Automatic quality testing using Aqua (#4)

* Add Aqua in testing

* Add analytical second derivative

* version bump to v0.3.1

Author: jishnub

Project.toml

@@ -1,17 +1,20 @@
 name = "LegendrePolynomials"
 uuid = "3db4a2ba-fc88-11e8-3e01-49c72059a882"
-version = "0.3"
+version = "0.3.1"
 
 [deps]
 OffsetArrays = "6fe1bfb0-de20-5000-8ca7-80f57d26f881"
 
 [compat]
+Aqua = "0.5"
+HyperDualNumbers = "4"
 OffsetArrays = "0.11, 1"
 julia = "1"
 
 [extras]
+Aqua = "4c88cf16-eb10-579e-8560-4a9242c79595"
 HyperDualNumbers = "50ceba7f-c3ee-5a84-a6e8-3ad40456ec97"
 Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40"
 
 [targets]
-test = ["HyperDualNumbers", "Test"]
+test = ["Aqua", "HyperDualNumbers", "Test"]

docs/Project.toml

@@ -1,8 +1,10 @@
 [deps]
 Documenter = "e30172f5-a6a5-5a46-863b-614d45cd2de4"
+DualNumbers = "fa6b7ba4-c1ee-5f82-b5fc-ecf0adba8f74"
 HyperDualNumbers = "50ceba7f-c3ee-5a84-a6e8-3ad40456ec97"
 LegendrePolynomials = "3db4a2ba-fc88-11e8-3e01-49c72059a882"
 
 [compat]
 Documenter = "0.26"
-HyperDualNumbers = "4"
+DualNumbers = "0.6"
+HyperDualNumbers = "4"

docs/src/derivatives.md

@@ -85,3 +85,50 @@ Pl_dPl_d2Pl (generic function with 1 method)
 julia> Pl_dPl_d2Pl(0.5, lmax = 3)
 ([1.0, 0.5, -0.125, -0.4375], [0.0, 1.0, 1.5, 0.375], [0.0, 0.0, 3.0, 7.5])
 ```
+
+# Analytical approach for higher derivatives
+
+Legendre polynomials satisfy the differential equation 
+
+```math
+\frac{d}{dx}\left[(1-x^2)\frac{d P_n}{dx} \right] + n(n+1) P_n(x) = 0
+```
+
+We may rearrange the terms to obtain 
+
+```math
+\frac{d^2 P_n}{dx^2} = \frac{1}{(1-x^2)}\left( 2x \frac{d P_n(x)}{dx} - n(n+1)P_n{x} \right)
+```
+
+We may therefore compute the second derivative from the function and its first derivative. Higher derivatives may further be computed in terms of the lower ones.
+
+We demonstrate the second-derivative computation using the package [`DualNumbers.jl`](https://github.com/JuliaDiff/DualNumbers.jl) v0.5: 
+
+```jldoctest dual
+julia> using DualNumbers
+
+julia> x = 0.5;
+
+julia> xd = Dual(x, one(x));
+
+julia> d2Pl(x, P, dP, n) = (2x * dP - n*(n+1) * P)/(1 - x^2);
+
+julia> function d2Pl(x, n)
+           xd = Dual(x, one(x))
+           y = Pl(xd, n)
+           P, dP = realpart(y), dualpart(y)
+           d2Pl(x, P, dP, n)
+       end;
+
+julia> d2Pl(x, 20)
+32.838787646905985
+```
+
+We may check that this matches the result obtained using `HyperDualNumbers`:
+
+```jldoctest hyperdual
+julia> ε₁ε₂part(Pl(xh, 20))
+32.838787646905985
+```
+
+Unfortunately at this point, higher derivatives need to be evaluated analytically and expresed in terms of lower derivatives.

docs/src/index.md

@@ -13,10 +13,10 @@ Compute [Legendre polynomials](https://en.wikipedia.org/wiki/Legendre_polynomial
 P_\ell(x) = \left((2\ell-1) x P_{\ell-1}(x) - (\ell-1)P_{\ell - 2}(x)\right)/\ell
 ```
 
-Currently this package evaluates the standard polynomials that satisfy ``P_\ell(1) = 1``. These are normalized as 
+Currently this package evaluates the standard polynomials that satisfy ``P_\ell(1) = 1`` and ``P_0(x) = 1``. These are normalized as 
 
 ```math
-\int P_m(x) P_n(x) dx = \frac{2}{2n+1} \delta_{mn}.
+\int_{-1}^1 P_m(x) P_n(x) dx = \frac{2}{2n+1} \delta_{mn}.
 ```
 
 There are two main functions: 

test/runtests.jl

@@ -2,6 +2,11 @@ using Test
 using LegendrePolynomials
 using OffsetArrays
 using HyperDualNumbers
+using Aqua
+
+@testset "Project quality assurance" begin
+    Aqua.test_all(LegendrePolynomials)
+end
 
 import LegendrePolynomials: LegendrePolynomialIterator