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exampletesting32intervals.m

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+%Ka Wa Yip (github: kwyip)
+%exp(-x).*cos(2*x) from 0 to 2pi divided into 32 intervals
+function exampletesting32intervals
+fprintf('testingforover32intervals\n');
+f = @(x)exp(-x).*cos(2*x);
+a = 0;
+b = 2*pi;
+
+fprintf('This is composite trapezoid rule over 32 intervals\n');
+n = 32;
+h = (b - a)/n;
+trapezoid = 0.5*(f(a) + f(b));
+for i = 1:n-1
+    x = a + i*h;
+    trapezoid = trapezoid + f(x);
+end
+trapezoid = trapezoid*h;
+fprintf('The integral under trapezoid is %f.\n', trapezoid);
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+fprintf('This is composite midpoint rule over 32 intervals\n');
+n = 32;
+h = (b - a)/n;
+midpoint = 0;
+m = a + 0.5*h;
+for i = 1:n
+    midpoint = midpoint + h*f(m);
+    m = m + h;
+end
+fprintf('The integral under midpoint is %f.\n', midpoint);
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+fprintf('This is composite three-point Gaussian over 32 intervals\n');
+n = 32;
+h = (b - a)/n;
+t0 = -sqrt(3/5);
+t1 = 0;
+t2 = +sqrt(3/5);
+A0 = 5/9;
+A1 = 8/9;
+A2 = 5/9;
+
+ai = a;
+int = 0.0;
+for k = 1:n
+    bi = ai + h;
+    inti = 0.0;
+    inti = A0 * f(0.5*(bi - ai)*t0 + 0.5*(bi + ai)) +...
+        A1 * f(0.5*(bi - ai)*t1 + 0.5*(bi + ai)) +...
+        A2 * f(0.5*(bi - ai)*t2 + 0.5*(bi + ai));
+    inti = 0.5*(bi - ai)*inti;
+    ai = bi; %reset ai for next loop
+    int = int + inti;
+end
+
+fprintf('The integral under three-point Gaussian is %f.\n', int);
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+fprintf('This is composite Simpson 13 rule over 32 intervals\n');
+n = 32;
+h=(b-a)/n; 
+xi=a:h:b;
+csimpson = h/3*(f(xi(1))+2*sum(f(xi(3:2:end-2)))+4*sum(f(xi(2:2:end)))+f(xi(end)));
+fprintf('The integral under Simpson 13 rule is %f.\n',csimpson);
+
+fprintf('The ideal solution is 0.199627.\n')
+figure
+xrange = linspace(0,2*pi,100);
+fplot(f,'-',[0 2*pi])
+
+title('Plotting the function $e^{-x} \cos(2x)$','Interpreter','latex')
+xlabel('x')
+ylabel('y')