"""
Helpers for matrix operations.
"""
from __future__ import absolute_import, print_function
import numpy as np
[docs]
def rank(A, atol=1e-13, rtol=0):
"""Estimate the rank (i.e. the dimension of the columnspace) of a matrix.
The algorithm used by this function is based on the singular value
decomposition of `A`.
Parameters
----------
A : ndarray
A should be at most 2-D. A 1-D array with length n will be treated
as a 2-D with shape (1, n)
atol : float
The absolute tolerance for a zero singular value. Singular values
smaller than `atol` are considered to be zero.
rtol : float
The relative tolerance. Singular values less than rtol*smax are
considered to be zero, where smax is the largest singular value.
If both `atol` and `rtol` are positive, the combined tolerance is the
maximum of the two; that is::
tol = max(atol, rtol * smax)
Singular values smaller than `tol` are considered to be zero.
Return value
------------
r : int
The estimated rank of the matrix.
See also
--------
numpy.linalg.matrix_rank
matrix_rank is basically the same as this function, but it does not
provide the option of the absolute tolerance.
"""
A = np.atleast_2d(A)
s = np.linalg.svd(A, compute_uv=False)
tol = max(atol, rtol * s[0])
rank = int((s >= tol).sum())
return rank
[docs]
def nullspace(A, atol=1e-13, rtol=0):
"""Compute an approximate basis for the nullspace of A.
The algorithm used by this function is based on the singular value
decomposition of `A`.
Parameters
----------
A : ndarray
A should be at most 2-D. A 1-D array with length k will be treated
as a 2-D with shape (1, k)
atol : float
The absolute tolerance for a zero singular value. Singular values
smaller than `atol` are considered to be zero.
rtol : float
The relative tolerance. Singular values less than rtol*smax are
considered to be zero, where smax is the largest singular value.
If both `atol` and `rtol` are positive, the combined tolerance is the
maximum of the two; that is::
tol = max(atol, rtol * smax)
Singular values smaller than `tol` are considered to be zero.
Return value
------------
ns : ndarray
If `A` is an array with shape (m, k), then `ns` will be an array
with shape (k, n), where n is the estimated dimension of the
nullspace of `A`. The columns of `ns` are a basis for the
nullspace; each element in numpy.dot(A, ns) will be approximately
zero.
"""
A = np.atleast_2d(A)
u, s, vh = np.linalg.svd(A)
tol = max(atol, rtol * s[0])
nnz = (s >= tol).sum()
ns = vh[nnz:].conj().T
return ns
[docs]
def rref(A):
"""Compute the reduced row echelon for the matrix A. Returns
returns a tuple of two elements. The first is the reduced row
echelon form, and the second is a list of indices of the pivot columns.
"""
# We import sympy here because it is slow to load and would slow down the initial
# start up of tellurium
import sympy
m = sympy.Matrix(A)
return m.rref()