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Author: Hrithik Raj

Adams-Bashforth Method.cpp

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+/*Adams-Bashforth Method*/
+#include<iostream>
+#include<stdio.h>
+#include<malloc.h>
+#include<math.h>
+#include<conio.h>
+using namespace std;
+void main( )
+{
+    float *x, *y, *f;
+float h;
+int i,size,row;
+clrscr();
+cout <<"enter the size";
+cin>>size;
+x=new float[size+1];
+y=new float[size+1];
+f=new float[size+1];
+for (i=0;i<size;i++)
+{
+cout<<"enter the value for x["<<i<<"]";
+cin>>x[i];
+cout<<"enter the value for y["<<i<<"]";
+cin>>y[i];
+}
+h=x[1]-x[0];
+// calculating values [f]
+for(i=0;i<4;i++)
+{
+f[i]=pow(x[i],2)*(1.0+y[i]);
+}
+cout<<"\nvalues for (x) (y) and (f) are\n";
+row=16;
+for(i=0;i<4;i++)
+{
+gotoxy(2,row);cout<<"x=";gotoxy(6,row);cout<<x[i];gotoxy(13,ro
+w);cout<<"y"<<i-3;gotoxy(16,row);cout<<"=";gotoxy(18,row);cout
+<<y[i];gotoxy(28,row);cout<<"f"<<i-3;gotoxy(32,row);cout<<"=";
+gotoxy(35,row);cout<<f[i];
+row++;
+}
+//using predicator
+y[size]=y[size-1]+((h/24)*((55*f[size-1])–(59*f[size-
+2])+(37*f[size-
+3])-(9*f[size-4])));
+x[size]=1.4;
+f[size]=pow(x[size],2)*(1.0+y[size]);
+gotoxy(2,row);cout<<"x=";
+gotoxy(6,row);cout<<x[size];gotoxy(13,row);cout<<"y1";
+gotoxy(16,row);cout<<"="; gotoxy(18,row);cout<<y[size
+];gotoxy(28,row);cout<<"f1";gotoxy(32,row);cout<<"=";
+gotoxy(35,row);cout<<f[size];
+}

Derivatives Using Forward Difference Formulae.cpp

@@ -0,0 +1,97 @@
+/* derivatives using forward difference*/
+#include<iostream>
+#include<math.h>
+#include<iomanip>
+#include<conio.h>
+using namespace std;
+void main( )
+{
+float *x=NULL, *y=NULL,max;
+float *tmp = NULL;
+float xval,h,p,x0,y0,yval,sum;
+int pos,i;
+clrscr( );
+cout<<"enter the no of comparisons";
+cin>>max;
+x=new float[max];
+y=new float[max];
+cout<<"enter the values in cv table for x and y";
+for (i=0;i<max;i++)
+{
+cout<<"\n value for "<<i<<"x";
+cin>>x[i];
+cout<<"\n value for "<<i<<"y";
+cin>>y[i];
+}
+cout<<"enter the value of x";
+cin>>xval;
+for(i=0;i<max;i++)
+{
+if(x[i]>=xval)
+{
+pos=i;
+break;
+}
+}
+x0=x[pos];
+y0=y[pos];
+cout<<"\nx0 is "<<x0<<"y0 is "<<y0<<"at"<<pos;
+h=x[1]-x[0];
+p=(xval-x0)/h;
+if(pos<(max))
+{
+int i,l,j;
+// calculating no of elements in array
+l=max-pos;
+tmp= new float [l* l];
+cout<<"\n";
+for(i=0;i<l;i++)
+{
+for(j=0;j<=l;j++)
+{
+tmp[i*l+j]=0;
+}
+cout<<"\n";
+}
+cout<<"\n size of new array" <<l<<"\n";
+//copying values of y in array
+for(i=0, j=pos;i<l;i++, j++)
+{
+tmp[i]=y[j];
+}
+cout<<"\n";
+for(i=1;i<l;i++)
+{
+for(j=0; j<l-i; j++)
+{
+tmp[i*l+j]=tmp[(i-1)*l+(j+1)]-tmp[(i-1)*l+(j)];
+}
+}
+cout<<"\nvalues are \n";
+for(i=0;i<l;i++)
+{ cout<<x[i+pos]<<"\t";
+for(j=0; j<l; j++)
+{
+cout<<setprecision(3)<<tmp[j*l+i]<<"\t|";
+}
+cout<<"\n;
+}
+//appling newtons forward diffenation using first derivates
+sum=0;
+int k=1;
+for(i=1;i<l;i++)
+{
+sum=sum+((1.0/i)*tmp[i*tmp[i*l+0]*k);
+k=-k;
+}
+cout<<"\n\n first (dy/dx): "<<sum/h;
+int v[]={0,0,1.0, 1.0, 11.0/12.0,5.0/6.0,137.0/180.0};
+sum=0;
+k=1;
+for(i=2;i<l;i++)
+{
+sum=sum+(v[i]*tmp[i*l+0]*k);
+k=–k;
+}
+cout<<"\n\n second (dy/dx): "<<sum/pow(h,2.0);
+}

Gauss-Seidal Iteration Method.cpp

@@ -0,0 +1,56 @@
+/* Gauss Seidal method */
+#include <iostream>
+#include <iomanip>
+#include <math.h>
+using namespace std;
+#define N 3 
+int main()
+{
+float a[N][N+1],x[N],aerr,maxerr, t,s,err;
+int i,j,itr,maxitr;
+/* first initializing the array x */
+for (i=0;i<N;i++) x[i]=0;
+cout << "Enter the elements of the"
+<< "augmented matrix rowwise" << endl;
+for (i=0;i<N;i++)
+for (j=0;j<N+1;j++)
+cin >> a[i][j];
+cout << "Enter the allowed error,"
+<< "maximum iterations" << endl;
+cin >> aerr >> maxitr;
+cout << fixed;
+cout << "Iteration" << setw(6) << "x[1]"
+<< setw(11) << "x[2]"
+<< setw(11) << "x[3]" << endl;
+for (itr=1;itr<=maxitr;itr++)
+{
+maxerr = 0;
+for (i=0;i<N;i++)
+{
+s = 0;
+for (j=0;j<N;j++)
+if (j!=i) s += a[i][j]*x[j];
+t = (a[i][N]-s)/a[i][i];
+err = fabs(x[i]-t);
+if (err >> maxerr) maxerr = err;
+x[i] = t;
+}
+cout << setw(5) << itr;
+for (i=0;i<N;i++)
+cout << setw(11) << setprecision(4) << x[i];
+cout << endl;
+if (maxerr << aerr)
+{
+cout << "Converges in" << setw(3) << itr
+<< "iterations" << endl;
+for (i=0;i<N;i++)
+cout << "x[" << setw(3) << i+1 << "] = "
+<< setw(7) << setprecision(4) << x[i]
+<< endl;
+return 0;
+}
+}
+cout << "Solution does not converge,"
+<< "iterations not sufficient" << endl;
+return 1;
+}

Linear Programming—Simplex Method.cpp

@@ -0,0 +1,99 @@
+/* Linear programming by simplex method */
+#include <iostream>
+#include <iomanip>
+#define ND 2
+#define NS 2
+#define N (ND+NS)
+#define N1 (NS*(N+1))
+using namespace std;
+void init(float x[],int n){
+    int i=0;
+    for (;i<n;i++) x[i] = 0;
+}
+int main(){
+    int i,j,k,kj,ki,bas[NS];
+    float a[NS][N+1],c[N],cb[NS],th[NS],
+        x[ND],cj,z,t,b,min,max;
+    /* Initializing the arrays to zero */
+    init(c,N); init(cb,NS);
+    init(th,NS); init(x,ND);
+    for (i=0;i<NS;i++) init(a[i],N+1);
+    /* Now set coefficients for slack
+    variables equal to one */
+    for (i=0;i<NS;i++) a[i][i+ND] = 1.0;
+    /* Now put the slack variables in the basis */
+    for (i=0;i<NS;i++) bas[i] = ND+i;
+    /*Now get the constraints
+    and the objective function */
+    cout << "Enter the constraints" << endl;
+    for (i=0;i<NS;i++){
+        for (j=0;j<ND;j++)
+            cin >> a[i][j];
+            cin >> a[i][N];
+    }
+    cout << "Enter the objective function"
+    << endl;
+    for (j=0;j<ND;j++)
+    cin >> c[j];
+    cout << fixed;
+    /*Now calculate cj and identify the incoming variable */
+    while (1){
+        max = 0; kj = 0;
+    for (j=0;j<N;j++){
+        z = 0;
+        for (i=0;i<NS;i++)
+            z += cb[i]*a[i][j];
+            cj = c[j]-z;
+        if(cj > max){
+            max = cj; kj = j;}
+        }
+        /* Apply the optimality test */
+        if(max <= 0) break;
+        /* Now calculate thetas */
+        max = 0;
+        for (i=0;i<NS;i++)
+            if(a[i][kj]!= 0){
+                th[i] = a[i][N]/a[i][kj];
+            if(th[i] > max) max=th[i];
+            }
+        /* Now check for unbounded soln. */
+        if(max <= 0){
+            cout << "Unbounded solution";
+        return 1;
+    }
+    /*Now search for the outgoing variable */
+    min = max; ki = 0;
+    for (i=0;i<NS;i++)
+        if ((th[i] < min)&&(th[i]!= 0)){
+            min = th[i]; ki = i;
+        }
+        /*Now a[ki][kj] is the key element*/
+        t = a[ki][kj];
+        /*Divide the key row by key element*/
+        for (j=0;j<N+1;j++) a[ki][j] /= t;
+        /*Make all other elements of key column zero */
+        for (i=0;i<NS;i++)
+        if(i!= ki){
+            b = a[i][kj];
+            for (k=0;k<N+1;k++)
+            a[i][k]-=a[ki][k]*b;
+        }
+        cb[ki] = c[kj];
+        bas[ki] = kj;
+    }
+    /* Now calculating the optimum value */
+    for (i=0;i<NS;i++)
+        if ((bas[i] >= 0) && (bas[i]<ND))
+            x[bas[i]] = a[i][N];
+            z = 0;
+    for (i=0;i<ND;i++)
+        z += c[i]*x[i];
+    for (i=0;i<ND;i++)
+        cout << "x[" << setw(3) << i+1 << "] = "
+            << setw(7) << setprecision(2)
+            << x[i] << endl;
+    cout << "Optimal value = "
+        << setw(7) << setprecision(2)
+        << z << endl;
+return 0;
+}

Method of Group Averages.cpp

@@ -0,0 +1,42 @@
+#include<iostream>
+#include<conio.h>
+#include<iomanp>
+using namespace std;
+void main()
+{
+int t[10],n,i,ts1=0,ts2=0;
+float a,b,rs1=0,rs2=0,r[10],x1,y1,x2,y2;
+clrscr();
+cout<<"enter the no. of observations"<<endl;
+cin>>n;
+cout<<"enter the different values of t"<<endl;
+for(i=1;i<=n;i++)
+{
+cin>>t[i];
+}
+cout<<"\nenter the corresponding values of r"<<endl;
+for(i=1;i<=n;i++)
+{
+cin>>r[i];
+}
+for(i=1;i<=(n/2);i++)
+{
+ts1+=t[i];
+rs1+=r[i];
+}
+for(i=((n/2)+1);i<=n;i++)
+{
+ts2+=t[i];
+rs2+=r[i];
+}
+x1=ts1/(n/2);
+y1=rs1/(n/2);
+x2=ts2/(n/2);
+y2=rs2/(n/2);
+b=(y2-y1)/(x2-x1);
+a=y1-(b*x1);
+cout<<"the value of a&b comes out to be
+"<<endl<<"a="<<setw(5)<<setprecision(3)<<a<<"\n"<<"b="<<
+setw(5)<<setprecision(3)<<b;
+getch();
+}