numpy的“智能"对称矩阵 [英] Numpy ‘smart’ symmetric matrix

问题描述

在numpy中是否存在一个智能且节省空间的对称矩阵，当写入[i][j]时，该矩阵会自动(透明地)填充[j][i]的位置?

Is there a smart and space-efficient symmetric matrix in numpy which automatically (and transparently) fills the position at [j][i] when [i][j] is written to?

import numpy
a = numpy.symmetric((3, 3))
a[0][1] = 1
a[1][0] == a[0][1]
# True
print(a)
# [[0 1 0], [1 0 0], [0 0 0]]

assert numpy.all(a == a.T) # for any symmetric matrix

自动Hermitian也会很好，尽管在撰写本文时我不需要.

An automatic Hermitian would also be nice, although I won’t need that at the time of writing.

推荐答案

如果在进行计算之前就可以对称化矩阵，那么以下操作应该相当快:

If you can afford to symmetrize the matrix just before doing calculations, the following should be reasonably fast:

def symmetrize(a):
    """
    Return a symmetrized version of NumPy array a.

    Values 0 are replaced by the array value at the symmetric
    position (with respect to the diagonal), i.e. if a_ij = 0,
    then the returned array a' is such that a'_ij = a_ji.

    Diagonal values are left untouched.

    a -- square NumPy array, such that a_ij = 0 or a_ji = 0, 
    for i != j.
    """
    return a + a.T - numpy.diag(a.diagonal())

这在合理的假设下有效(例如在运行symmetrize之前不同时执行a[0, 1] = 42和矛盾的a[1, 0] = 123).

This works under reasonable assumptions (such as not doing both a[0, 1] = 42 and the contradictory a[1, 0] = 123 before running symmetrize).

如果您确实需要透明的对称化，则可以考虑子类化numpy.ndarray并简单地重新定义__setitem__:

If you really need a transparent symmetrization, you might consider subclassing numpy.ndarray and simply redefining __setitem__:

class SymNDArray(numpy.ndarray):
    """
    NumPy array subclass for symmetric matrices.

    A SymNDArray arr is such that doing arr[i,j] = value
    automatically does arr[j,i] = value, so that array
    updates remain symmetrical.
    """

    def __setitem__(self, (i, j), value):
        super(SymNDArray, self).__setitem__((i, j), value)                    
        super(SymNDArray, self).__setitem__((j, i), value)                    

def symarray(input_array):
    """
    Return a symmetrized version of the array-like input_array.

    The returned array has class SymNDArray. Further assignments to the array
    are thus automatically symmetrized.
    """
    return symmetrize(numpy.asarray(input_array)).view(SymNDArray)

# Example:
a = symarray(numpy.zeros((3, 3)))
a[0, 1] = 42
print a  # a[1, 0] == 42 too!

(或等价的矩阵而不是数组，具体取决于您的需求).这种方法甚至可以处理更复杂的分配，例如a[:, 1] = -1，可以正确设置a[1, :]元素.

(or the equivalent with matrices instead of arrays, depending on your needs). This approach even handles more complicated assignments, like a[:, 1] = -1, which correctly sets a[1, :] elements.

请注意，Python 3消除了编写def …(…, (i, j),…)的可能性，因此在使用Python 3运行之前，必须对代码稍作修改:def __setitem__(self, indexes, value): (i, j) = indexes…

Note that Python 3 removed the possibility of writing def …(…, (i, j),…), so the code has to be slightly adapted before running with Python 3: def __setitem__(self, indexes, value): (i, j) = indexes…

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