numerical_integration.cpp

Maths/numerical_integration.cpp

@@ -0,0 +1,295 @@
+#include <iostream>
+#include <math.h>
+#include <cstdlib>
+#include <cstdio>
+#include <time.h>
+#define PI 3.14159265359
+using namespace std;
+typedef double dd;
+
+// global array to store coefficient and power of polynomial
+
+dd coeff[1000], power[1000][10];
+
+// 2d array for eqations like xsin^4xcos^5x 
+
+char equation[1000][10];
+
+int size;
+
+// If the function cannot be given as input then hardcode the function
+
+/*
+ dd EQUATION(dd x) {
+     dd value = 0;
+     value = pow(2.71828, -(x*x)/2) / sqrt(2*PI);
+     return value;
+ }
+*/
+
+// This is how the function looks like
+
+void visualise() {
+    cout << "F(x) = ";
+
+    for (int i = 0; i < size; ++i) {
+
+    	if (coeff[i] > 0)
+	        cout << "+";
+	    cout << coeff[i];
+    	for (int j = 0; j < 10; ++j) {
+
+	        if (equation[i][j] == 'x')
+	            cout << "x^"<< power[i][j];
+
+	        if (equation[i][j] == 's')
+	            cout << "sin^" <<power[i][j]<<"(x)";
+
+	        if (equation[i][j] == 'c')
+	            cout << "cos^" << power[i][j] << "(x)";
+
+	        if (equation[i][j] == 't')
+	            cout << "tan^" << power[i][j] << "(x)";
+
+	        if (equation[i][j] == 'l')
+	            cout << "log^" << power[i][j] << "(x)";
+
+	        if (equation[i][j] == 'e')
+	            cout << "e^" << power[i][j] << "x";
+
+	        if (equation[i][j] == '1')
+	            cout <<"";
+
+	    }
+	}
+}
+
+//evaluate the value of function
+dd EQUATION(dd x) {
+    dd value = 0;
+    dd factor;
+
+    for (int i = 0; i < size; ++i) {
+        factor = coeff[i];
+        for (int j = 0; j < 10; j++) {
+
+            if (equation[i][j] == 'x')
+                factor = factor*pow(x, power[i][j]);
+
+            if(equation[i][j] == 's')
+                factor = factor*pow(sin(x), power[i][j]);
+
+            if(equation[i][j] == 'c')
+                factor = factor*pow(cos(x), power[i][j]);
+
+            if(equation[i][j] == 't') {
+
+                factor = factor*pow(tan(x), power[i][j]);
+            }
+
+            if(equation[i][j] == 'e')
+                factor = factor*pow(exp(x), power[i][j]);
+
+            if(equation[i][j] == 'l') {
+                factor = factor*pow(log(x), power[i][j]);
+            }
+
+            if(equation[i][j] == '1')
+                factor = factor * 1;
+        }
+        value = value + factor;
+    }
+    return value;
+}
+//===============================================================================================================//
+
+// composite midpoint theorem implementation
+dd MIDPOINT_EQUATION(dd a, dd b, dd n) {
+    dd area = 0;
+    dd h = (b-a)/n;
+
+    for (dd i = 1; i <= n; i++) {
+        area = area + h*EQUATION(a+(i-0.5)*h);
+    }
+    return area ;
+}
+
+// composite rectangle method
+dd RECTANGULAR_EQUATION(dd a, dd b, dd n) {
+    dd area = 0;
+    dd h = (b-a) / n;
+    for (int i = 0; i < n; ++i) {
+        area = area + h*EQUATION(a+i*h);
+    }
+    return area;
+}
+
+// composite trapezoidal theorem implementation
+dd TRAPEZOIDAL_EQUATION(dd a, dd b, dd n) {
+    dd temp = 0, area = 0;
+    dd h = (b-a)/n;
+    temp = temp + EQUATION(a) + EQUATION(b);
+        for (dd i = 1; i < n; i++)
+    temp = temp + 2 * EQUATION(a + i*h);
+    area = temp * (b-a)/(2*n);
+    return area;
+}
+
+// composite simpson's theorem implementation
+dd SIMPSON_EQUATION(dd a, dd b, dd n) {
+    dd area;
+    dd temp;
+    dd h = (b-a)/n;
+    temp = EQUATION(a) + EQUATION(b);
+    dd odd_sum = 0, even_sum = 0;
+    for (int i = 1; i < n; ++i) {
+        if (i%2)
+            odd_sum += EQUATION(a + i*h);
+        else
+            even_sum += EQUATION(a + i*h);
+    }
+    odd_sum = 4 * odd_sum;
+    even_sum = 2 * even_sum;
+    area = h/3*(temp + odd_sum + even_sum);
+    return area;
+}
+
+// romberg theorem implementation
+dd ROMBERG_EQUATION(dd a, dd b) {
+    dd r[10][10]; // note here n is fixed to 10 for computation purpose you can change the size of the array r size
+    r[1][1] = 0.5 * (b-a) * (EQUATION(a) + EQUATION(b));
+    for (int i = 2; i < 10; ++i) {
+    dd area = 0;
+    for (int k = 1; k <= pow(2, i-2); ++k)
+       area = area + EQUATION(a + (2*k-1)*(b-a)/pow(2, i-1));
+        r[i][1] = 0.5 * r[i-1][1] + (b-a)/pow(2, i-1) * area;
+    }
+
+
+    for (int i = 2; i < 10; ++i)
+        for (int j = 2; j <= i; ++j)
+            r[i][j] = r[i][j - 1] + (r[i][j - 1] - r[i - 1][j - 1]) / (pow(4, j - 1) - 1);
+    return r[9][9];
+}
+
+int main() {
+
+    for (int i = 0; i < 1000; i++)
+        for (int j = 0; j < 10; j++)
+            equation[i][j] = '1';
+
+    char ch, flush;
+
+    cout << endl;
+    cout << "Program supports Integration of two terms" << endl;
+    cout << "Usage:" << endl;
+    cout << "To enter F(x) = 3sin^5(x)cos^7(x) give input 3 s 5 c 7" << endl;
+    cout << endl;
+
+    dd a, b; //enter lower and upper bound
+    int n, opt; // number of iterations
+    clock_t t; // computational time
+    dd time_taken;
+    int tan_flag;
+    while (1) {
+    	tan_flag = 0;
+        /* polynomial input*/
+
+        cout << endl;
+        cout << "Enter the number of terms in the polynomial" << endl;
+        cin >> size;
+        cout << "Enter the coefficient, function and power of the equation" << endl;
+
+        /*
+        if the polynomial is ax^2 + b^x + c;
+        enter in the format a 2 b 1 c 0;
+        */
+
+        for (int i = 0; i < size; ++i) {
+            cin >> coeff[i];
+            for (int j = 0; j < 10; ++j) {
+                flush = getchar();
+                if (flush == '\n')
+                    break;
+                cin >> equation[i][j];
+                if (equation[i][j] == 't')
+                	tan_flag = 1;
+                cin >> power[i][j];
+                // cout << equation[i][j] << endl;
+                // if (equation[i][j] == '\n')
+                    // break;
+            }
+        }
+        // Calling function to see how the function actually looks like
+        visualise();
+
+        LOOP:
+            cout << endl ;
+            cout << "Enter lower bound: \n";
+            cin >> a;
+            cout << "Enter upper bound: \n";
+            cin >> b;
+            cout << "Enter number of iterations: \n";
+            cin >> n;
+
+
+            if ((int(floor((b*2)/PI) - floor((a*2)/PI)) % 2) && tan_flag == 1) {
+                cout << "Divergent";
+                goto LOOP;
+            }
+            dd h = (b-a)/n;
+
+            cout << endl;
+            cout << "Numerical Methods of Integration: " << endl;
+
+            cout << "1. Midpoint Rule: ";
+            t = clock();
+            cout << MIDPOINT_EQUATION(a, b, n) << endl;
+            t = clock() - t;
+            time_taken = ((double)t)/CLOCKS_PER_SEC;
+            cout << "   Computation Time: " << time_taken << endl << endl;
+
+            cout << "2. Rectangle Rule: ";
+            t = clock();
+            cout << RECTANGULAR_EQUATION(a, b, n) << endl;
+            t = clock() - t;
+            time_taken = ((double)t)/CLOCKS_PER_SEC;
+            cout << "   Computation Time: " << time_taken << endl << endl;
+
+            cout << "3. Trapezoidal Rule: ";
+            t = clock();
+            cout << TRAPEZOIDAL_EQUATION(a, b, n) << endl << endl;
+            t = clock() - t;
+            time_taken = ((double)t)/CLOCKS_PER_SEC;
+            cout << "   Computation Time: " << time_taken << endl << endl;
+
+            cout << "4. Simpsons Rule: ";
+            t = clock();
+            cout << SIMPSON_EQUATION(a, b, n) << endl;
+            t = clock() - t;
+            time_taken = ((double)t)/CLOCKS_PER_SEC;
+            cout << "   Computation Time: " << time_taken << endl << endl;
+
+            cout << "5. Romberg Rule: ";
+            t = clock();
+            cout << ROMBERG_EQUATION(a, b) << endl;
+            t = clock() - t;
+            time_taken = ((double)t)/CLOCKS_PER_SEC;
+            cout << "   Computation Time: " << time_taken << endl << endl;
+
+        cout << endl;
+        cout << "Press 0 to exit" << endl;
+        cout << "Press 1 to change integral bounds" << endl;
+        cout << "Press 2 to change the function"<< endl;
+
+        cin >> opt;
+        if(opt == 1)
+            goto LOOP;
+        if(opt == 2)
+            continue;
+        else
+            exit(0);
+        }
+
+    return 0;	
+}