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38 lines
1.0 KiB
Python
38 lines
1.0 KiB
Python
import numpy as np
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def lanczos(a: np.ndarray) -> tuple[list[float], list[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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rng = np.random.default_rng()
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v[:, 0] = rng.standard_normal(n)
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v[:, 0] /= np.linalg.norm(v[:, 0])
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alpha: list[float] = []
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beta: list[float] = []
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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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