This document contains C++ code implementing Simpson's rule and Lagrange interpolation. It includes functions for: 1) Calculating the integral of a function using Simpson's rule 2) Filling an array with function values at points and calculating the Lagrange polynomial 3) Drawing the function graphically The main function provides menus to call the Simpson's rule or Lagrange interpolation methods and draw the results.

1. Damascus University
Faculty of Information Technology Engineering
Simpson and Lagranje Dalambair math methods

2. CODE:
#include<iostream>
#include<cmath>
#include"graphics.h"
using namespace std;
void simpson(double x0,double xn,int en, double f[100][100],double& sum1)
{
double h;
double* b= new double[en] ; // F( Xi)
h=(xn-x0)/en; // h
//Find points
cout<<"Now put for each point Xi .. its F(xi): n;"
for(int i=0; i<=en;i)++
{
cout<<"F( "<<x0+(i*h;" = )"<<)
cin>>b[i;]
}
//calculate the INTEGRATION!!!!
//double sum1;
sum1=b[0]+b[en]; // sum=f0 + fn
for(int i=1; i<en; i)++
{
if(i % 2 !=0)
sum1=sum1+4*b[i;]
else
sum1=sum1+2*b[i;]
}
sum1=sum1*(h/3) ;// I= h/3 [f0 + 4f1+ 2f2 + 4f3+ 2f4+ .... +fn]
// copy to the F array
for(int i=0;i<=en;i)++
{
f[1][i]=b[i;]
f[0][i]=x0+i*h;
}
}
void fillarr(double a[][100] , int n)
{
for(int i=0; i<=1; i++) // first row X0
for(int j=0; j<=n ; j++) //second row Y0
{

3. if( i==0)
cout<<"X "<<j;" = "<<
else
cout<<"Y "<<j;" = "<<
cin>>a[i][j;]
}
}
double p(double a[][100],int n , double x)
{
double res,sum,temp;
sum=0;
for(int i=0; i<=n; i)++
{
res=1;
for(int j=0; j<=n ; j)++
{
if(j != i)
res=res*(x-a[0][j;)]
}
temp=1;
for(int k=0;k<=n; k)++
{
if(k!=i)
temp=temp*(a[0][i]-a[0][k;)]
}
res=res/temp ;
res=res*a[1][i]; // * (y0,y1...,
sum=sum+res;
}
return sum;
}
void print(double a[][100] , int n)
{
cout<<"P"<<n<<"(x;" =)
double temp;
for(int i=0; i<=n;i)++
{
temp=1;
for(int j=0; j<=n;j)++
{
if(j!=i)

4. temp=temp*(a[0][i]-a[0][j;)]
}
temp=a[1][i]/temp;
if(temp>0 && i!=0)
cout<<"+"<<temp;
else if (temp==0)
cout<<"0;"
else
cout<<temp;
if(temp!=0)
{
for(int k=0; k<= n ; k)++
{
if(k!= i)
{
if(a[0][k]<0)
cout<<"(x + "<<fabs(a[0][k;" )"<<)]
else if(a[0][k]>0)
cout<<"(x - "<<a[0][k;" )"<<]
else
cout<<"( x;")
}
}
}
}
cout<<endl;
}
void get_xy(double wxb ,double wyb ,double wxt ,double wyt ,int vxt ,int vyt ,int vxb ,int vyb ,double wx ,double wy ,int& vx, int & vy)
{
double sx,sy;
sx=abs((vxb-vxt)/(wxt-wxb;) )
sy=abs((vyb-vyt)/(wyt-wyb;) )
vx=(vxt+( wx-wxb)*sx;)
vy=(vyb-( wy-wyb)*sy;)
}
void draw_function (double a[][100] , int n)
{
// Variables for graphics
initwindow(1000, 650, "Function Ploting"); //initilize windows
int maxx=getmaxx;)(
int maxy=getmaxy;)(
double x1,x2,y1,y2;
int nx1,nx2,ny1,ny2;

5. double x00=-70 ,xm=70, n_graph=2000; //domain of x [x00, xm]
double wyb=-49.8, wyt=50;
double dx; dx=abs(xm-x00)/n_graph;
x1=x00 ;
y1=p(a,n-1,x1;)
get_xy(x00,wyb,xm,wyt,0,0,maxx,maxy,x1,y1,nx1,ny1); // get x y after appropriate coordinates
for(int i=1; i<=n_graph; i)++
{
x2=x00+(i*dx;)
y2=p(a,n-1,x2;)
get_xy(x00,wyb,xm,wyt,0,0,maxx,maxy,x2,y2,nx2,ny2;)
line(nx1,ny1,nx2,ny2;)
nx1=nx2;
ny1=ny2;
}
outtextxy(30,0,"Approximetely Drawing");outtextxy(maxx-200,maxy-30,"Student:kinan-keshkeh;)"
*/ draw Ordinates */ setcolor(10); line(0, maxy/ 2,maxx,maxy / 2); line(maxx / 2,0,maxx / 2,maxy ;)
}
void main_menu(int & q)
{
cout<<"|----------------------------------------------------------------------|n;"
cout<<"| CHOICE MENU |n; "
cout<<"|-----|----|-----------------------------------------------------------|n;"
cout<<"| |(1)-| (Interpolation)Lagranje Dalambair 'press (1) ' |n;"
cout<<"| --|----|------------------------------------------- |n;"
cout<<"| |(2)-| (Integration )Simpson 'press (2) ' |n;"
cout<<"| --|----|------------------------------------------- |n;"
cout<<"| |(0)-| Exit 'press (0) ' |n ;"
cout<<"| --|----|------------------------------------------- |n;"
cin>>q;
}
void menu2(int & q1)
{
cout<<"|----------------------------------------------------------------------|n;"

6. cout<<"| CHOICE MENU |n; "
cout<<"|-----|----|-----------------------------------------------------------|n;"
cout<<"| |(1)-| calculate (Integration )Simpson 'press (1) ' |n;"
cout<<"| --|----|------------------------------------------- |n;"
cout<<"| |(2)-| Draw F(x) 'press (2) ' |n;"
cout<<"| --|----|------------------------------------------- |n;"
cout<<"| |(00)| to Main_Menu.. 'press (00) ' |n;"
cout<<"| --|----|------------------------------------------- |n;"
cin>>q1;
}
void menu(int & y)
{
cout<<"|----------------------------------------------------------------------|n;"
cout<<"| CHOICE MENU |n; "
cout<<"|-----|----|-----------------------------------------------------------|n;"
cout<<"| |(1)-| Put the Xi / Yi 'press (1) ' |n;"
cout<<"| --|----|------------------------------------------- |n;"
cout<<"| |(2)-| print the Lagranj Function for this points 'press (2) ' |n;"
cout<<"| --|----|------------------------------------------- |n;"
cout<<"| |(3)-| Find Pn(b) 'press (3) ' |n;"
cout<<"| --|----|------------------------------------------- |n;"
cout<<"| |(4)-| Draw F(x) 'press (4) ' |n;"
cout<<"| --|----|------------------------------------------- |n;"
cout<<"| |(0)-| to Main_Menu.. 'press (0) ' |n;"
cout<<"|-----|----|-----------------------------------------------------------|n;"
cout<<"|----------------------------------------------------------------------|n;"
cout<<"enter your choice;" :
cin>>y;
}
void main)(
{
//double v1,v2; //for draw domain [v1,v2]
cout.setf(ios::fixed); // to print just 4 numbers after the Point
cout.setf(ios::showpoint;)
cout.precision(2;)
int q; main_menu(q;)
while (q!=0)
{
switch(q)
{

7. case 1{ :
//////////////////////////////////////////////////////Lagranj/////////////////////////////////////////////////////
int n,y; double a[100][100;]
double x;
menu(y); //Print the menu and get choice y
while (y!=0)
{
switch(y)
{
case 1 : { cout<<" How many points?? n;"
cout<<"put the number of points : N= n; "
cin>>n;
fillarr(a,n-1;)
menu(y ;)
break;
}
case 2{ :
cout<<"Two numbers after POINT: ex (0.00) (!!more than 2 may not seen!!! )n;"
print(a,n-1 ;)
menu(y ;)
break;
}
case 3{ :
cout<<"Put the X0 : f(x0)....n"; cin>>x;
cout<<"P"<<n-1<<"("<<x<<") = "<<p(a,n-1,x)<<endl;
menu(y ;)
break;
}
case 4:
{
draw_function(a,n-1;)
menu(y ;)
break;
}
case 0 : { cout<<" :) :) End program My wishes :) :) !! n;"
menu(y ;)
break } ;
default:{ cout<<"Error in choice !!! n ;"
menu(y); //Print the menu and get choice y

8. break} ;
// }switch
cout<<".................................... n;"
}
main_menu(q;)
break;}//case 1
//////////////////////////////////////////////////////Simpson/////////////////////////////////////////////////////
case 2:
{
int q1; double x0,xn; int en; double sum1,sum2;
double f[100][100;]
menu2(q1;)
while (q1!=0)
{
switch(q1)
{
case 1:
{
cout<<" Put the Domain [ X0, Xn ]n;"
cout<<" X0 = "; cin>>x0; cout<<" Xn = "; cin>>xn;
cout<<" Put the NUmber of Domains u want to Divide into them(even) : n"; cin>>en;
if(en %2 !=0) // en isn't Even
{
cout<<"YOUR "<<en<<" is not Even and i will calculat by another way !! n;"
cout<<"simpson(fo -->fn-1) + simpson(fn-1-->fn)n;"
simpson(x0,xn-((xn-x0)/en),en-1,f,sum1); //fo -->fn-1
simpson(xn-((xn-x0)/en),xn,en-1,f,sum2); //fn-1-->fn
cout<<"The Integration fo -->fn-1 = I1 = "<<sum1<<endl;
cout<<"The Integration fn-1-->fn = I2 = "<<sum2<<endl;
cout<<"The Integration = I = "<<sum1+sum2<<endl;
}
else
{
simpson(x0,xn,en,f,sum1;)
cout<<"The Integration = I = "<<sum1<<endl;
}
menu2(q1;)
break;
}

9. case 2: { draw_function(f,en-1);menu2(q1); break};
case 00 : { break} ;
default:{ cout<<"Error choice!!!n"; menu2(q1); break} ;
//}switch
}
main_menu(q;)
break;
//}case 2
case 0 :
{
cout<<":) Best wishes >>>End Program>>>>>>>>>>>>>>>> n;"
break};
default:{ cout<<"Error In choice!!!!!!!!! n"; main_menu(q); break } ;
}
}
//while q != 0
system("pause;)"
}