Write a program to solve first order ordinary differential equations (initial value problem) using adaptive Runge-Kutta method .

#include<stdio.h>
#include<conio.h>
#include<math.h>
#include<stdlib.h>
float fun(float x,float y)
{
    return(x*y);
}
void main()
{
    float y,x,h,e,k1,k2,k3,k4,k5,k;
    int n,i;
    clrscr();
    printf("\n enter the value of x0:");
    scanf("%f",&x);
    printf("\n enter the value of y0:");
    scanf("%f",&y);
    printf("\n enter step length h: ");
    scanf("%f",&h);
    printf("\n enter the vlaue of x at which y is needed :");
    scanf("%f",&e);
    n=(e-x)/h;
    for(i=1;i<=n;i++)
          {
            k1=h*fun(x,y);
            k2=h*fun(x+h/4,y+k1/4);
            k3=h*fun(x+(3*h)/8,y+(3*k1)/32+(9*k2)/32);
            k4=h*fun(x+(12*h)/13,y+(1932*k1/21971)+(7200*k2/2197)+(7296*k3/2197));
            k5=h*fun(x+h,y+(439*k1/216)-8*k2+(3600*k3/513)-(845*k4/4104));
            k=((25*k1/216)+(1048*k3/2565)+(2197*k4/4104)-k5/5);
            y=y+k;
            x=x+h;
            printf("the value of y at x=%f is %f ",x,y);
           }
   getch();
}

Write a program to solve first and second order ordinary differential equations (initial value problem) using Runge-Kutta fourth order method.

#include<stdio.h>
#include<conio.h>
#include<math.h>
#include<stdlib.h>
float fun(float x,float y)
{
 return(x+fabs(sqrt(y)));
}
void main()
{
 float y,x,h,e,k1,k2,k3,k4,k;
 int n,i;
 clrscr();
 printf("\n enter the value of x0:");
 scanf("%f",&x);
 printf("\n enter the value of y0:");
 scanf("%f",&y);
 printf("\n enter step length h: ");
 scanf("%f",&h);
 printf("\n enter the vlaue of x at which y is needed :");
 scanf("%f",&e);
 n=(e-x)/h;
 for(i=1;i<=n;i++)
 {
  printf("%d the iteration \n",i);
  k1=h*fun(x,y);
  k2=h*fun(x+h/2,y+k1/2);
  k3=h*fun(x+h/2,y+k2/2);
  k4=h*fun(x+h,y+k3);
  k=(k1+2*k2+2*k3+k4)/6;
  y=y+k;
  x=x+h;
  printf("\n the value of y at x=%f is %f",x,y);
  }
  getch();
  }

Write a program to solve the system of equations Ax = b using Gauss-Seidel method

#include<stdio.h>
#include<conio.h>
#include<math.h>
#include<stdlib.h>
void main()
{
 float a[20][20],x[20],e,big,temp,relerror,sum;
 int n,i,j,maxit,itr;
 char ch;
 clrscr();
 printf("enter the size of the equation:");
 scanf("%d",&n);
 top:for(i=1;i<=n;i++)
    {
    printf("\n enter the co-efficient of the equation %d and rhs:\n",i);
    for(j=1;j<=n+1;j++)
    scanf("%f",&a[i][j]);
    }
 printf("\n enter the relative error and number of iteration:");
 scanf("%f%d",&e,&maxit);
  for(i=1;i<=n;i++)
  x[i]=0;
  for(itr=1;itr<=maxit;itr++)
  {
   big=0;
   for(i=1;i<=n;i++)
   {
    sum=0;
    for(j=1;j<=n;j++)
    {
     if(j!=i)
     sum=sum+a[i][j]*x[j];
     }
    temp=(a[i][n+1]-sum)/a[i][i];
    relerror=fabs((x[i]-temp)/temp);
    if(relerror>big)
    big=relerror;
    x[i]=temp;
    }
    if(big<=e)
    {
     printf("\n converges to a solution \n");
     for(i=1;i<=n;i++)
     printf("%f\t",x[i]);
     getch();
     exit(1);
     }
    }
   printf("\n does not converge is %d iteration \n",maxit);
   printf("\n please try by interchanging any two equation \n");
   printf("make diagonal entries pivotal \n ");
   printf("\n do you want to try(y/n):");
   fflush(stdin);
   ch=getchar();
   if(ch=='y')
   goto top;
   for(i=1;i<=n;i++)
   printf("%f\t",x[i]);
   getch();
  }

Write a program to solve the system of equations Ax = b using Gauss elimination method.

#include<stdio.h>
#include<conio.h>
#include<math.h>
#include<stdlib.h>
void main()
{
       float u,e,z,a[20][20],temp,x[20],sum,max;
       int m,n,j,i,p,k,q;
       clrscr();
       printf("enter the size of the equation");
       scanf("%d",&n);
       for(i=1;i<=n;i++)
             {
              printf("\n enter the co-eficient of the equation %d and RHS:\n",i);
              for(j=1;j<=n+1;j++)
              scanf("%f",&a[i][j]);
              }
       printf("\n enter the error allowed : \n");
       scanf("%f",&e);
       for(k=1;k<=n-1;k++)
         {
              max=abs(a[k][k]);
              p=k;
        for(m=k+1;m<=n;m++)
           {
              z=fabs(a[m][k]);
        if(z>max)
            {
               max=abs(a[m][k]);
               p=m;
             }
         }
   if(max<=e)
   {
          printf("ill conditional equation:");
          getch();
          exit(1);
   }
         if(p==k)goto cont;
         for(q=k;q<=n+1;q++)
               {
                temp=a[k][q];
                a[k][q]=a[p][q];
                a[p][q]=temp;
               }
        cont:for(i=k+1;i<=n;i++)
              {
               u=a[i][k]/a[k][k];
               for(j=k;j<=n+1;j++)
               a[i][j]=a[i][j]-u*a[k][j];
               }
               }
         x[n]=a[n][n+1]/a[n][n];
         printf("\nthe value of x%d is %f \n",n,x[n]);
         for(i=n-1;i>=1;i--)
               {
                sum=0;
          for(j=i+1;j<=n;j++)
          {
           sum=sum+a[i][j]*x[j];
           }
          x[i]=(a[i][n+1]-sum)/a[i][i];
          printf("\nthe value of x%2d is %f\n",i,x[i]);
          }
    getch();
}

Write a program to solve the system of equations Ax = b in tridiagonal form using Thomas Algorithm.

#include<stdio.h>
#include<conio.h>
#include<math.h>
#define MAX_N 20
void tridge(float a[],float b[],float c[],float f[],int i,int iflag);
void main()
{
 float a[MAX_N+1],b[MAX_N+1],c[MAX_N+1],f[MAX_N+1];
 int n,i,j,iflag;
 clrscr();
 printf("\n what is the order n of system ?");
 scanf("%d",&n);
 printf("\n give b[1],c[1],rhs[1] for equation 1:");
 scanf("%f%f%f",&b[1],&c[1],&f[1]);
 for(i=2;i<=n-1;i++)
 {
 printf("\n give a[%d],b[%d],c[%d],rhs[%d] for the equation %d\n",i,i,i,i,i);
 scanf("%f%f%f%f",&a[i],&b[i],&c[i],&f[i]);
 }
 printf("\n give a[n],b[n],rhs[n] for equation %d:",n);
 scanf("%f%f%f",&a[n],&b[n],&f[n]);
 iflag=0;
 tridge(a,b,c,f,n,iflag);
 printf("solution");
 printf("\ni a[i]");
 for(j=1;j<=n;j++)
 printf("\n%d\t%f",j,f[j]);
 printf("\n");
 getch();
 return;
 }
 void tridge(float a[],float b[],float c[],float f[],int n,int iflag)
 {
  const float zero=0.0;
  int j;
  if(iflag==0)
  {
   for(j=2;j<=n;j++)
   {
    if(b[j-1]==zero)
    {
     return;
     }
    a[j]=a[j]/b[j-1];
    b[j]=b[j]-a[j]*c[j-1];
    }
    if(b[n]==zero)
    {
     return;
     }
    }
    for(j=2;j<=n;j++)
    f[j]=f[j]-a[j]*f[j-1];
    f[n]=f[n]/b[n];
    for(j=n-1;j>=1;j--)
    f[j]=(f[j]-c[j]*f[j+1])/b[j];
    return;
    }

Write a program to find the integral of a function using Simpson’s 1/3rd and 3/8th rule using switch case.

#include<stdio.h>
#include<conio.h>
#include<math.h>
float sim_1(float,float,int);
float sim_2(float,float,int);
int i;
float sum;
float fun(float x)
{
 return(1/(1+pow(x,2)));
 }
 void main()
 {
  float x0,x1,sum,result;
  int n,cho=0;
  clrscr();
  printf("enter the lower and upper limit \n");
  scanf("%f%f",&x0,&x1);
  printf("enter number of intervals:\n");
  scan:scanf("%d",&n);
  if(cho==0)
  {
   top:printf("\t enter choice \n");
   printf("\n 1.for simpsons 1/3 rule\n");
   printf("\n 2.for simpsons 3/8 rule\n");
   scanf("%d",&cho);
   }
   switch(cho)
   {
    case 1:if(n%2==0)
    result=sim_1(x0,x1,n);
    else
    {
    printf("wrong choice of interval");
    printf("please enter even number");
    goto scan;
    }
    break;
    case 2:if(n%3==0)
    result=sim_2(x0,x1,n);
    else
    {
     printf("wrong choice of intervals \n ");
     printf("\n please enter a multiple of 3 \n");
     goto scan;
     }
     break;
  default:printf("\n wrong choice enter again:");
    goto top;
  }
  printf("\n the result=%f",result);
  getch();
  }
  float sim_1(float x0,float x1,int n)
  {
   float result,h;
   h=(x1-x0)/n;
   sum=fun(x0)+fun(x1);
   for(i=1;i<n;i++)
   {
    if(i%2==0)
    sum=sum+2*fun(x0+i*h);
    else
    sum=sum+4*fun(x0+i*h);
   }
   result=sum*(h/3);
   return(result);
   }
   float sim_2(float x0,float x1,int n)
  {
   float result,h;
   h=(x1-x0)/n;
   sum=fun(x0)+fun(x1);
   for(i=1;i<n;i++)
   {
    if(i%3==0)
    sum=sum+2*fun(x0+i*h);
    else
    sum=sum+3*fun(x0+i*h);
   }
   result=sum*(3*h/8);
   return(result);
   }

Write a program to find the integral of a function using Trapezoidal rule.

#include<stdio.h>
#include<conio.h>
#include<math.h>
float trap(float,float,int);
int i;
float sum;
float fun(float);
void main()
{
 float x0,x1,res;
 int n;
 clrscr();
 printf("\n enter the lower and upper limits :");
 scanf("%f%f",&x0,&x1);
 printf("\n enter the number of intervals :");
 scanf("%d",&n);
 res=trap(x0,x1,n);
 printf("\n TRAPEZOIDAL RULE =%f \n",res);
 getch();
 }
 float trap(float x0,float x1,int n)
 {
  float h,result;
  h=(x1-x0)/n;
  sum=fun(x0)+fun(x1);
  for(i=1;i<n;i++)
  sum=sum+2*fun(i*h);
  result=(sum*h)/2;
  return(result);
  }
  float fun(float x)
  {
   return(1/(1+pow(x,2)));
  }

Write a program to find the simple/multiple roots of f (x) = 0 using Newton – Raphson method.

#include<stdio.h>
#include<conio.h>
#include<math.h>
main()
{
 float a,x1,x2;
 int i;
 float func(float);
 float func1(float);
 clrscr();
 printf("\n enter the initial value");
 scanf("%f",&a);
 x1=a;
 printf("a=%f",x1);
 x2=(x1-(func(x1)/func1(x1)));
 printf("\nx2=%f",x2);
 for(i=1;x1!=x2;i++)
 {
  printf("\n(%d)%f",i,x2);
  x1=x2;
  x2=x1-(func(x1)/func1(x1));
  }
  printf("\n the root is %f converge is %d approximately",x2,i);
  getch();
  return(0);
  }
  float func(float x)
  {
   return(pow(x,3)+(-2*x-5));
   }
   float func1(float x)
   {
    return((3*pow(x,2)-2));
   }

Write a program to find the roots of an equation f (x) = 0 using Bisection method.

#include<stdio.h>
#include<conio.h>
#include<math.h>
float fun(float x)
{
     return(pow(x,3)-x-1);
}
 void main()
 {
         float a,b,eps,x;
         int i=0;
         clrscr();
         printf("\n enter the lower and upper limit ");
         scanf("%f%f",&a,&b);
         printf("\n enter the epsilon value");
         scanf("%f",&eps);
         if((fun(a)*fun(b))>0)
         printf("\n starting value is unsuitable ");
         else
         while(fabs((b-a)/b)>eps)
                {
                    x=(a+b)/2;
                    i++;
                    if((fun(x)*fun(a))>0)
                    a=x;
                    else
                    b=x;
                    }
        printf("\n solution converges to a root \n");
        printf("\n number of iterationo=%d\n",i);
        printf("%f\n",x);
        getch();
   }