Source code for quantecon._rank_nullspace
import numpy as np
from numpy.linalg import svd
[docs]def rank_est(A, atol=1e-13, rtol=0):
"""
Estimate the rank (i.e. the dimension of the nullspace) of a matrix.
The algorithm used by this function is based on the singular value
decomposition of `A`.
Parameters
----------
A : array_like(float, ndim=1 or 2)
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 : scalar(float), optional(default=1e-13)
The absolute tolerance for a zero singular value. Singular
values smaller than `atol` are considered to be zero.
rtol : scalar(float), optional(default=0)
The relative tolerance. Singular values less than rtol*smax are
considered to be zero, where smax is the largest singular value.
Returns
-------
r : scalar(int)
The estimated rank of the matrix.
Note: If both `atol` and `rtol` are positive, the combined tolerance
is the maximum of the two; that is:
tol = max(atol, rtol * smax)
Note: Singular values smaller than `tol` are considered to be zero.
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 = 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 : array_like(float, ndim=1 or 2)
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 : scalar(float), optional(default=1e-13)
The absolute tolerance for a zero singular value. Singular
values smaller than `atol` are considered to be zero.
rtol : scalar(float), optional(default=0)
The relative tolerance. Singular values less than rtol*smax are
considered to be zero, where smax is the largest singular value.
Returns
-------
ns : array_like(float, ndim=2)
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.
Note: If both `atol` and `rtol` are positive, the combined tolerance
is the maximum of the two; that is:
tol = max(atol, rtol * smax)
Note: Singular values smaller than `tol` are considered to be zero.
"""
A = np.atleast_2d(A)
u, s, vh = svd(A)
tol = max(atol, rtol * s[0])
nnz = (s >= tol).sum()
ns = vh[nnz:].conj().T
return ns