June 2018 Update

Task/Numerical-integration-Gauss-Legendre-Quadrature/MATLAB/numerical-integration-gauss-legendre-quadrature.m

@@ -0,0 +1,74 @@
+%Integration using Gauss-Legendre quad
+%Does almost the same as 'integral' in MATLAB
+function y=GLGD_int(fun,xmin,xmax,n)
+%fun: the intergrand as a function handle
+%xmin: lower boundary of integration
+%xmax: upper boundary of integration
+%n: order of polynomials used (number of integration ponts)
+[x_IP,weight]=GLGD_para(n);
+%assign global coordinates to the integraton points
+x_eval=x_IP*(xmax-xmin)/2+(xmax+xmin)/2;
+y=0;
+for aa=1:n
+    y=y+feval(fun,x_eval(aa))*weight(aa)*(xmax-xmin)/2;
+end
+end
+
+function [x_IP,weight]=GLGD_para(n)
+%n: the order of the polynomial
+x_IP=legendreRoot(n,10^(-16));
+weight=2./(1-x_IP.^2)./diff_legendrePoly(x_IP,n).^2;
+end
+
+%roots of the Legendre Polynomial using Newton-Raphson
+function x_IP=legendreRoot(n,tol)
+%n: order of the polynomial
+%tol: tolerence of the error
+if n<2
+    disp('No root can be found');
+else
+    root=zeros(1,floor(n/2));
+    for aa=1:floor(n/2) %iterate to find half of the roots
+        x=cos(pi*(aa-0.25)/(n+0.5));
+        err=10*tol;
+        iter=0;
+        while (err>tol)&&(iter<1000)
+            dx=-legendrePoly(x,n)/diff_legendrePoly(x,n);
+            x=x+dx;
+            iter=iter+1;
+            err=abs(legendrePoly(x,n));
+        end
+        root(aa)=x;
+    end
+    if mod(n,2)==0
+        x_IP=[-1*root,root];
+    else
+        x_IP=[-1*root,0,root];
+    end
+    x_IP=sort(x_IP);
+end
+end
+
+%derivative of the Legendre Polynomial
+function y=diff_legendrePoly(x_IP,n)
+%n: order of the polynomial
+%x_IP: coordinates of the integration points
+if n==0
+    y=0;
+else
+    y=n./(x_IP.^2-1).*(x_IP.*legendrePoly(x_IP,n)-legendrePoly(x_IP,n-1));
+end
+end
+
+%Produces Legendre Polynomials
+function y=legendrePoly(x,n)
+%n: order of polynomial
+%x: input x
+if n==0
+    y=1;
+elseif n==1
+    y=x;
+else
+    y=((2*n-1).*x.*legendrePoly(x,n-1)-(n-1)*legendrePoly(x,n-2))/n;
+end
+end