rank_nullspace¶
- quantecon.rank_nullspace.nullspace(A, atol=1e-13, rtol=0)[source]¶
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:
- Aarray_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)
- atolscalar(float), optional(default=1e-13)
The absolute tolerance for a zero singular value. Singular values smaller than atol are considered to be zero.
- rtolscalar(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:
- nsarray_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.
- quantecon.rank_nullspace.rank_est(A, atol=1e-13, rtol=0)[source]¶
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:
- Aarray_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)
- atolscalar(float), optional(default=1e-13)
The absolute tolerance for a zero singular value. Singular values smaller than atol are considered to be zero.
- rtolscalar(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:
- rscalar(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.