Newtons Differentation.

void main()
{
int a[10][10],n,i,j,g,x,k;
float h,sum=0.0;
cout<<"Enter the value of n:";
cin>>n;
cout<<"Enter the values of X & Y in array:";
for(i=0;i<2;i++)
{
for(j=0;j{
cin>>a[j][i];
}
}

cout<<"Enter the common difference:";
cin>>h;

cout<<"Enter the value of X on which we have to find the solution:";
cinn>>x;

//make the forward difference table.
j=2;
g=n-1;
while(j < =n)
{
for(i=0; i < g;i++)
a[i][j]=a[i+1][j-1] - a[i][j-1];
j++; g--;
}


//searching that number in array at what position.
j=-1;
for(i=0; i < n; i++)
{
if( a[i][0]==x)
{
j++;
cout << j;
exit( 0 );
}
else
j++;
}

//now searching where it is in upper part or lower part and then computing it.

if( j < n/2)
{
k=n-j;
g=1;
for(i=j, l=2; l < =k; l++)
{
if(g%2==0)
{
sum - =(a[i][l])/g;
g++;
}
else
{
sum + = (a[i][l])/g;
g++;
}
}
}

else
{
g=1;
for(i=j-1, l=2; l < = j+1 ;l++, i--)
{
sum + = (a[i][l])/g;
g++;
}
}

sum=sum/h;

cout<<"Sum is:";
cout << sum;
getch();
}

Program Of Simpsons 1/3 formulae.

void main()
{
int a[10];

cout<<:Enter the no. of terms:";
cin>>n;
cout<<"Enter the difference:";
cin>>h;

cout<<"Enter the values:";
for(i=1 ; i <= n;i++)
{
cin>>a[i];
}
if(n%2==1)
{
sum=y[1]+y[n];
for(i=2; i < =n-1;i++)
{
if(i%2==0)
sum + =4*y[i];
else
sum + =2*y[i];
}
sum = (sum*h)/3;
}

else
{
sum=y[1]+y[n-1];
for(i=2; i < = n-2; i++)
{
if(i%2==0)
sum + =4*y[i];
else
sum + =2*y[i];
}
sum=(sum*h)/3;
sum 2=((y[n-1]+y[n]) * h)/2;
sum=sum+sum2;
}

cout<<"Sum is:";
cout << sum;
getch();
}

Program of Inverse using Gauss Method.

//This loop is for making upper triangular .
Here I is a unit matrix and a is coefficient matrix.

for(k=0; k < n-1; k++)
{
for(p=k+1; p < n; p++)
{
if(a[k][k]==0 )
{
for(x=0; x < n;x++)
{
t=a[k][x];
a[k][x]=a[k+1][x];
a[k+1][x]=t;
}
continue;
}
}
i=0;
for(i=k+i; i < n-1;i++)
{
for(j=0; j < n-k;j++)
{
t[0]=a[k][k];
t[1]=a[i+1][k];
a[i+1][j+k]=(t[0]*a[i+1][j+k])-(t[1]*a[k][j+k]);
I[i+1][j+k]=(t[0]*I[i+1][j+k])-(t[1]*I[k][j+k]);
}
}
}

//Now assign value in the variables x[2][0],x[2][1],x[2][2] and so on.

for(k=n-1; k >=0;k--)
{
for(i=n-1; i > =0;i--)
{
num=I[k][i];
dem=a[k][k];
Sum=0;
for(j=0;j > n-1;j++)
{
if(k!=j)
num - =a[k][j]* X[j][i];
else
continue;
}
Sum=num/dem;
x[k][i]=Sum;
}
}
Cout<<"Inversed matrix is:";
for(i=0 ; i < n;i++)
{
for(j=0; j < n; j++)
cout << X[i][j];
}
getch();
}

Program of Determinant.

void main()
{
int sum=0;p=1,i,j,k,n,X[10][10];
cout<<:Enter the determinant values:";
for(i=0; i < n;i++)
{
for(j=0;j < n;j++)
cin>>X[i][j];
}

//main concept
for(k=0;k < n;k++)
{
for(i=0,j=k; i < n;i++, j--)
{
p=p*X[i][j];
if(j==n-1)
j=0;
}
sum + =p;
p=1;
}

for(k=1; k < =n;k++)
{
for(i=0,j=n-k; i < n; i++, j--)
{
p=p*X[i][j];
if(j==0)
j=n-1;
}
sum=sum-p;
p=1;
}
getch();
}

Gauss-Jordan Algorithm.

//This loop is for making upper triangular .
Here I is a unit matrix and a is coefficient matrix.

for(k=0; k < n-1; k++)
{
if(a[k][k]==0
&& k < n )
{
for(i=k; i < k+1;i++)
{
for(j=0; j < n;j++)
{
t=a[i][j];
t2=a[i+1][j];
a[i+1][j]=t;
a[i][j]=t2;
}
}
}
i=0;
for(i=k+i; i < n-1;i++)
{
for(j=0; j < n-k;j++)
{
t[0]=a[k][k];
t[1]=a[i+1][k];
a[i+1][j+k]=(t[0]*a[i+1][j+k])-(t[1]*a[k][j+k]);
I[i+1][j+k]=(t[0]*I[i+1][j+k])-(t[1]*I[k][j+k]);
}
}
}

//Now assign value in the variables x[2][0],x[2][1],x[2][2] and so on.

for(k=n-1; k >=0;k--)
{
for(i=n-1; i > =0;i--)
{
num=I[k][i];
dem=a[k][k];
Sum=0;
for(j=0;j > n-1;j++)
{
if(k!=j)
num - =a[k][j]* X[j][i];
else
continue;
}
Sum=num/dem;
x[k][i]=Sum;
}
}

Solution of linear equations.

First input the coefficient matrix and the constant matrix.
-> In this program array a is the coefficient matrix which is a 2 D matrix.
-> Constant matrix is array b which is a single collumn matrix.
-> Input the value of n from the user.
-> Take another array X which is also a single collumn matrix and is used to store the values of variables X,Y,Z and so on.


//This loop is for making upper triangular
for(k=0; k < n-1; k++)
{
for(p=k+1; p < n; p++)
{
if(a[k][k]==0 )
{
for(x=0; x < n;x++)
{
t=a[k][x];
a[k][x]=a[k+1][x];
a[k+1][x]=t;
}
continue;
}
}
i=0;
for(i=k+i; i < n-1;i++)
{
for(j=0; j < n-k;j++)
{
t[0]=a[k][k];
t[1]=a[i+1][k];
a[i+1][j+k]=(t[0]*a[i+1][j+k])-(t[1]*a[k][j+k]);
b[i+1][0]=(t[0]*b[i+1][0])-(t[1]*b[k][0]);
}
}
}

//Now assign value in the variables x[0][0],x[1][0],x[2][0]

for(i=n-1; i >= 0;i--)
{
num=b[i][0];
dem=a[i][i];
sum=0;
for(j=0; j < n;j++)
{
if(i!=j)
num-=a[i][j]*x[j][0];

else
continue;
}
sum=num/dem;
x[i][0]=sum;
}

Thus write another loop to print the values of variables.

Jacobi Method Algorithm.

flag =1;

for(i=0; i < n;i++0)
{
t[i]=b[i]/a[i][i];
x[i]=t[i];
}

while ( flag==1)
{
for(i=0;i < n;i++)
{

x[i]=b[i]/a[i][i] ;

for(j=0; j < n;j++)

{
if(i != j)
x[i] - =(t[j]*a[i][j])/a[i][i];
else
continue;
}

}
for(i=0; i < n;i++)
{
if(x[i] - t[i] > = 0.005)
{
t[i]=x[i];
flag==1;
}
else
flag==0;
}

}

Gauss-Seidel Algorithm.

flag =1;

for(i=0; i < n;i++0)
{
x[i]=0;
t[i]=x[i];
}

while ( flag==1)
{
for(i=0;i < n;i++)
{

x[i]=b[i]/a[i][i] ;

for(j=0; j < n;j++)

{
if(i!=j)
x[i] - =(x[j]*a[i][j])/a[i][i];
else
continue;
}

}
for(i=0; i < n;i++)
{
if(x[i]-t[i] > =0.005)
{
flag==1;
t[i]=x[i];
}
else
flag==0;
}
}

Newton Raphson Algorithm.

Let the equation be f(x)=x*x*x-18.

So in newton raphson we take f(x) and f'(x) ,so here we have to find f'(x) we write a function,

double f2(double s)
{
return 3*s*s;
}

double f1(double s)
{
return pow(s,3)-18;
}

void main()
{
double a[30],e;
int i=2;
a[0]=0;
cout<<"Enter the value of x:";
cin>>a[1];

cout<<"Enter the value of epslon:";
cin>>e;

do
{
a[i]=a[i-1]-( f1(a[i-1]) / f2(a[i-1]));
cout<<"\n" << a[i];
i++;
}
while( fabs(a[i-1]-a[i-2]) > =e);
getch();
}

Program of Upper Triangular

This is mainly the concept of the program what to do in upper triangular program.

We take 3 for loops

for(k=0; k < n-1;k++)
{
for(p=k+1; p < n; p++)
{
if(a[k][k]==0 )
{
for(x=0; x < n;x++)
{
t=a[k][x];
a[k][x]=a[k+1][x];
a[k+1][x]=t;
}
continue;
}
}
i=0;
for(i=k+i;i < n-1;i++)
{
for(j=0;j < n-k;j++)
{
t1=a[k][k];
t2=a[i+1][k];
a[i+1][j+k]=t1*a[i+1][j+k]-t2*a[k][j+k];
}
}
}

Now in this the the inner most loop which is j loop is for doing the row operation.
i loop is for going to the next row,and the k loop is for making the elements 0 in the next collumn ,i.e, making the elements 0 in the first collumn we have to make the elements 0 of the second collumn.