vander

→Tensor

vander(x: Tensor, N: int | None = None, increasing: bool = False)

source

Construct a Vandermonde matrix from a 1-D vector.

Given a 1-D input $x = (x_1, \ldots, x_n)$ , returns the $n \times N$ matrix whose $j$ -th column is a power of $x$ :

V_{ij} \,=\, x_i^{\,j} \quad (\text{increasing}) \qquad\text{or}\qquad V_{ij} \,=\, x_i^{\,N-1-j} \quad (\text{decreasing, default}).

Vandermonde matrices arise naturally in polynomial fitting and interpolation: the columns are the basis $\{1, x, x^2, \ldots\}$ evaluated at the data points.

Parameters

xTensor

1-D input of length

n

Nint or None= None

Number of columns. Defaults to

n

(square output).

increasingbool= False

If True powers increase left-to-right (column 0 is

x^0

). Default False matches the classical convention used in polynomial regression.

Returns

Tensor

$n \times N$ Vandermonde matrix.

Notes

Vandermonde matrices become highly ill-conditioned as $N$ grows (condition number grows exponentially) — for polynomial regression beyond degree $\sim 10$ prefer an orthogonal-polynomial basis or QR-based fitting.

Examples

>>> import lucid
>>> from lucid.linalg import vander
>>> vander(lucid.tensor([1.0, 2.0, 3.0]), N=3, increasing=True)
Tensor([[1.0000, 1.0000, 1.0000],
        [1.0000, 2.0000, 4.0000],
        [1.0000, 3.0000, 9.0000]])