numpy.polynomial.polynomial.polyvander

polynomial.polynomial.polyvander(x, deg)

Vandermonde matrix of given degree.

Returns the Vandermonde matrix of degree deg and sample points x. The Vandermonde matrix is defined by

\[V[..., i] = x^i,\]

where 0 <= i <= deg. The leading indices of V index the elements of x and the last index is the power of x.

If c is a 1-D array of coefficients of length n + 1 and V is the matrix V = polyvander(x, n), then np.dot(V, c) and polyval(x, c) are the same up to roundoff. This equivalence is useful both for least squares fitting and for the evaluation of a large number of polynomials of the same degree and sample points.

Parameters:

xarray_like

Array of points. The dtype is converted to float64 or complex128 depending on whether any of the elements are complex. If x is scalar it is converted to a 1-D array.

degint

Degree of the resulting matrix.

Returns:

vanderndarray.

The Vandermonde matrix. The shape of the returned matrix is x.shape + (deg + 1,), where the last index is the power of x. The dtype will be the same as the converted x.

See also

polyvander2d, polyvander3d

Examples

The Vandermonde matrix of degree deg = 5 and sample points x = [-1, 2, 3] contains the element-wise powers of x from 0 to 5 as its columns.

>>> from numpy.polynomial import polynomial as P
>>> x, deg = [-1, 2, 3], 5
>>> P.polyvander(x=x, deg=deg)
array([[  1.,  -1.,   1.,  -1.,   1.,  -1.],
       [  1.,   2.,   4.,   8.,  16.,  32.],
       [  1.,   3.,   9.,  27.,  81., 243.]])