Numerical Method Problems done in Pulchowk Campus (063)

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curve fitting:

Code:

# include
# include
# include
# include
void main()
{
int n,i,j;
float x[10],y[10],sumx=0,sumy=0,sumx2=0,sumxy=0,a,b,x0,fx;

clrscr();
printf("\nEnter the no of points:");
scanf("%d",&n);

printf("\nEnter the points(x,y):\n");
for(i=0;i
scanf("%f,%f",&x[i],&y[i]);

for(i=0;i
{
sumx+=x[i];
sumx2+=x[i]*x[i];
sumy+=y[i];
sumxy+=x[i]*y[i];
}

printf("\nsumx=%f\tsumx2=%f\tsumy=%f\tsumxy=%f\n",sumx,sumx2,sumy,sumxy);

a=(n*sumxy-sumx*sumy)/(n*sumx2-sumx*sumx);
b=(sumy*sumx2-sumx*sumxy)/(n*sumx2-sumx*sumx);

printf("The coefficients are a=%f\t b=%f\n",a,b);

printf("\nEnter the point x:");
scanf("%f",&x0);

fx=a*x0+b;
printf("\nf(%f)=%f",x0,fx);

getch();
}

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gauss elimination

Code:

#include<stdio.h>
#include<math.h>

#include<stdlib.h>
#include<conio.h>

void main()
{
int i,j,k,n=0;
float a[10][10],l[10][10]={0},u[10][10]={0},p,b[10],z[10],x[10];

clrscr();
printf("\nDimension of square matrix:");
scanf("%d",&n);
printf("\nEnter the elements of matrix:\n");

for(i=0;i<n;i++)
{
for(j=0;j<n;j++)
{
printf("\na[%d,%d]:",i+1,j+1);
scanf("%f",&a[i][j]);
}
}
printf("\nEnter the matrix B:\n");
for(i=0;i<n;i++)
{
printf("\nb[%d]:",i+1);
scanf("%f",&b[i]);
}

for(i=0;i<n;i++)
for(j=0;j<n;j++)
if(i==j)
u[i][j]=1;

for(i=0;i<n;i++)
l[i][0]=a[i][0];

for(j=0;j<n;j++)
u[0][j]=a[0][j]/l[0][0];

p=0;
for(j=1;j<n;j++)
{
for(i=j;i<n;i++)
{
p=0;
for(k=0;k<=j-1;k++)
p+=l[i][k]*u[k][j];

l[i][j]=a[i][j]-p;
}
for(i=j+1;i<n;i++)
{
p=0;
for(k=0;k<=j-1;k++)
p+=l[j][k]*u[k][i];

u[j][i]=(a[j][i]-p)/l[j][j];
}
}

printf("\n\nThe lower triangular matrix:\n");
for(i=0;i<n;i++)
{
for(j=0;j<n;j++)
printf("%f\t",l[i][j]);
printf("\n");
}

printf("\n\nThe upper triangular matrix:\n");

for(i=0;i<n;i++)
{
for(j=0;j<n;j++)
printf("%f\t",u[i][j]);
printf("\n");
}

z[0]=b[0]/l[0][0];

for(i=1;i<n;i++)
{
p=0;
for(j=0;j<=i-1;j++)
p+=l[i][j]*z[j];
z[i]=(b[i]-p)/l[i][i];
}

x[n-1]=z[n-1];
for(i=n-2;i>=0;i--)
{
p=0;
for(j=i+1;j<n;j++)
p+=u[i][j]*x[j];
x[i]=z[i]-p;
}

printf("\n");
for(i=0;i<n;i++)
printf("x[%d]=%f\t",i+1,x[i]);

printf("\n");
getch();
}

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shoot

Code:

#include<stdio.h>
#include<conio.h>
#include<stdlib.h>
#include<math.h>

float fz(float x,float z);
float fy(float x,float y,float z);

void main()
{
int n,i,j=0;
float x[10],y[10],z[10],sz1,sz2,sy1,sy2,sz,sy,guess[10],b_gss[10],xn,f_lmt,h;

clrscr();
//read x[0],y[0]
printf("\nx[0]=");
scanf("%f",&x[0]);
printf("\ny(x[0])=");
scanf("%f",&y[0]);
printf("\nx[n]=");
scanf("%f",&xn);
printf("\ny(x[n]))=");
scanf("%f",&f_lmt);
printf("\nStep size:");
scanf("%f",&h);
n=(int)((xn-x[0])/h);
j=0;
do
{
//read z[0]
if(j<2)
{
printf("\nInitial guess,z[0]=");
scanf("%f",&z[0]);
}
guess[j]=z[0];
for(i=0;i<n;i++)
{
sz1=fz(x[i],z[i]);
sz2=fz(x[i]+h,z[i]+h*sz1);
sz=(sz1+sz2)/2;
z[i+1]=z[i]+h*sz;
//printf("\n*******z=%f\n",z[i+1]);

sy1=fy(x[i],y[i],z[i]);
sy2=fy(x[i]+h,y[i]+h*sy1,z[i]+h*sz1);
sy=(sy1+sy2)/2;
y[i+1]=y[i]+h*sy;
//printf("\n*******y=%f\n",y[i+1]);
x[i+1]=x[i]+h;
//printf("\n*******y=%f\n",x[i+1]);
}
b_gss[j]=y[n];
j++;

if(j>=2)
{
//printf("\n%f**************%f\n",b_gss[j-2],b_gss[j-1]);
//printf("\n********************************\n");
z[0]=guess[j-2]+(guess[j-1]-guess[j-2])*(f_lmt-b_gss[j-2])/(b_gss[j-1]-b_gss[j-2]);
}
}while(fabs(b_gss[j-1]-f_lmt)>0.000001);

for(i=0;i<=n;i++)
printf("\nx=%f\ty[x]=%f",x[i],y[i]);

getch();
}

float fz(float x,float z)
{
return 6*x;
}
float fy(float x,float y,float z)
{
return z;
}

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lagrange

Code:

include<stdio.h>
#include<conio.h>
#include<stdlib.h>
#include<math.h>

void main()
{
int i,j,n=0;
char ans;
float x[20],f[20],x0;
float sum,prod;

clrscr();
printf("\nEnter the no of sample points:");
scanf("%d",&n);

printf("\nEnter the sample points(x,f(x)):\n");
for(i=0;i<n;i++)
scanf("%f,%f",&x[i],&f[i]);
do{
printf("\nEnter the point for approximation:");
scanf("%f",&x0);

sum=0;
for(i=0;i<n;i++)
{
prod=1;
for(j=0;j<n;j++)
{
if(j!=i)
prod*=(x0-x[j])/(x[i]-x[j]);
}
sum+=prod*f[i];
}
printf("\nThe functional value is f(%f)=%f",x0,sum);
printf("\nDo you want to continue(y/n)?");
scanf("%c",&ans);
}while(ans=='y');

getch();
}

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jordan

Code:

#include<stdio.h>
#include<math.h>

#include<stdlib.h>
#include<conio.h>

void main()
{
int i,j,k,l,n=0;
float a[20][20],x[20],p,w,u,b[20][20];

clrscr();
printf("\nNo.of equations:");
scanf("%d",&n);

n--;

for(i=0;i<=n;i++)
{
for(j=0;j<=n;j++)
{
printf("\na[%d,%d]:",i+1,j+1);
scanf("%f",&a[i][j]);
}
printf("\nb[%d]:",i+1);
scanf("%f",&a[i][n+1]);
printf("\n");
}

for(i=0;i<=n;i++)
{
for(j=0;j<=n;j++)
{
printf("\t%f",a[i][j]);
}
printf("\t%f",a[i][n+1]);
printf("\n");
}
printf("\n");

for(i=0;i<=n;i++)
{

w=0;
for(j=i;j<=n;j++)
{
if(fabs(a[j][i]>w))
{
u=j;
w=fabs(a[j][i]);
}
}
if(u!=i)
{
for(j=i;j<=n+1;j++)
{
p=a[u][j];
a[u][j]=a[i][j];
a[i][j]=p;
}
}

for(j=0;j<=n;j++)
{
p=a[j][i]/a[i][i];
for(k=0;k<=n+1;k++)
{
if(j!=i)
a[j][k]=a[j][k]-p*a[i][k];
}
}
}
/*
for(i=1;i<=n;i++)
{
for(j=i-1;j>=0;j--)
{
p=a[j][i]/a[i][i];
for(k=i;k<=n+1;k++)
{

b[j][k]=a[j][k]-p*a[i][k];
a[j][k]=b[j][k];
}
}
}
*/
for(i=0;i<=n;i++)
{
for(j=0;j<=n;j++)
{
printf("\t%f",a[i][j]);
}
printf("\t%f",a[i][n+1]);
printf("\n");
}

for(i=0;i<=n;i++)
{
x[i]=a[i][n+1]/a[i][i];

}

for(i=0;i<=n;i++)
{
printf("\nx[%d]=%f",i+1,x[i]);
}

getch();
}

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gaussian

Code:

void main()
{
int i,j,k,l,n=0;
float a[20][20],x[20],p,w,u;
clrscr();
printf("\nNo.of equations:");
scanf("%d",&n);

n--;

for(i=0;i<=n;i++)
{
for(j=0;j<=n;j++)
{
printf("\na[%d,%d]:",i+1,j+1);
scanf("%f",&a[i][j]);
}
printf("\nb[%d]:",i+1);
scanf("%f",&a[i][n+1]);
printf("\n");
}

for(i=0;i<=n;i++)
{
for(j=0;j<=n;j++)
{
printf("\t%f",a[i][j]);
}
printf("\t%f",a[i][n+1]);
printf("\n");
}
printf("\n");

for(i=0;i<=n-1;i++)
{
w=0;
/*if(a[i][i]==0)
{
for(j=i+1;j<=n;j++)
{
if(fabs(a[j][i]>w))
{
u=j;
w=fabs(a[j][i]);
}
}
for(j=0;j<=n+1;j++)
{
p=a[u][j];
a[u][j]=a[i][j];
a[i][j]=p;
}
} */

for(j=i;j<=n;j++)
{
if(fabs(a[j][i]>w))
{
u=j;
w=fabs(a[j][i]);
}
}
if(u!=i)
{
for(j=i;j<=n+1;j++)
{
p=a[u][j];
a[u][j]=a[i][j];
a[i][j]=p;
}
}

for(j=i+1;j<=n;j++)
{
p=a[j][i]/a[i][i];
for(k=i;k<=n+1;k++)
{
a[j][k]=a[j][k]-p*a[i][k];
}
}
}

for(i=0;i<=n;i++)
{
for(j=0;j<=n;j++)
{
printf("\t%f",a[i][j]);
}
printf("\t%f",a[i][n+1]);
printf("\n");
}

for(i=n;i>=0;i--)
{
p=a[i][n+1];
if(i!=n)
{
for(j=i+1;j<=n;j++)
{
p=p-a[i][j]*x[j];
}
}
x[i]=p/a[i][i];
}

for(i=0;i<=n;i++)
{
printf("\nx[%d]=%f",i+1,x[i]);
}

getch();
}

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spline

Code:

void gaussian(float a[10][10],float s[10],int n);

void main()
{
char ans='n';
int n,i,j,ind;

float x[10],y[10],h[10],s[10]={0},at[10][10]={0},f[10],x0;
float a,b,c,d,gx; //coefficients for interpolating polynomial

clrscr();

//read the points(x,y)-----------------------------------
printf("\nEnter the no of points:");
scanf("%d",&n);

printf("\nEnter the points(x,y):\n");
for(i=0;i<n;i++)
scanf("%f,%f",&x[i],&y[i]);
//-------------------------------------------------------

for(i=0;i<n-1;i++)
{
f[i]=(y[i+1]-y[i])/(x[i+1]-x[i]); //First divided difference
h[i]=x[i+1]-x[i]; //calculation of h[i]
}
//--------------------------------------------------------
//formation of the tridiagonal matrix for s
for(i=0;i<n-2;i++)
{
for(j=0;j<n-2;j++)
{
if(i==j)
at[i][j]=2*(h[i]+h[i+1]);
if(j==i+1)
{
at[i][j]=h[j];
at[j][i]=at[i][j];
}

}
at[i][n-2]=6*(f[i+1]-f[i]);
}
//----------------------------------------------------
/*
printf("\nFDD's\n");
for(i=0;i<n-1;i++)
{
printf("%f\t",f[i]);
}
*/
//---------------------------------------------------
printf("\n\nThe tridiagonal matrix is\n");
for(i=0;i<n-2;i++)
{
for(j=0;j<n-2;j++)
printf("%f\t",at[i][j]);
printf("\t%f",at[i][n-2]);
printf("\n");
}
//--------------------------------------------------
//solve for s[i] using gaussian elimination
gaussian(at,s,n-2);

printf("\nThe values of s[i] are:\n");
for(i=0;i<n;i++)
{
printf("\ns[%d]=%f",i,s[i]);
}
//------------------------------------------------
//start the interpolation process
do{
printf("\nEnter the point for interpolation:");
scanf("%f",&x0);

for(i=0;i<n-1;i++)
{
if(x0>=x[i] && x0<=x[i+1])
ind=i;
}
//printf("\n%d",ind);

//calculation of coefficients
a=(s[ind+1]-s[ind])/(6*h[ind]);
b=s[ind]/2;
c=((y[ind+1]-y[ind])/h[ind])-(2*h[ind]*s[ind]+h[ind]*s[ind+1])/6;
d=y[ind];

printf("\nCoefficients:a[%d]=%f,b[%d]=%f,c[%d]=%f,d[%d]=%f",ind,a,ind,b,ind,c,ind,d);

gx=a*pow(x0-x[ind],3)+b*pow(x0-x[ind],2)+c*(x0-x[ind])+d;
printf("\nFunctional value: f(%f)=%f",x0,gx);

printf("\nDo you want to continue(y/n)?");
fflush(stdin);
ans=getche();
}while(ans=='y');

getch();
}

void gaussian(float a[10][10],float s[10],int n)
{
int i,j,k,u;
float w,p,dum[100];

n--;
//forward elimination steps
for(i=0;i<=n-1;i++)
{
//code for pivoting
w=0;
for(j=i;j<=n;j++)
{
if(fabs(a[j][i]>w))
{
u=j;
w=fabs(a[j][i]);
}
}
if(u!=i)
{
for(j=i;j<=n+1;j++)
{
p=a[u][j];
a[u][j]=a[i][j];
a[i][j]=p;
}
}

for(j=i+1;j<=n;j++)
{
p=a[j][i]/a[i][i];
for(k=i;k<=n+1;k++)
{
a[j][k]=a[j][k]-p*a[i][k];
}
}
}

printf("\nThe final result of gaussian elimination is:\n");
for(i=0;i<=n;i++)
{
for(j=0;j<=n;j++)
{
printf("\t%f",a[i][j]);
}
printf("\t%f",a[i][n+1]);
printf("\n");
}
//backward substitution steps
for(i=n;i>=0;i--)
{
p=a[i][n+1];
if(i!=n)
{
for(j=i+1;j<=n;j++)
{
p=p-a[i][j]*dum[j];
}
}
dum[i]=p/a[i][i];
}
for(i=1;i<=n+1;i++)
{
s[i]=dum[i-1];
}
}

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num_diff

Code:

include<stdio.h>
#include<conio.h>
#include<stdlib.h>
#include<math.h>

float f(float x);

void main()
{
int i;
float x,df1,df2,h;
char ans;

clrscr();

printf("\nEnter the value of x:");
scanf("%f",&x);
do{
printf("\nEnter the value of h:");
scanf("%f",&h);

df1=(f(x+h)-f(x))/h;
df2=(f(x+h)-f(x-h))/(2*h);
printf("\nUsing two point formula:");
printf("\nf'(%f)=%f",x,df1);

printf("\nUsing three point central difference formula:");
printf("\nf'(%f)=%f",x,df2);

printf("\nTry for another value of h?(y/n)");
ans=getche();

}while(ans=='y');

getch();
}

float f(float x)
{
return exp(x)*sin(x);
}

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Code:

num_int

#include<stdio.h>
#include<conio.h>
#include<stdlib.h>
#include<math.h>

float f(float x);

void main()
{
int i=0;
float h,integral1=0,integral2=0,integral3=0,integral4=0,a,b,p,q,temp;
char ans;

clrscr();
printf("\nEnter the limit of integration(a,b):");
scanf("%f,%f",&a,&b);
do{
printf("\nEnter the value of h:");
scanf("%f",&h);
if(a>b)
{
temp=b;
b=a;
a=temp;
}
printf("\nUsing trapezoid rule:");
i=0;
integral1=0;
while((a+i*h)<b-h)
integral1+=(h/2)*(f(a+i*h)+f(a+(++i)*h));
printf("\nThe integral is:%f\n",integral1);
printf("\n\n");

printf("\nUsing Simpson's 1/3 rule:");
i=0;
integral2=0;
while((a+i*h)<b-h)
{
integral2+=(h/3)*(f(a+i*h)+4*f(a+(++i)*h)+f(a+(++i)*h));

}
printf("\nThe integral is:%f\n",integral2);

printf("\nUsing Gaussian two point formula:");

p=(b-a)/2;
q=(b+a)/2;
integral3=p*(f(-.577350*p+q)+f(.577350*p+q));
printf("\nThe integral is:%f\n",integral3);

printf("\nUsing Gaussian three point formula:");
integral4=p*((5.0/9)*f(-.774596*p+q)+(8.0/9)*f(q)+(5.0/9)*f(.774596*p+q));
printf("\nThe integral is:%f",integral4);

printf("\nTry for another value of h?(y/n)");
ans=getche();

}while(ans=='y');
getch();
}

float f(float x)
{
return x*exp(x);
}

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Code:

lu decomposition
#include<stdio.h>
#include<math.h>

#include<stdlib.h>
#include<conio.h>

void main()
{
int i,j,k,n=0;
float a[10][10],l[10][10]={0},u[10][10]={0},p,b[10],z[10],x[10];

clrscr();
printf("\nDimension of square matrix:");
scanf("%d",&n);
printf("\nEnter the elements of matrix:\n");

for(i=0;i<n;i++)
{
for(j=0;j<n;j++)
{
printf("\na[%d,%d]:",i+1,j+1);
scanf("%f",&a[i][j]);
}
}
printf("\nEnter the matrix B:\n");
for(i=0;i<n;i++)
{
printf("\nb[%d]:",i+1);
scanf("%f",&b[i]);
}

for(i=0;i<n;i++)
for(j=0;j<n;j++)
if(i==j)
u[i][j]=1;

for(i=0;i<n;i++)
l[i][0]=a[i][0];

for(j=0;j<n;j++)
u[0][j]=a[0][j]/l[0][0];

p=0;
for(j=1;j<n;j++)
{
for(i=j;i<n;i++)
{
p=0;
for(k=0;k<=j-1;k++)
p+=l[i][k]*u[k][j];

l[i][j]=a[i][j]-p;
}
for(i=j+1;i<n;i++)
{
p=0;
for(k=0;k<=j-1;k++)
p+=l[j][k]*u[k][i];

u[j][i]=(a[j][i]-p)/l[j][j];
}
}

printf("\n\nThe lower triangular matrix:\n");
for(i=0;i<n;i++)
{
for(j=0;j<n;j++)
printf("%f\t",l[i][j]);
printf("\n");
}

printf("\n\nThe upper triangular matrix:\n");

for(i=0;i<n;i++)
{
for(j=0;j<n;j++)
printf("%f\t",u[i][j]);
printf("\n");
}

z[0]=b[0]/l[0][0];

for(i=1;i<n;i++)
{
p=0;
for(j=0;j<=i-1;j++)
p+=l[i][j]*z[j];
z[i]=(b[i]-p)/l[i][i];
}

x[n-1]=z[n-1];
for(i=n-2;i>=0;i--)
{
p=0;
for(j=i+1;j<n;j++)
p+=u[i][j]*x[j];
x[i]=z[i]-p;
}

printf("\n");
for(i=0;i<n;i++)
printf("x[%d]=%f\t",i+1,x[i]);

printf("\n");
getch();
}

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