# pyerrors.mpm

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     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    if isinstance(corrs[0], Obs):
    26        data = [corrs]
    27    else:
    28        data = corrs
    29
    30    lengths = [len(d) for d in data]
    31    if lengths.count(lengths[0]) != len(lengths):
    32        raise Exception('All datasets have to have the same length.')
    33
    34    data_sets = len(data)
    35    n_data = len(data[0])
    36
    37    if p is None:
    38        p = max(n_data // 2, k)
    39    if n_data <= p:
    40        raise Exception('The pencil p has to be smaller than the number of data samples.')
    41    if p < k or n_data - p < k:
    42        raise Exception('Cannot extract', k, 'energy levels with p=', p, 'and N-p=', n_data - p)
    43
    44    # Construct the hankel matrices
    45    matrix = []
    46    for n in range(data_sets):
    47        matrix.append(scipy.linalg.hankel(data[n][:n_data - p], data[n][n_data - p - 1:]))
    48    matrix = np.array(matrix)
    49    # Construct y1 and y2
    50    y1 = np.concatenate(matrix[:, :, :p])
    51    y2 = np.concatenate(matrix[:, :, 1:])
    52    # Apply SVD to y2
    53    u, s, vh = svd(y2, **kwargs)
    54    # Construct z from y1 and SVD of y2, setting all singular values beyond the kth to zero
    55    z = np.diag(1. / s[:k]) @ u[:, :k].T @ y1 @ vh.T[:, :k]
    56    # Return the sorted logarithms of the real eigenvalues as Obs
    57    energy_levels = np.log(np.abs(eig(z, **kwargs)))
    58    return sorted(energy_levels, key=lambda x: abs(x.value))

def matrix_pencil_method(corrs, k=1, p=None, **kwargs): View Source

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

Matrix pencil method to extract k energy levels from data

Implementation of the matrix pencil method based on eq. (2.17) of Y. Hua, T. K. Sarkar, IEEE Trans. Acoust. 38, 814-824 (1990)

###### Parameters

- data (list): can be a list of Obs for the analysis of a single correlator, or a list of lists of Obs if several correlators are to analyzed at once.
- k (int): Number of states to extract (default 1).
- p (int): matrix pencil parameter which filters noise. The optimal value is expected between len(data)/3 and 2*len(data)/3. The computation is more expensive the closer p is to len(data)/2 but could possibly suppress more noise (default len(data)//2).