nt lab manual

Prepared By:

AVANTIKA YADAV

Numerical Techniques Lab Manual


Avantika Yadav KEC Page 1

List of Programs

S.NO. ALGORITHM/ FLOW CHART/ PROGRAM PAGE NO.

1. To deduce error involved in polynomial equation.

2. To Find out the root of the Algebraic and Transcendental equations using Bisection method.

3. To Find out the root of the Algebraic and Transcendental equations using Regula-Falsi method.

4. To Find out the root of the Algebraic and Transcendental equations using Newton-Raphson method.

5. To Find out the root of the Algebraic and Transcendental equations using Iterative method.

6. To implement Numerical Integration using Trapezoidal rule.

7. To implement Numerical Integration using Simpson 1/3 rule.

8. To implement Numerical Integration Simpson 3/8 rule.

9. To implement Newtons Forward Interpolation formula.

10. To implement Newtons Backward Interpolation formula.

11. To implement Gauss Forward Interpolation formula.

12. To implement Gauss Backward Interpolation formula.

13. To implement Bessels Interpolation formula.

14. To implement Sterlings Interpolation formula.

15. To implement Newtons Divided Difference formula.

16. To implement Langranges Interpolation formula.

17. To implement Numerical Differentiations.

18. To implement Least Square Method for curve fitting.

19. To draw frequency chart like histogram, frequency curve and pie-chart etc.

20. To estimate regression equation from sampled data and evaluate values of standard deviation, t-statistics, regression coefficient,

value of R^2 for at least two independent variables.



1. Algorithm to deduce error involved in polynomial equation.

Step-1. Start of the program.

Step-2. Input the variable t_val, a_value.

Step-3. Calculate absolute error as

abs_err=|t_val-a_val|

Step-4. Calculate relative error as

rel_err=abs_err/t_val

Step-5. Calculate percentage relative error as

p_rel_err=rel_err*100

Step-6. PRINT abs_err, rel_err and p_rel_err

Step-7. STOP



Program to deduce error involved in polynomial equation

#include

#include

#include

void main()

{

double abs_err, rel_err, p_rel_err, t_val, a_val;

printf(\n INPUT TRUE VALUE:);

scanf(%lf, &t_val);

printf(\n INPUT APPROXIMATE VALUE:);

scanf(%lf, &a_val);

abs_err=fabs(t_val-a_val);

rel_err=abs_err/t_val;

p_rel_err=rel_err*100;

printf(\nABSOLUTE ERROR= %lf, abs_err);

printf(\nRELATIVE ERROR= %lf, rel_err);

printf(\nPERCENTAGE RELATIVE ERROR= %lf, p_rel_err);

getch();

}



2. Algorithm of BISECTION METHOD.


Step-2. Input the variable x1, x2 for the task.

Step-3. Check f(x1)*f(x2)



PROGRAM: BISECTION METHOD.

#include

#include

#include

#include

#include

#define EPS 0.00000005

#define F(x) (x)*log10(x)-1.2

void Bisect();

int count=1,n;

float root=1;

void main()

{

clrscr();

printf("\n Solution by BISECTION method \n");

printf("\n Equation is ");

printf("\n\t\t\t x*log(x) - 1.2 = 0\n\n");

printf("Enter the number of iterations:");

scanf("%d",&n);

Bisect();

getch();

}

void Bisect()

{

float x0,x1,x2;

float f0,f1,f2;

int i=0;

for(x2=1;;x2++)

{

f2=F(x2);

if (f2>0)

{

break;

}

}

for(x1=x2-1;;x2--)

{

f1=F(x1);

if(f1



printf("\n\t\t ITERATIONS\t\t ROOTS\n");

printf("\t\t-----------------------------------------");

for(;count



3. Algorithm of FALSE POSITION or REGULA-FALSI METHOD.


Step-2. Input the variable x0, x1,e, n for the task.

Step-3. f0=f(x0)

Step-4. f2=f(x2)

Step-5. for i=1 and repeat if i



PROGRAM: FALSE POSITION or REGULA-FALSI METHOD.

#include

#include

#include

#include

#include

#define EPS 0.00005

#define f(x) 3*x+sin(x)-exp(x)

void FAL_POS();

void main()

{

clrscr();

printf("\n Solution by FALSE POSITION method\n");

printf("\n Equation is ");

printf("\n\t\t\t 3*x + sin(x)-exp(x)=0\n\n");

FAL_POS();

}

void FAL_POS()

{

float f0,f1,f2;

float x0,x1,x2;

int itr;

int i;

printf("Enter the number of iteration:");

scanf("%d",&itr);

for(x1=0.0;;)

{

f1=f(x1);

if(f1>0)

{

break;

}

else

{

x1=x1+0.1;

}

}

x0=x1-0.1;

f0=f(x0);

printf("\n\t\t-----------------------------------------");

printf("\n\t\t ITERATION\t x2\t\t F(x)\n");

printf("\t\t--------------------------------------------");

for(i=0;i



{

x2=x0-((x1-x0)/(f1-f0))*f0;

f2=f(x2);

if(f0*f2>0)

{

x1=x2;

f1=f2;

}

else

{

x0=x2;

f0=f2;

}

if(fabs(f(2))>EPS)

{

printf("\n\t\t%d\t%f\t%f\n",i+1,x2,f2);

}

}

printf("\t\t--------------------------------------------");

printf("\n\t\t\t\tRoot=%f\n",x2);

printf("\t\t-------------------------------------------");

getch();

}



4. Algorithm of NEWTON-RAPHSON METHOD.


Step-2. input the variables x0, n for the task.

Step-3. input Epsilon & delta

Step-4. for i= 1 and repeat if i



PROGRAM: NEWTON RAPHSON METHOD.

# include

# include

# include

# include

# include

# define f(x) 3*x -cos(x)-1

# define df(x) 3+sin(x)

void NEW_RAP();

void main()

{

clrscr();

printf ("\n Solution by NEWTON RAPHSON method \n");

printf ("\n Equation is: ");

printf ("\n\t\t\t 3*X - COS X - 1=0 \n\n ");

NEW_RAP();

getch();

}

void NEW_RAP()

{

long float x1,x0;

long float f0,f1;

long float df0;

int i=1;

int itr;

float EPS;

float error;

for(x1=0;;x1 +=0.01)

{

f1=f(x1);

if (f1 > 0)

{

break;

}

}

x0=x1-0.01;

f0=f(x0);

printf(" Enter the number of iterations: ");

scanf(" %d",&itr);

printf(" Enter the maximum possible error: ");

scanf("%f",&EPS);

if (fabs(f0) > f1)

{

printf("\n\t\t The root is near to %.4f\n",x1);

}



if(f1 > fabs(f(x0)))

{

printf("\n\t\t The root is near to %.4f\n",x0);

}

x0=(x0+x1)/2;

for(;i



5. ALGORITHM FOR ITERATION METHOD.

Algorithm 1

Step-1. Read x0, e, n

x0 is the initial guess, e is the allowed error in root, n is total iterations to be

allowed for convergence.

Step-2. x1 g(x0)

Steps 4 to 6 are repeated until the procedure converges to a root or iterations

reach n.

Step-3. For i = 1 to n in steps of 1 do

Step-4. x0 x1

Step-5. x1 g(x0)

Step-6. If e then, GO TO 9

Step-7. end for.

Step-8. Write Does not converge to a root, x0, x1

Step-9. Stop

Step-10. Write converges to a root, i, x1

Step-11. Stop.



Algorithm 2 (Iteration Method)

Step-1. Define function f(x)

Step-2. Define function df(x)

Step-3. Get the value of a, max_err.

Step-4. Initialize j

Step-5. If df(a) < 1 then b = 1, a = f(a)

Step-6. Print root after j, iteration is f(a)

Step-7. If fabs(b a) > max_err then

Step-8. j++, goto (5)

Step-9. End if

Step-10. Else print root doesnt exist

Step-11. End.



PROGRAM: ITERATION METHOD.

#include

#include

#include

#define EPS 0.00005

#define F(x) (x*x*x + 1)/2

#define f(x) x*x*x - 2*x + 1

void ITER();

void main ()

{

clrscr();

printf("\n\t Solution by ITERATION method - ");

printf("\n\t Equation is - ");

printf("\n\t\t\t\t X*X*X - 2*X + 1 = 0\n");

ITER();

getch();

}

void ITER()

{

float x1,x2,x0,f0,f1,f2,error;

int i=0,n;

for(x1=1;;x1++)

{

f1=F(x1);

if (f1>0)

break;

}

for(x0=x1-1;;x0--)

{

f0=f(x0);

if(f0



x2=F(x2);

error=fabs(f2-f1);

if(errorEPS)

printf("\n\n\t NOTE:- The number of iterations are not sufficient.");

printf("\n\n\n\t\t\t------------------------------");

printf("\n\t\t\t The root is %.4f",f2);

printf("\n\t\t\t-----------------------------");

}



6. ALGORITHM OF TRAPEZOIDAL RULE


Step-2. Input Lower limit a

Step-3. Input Upper Limit b

Step-4. Input number of sub intervals n

Step-5. h=(b-a)/n

Step-6. sum=0

Step-7. sum=fun(a)+fun(b)

Step-8. for i=1; i



PROGRAM: TRAPEZOIDAL METHOD.

# include

# include

# include

# include

# include

float fun(float);

void main()

{

float result=1;

float a,b;

float h,sum;

int i,j;

int n;

clrscr();

printf("\n\n Enter the range - ");

printf("\n\n Lower Limit a - ");

scanf("%f" ,&a);

printf("\n\n Upper Limit b - ");

scanf("%f" ,&b);

printf("\n\n Enter number of subintervals - ");

scanf("%d" ,&n);

h=(b-a)/n;

sum=0;

sum=fun(a)+fun(b);

for(i=1;i



7. ALGORITHM OF SIMPSONS 1/3rd RULE



Step-3. Input Upper limit b

Step-4. Input number of subintervals n

Step-5. h=(ba)/n

Step-6. sum=0

Step-7. sum=fun(a)+4*fun(a+h)+fun(b)

Step-8. for i=3; i



PROGRAM: SIMPSONS 1/3rd

METHOD OF NUMERICAL INTEGRATION

#include

#include

#include

#include

#include

float fun(float);

void main()

{

float result=1;

float a,b;

float sum,h;

int i,j,n;

clrscr();

printf("\n Enter the range - ");

printf("\n Lower Limit a - ");

scanf("%f",&a)

;printf("\n Upper limit b - ");

scanf("%f",&b);

printf("\n\n Enter number of subintervals - ");

scanf("%d",&n);

h=(b-a)/n;

sum=0;

sum=fun(a)+4*fun(a+h)+fun(b);

for(i=3;i



8. ALGORITHM OF SIMPSONS 3/8th RULE



Step-3. Input Upper limit b

Step-4. Input number of sub itervals n

Step-5. h = (b a)/n

Step-6. sum = 0

Step-7. sum = fun(a) + fun (b)

Step-8. for i = 1; i < n; i++

Step-9. if i%3=0:

Step-10. sum + = 2*fun(a + i*h)

Step-11. else:

Step-12. sum + = 3*fun(a+(i)*h)

Step-13. End of loop i

Step-14. result = sum*3*h/8

Step-15. Print Output result

Step-16. End of Program

Step-17. Start of Section fun

Step-18. temp = 1/(1+(x*x))

Step-19. Return temp

Step-20. End of section fun



PROGRAM: SIMPSONS 3/8th

METHOD OF NUMERICAL INTEGRATION

#include

#include

float fun(int);

void main()

{

int n,a,b,i;

float h, sum=0, result;

//clrscr();

printf("enter range");

scanf("%d",&n);

printf("enter lower limit");

scanf("%d",&a);

printf("enter upper limit");

scanf("%d",&b);

h=(b-a)/n;

sum=fun(a)+fun(b);

for(i=0;i



9. Algorithm for Newtons Forward method of interpolation

Step-1. Start of the program

Step-2. Input number of terms n

Step-3. Input the array ax

Step-4. Input the array ay

Step-5. h=ax[1] ax[0]

Step-6. for i=0; i



PROGRAM: NEWTONS FORWORD METHOD OF INTERPOLATION

# include

# include

# include

# include

# include

void main()

{

int n;

int i,j;

float ax[10];

float ay[10];

float x;

float y = 0;

float h;

float p;

float diff[20][20];

float y1,y2,y3,y4;

clrscr();

printf("\n Enter the number of terms - ");

scanf("%d",&n);

printf("Enter the value in the form of x - ");

for (i=0;i



diff[i][j]=diff[i+1][j-1]-diff[i][j-1];

}

}

i=0;

do

{

i++;

}

while(ax[i]



10. Algorithm for Newtons Backward method of interpolation





Step-5. h=ax[1]-ax[0]

Step-6. for i=0; i



PROGRAM: NEWOTNS BACKWARD METHOD OF INTERPOLATION

#include

#include

#include

#include

#include

void main()

{

int n,i,j,k;

float mx[10],my[10],x,x0=0,y0,sum,h,fun,p,diff[20][20],y1,y2,y3,y4;

clrscr();

printf("\n enter the no. of terms - ");

scanf("%d",&n);

printf("\n enter the value in the form of x - ");

for(i=0;i



}

x0=mx[i];

sum=0;

y0=my[i];

fun=1;

p=(x-x0)/h;

sum=y0;

for(k=1;k



11. Algorithm of GAUSS FORWARD interpolation formula






Step-6. for i=0;i



PROGRAM: GAUSSS FORWORD METHOD OF INTERPOLATION

# include

# include

# include

# include

# include

void main()

{

int n;

int i,j;

float ax[10];

float ay[10];

float x;

float nr,dr;

float y=0; float h;

float p;

float diff[20][20];

float y1,y2,y3,y4;

clrscr();

printf(" Enter the number of terms - ");

scanf("%d",&n);

printf("\n Enter the value in the form of x - ");

for (i=0;i




}

}

i=0;

do {

i++;

}

while(ax[i]



12. Algorithm of Gausss Backward interpolation formula






Step-6. for i=0;i



PROGRAM: GAUSSS BACKWARD METHOD OF INTERPOLATION.

# include

# include

# include

# include

# include

void main()

{

int n;

int i,j; float ax[10];

float ay[10];

float x;

float y=0;

float h;

float p;

float diff[20][20];

float y1,y2,y3,y4;

clrscr();

printf("\n Enter the number of terms - ");

scanf("%d",&n);

printf("\n Enter the value in the form of x - ");

for (i=0;i




}

}

i=0;

do {

i++;

}

while (ax[i]



13. Algorithm to implement BESSELS INTERPOLATION FORMULA



Program to implement BESSELS INTERPOLATION FORMULA in C.

#include

#include

#include

#include

#include

void main()

{

int n; // no. of terms.

int i,j; // Loop variables

float ax[10]; // 'X' array limit 9

float ay[10]; // 'Y' array limit 9

float x; // User Query for what value of X

float y; // Calculated value for coressponding X.

float h; // Calc. Section

float p; // Calc. Section

float diff[20][20]; // to store Y

float y1,y2,y3,y4; // Formulae variables.

clrscr();

printf("\t\t !! BESSELS INTERPOLATION FORMULA!! ");

// Input section.

printf("\n\n Enter the no. of terms -> ");

scanf("%d",&n);

// Input Sequel for array X

printf("\n\n Enter the value in the form of x -> ");

// Input loop for X.

for(i=0;i



printf("\n\n Enter the value of x for ");

printf("\n which u want the value of y -> ");

scanf("%f",&x);

// Calculation and processing section.

h=ax[1]-ax[0];

for(i=0;i



14. Algorithm of Sterlings Interpolation Formula





Step-5. h = ax[1]-ax[0]

Step-6. for i = 1;i < n-1; i++

Step-7. diff [i][1] = ay[i + 1]-ay[i]

Step-8. End loop i

Step-9. for j = 2; j < = 4; j++

Step-10. for i = 0; i < n-j; i++

Step-11. diff[i][j] = diff[i + 1][j-1]-diff[i][j-1]

Step-12. End loop i

Step-13. End loop j

Step-14. i = 0

Step-15. Repeat step 16 until ax[i] < x

Step-16. i = i + 1

Step-17. i = i-1;

Step-18. p = (x-ax[i])/h

Step-19. y1= p*(diff[i][1] + diff[i-1][1])/2

Step-20. y2 = p*p*diff[i-1][2]/2

Step-21. y3 = p*(p*p-1)*(diff[i-1][3]+diff[i-2][3])/6

Step-22. y4 = p*p*(p*p-1)*diff[i-2][4]/24

Step-23. y = ay[i]+y1 + y2 + y3 + y4

Step-24. Print output

Step-25. End of program



PROGRAM: STERLINGS METHOD OF INTERPOLATION.

#include

#include

#include

#include

void main()

{

int n;

int i,j;

float ax[10];

float ax[10];

float h;

float p;

float diff[20][20];

float x,y;

float y1,y2,y3,y4;

clrscr();

printf("\n Enter the value of terms");

scanf("%d",%n);

printf(\n Enter the values for x \n);

for(i=0;i



}

i=0;

do {

i++;

}

while(ax[i]



15. Algorithm to implement Newtons Divided Difference formula.



Program to implement Newtons Divided Difference formula.

#include

#include

#include

void main()

{

float x[10],y[10][10],sum,p,u,temp;

int i,n,j,k=0,f,m;

float fact(int);

clrscr();

printf("\nhow many record you will be enter: ");

scanf("%d",&n);

for(i=0; i



}

i=0;

do

{

if(x[i]



16. Algorithm of LAGRANGES INTERPOLATION FORMULA.

Step-1. Start of the program




Step-5. for i=0; i



PROGRAM: LAGRANGES INTERPOLATION FORMULA.

#include

#include

#define MAX 10

void main()

{

float x[MAX],y[MAX],k=0,z,nr,dr;

int i,j,m;

//clrscr();

printf("\n enter the range ");

scanf("%d",&m);

printf("\n enter the x value ");

for(i=0;i



17. Algorithm to implement Numerical Differentiations.



Program to implement Numerical Differentiations.

#include

#include

#include

float F (float x);

void deriv (float f(float), float x, int n, float h, float **D)

{

int i, j;

unsigned powerof4;

for (i = 0; i



/**********************************************************************/

/* Programmer's comment: */

/* if only one function will be called, there's no need to pass the */

/* function as argument. In that case, the procedure deriv will have */

/* this format: void deriv(float, int, float, float **) */

/* */

/**********************************************************************/

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

for (j = 0; j



18. Algorithm to implement Least Square Method for curve fitting.



Program to implement Least Square Method for curve fitting.

#include

#include

#include

main()

{

float x[10],y[10],a[10][10];

int i,j,k,n,itr;

printf("\n ENTER THE SIZE OF MATRIX n:");

scanf("%d",&n);

printf("\n ENTER MATRIX ELEMENTS AND RHS:\n");

for(i=1;i



}

goto top;

}

else

continue;

return;

}



19. Algorithm to draw frequency chart like histogram, frequency curve and pie-chart etc.



Program to draw frequency chart like histogram, frequency curve and pie-chart etc.

nt lab manual

Documents

val rel

val step

val abs

bessels interpolation

numerical differentiations

algorithm of bisection

printfnrelative error

printfnabsolute error