def logm(ctx, A):
        r"""
        Computes a logarithm of the square matrix `A`, i.e. returns
        a matrix `B = \log(A)` such that `\exp(B) = A`. The logarithm
        of a matrix, if it exists, is not unique.

        **Examples**

        Logarithms of some simple matrices::

            >>> from mpmath import *
            >>> mp.dps = 15; mp.pretty = True
            >>> X = eye(3)
            >>> logm(X)
            [0.0  0.0  0.0]
            [0.0  0.0  0.0]
            [0.0  0.0  0.0]
            >>> logm(2*X)
            [0.693147180559945                0.0                0.0]
            [              0.0  0.693147180559945                0.0]
            [              0.0                0.0  0.693147180559945]
            >>> logm(expm(X))
            [1.0  0.0  0.0]
            [0.0  1.0  0.0]
            [0.0  0.0  1.0]

        A logarithm of a complex matrix::

            >>> X = matrix([[2+j, 1, 3], [1-j, 1-2*j, 1], [-4, -5, j]])
            >>> B = logm(X)
            >>> nprint(B)
            [ (0.808757 + 0.107759j)    (2.20752 + 0.202762j)   (1.07376 - 0.773874j)]
            [ (0.905709 - 0.107795j)  (0.0287395 - 0.824993j)  (0.111619 + 0.514272j)]
            [(-0.930151 + 0.399512j)   (-2.06266 - 0.674397j)  (0.791552 + 0.519839j)]
            >>> chop(expm(B))
            [(2.0 + 1.0j)           1.0           3.0]
            [(1.0 - 1.0j)  (1.0 - 2.0j)           1.0]
            [        -4.0          -5.0  (0.0 + 1.0j)]

        A matrix `X` close to the identity matrix, for which
        `\log(\exp(X)) = \exp(\log(X)) = X` holds::

            >>> X = eye(3) + hilbert(3)/4
            >>> X
            [              1.25             0.125  0.0833333333333333]
            [             0.125  1.08333333333333              0.0625]
            [0.0833333333333333            0.0625                1.05]
            >>> logm(expm(X))
            [              1.25             0.125  0.0833333333333333]
            [             0.125  1.08333333333333              0.0625]
            [0.0833333333333333            0.0625                1.05]
            >>> expm(logm(X))
            [              1.25             0.125  0.0833333333333333]
            [             0.125  1.08333333333333              0.0625]
            [0.0833333333333333            0.0625                1.05]

        A logarithm of a rotation matrix, giving back the angle of
        the rotation::

            >>> t = 3.7
            >>> A = matrix([[cos(t),sin(t)],[-sin(t),cos(t)]])
            >>> chop(logm(A))
            [             0.0  -2.58318530717959]
            [2.58318530717959                0.0]
            >>> (2*pi-t)
            2.58318530717959

        For some matrices, a logarithm does not exist::

            >>> logm([[1,0], [0,0]])
            Traceback (most recent call last):
              ...
            ZeroDivisionError: matrix is numerically singular

        Logarithm of a matrix with large entries::

            >>> logm(hilbert(3) * 10**20).apply(re)
            [ 45.5597513593433  1.27721006042799  0.317662687717978]
            [ 1.27721006042799  42.5222778973542   2.24003708791604]
            [0.317662687717978  2.24003708791604    42.395212822267]

        """
        A = ctx.matrix(A)
        prec = ctx.prec
        try:
            ctx.prec += 10
            tol = ctx.eps * 128
            I = A**0
            B = A
            n = 0
            while 1:
                B = ctx.sqrtm(B)
                n += 1
                if ctx.mnorm(B-I, 'inf') < 0.125:
                    break
            T = X = B-I
            L = X*0
            k = 1
            while 1:
                if k & 1:
                    L += T / k
                else:
                    L -= T / k
                T *= X
                if ctx.mnorm(T, 'inf') < tol:
                    break
                k += 1
                if k > ctx.prec:
                    raise ctx.NoConvergence
        finally:
            ctx.prec = prec
        L *= 2**n
        return L
dataset/ETHPy150Open fredrik-johansson/mpmath/mpmath/matrices/calculus.py/MatrixCalculusMethods.logm/original