From f60862267a551f469283b0465862baa31b0d9bea Mon Sep 17 00:00:00 2001
From: ludoro <ludovicobessi@gmail.com>
Date: Mon, 17 Dec 2018 22:02:33 +0100
Subject: [PATCH 1/2] First try at implementing evaluation of
 legendrePolynomial

---
 src/legendrePolynomial.jl | 9 +++++++++
 1 file changed, 9 insertions(+)
 create mode 100644 src/legendrePolynomial.jl

diff --git a/src/legendrePolynomial.jl b/src/legendrePolynomial.jl
new file mode 100644
index 00000000..a84b2b7a
--- /dev/null
+++ b/src/legendrePolynomial.jl
@@ -0,0 +1,9 @@
+function legendrePolynomial(x,n)
+    #Evaluating the legendre polynomial of degree n in the point x.
+    sum = 0
+    for k = 0:n
+        sum += (factorial(n)/(factorial(n-k)*factorial(k)))^2*(x-1)^(n-k)*(x+1)^k
+    end
+    evaluation = 1/(2^n)*sum
+    return evaluation
+end

From 88ead3e304f41536c00f19b2ed00c636cc582364 Mon Sep 17 00:00:00 2001
From: ludoro <ludovicobessi@gmail.com>
Date: Thu, 27 Dec 2018 13:46:09 +0100
Subject: [PATCH 2/2] Worked on recursion algorithm for legendre polynomial
 coefficients

---
 src/legendrePolynomial.jl | 28 +++++++++++++++++++++-------
 1 file changed, 21 insertions(+), 7 deletions(-)

diff --git a/src/legendrePolynomial.jl b/src/legendrePolynomial.jl
index a84b2b7a..97c3bac6 100644
--- a/src/legendrePolynomial.jl
+++ b/src/legendrePolynomial.jl
@@ -1,9 +1,23 @@
-function legendrePolynomial(x,n)
-    #Evaluating the legendre polynomial of degree n in the point x.
-    sum = 0
-    for k = 0:n
-        sum += (factorial(n)/(factorial(n-k)*factorial(k)))^2*(x-1)^(n-k)*(x+1)^k
+function legendrePolynomial(n)
+    #Storing the coefficients of the Legendre Polynomial up
+    # to n degree in a n+1 x n+1 matrix. We need to store all the coefficients
+    # because the loops are expression of the following recursive relation (Bonnet):
+    #(n+1)*P_{n+1}(x) = (2n+1)*x*P_{n}(x) - n*P_{n-1}(x)
+    c = zeros((n+1, n+1))
+    c[1,1] = 1.0
+
+    if n <= 0
+        return c
     end
-    evaluation = 1/(2^n)*sum
-    return evaluation
+    c[2,2] = 1.0
+
+    for i = 2:n
+        for j=0:i-1
+            c[i+1,j+1] = (- i + 1) * c[i-1,j+1] / i
+        end
+        for j=1:i
+            c[i+1,j+1] = c[i+1,j+1] + (i+i - 1) * c[i,j] / i
+        end
+    end
+    return c
 end