HERMITE_PRODUCT_POLYNOMIAL, a MATLAB library 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.
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 ...
The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.
HERMITE_PRODUCT_POLYNOMIAL is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version.
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