hermite_product_polynomial

hermite_product_polynomial, an Octave which defines a Hermite product polynomial (HePP), creating a multivariate polynomial as the product of univariate Hermite polynomials.

The Hermite polynomials are a polynomial sequence He(i,x), with polynomial I having degree I.

The first few Hermite polynomials He(i,x) are

        0: 1
        1: x
        2: x^2 -  1
        3: x^3 -  3 x
        4: x^4 -  6 x^2 + 3
        5: x^5 - 10 x^3 + 15 x
      

A Hermite product polynomial may be defined in a space of M dimensions by choosing M indices. To evaluate the polynomial at a point X, compute the product of the corresponding Hermite polynomials, with each the I-th polynomial evaluated at the I-th coordinate:

        He((I1,I2,...IM),X) = He(1,X(1)) * He(2,X(2)) * ... * He(M,X(M)).
      

Families of polynomials which are formed in this way can have useful properties for interpolation, derivable from the properties of the 1D family.

While it is useful to generate a Hermite product polynomial from its index set, and it is easy to evaluate it directly, the sum of two Hermite product polynomials cannot be reduced to a single Hermite product polynomial. Thus, it may be useful to generate the Hermite product polynomial from its indices, but then to convert it to a standard polynomial form.

The representation of arbitrary multivariate polynomials can be complicated. In this library, we have chosen a representation involving the spatial dimension M, and three pieces of data, O, C and E.

• O is the number of terms in the polynomial.
• C() is a real vector of length O, containing the coefficients of each term.
• E() is an integer vector of length O, which defines the index (the exponents of X(1) through X(M)) of each term.

The exponent indexing is done in a natural way, suggested by the following indexing for the case M = 2:

        1: x^0 y^0  
        2: x^0 y^1
        3: x^1 y^0
        4: x^0 y^2
        5: x^1 y^1
        6; x^2 y^0
        7: x^0 y^3
        8: x^1 y^2
        9: x^2 y^1
       10: x^3 y^0
       ...
      

Licensing:

The computer code and data files described and made available on this web page are distributed under the MIT license

Languages:

hermite_product_polynomial is available in a C version and a C++ version and a Fortran90 version and a MATLAB version and an Octave version.

Related Data and Programs:

hermite_product_polynomial_test

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legendre_product_polynomial, an Octave which defines Legendre product polynomials, creating a multivariate polynomial as the product of univariate Legendre polynomials.

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Source Code:

• comp_enum.m, enumerates the compositions of an integer into K parts.
• comp_next_grlex.m, returns the next composition of an integer into K parts, using grlex order.
• comp_random_grlex.m, returns a random composition of an integer into K parts, with the integer between 0 and N.
• comp_rank_grlex.m, ranks a composition of an integer into K parts, using grlex order.
• comp_unrank_grlex.m, returns the composition of an integer into K parts of a given rank, using grlex order.
• hep_coefficients.m, returns the coefficients of a Hermite polynomial
• hep_value.m, evaluates a Hermite polynomial at a point.
• hep_values.m, returns a table of sample values of Hermite polynomials.
• hepp_to_polynomial.m, converts a Hermite Product Polynomial to standard polynomial form.
• hepp_value.m, evaluates a Hermite Product Polynomial at a point.
• i4_uniform_ab.m, returns a random integer in a given range.
• mono_rank_grlex.m, returns the grlex rank of a monomial in the sequence of all monomials in D dimensions of degree N or less.
• mono_unrank_grlex.m, given the grlex rank, returns the corresponding monomial in the sequence of all monomials in M dimensions.
• mono_value.m, evaluates a monomial.
• polynomial_compress.m, "compresses" a polynomial by merging coefficients associated with the same monomial.
• polynomial_print.m, prints a polynomial.
• polynomial_sort.m, sorts the terms in a polynomial.
• polynomial_value.m, evaluates a polynomial.