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