechsel ist es, das Intervall, auf welchem die Funktion definiert ist von $[0, 1]$ nach $[x_{j - 1}, x_j]$ zu verschieben. + +\innumpy Stückweise lineare interpolation lässt sich leicht umsetzen via numpy \verb|piecewise| Funktionen: + +\begin{code}{python} +def slope(x: np.ndarray, j: int, x_data: np.ndarray, y: np.ndarray): + """ Slope on j-th sub-interval, for x falling inside that sub-interval """ + h = np.abs(x_data[j] - x_data[j-1]) # size of sub-interval + return y[j-1]*(x_data[j] - x)/h + y[j]*(x - x_data[j-1])/h + + +def eval_linear_interp(x: np.ndarray, x_data: np.ndarray, y: np.ndarray): + """ Evaluate linear interpolation on x, using data points (x_data, y) """ + n = x_data.size; m = x.size + + condlist = [ + (x_data[j-1] <= x) & (x <= x_data[j]) + for j in range(1, n) + ] + funclist = [ + lambda x, ind=j: slope(x, ind, x_data, y) + for j in range(1, n) + ] + + return np.piecewise(x, condlist, funclist) +\end{code} \ No newline at end of file

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