differentiation breaks for multi-value coefs

Description of the problem

If one uses the differentiation function on Newton polynomials with vector-valued coefficients, an error is thrown

MWE

import numpy as np
import minterpy as mp

print("mp version: ", minterpy.__version__)

mi = mp.MultiIndexSet.from_degree(2,2,np.inf)
len(mi)

c = np.random.rand(len(mi),2)
poly = mp.NewtonPolynomial(mi,c)

poly.diff([0,1])

mp version:  0.2.0a0
---------------------------------------------------------------------------
ValueError                                Traceback (most recent call last)
Input In [2], in <cell line: 12>()
      9 c = np.random.rand(len(mi),2)
     10 poly = mp.NewtonPolynomial(mi,c)
---> 12 poly.diff([0,1])

File ~/Work/pyProjects/interpol/minterpy/src/minterpy/core/ABC/multivariate_polynomial_abstract.py:826, in MultivariatePolynomialSingleABC.diff(self, order)
    822 if len(order) != self.spatial_dimension:
    823     raise ValueError(f"inconsistent number of elements in 'order' <{len(order)}>,"
    824                      f"expected <{self.spatial_dimension}> corresponding to each spatial dimension")
--> 826 return self._diff(self, order)

File ~/Work/pyProjects/interpol/minterpy/src/minterpy/polynomials/newton_polynomial.py:79, in _newton_diff(poly, order)
     70 """ Partial differentiation in Newton basis.
     71 
     72 Notes
     73 -----
     74 Performs a transformation from Lagrange to Newton using DDS.
     75 """
     77 # When you evaluate the derivatives on the unisolvent nodes, you get the coefficients for
     78 # the derivative polynomial in Lagrange basis.
---> 79 lag_coeffs = deriv_newt_eval(poly.grid.unisolvent_nodes, poly.coeffs,
     80                              poly.grid.multi_index.exponents,
     81                              poly.grid.generating_points, order)
     83 # DDS returns a 2D array, converting it to 1d
     84 newt_coeffs = dds(lag_coeffs, poly.grid.tree).reshape(-1)

File ~/Work/pyProjects/interpol/minterpy/src/minterpy/polynomials/utils.py:110, in deriv_newt_eval(x, coefficients, exponents, generating_points, derivative_order_along)
    107                     newt_mon_val = 0.0
    108         monomial_vals[j] = newt_mon_val
--> 110     results[point_nr] = np.dot(coefficients, monomial_vals)
    112 return results

File <__array_function__ internals>:180, in dot(*args, **kwargs)

ValueError: shapes (9,2) and (9,) not aligned: 2 (dim 1) != 9 (dim 0)

Similar ValueErrors appear, if one uses CanonicalPolynomials:

import numpy as np
import minterpy as mp

print("mp version: ", mp.__version__)

mi = mp.MultiIndexSet.from_degree(2,2,np.inf)
len(mi)

c = np.random.rand(len(mi),2)
poly = mp.CanonicalPolynomial(mi,c)

poly.diff([0,1])

mp version:  0.2.0a0
---------------------------------------------------------------------------
ValueError                                Traceback (most recent call last)
Input In [3], in <cell line: 12>()
      9 c = np.random.rand(len(mi),2)
     10 poly = mp.CanonicalPolynomial(mi,c)
---> 12 poly.diff([0,1])

File ~/Work/pyProjects/interpol/minterpy/src/minterpy/core/ABC/multivariate_polynomial_abstract.py:826, in MultivariatePolynomialSingleABC.diff(self, order)
    822 if len(order) != self.spatial_dimension:
    823     raise ValueError(f"inconsistent number of elements in 'order' <{len(order)}>,"
    824                      f"expected <{self.spatial_dimension}> corresponding to each spatial dimension")
--> 826 return self._diff(self, order)

File ~/Work/pyProjects/interpol/minterpy/src/minterpy/polynomials/canonical_polynomial.py:316, in _canonical_diff(poly, order)
    313     raise ValueError(f"Cannot differentiate as some of the required multi indices are not present.")
    315 # coefficients of the differentiated polynomial
--> 316 diff_coeffs = coeffs[diff_exp_mask] * np.prod(factorial(exponents[diff_exp_mask]) / factorial(diff_exponents),
    317                                               axis=1)
    319 # The differentiated polynomial being expressed wrt multi indices of the given poly
    320 # NOTE: 'find_match_between' assumes 'exponents' is lexicographically ordered
    321 map_pos = find_match_between(diff_exponents, exponents)

ValueError: operands could not be broadcast together with shapes (6,2) (6,)

differentiation breaks for multi-value coefs

Description of the problem

MWE

Suggested solution