canonical#

A module for compiled code for polynomial manipulation in the canonical basis.

Notes

  • The most “fine-grained” functions must be defined first in order for Numba to properly infer the function types.

minterpy.jit_compiled.canonical.can_eval_mult(x_multiple, coeffs, exponents, result_placeholder)[source]#

Naive evaluation of polynomials in canonical basis.

  • m spatial dimension

  • k number of points

  • N number of monomials

  • p number of polynomials

Parameters:

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

  • coeffs – numpy array of polynomial coefficients in canonical basis. The shape has to be (N x p).

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

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

Notes

This is a naive evaluation; a more numerically accurate approach would be to transform to Newton basis and using the newton evaluation scheme.

Multiple polynomials in the canonical basis can be evaluated at once by having a 2D coeffs array. It is assumed that they all have the same set of exponents.

minterpy.jit_compiled.canonical.compute_coeffs_poly_prod(exponents_1, coeffs_1, exponents_2, coeffs_2, exponents_prod, coeffs_prod)[source]#

Compute the coefficients of polynomial product in the canonical basis.

For example, suppose: \(A = \{ (0, 0) , (1, 0), (0, 1) \}\) with coefficients \(c_A = (1.0 , 2.0, 3.0)\) is multiplied with \(B = \{ (0, 0) , (1, 0) \}\) with coefficients \(c_B = (1.0 , 5.0)\). The product multi-index set is \(A \times B = \{ (0, 0) , (1, 0), (2, 0), (0, 1), (1, 1) \}\).

The corresponding coefficients of the product are:

  • \((0, 0)\) is coming from \((0, 0) + (0, 0)\), the coefficient is \(1.0 \times 1.0 = 1.0\)

  • \((1, 0)\) is coming from \((0, 0) + (1, 0)\) and \((1, 0) + (0, 0)\), the coefficient is \(1.0 \times 5.0 + 2.0 \times 1.0 = 7.0\)

  • \((2, 0)\) is coming from \((1, 0) + (1, 0)\), the coefficient is \(2.0 \times 5.0 = 10.0\)

  • \((0, 1)\) is coming from \((0, 1) + (0, 0)\), the coefficient is \(3.0 \times 1.0 = 3.0\)

  • \((1, 1)\) is coming from \((0, 1) + (1, 0)\), the coefficient is \(3.0 \times 5.0 = 15.0\)

or \(c_{A \times B} = (1.0, 7.0, 10.0, 3.0, 15.0)\).

Parameters:

  • exponents_1 (numpy.ndarray) – The multi-indices exponents of the first multidimensional polynomial operand in the multiplication expression.

  • coeffs_1 (numpy.ndarray) – The coefficients of the first multidimensional polynomial operand.

  • exponents_2 (numpy.ndarray) – The multi-indices exponents of the second multidimensional polynomial.

  • coeffs_2 (numpy.ndarray) – The coefficients of the second multidimensional polynomial operand.

  • exponents_prod (numpy.ndarray) – The multi-indices exponents that are the product between exponents_1 and exponents_2 (i.e., the sum of cartesian products).

  • coeffs_prod (numpy.ndarray) – The placeholder for the corresponding product coefficients.

Notes

  • exponents_1, exponents_2, exponents_prod are assumed to be two-dimensional integer arrays that are sorted lexicographically.

  • exponents_prod is assumed to be the result of multiplying exponents_1 and exponents_2 as multi-indices.

  • coeffs_1, coeffs_2, and coeffs_prod are assumed to be two-dimensional float arrays. Their number of columns must be the same.

  • coeffs_prod is a placeholder array to store the results; it must be initialized with zeros.

  • The function does not check whether the above assumptions are fulfilled; the caller is responsible to make sure of that. If the assumptions are not fulfilled, the function may not raise any exception but produce the wrong results.