newton_diff#

A module for compiled code for polynomial differentiation in the Newton basis.

Notes

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

minterpy.jit_compiled.newton.diff.create_lut_difference(x, generating_points)[source]#

Create a look-up table containing the differences between a query point and the generating points in each dimension.

The table (matrix) contains \((x_j - p_{i, j})\) where \(p_{i, j}\) is the \(i\)-th generating point of the \(j\)-th dimension.

Parameters:

x (numpy.ndarray) – The query point, a one-dimensional array of length m, where m is the spatial dimension of the polynomial.
generating_points (numpy.ndarray) – Interpolation points for each dimension given as a two-dimensional array of shape (n + 1, m), where n is the maximum polynomial degree in all dimensions.

Returns:

A look-up table (LUT) of shape (n, m) containing the difference between the query point and the generating points (size n) in each dimension (size m).

Return type:

numpy.ndarray

Notes

This function is compiled (NJIT-ted) with the help of Numba.

minterpy.jit_compiled.newton.diff.create_lua_differentiated(combinations, lua_difference, prev_lua_prod, lua_prod)[source]#

Create a differentiated product look-up array (LUA) in one-dimension for a single query point, a given monomial degree, and order of derivative.

Due to chain rule, the differentiated one-dimensional Newton monomials evaluated at a query point consists of sum of products of difference between the query point and the generating points.

The products that appears in the sum of a given degree may be recycled in the computation of the products of the next higher degree to avoid the re-computations from scratch.

Therefore, this function both returns the sum of products of a given degree and modifies in-place the product array to be re-used.

Parameters:

combinations (numpy.ndarray) – The combination of terms to multiply, the information regarding the degree of monomials and the order of derivative is encoded here and does not appear explicitly in this function.
lua_difference (numpy.ndarray) – The one-dimensional array containing the difference between a query point and the generating points in one dimension.
prev_lua_prod (numpy.ndarray) – The products of difference from previous computation (degree).
lua_prod (numpy.ndarray) – The products of difference for the current computation; it is passed to avoid reinitialization.

Returns:

The sum of products of difference between the query point and the generating points for a differentiated one-dimensional Newton monomial of a given degree and a given order of derivative.

Return type:

float

Notes

At the end of the computation, the current products are stored in the previous products array to be used in the next call.
The parameter (array) lua_prod is not actually used in the subsequent computation but is passed to avoid re-initialization of the array.
This function is compiled (NJIT-ted) with the help of Numba.

minterpy.jit_compiled.newton.diff.create_lut_differentiated(x, max_exponents, generating_points, derivative_order_along, products_placeholder)[source]#

Create a look-up table of one-dimensional Newton monomials derivatives evaluated on a single query point for all dimensions.

The following symbols are used in the following as shortcuts:

m: spatial dimension
n: polynomial degree
n_max: maximum exponent in all dimension

Parameters:

x (numpy.ndarray) – A single query point at which the derivative is evaluated; the values are given in a one-dimensional array of length m.
max_exponents (numpy.ndarray) – The maximum exponents in the multi-index set for each dimension given as one-dimensional non-negative integer array of length m.
generating_points (numpy.ndarray) – Interpolation points for each dimension given as a two-dimensional array of shape (m, n + 1).
derivative_order_along (numpy.ndarray) – Specification of the orders of derivative along each dimension given as a one-dimensional non-negative integer array of length m. For example, the array np.array([2, 3, 1]) specifies the 2nd-order, 3rd-order, and 1st-order derivatives along the 1st, 2nd, and 3rd dimension, respectively.
products_placeholder (numpy.ndarray) – A placeholder for a look-up table containing differentiated Newton monomials per dimension; the differentiated monomials are products of difference terms.

Returns:

The look-up table (LUT) of shape (n_max, m) where each column consists of the derivative value of one-dimensional Newton monomials at the (single) query point.

Return type:

numpy.ndarray

Notes

This function is compiled (NJIT-ted) with the help of Numba.

See also

minterpy.jit_compiled.newton.diff.compute_from_lut: Based on the look-up table (LUT) created by the current function, compute the multi-dimensional differentiated Newton monomials on a single query point for all elements in the multi-index set.

minterpy.jit_compiled.newton.diff.compute_monomials_from_lut(exponents, lut_diff, derivative_order_along, monomials_placeholder)[source]#

Evaluate the derivatives of multi-dimensional Newton monomials evaluated on a query point from a look-up table of differentiated monomial terms.

The following symbols are used in the following as shortcuts:

m: spatial dimension
N: number of coefficients or the cardinality of the multi-index set
n_max: maximum exponent in all dimension

Parameters:

exponents (numpy.ndarray) – Set of exponents given as a two-dimensional non-negative integer array of shape (N, m).
lut_diff (numpy.ndarray) – A look-up table that consists of the one-dimensional differentiated Newton monomials evaluated on a single query point. The table is a two-dimensional array of shape (n_max, m).
derivative_order_along (numpy.ndarray) – Specification of the orders of derivative along each dimension given as a one-dimensional non-negative integer array of length m. For example, the array np.array([2, 3, 1]) specifies the 2nd-order, 3rd-order, and 1st-order derivatives along the 1st, 2nd, and 3rd dimension, respectively.
monomials_placeholder (numpy.ndarray) – The placeholder for all the differentiated monomials values evaluated at a single query point. The placeholder is a one-dimensional array of length N.

Notes

This function is compiled (NJIT-ted) with the help of Numba.

minterpy.jit_compiled.newton.diff.eval_monomials_single_query(x, exponents, max_exponents, generating_points, derivative_order_along, products_placeholder, monomials_placeholder)[source]#

Evaluate the derivative of a Newton poly(s) on a single query point.

The following symbols are used in the subsequent description as shortcuts:

m: spatial dimension
n: polynomial degree
N: number of coefficients or the cardinality of the multi-index set
np: number of polynomials (i.e., number of coefficient sets)

Parameters:

x (numpy.ndarray) – A single query point at which the derivative is evaluated; the values are given in a one-dimensional array of length m.
exponents (numpy.ndarray) – Set of exponents given as a two-dimensional non-negative integer array of shape (N, m).
max_exponents (numpy.ndarray) – The maximum exponent values in each dimension as a one-dimensional array of length m; this is to avoid re-computation.
generating_points (numpy.ndarray) – Interpolation points for each dimension given as a two-dimensional array of shape (m, n + 1).
derivative_order_along (numpy.ndarray) – Specification of the orders of derivative along each dimension given as a one-dimensional non-negative integer array of length m. For example, the array np.array([2, 3, 1]) specifies the 2nd-order, 3rd-order, and 1st-order derivatives along the 1st, 2nd, and 3rd dimension, respectively.
products_placeholder (numpy.ndarray) – A placeholder for a look-up table containing differentiated Newton monomials per dimension; the differentiated monomials are products of difference terms.
monomials_placeholder (numpy.ndarray) – The differentiated Newton monomials evaluated on a single query point. The array is of shape N.

Returns:

The value of the derivative at the query point given as a one-dimensional array of length np.

Return type:

numpy.ndarray

Notes

This is a direct differentiation and evaluation of Newton monomials on a single query point via chain rule without any transformation to another (e.g., canonical) basis.
This function is compiled (NJIT-ted) with the help of Numba.

See also

minterpy.jit_compiled.newton.diff.eval_multiple_query: Evaluation of the derivative of polynomial(s) in the Newton basis for multiple query points.

minterpy.jit_compiled.newton.diff.eval_multiple_query(xx, coefficients, exponents, generating_points, derivative_order_along)[source]#

Evaluate the derivative of Newton poly(s) on multiple query points.

The following symbols are used in the subsequent description as shortcuts:

k: number of query points
m: spatial dimension
n: polynomial degree
N: number of coefficients or the cardinality of the multi-index set
np: number of polynomials (i.e., number of coefficient sets)

Parameters:

xx (numpy.ndarray) – The set of query points at which the derivative is evaluated; the values are given in a two-dimensional array of shape (k, m).
coefficients (numpy.ndarray) – The coefficients of the Newton polynomial; the values are given in a two-dimensional array of shape (N, np).
exponents (numpy.ndarray) – Set of exponents given as a two-dimensional non-negative integer array of shape (N, m).
generating_points (numpy.ndarray) – Interpolation points for each dimension given as a two-dimensional array of shape (m, n + 1).
derivative_order_along (numpy.ndarray) – Specification of the orders of derivative along each dimension given as a one-dimensional non-negative integer array of length m. For example, the array np.array([2, 3, 1]) specifies the 2nd-order, 3rd-order, and 1st-order derivatives along the 1st, 2nd, and 3rd dimension, respectively.

Returns:

The value of the derivative at the query point given as a two-dimensional array of shape (k, np).

Return type:

numpy.ndarray

Notes

This is a direct differentiation and evaluation of polynomial(s) in the Newton basis on multiple query points via chain rule without any transformation to another (e.g., canonical) basis.
This function is compiled (NJIT-ted) with the help of Numba.

See also

minterpy.jit_compiled.newton.diff.eval_monomials_single_query: Evaluation of the derivative of monomials in the Newton form for a single query point.

minterpy.jit_compiled.newton.diff.eval_multiple_query_par(xx, coefficients, exponents, generating_points, derivative_order_along)[source]#

Evaluate the derivative of Newton polynomial(s) on multiple query points in parallel.

The following symbols are used in the subsequent description as shortcuts:

k: number of query points
m: spatial dimension
n: polynomial degree
N: number of coefficients or the cardinality of the multi-index set
np: number of polynomials (i.e., number of coefficient sets)

Parameters:

xx (numpy.ndarray) – The set of query points at which the derivative is evaluated; the values are given in a two-dimensional array of shape (k, m).
coefficients (numpy.ndarray) – The coefficients of the Newton polynomial; the values are given in a two-dimensional array of shape (N, np).
exponents (numpy.ndarray) – Set of exponents given as a two-dimensional non-negative integer array of shape (N, m).
generating_points (numpy.ndarray) – Interpolation points for each dimension given as a two-dimensional array of shape (m, n + 1).
derivative_order_along (numpy.ndarray) – Specification of the orders of derivative along each dimension given as a one-dimensional non-negative integer array of length m. For example, the array np.array([2, 3, 1]) specifies the 2nd-order, 3rd-order, and 1st-order derivatives along the 1st, 2nd, and 3rd dimension, respectively.

Returns:

The value of the derivative at the query point given as a two-dimensional array of shape (k, np).

Return type:

numpy.ndarray

Notes

This is a direct differentiation and evaluation of polynomial in the Newton basis via chain rule without any transformation to another (e.g., canonical) basis.
This function is compiled (NJIT-ted) and executed in parallel with the help of Numba.

See also

minterpy.jit_compiled.newton.diff.eval_monomials_single_query: Evaluation of the derivative of monomials in the Newton form for a single query point.
minterpy.jit_compiled.newton.diff.eval_multiple_query: Evaluation of the derivative of polynomial(s) in the Newton basis for multiple query points on a single CPU.