legendre_product_polynomial, a Python code which can define a Legendre product polynomial (LPP), creating a multivariate polynomial as the product of univariate Legendre polynomials.
The Legendre polynomials are a polynomial sequence L(I,X), with polynomial I having degree I.
The first few Legendre polynomials are
0: 1
1: x
2: 3/2 x^2 - 1/2
3: 5/2 x^3 - 3/2 x
4: 35/8 x^4 - 30/8 x^2 + 3/8
5: 63/8 x^5 - 70/8 x^3 + 15/8 x
A Legendre 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 Legendre polynomials, with each the I-th polynomial evaluated at the I-th coordinate:
L((I1,I2,...IM),X) = L(1,X(1)) * L(2,X(2)) * ... * L(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 Legendre product polynomial from its index set, and it is easy to evaluate it directly, the sum of two Legendre product polynomials cannot be reduced to a single Legendre product polynomial. Thus, it may be useful to generate the Legendre 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 code, 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 MIT license
legendre_product_polynomial is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version and a Python version.
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