This is a C++ program which uses polynomial interpolation to approximate the graph of the function y=sinx. Radhika Prasad

1. #include<iostream.h>
#include<conio.h>
#include<graphics.h>
#include<math.h>
#define pi 3.1415926
const int n = 5; //number of input data points
double sum1 = 0, sum2 = 0, coeff = 1, p, X1, X2;
int h;
int factorial(int n)
{
int x=1;
while(n>=1)
{
x=x*n;
n--;
}
return x;
}
void forward_diff_table(double Y[][n])
{
int i, j;
for(j=1; j<n; j++) //compute values in the table
{
for(i=0; i<n-j; i++)
{
Y[i][j] = Y[i+1][j-1] - Y[i][j-1];
}
}
}
double forward_interpolation(double Y[][n])
{
double sum = 0;
for(int i=1; i<n; i++)
{
coeff = 1;
for(int j=1; j<=i; j++)
{
coeff = coeff*(p + 1 - j);
}
sum = sum + (coeff*Y[0][i])/factorial(i); //sum gives the
interpolated value at a given point
}
return sum;
}
void main()
{
clrscr();
double x[n]={10, 20, 30, 40, 50}; //x: input values in degrees
double Y[n][n]={{0.1736},{0.342},{0.5},{0.6428},{0.766}}; //Y stores
output values corresponding to the input AS WELL AS the difference table

2. h = x[1] - x[0]; //step size
forward_diff_table(Y);
int gdriver = DETECT, gmode;
initgraph(&gdriver, &gmode, "C:turboc3bgi"); //initialising
graphics
int max_x, max_y; //max_x = 639, max_y = 479
max_x = getmaxx();
max_y = getmaxy();
setcolor(15); //white colour
line(max_x/2,0 , max_x/2,max_y); //y axis
line(0,max_y/2 , max_x,max_y/2); //x axis
//plotting interpolating polynomial
setcolor(4); //red colour
setlinestyle(0,0,3);
int comp_y1, comp_y2;
for(int comp_x=0; comp_x<=max_x; comp_x++)
{
X1 = (comp_x - max_x/2.0)*360.0/(max_x/2.0); //cartesian x
(coordinate transformation from computer to cartesian)
p = (X1 - x[0])/h;
sum1 = forward_interpolation(Y) + Y[0][0]; //cartesian y
comp_y1 = max_y * (-sum1 + 3.0)/6.0; //computer y
(coordinate transformation from cartesian to computer)
X2 = (comp_x + 1 -max_x/2.0)*360.0/(max_x/2.0);
p = (X2 - x[0])/h;
sum2 = forward_interpolation(Y) + Y[0][0];
comp_y2 = max_y * (-sum2 + 3.0)/6.0;
line(comp_x,comp_y1 , comp_x+1,comp_y2); //drawing line
between two very very close points
}
//y = sinx
setcolor(14); //yellow colour
int X, y1, y2; //computer
coordinates
double cartesian_x1, cartesian_y1, cartesian_x2, cartesian_y2;
//cartesian coordinates
for(X=0; X<=max_x; X++)
{
cartesian_x1 = (X - max_x/2.0)*(2*pi)/(max_x/2.0); //coordinate
transformations (X,y1,y2: computer coordinates)
cartesian_y1 = sin(cartesian_x1); //f(x) = sinx
y1 = max_y*(-cartesian_y1 + 3.0)/6.0; //coordinate
transformations
cartesian_x2 = (X + 1 - max_x/2.0)*(2*pi)/(max_x/2.0);

3. cartesian_y2 = sin(cartesian_x2);
y2 = max_y*(-cartesian_y2 + 3.0)/6.0;
line(X,y1, X+1,y2); //draw line b/w (x,sinx) &
(x+1,sin(x+1))
}
setlinestyle(0,0,1);
setcolor(15); line(500,30 , 520,30); outtextxy(530,25,
"axes");
setlinestyle(0,0,3);
setcolor(14); line(500,60 , 520,60); outtextxy(530,55, "y =
sinx");
setcolor(4); line(500,90 , 520,90); outtextxy(530,85,
"interpolating"); outtextxy(530,95, "polynomial");
setcolor(1); fillellipse(502,120,2,2); outtextxy(520,115,
"input data"); outtextxy(520,125, "points");
setcolor(1);
for(int i=0; i<5; i++)
fillellipse(max_x*(x[i]+360)/(2.0*360), max_y*(-Y[i][0]+3.0)/6.0,
2,2); //plotting input data points
getch();
cleardevice();
}