pyerrors.mpm

 1import numpy as np
 2import scipy.linalg
 3from .obs import Obs
 4from .linalg import svd, eig
 5
 6
 7def matrix_pencil_method(corrs, k=1, p=None, **kwargs):
 8    """Matrix pencil method to extract k energy levels from data
 9
10    Implementation of the matrix pencil method based on
11    eq. (2.17) of Y. Hua, T. K. Sarkar, IEEE Trans. Acoust. 38, 814-824 (1990)
12
13    Parameters
14    ----------
15    data : list
16        can be a list of Obs for the analysis of a single correlator, or a list of lists
17        of Obs if several correlators are to analyzed at once.
18    k : int
19        Number of states to extract (default 1).
20    p : int
21        matrix pencil parameter which filters noise. The optimal value is expected between
22        len(data)/3 and 2*len(data)/3. The computation is more expensive the closer p is
23        to len(data)/2 but could possibly suppress more noise (default len(data)//2).
24
25    Returns
26    -------
27    energy_levels : list[Obs]
28        Extracted energy levels
29    """
30    if isinstance(corrs[0], Obs):
31        data = [corrs]
32    else:
33        data = corrs
34
35    lengths = [len(d) for d in data]
36    if lengths.count(lengths[0]) != len(lengths):
37        raise Exception('All datasets have to have the same length.')
38
39    data_sets = len(data)
40    n_data = len(data[0])
41
42    if p is None:
43        p = max(n_data // 2, k)
44    if n_data <= p:
45        raise Exception('The pencil p has to be smaller than the number of data samples.')
46    if p < k or n_data - p < k:
47        raise Exception('Cannot extract', k, 'energy levels with p=', p, 'and N-p=', n_data - p)
48
49    # Construct the hankel matrices
50    matrix = []
51    for n in range(data_sets):
52        matrix.append(scipy.linalg.hankel(data[n][:n_data - p], data[n][n_data - p - 1:]))
53    matrix = np.array(matrix)
54    # Construct y1 and y2
55    y1 = np.concatenate(matrix[:, :, :p])
56    y2 = np.concatenate(matrix[:, :, 1:])
57    # Apply SVD to y2
58    u, s, vh = svd(y2, **kwargs)
59    # Construct z from y1 and SVD of y2, setting all singular values beyond the kth to zero
60    z = np.diag(1. / s[:k]) @ u[:, :k].T @ y1 @ vh.T[:, :k]
61    # Return the sorted logarithms of the real eigenvalues as Obs
62    energy_levels = np.log(np.abs(eig(z, **kwargs)))
63    return sorted(energy_levels, key=lambda x: abs(x.value))