diff options
Diffstat (limited to 'numpy/linalg/linalg.py')
| -rw-r--r-- | numpy/linalg/linalg.py | 81 |
1 files changed, 41 insertions, 40 deletions
diff --git a/numpy/linalg/linalg.py b/numpy/linalg/linalg.py index aba656b5e..f25452064 100644 --- a/numpy/linalg/linalg.py +++ b/numpy/linalg/linalg.py @@ -250,15 +250,15 @@ def solve(a, b): Parameters ---------- - a : array_like, shape (M, M) + a : (M, M) array_like Coefficient matrix. - b : array_like, shape (M,) or (M, N) + b : {(M,), (M, N)}, array_like Ordinate or "dependent variable" values. Returns ------- - x : ndarray, shape (M,) or (M, N) depending on b - Solution to the system a x = b + x : {(M,), (M, N)} ndarray + Solution to the system a x = b. Returned shape is identical to `b`. Raises ------ @@ -410,12 +410,12 @@ def inv(a): Parameters ---------- - a : array_like, shape (M, M) + a : (M, M) array_like Matrix to be inverted. Returns ------- - ainv : ndarray or matrix, shape (M, M) + ainv : (M, M) ndarray or matrix (Multiplicative) inverse of the matrix `a`. Raises @@ -459,14 +459,15 @@ def cholesky(a): Parameters ---------- - a : array_like, shape (M, M) + a : (M, M) array_like Hermitian (symmetric if all elements are real), positive-definite input matrix. Returns ------- - L : ndarray, or matrix object if `a` is, shape (M, M) - Lower-triangular Cholesky factor of a. + L : {(M, M) ndarray, (M, M) matrix} + Lower-triangular Cholesky factor of `a`. Returns a matrix object + if `a` is a matrix object. Raises ------ @@ -709,12 +710,12 @@ def eigvals(a): Parameters ---------- - a : array_like, shape (M, M) + a : (M, M) array_like A complex- or real-valued matrix whose eigenvalues will be computed. Returns ------- - w : ndarray, shape (M,) + w : (M,) ndarray The eigenvalues, each repeated according to its multiplicity. They are not necessarily ordered, nor are they necessarily real for real matrices. @@ -815,7 +816,7 @@ def eigvalsh(a, UPLO='L'): Parameters ---------- - a : array_like, shape (M, M) + a : (M, M) array_like A complex- or real-valued matrix whose eigenvalues are to be computed. UPLO : {'L', 'U'}, optional @@ -824,7 +825,7 @@ def eigvalsh(a, UPLO='L'): Returns ------- - w : ndarray, shape (M,) + w : (M,) ndarray The eigenvalues, not necessarily ordered, each repeated according to its multiplicity. @@ -910,18 +911,17 @@ def eig(a): Parameters ---------- - a : array_like, shape (M, M) + a : (M, M) array_like A square array of real or complex elements. Returns ------- - w : ndarray, shape (M,) + w : (M,) ndarray The eigenvalues, each repeated according to its multiplicity. The eigenvalues are not necessarily ordered, nor are they necessarily real for real arrays (though for real arrays complex-valued eigenvalues should occur in conjugate pairs). - - v : ndarray, shape (M, M) + v : (M, M) ndarray The normalized (unit "length") eigenvectors, such that the column ``v[:,i]`` is the eigenvector corresponding to the eigenvalue ``w[i]``. @@ -1077,7 +1077,7 @@ def eigh(a, UPLO='L'): Parameters ---------- - a : array_like, shape (M, M) + a : (M, M) array_like A complex Hermitian or real symmetric matrix. UPLO : {'L', 'U'}, optional Specifies whether the calculation is done with the lower triangular @@ -1085,11 +1085,12 @@ def eigh(a, UPLO='L'): Returns ------- - w : ndarray, shape (M,) + w : (M,) ndarray The eigenvalues, not necessarily ordered. - v : ndarray, or matrix object if `a` is, shape (M, M) + v : {(M, M) ndarray, (M, M) matrix} The column ``v[:, i]`` is the normalized eigenvector corresponding - to the eigenvalue ``w[i]``. + to the eigenvalue ``w[i]``. Will return a matrix object if `a` is + a matrix object. Raises ------ @@ -1338,7 +1339,7 @@ def cond(x, p=None): Parameters ---------- - x : array_like, shape (M, N) + x : (M, N) array_like The matrix whose condition number is sought. p : {None, 1, -1, 2, -2, inf, -inf, 'fro'}, optional Order of the norm: @@ -1424,9 +1425,9 @@ def matrix_rank(M, tol=None): Parameters ---------- - M : array_like + M : {(M,), (M, N)} array_like array of <=2 dimensions - tol : {None, float} + tol : {None, float}, optional threshold below which SVD values are considered zero. If `tol` is None, and ``S`` is an array with singular values for `M`, and ``eps`` is the epsilon value for datatype of ``S``, then `tol` is @@ -1489,7 +1490,7 @@ def pinv(a, rcond=1e-15 ): Parameters ---------- - a : array_like, shape (M, N) + a : (M, N) array_like Matrix to be pseudo-inverted. rcond : float Cutoff for small singular values. @@ -1499,7 +1500,7 @@ def pinv(a, rcond=1e-15 ): Returns ------- - B : ndarray, shape (N, M) + B : (N, M) ndarray The pseudo-inverse of `a`. If `a` is a `matrix` instance, then so is `B`. @@ -1647,14 +1648,19 @@ def det(a): Parameters ---------- - a : array_like, shape (M, M) + a : (M, M) array_like Input array. Returns ------- - det : ndarray + det : float Determinant of `a`. + See Also + -------- + slogdet : Another way to representing the determinant, more suitable + for large matrices where underflow/overflow may occur. + Notes ----- The determinant is computed via LU factorization using the LAPACK @@ -1668,11 +1674,6 @@ def det(a): >>> np.linalg.det(a) -2.0 - See Also - -------- - slogdet : Another way to representing the determinant, more suitable - for large matrices where underflow/overflow may occur. - """ sign, logdet = slogdet(a) return sign * exp(logdet) @@ -1693,9 +1694,9 @@ def lstsq(a, b, rcond=-1): Parameters ---------- - a : array_like, shape (M, N) + a : (M, N) array_like "Coefficient" matrix. - b : array_like, shape (M,) or (M, K) + b : {(M,), (M, K)} array_like Ordinate or "dependent variable" values. If `b` is two-dimensional, the least-squares solution is calculated for each of the `K` columns of `b`. @@ -1706,18 +1707,18 @@ def lstsq(a, b, rcond=-1): Returns ------- - x : ndarray, shape (N,) or (N, K) + x : {(M,), (M, K)} ndarray Least-squares solution. The shape of `x` depends on the shape of `b`. - residues : ndarray, shape (), (1,), or (K,) - Sums of residues; squared Euclidean 2-norm for each column in + residuals : {(), (1,), (K,)} ndarray + Sums of residuals; squared Euclidean 2-norm for each column in ``b - a*x``. If the rank of `a` is < N or > M, this is an empty array. If `b` is 1-dimensional, this is a (1,) shape array. Otherwise the shape is (K,). rank : int Rank of matrix `a`. - s : ndarray, shape (min(M,N),) + s : (min(M, N),) ndarray Singular values of `a`. Raises @@ -1849,7 +1850,7 @@ def norm(x, ord=None): Parameters ---------- - x : array_like, shape (M,) or (M, N) + x : {(M,), (M, N)} array_like Input array. ord : {non-zero int, inf, -inf, 'fro'}, optional Order of the norm (see table under ``Notes``). inf means numpy's |