newton.eval#

minterpy.jit_compiled.newton.eval.eval_newton_monomials_single(x_single, exponents, generating_points, max_exponents, products_placeholder, monomials_placeholder)[source]#

Precomputes the value of all given Newton basis polynomials at a point.

Core of the fast polynomial evaluation algorithm. - m spatial dimension - N number of monomials - n maximum exponent in each dimension

Parameters:

  • x_single – coordinates of the point. The shape has to be m.

  • exponents – numpy array with exponents for the polynomial. The shape has to be (N x m).

  • generating_points – generating points used to generate the grid. The shape is (n x m).

  • max_exponents – array with maximum exponent in each dimension. The shape has to be m.

  • products_placeholder – a numpy array for storing the (chained) products.

  • monomials_placeholder – a numpy array of length N for storing the values of all Newton basis polynomials.

Return type:

None

Notes

  • This is a Numba-accelerated function.

  • The function precompute all the (chained) products required during Newton evaluation for a single query point with complexity of O(mN).

  • The (pre-)computation of Newton monomials is coefficient agnostic.

  • Results are stored in the placeholder arrays. The function returns None.

minterpy.jit_compiled.newton.eval.eval_newton_monomials_multiple(xx, exponents, generating_points, max_exponents, products_placeholder, monomials_placeholder, triangular)[source]#

Evaluate the Newton monomials at multiple query points.

The following notations are used below:

  • \(m\): the spatial dimension of the polynomial

  • \(p\): the (maximum) degree of the polynomial in any dimension

  • \(n\): the number of elements in the multi-index set (i.e., monomials)

  • \(\mathrm{nr_{points}}\): the number of query (evaluation) points

  • \(\mathrm{nr_polynomials}\): the number of polynomials with different coefficient sets of the same multi-index set

Parameters:

  • xx (ndarray) – numpy array with coordinates of points where polynomial is to be evaluated. The shape has to be (k x m).

  • exponents (ndarray) – numpy array with exponents for the polynomial. The shape has to be (N x m).

  • generating_points (ndarray) – generating points used to generate the grid. The shape is (n x m).

  • max_exponents (ndarray) – array with maximum exponent in each dimension. The shape has to be m.

  • products_placeholder (ndarray) – a numpy array for storing the (chained) products.

  • monomials_placeholder (ndarray) – placeholder numpy array where the results of evaluation are stored. The shape has to be (k x p).

  • triangular (bool) – whether the output will be of lower triangular form or not. -> will skip the evaluation of some values

Returns:

the value of each Newton polynomial at each point. The shape will be (k x N).

Return type:

None

Notes

  • This is a Numba-accelerated function.

  • The memory footprint for evaluating the Newton monomials iteratively with a single query point at a time is smaller than evaluating all the Newton monomials on all query points. However, when multiplied with multiple coefficient sets, this approach will be faster.

  • Results are stored in the placeholder arrays. The function returns None.