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37 lines
931 B
Python
37 lines
931 B
Python
import numpy as np
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def lanczos(A: np.ndarray) -> ([float], [float]):
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"""
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Implements the Lanczos algorithm for a symmetric matrix.
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Parameters:
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-----------
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matrix : numpy.ndarray
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Symmetric matrix of size (n, n).
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Returns:
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--------
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alpha : [float]
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List of diagonal elements of the resulting tridiagonal matrix.
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beta : [float]
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List of off-diagonal elements of the resulting tridiagonal matrix.
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"""
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n = A.shape[0]
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V = np.zeros((n, n))
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V[:, 0] = np.random.randn(n)
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V[:, 0] /= np.linalg.norm(V[:, 0])
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alpha = []
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beta = []
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for j in range(n):
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w = np.dot(A, V[:, j])
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alpha.append(np.dot(w, V[:, j]))
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if j == n - 1:
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break
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w -= alpha[j] * V[:, j]
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if j > 0:
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w -= beta[j - 1] * V[:, j - 1]
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beta.append(np.linalg.norm(w))
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V[:, j + 1] = w / beta[j]
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return alpha, beta
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