def vandermonde(x, N=None, increasing=False): x = asarray(x) if N is None: N = len(x) v = empty((len(x), N), dtype=promote_types(x.dtype, int)) tmp = v[:, ::-1] if not increasing else v if N > 0: tmp[:, 0] = 1 if N > 1: tmp[:, 1:] = x[:, None] multiply.accumulate(tmp[:, 1:], out=tmp[:, 1:], axis=1) return v
def vander(x, N=None, increasing=False): """ Generate a Vandermonde matrix. The columns of the output matrix are powers of the input vector. The order of the powers is determined by the `increasing` boolean argument. Specifically, when `increasing` is False, the `i`-th output column is the input vector raised element-wise to the power of ``N - i - 1``. Such a matrix with a geometric progression in each row is named for Alexandre- Theophile Vandermonde. Parameters ---------- x : array_like 1-D input array. N : int, optional Number of columns in the output. If `N` is not specified, a square array is returned (``N = len(x)``). increasing : bool, optional Order of the powers of the columns. If True, the powers increase from left to right, if False (the default) they are reversed. .. versionadded:: 1.9.0 Returns ------- out : ndarray Vandermonde matrix. If `increasing` is False, the first column is ``x^(N-1)``, the second ``x^(N-2)`` and so forth. If `increasing` is True, the columns are ``x^0, x^1, ..., x^(N-1)``. See Also -------- polynomial.polynomial.polyvander Examples -------- >>> x = np.array([1, 2, 3, 5]) >>> N = 3 >>> np.vander(x, N) array([[ 1, 1, 1], [ 4, 2, 1], [ 9, 3, 1], [25, 5, 1]]) >>> np.column_stack([x**(N-1-i) for i in range(N)]) array([[ 1, 1, 1], [ 4, 2, 1], [ 9, 3, 1], [25, 5, 1]]) >>> x = np.array([1, 2, 3, 5]) >>> np.vander(x) array([[ 1, 1, 1, 1], [ 8, 4, 2, 1], [ 27, 9, 3, 1], [125, 25, 5, 1]]) >>> np.vander(x, increasing=True) array([[ 1, 1, 1, 1], [ 1, 2, 4, 8], [ 1, 3, 9, 27], [ 1, 5, 25, 125]]) The determinant of a square Vandermonde matrix is the product of the differences between the values of the input vector: >>> np.linalg.det(np.vander(x)) 48.000000000000043 # may vary >>> (5-3)*(5-2)*(5-1)*(3-2)*(3-1)*(2-1) 48 """ x = asarray(x) if x.ndim != 1: raise ValueError("x must be a one-dimensional array or sequence.") if N is None: N = len(x) v = empty((len(x), N), dtype=promote_types(x.dtype, int)) tmp = v[:, ::-1] if not increasing else v if N > 0: tmp[:, 0] = 1 if N > 1: tmp[:, 1:] = x[:, None] multiply.accumulate(tmp[:, 1:], out=tmp[:, 1:], axis=1) return v
def vander(x, N=None, increasing=False): """ Generate a Vandermonde matrix. The columns of the output matrix are powers of the input vector. The order of the powers is determined by the `increasing` boolean argument. Specifically, when `increasing` is False, the `i`-th output column is the input vector raised element-wise to the power of ``N - i - 1``. Such a matrix with a geometric progression in each row is named for Alexandre- Theophile Vandermonde. Parameters ---------- x : array_like 1-D input array. N : int, optional Number of columns in the output. If `N` is not specified, a square array is returned (``N = len(x)``). increasing : bool, optional Order of the powers of the columns. If True, the powers increase from left to right, if False (the default) they are reversed. .. versionadded:: 1.9.0 Returns ------- out : ndarray Vandermonde matrix. If `increasing` is False, the first column is ``x^(N-1)``, the second ``x^(N-2)`` and so forth. If `increasing` is True, the columns are ``x^0, x^1, ..., x^(N-1)``. See Also -------- polynomial.polynomial.polyvander Examples -------- >>> x = np.array([1, 2, 3, 5]) >>> N = 3 >>> np.vander(x, N) array([[ 1, 1, 1], [ 4, 2, 1], [ 9, 3, 1], [25, 5, 1]]) >>> np.column_stack([x**(N-1-i) for i in range(N)]) array([[ 1, 1, 1], [ 4, 2, 1], [ 9, 3, 1], [25, 5, 1]]) >>> x = np.array([1, 2, 3, 5]) >>> np.vander(x) array([[ 1, 1, 1, 1], [ 8, 4, 2, 1], [ 27, 9, 3, 1], [125, 25, 5, 1]]) >>> np.vander(x, increasing=True) array([[ 1, 1, 1, 1], [ 1, 2, 4, 8], [ 1, 3, 9, 27], [ 1, 5, 25, 125]]) The determinant of a square Vandermonde matrix is the product of the differences between the values of the input vector: >>> np.linalg.det(np.vander(x)) 48.000000000000043 >>> (5-3)*(5-2)*(5-1)*(3-2)*(3-1)*(2-1) 48 """ x = asarray(x) if x.ndim != 1: raise ValueError("x must be a one-dimensional array or sequence.") if N is None: N = len(x) v = empty((len(x), N), dtype=promote_types(x.dtype, int)) tmp = v[:, ::-1] if not increasing else v if N > 0: tmp[:, 0] = 1 if N > 1: tmp[:, 1:] = x[:, None] multiply.accumulate(tmp[:, 1:], out=tmp[:, 1:], axis=1) return v