Symbolic Legendre Polynomials

General Usage
1

Is there an implemented version of the legendre Polynomials with symbolic variables?

I have checked

But they either don’t implement the Legendre Polynomials or a symbolic equivalent.

What would be the most convenient way to implement them?

1 Like
2

Their combination does:

julia> using ClassicalOrthogonalPolynomials, Symbolics

julia> @variables x;

julia> p4 = legendrep(4, x)
0.375 + 1.75x*(1.6666666666666667x*(1.5(x^2) - 0.5) - 0.6666666666666666x) - 1.125(x^2)

julia> expand(p4)
0.375 + 4.375(x^4) - 3.75(x^2)

Gotta love composability!

8 Likes
3

Following up on @stevengj’s enthusiasm for composability, if you needed exact coefficients, you might mix in SpecialPolynomials:

julia> using Symbolics, SpecialPolynomials

julia> @variables x
1-element Vector{Num}:
 x

julia> basis(Legendre{Rational{Int}}, 4)(x) |> expand
(3//8) + (35//8)*(x^4) - (15//4)*(x^2)
7 Likes
4

Excelent! As a rule of thumb, which functions should I expect to work with Symbolics.jl?

1 Like
5

Really nice! But this option seems to overflow for degrees n > 25

basis(Legendre{Rational{Int}}, 25)(x) |> expand
ERROR: OverflowError: -148257227381443 * 74290 overflowed for type Int64
[...]

Replacing Int for Int128 does not seem to work, because I’ll need to work with polynomials of degree 50 or more

6

Use BigInt if you need exact rational coefficients for high degrees?

3 Likes