DEPARTMENT OF COMPUTER AND
COMMUNICATION ENGINEERING
NUMERICAL METHODS A N D S T A T I S T I C A L
LABORATORY
STUDENT NAME :
REGISTER NO :
YEAR/SEMESTER/SECTION :
BATCH :
TABLE OF CONTENTS
Page Staff
S.N Date Title of the Exercises Mark no
o s signatu
re
1 ROOTS OF NON – LINEAR EQUATION
USING BISECTION METHOD
2
ROOTS OF NON – LINEAR EQUATION
USING NEWTON’S METHOD
3 SOLVE THE SYSTEM OF LINEAR
EQUATIONS USING GAUSS -
ELIMINATION METHOD
4 SOLVE THE SYSTEM OF LINEAR
EQUATIONS USING GAUSS - JORDAN
METHOD
5 SOLVE THE SYSTEM OF LINEAR
EQUATIONS USING GAUSS - SEIDAL
ITERATION METHOD.
6
FIND AREA BY USING TRAPEZOIDAL
RULE
7 FIND AREA BY USING SIMPSON’S 1/3
RULE
8 FIND MEAN, MEDIAN AND MODE
9 FIND QUARTILE DEVIATION
10 FIND PEARSON’S COEFFICIENT OF
SKEWNESS
EX.NO:1
ROOTS OF NON – LINEAR EQUATION USING BISECTION METHOD
DATE:
AIM : To write a program in “C” Language to find out the root of the Algebraic and
Transcendental equations using Bisection Method.
ALGORITHM:
Step - 1 Start of the program.
Step - 2. Input the variable x1, x2 for
the task. Step - 3. Check f(x1)*f(x2)<0
Step - 4. If yes proceed
Step - 5. If no exit and print error
message Step - 6. Repeat 7-11 if
condition not satisfied Step - 7.
X0=(x1+x2)/2
Step - 8. If
f(x0)*f(x1)<0 Step -
9. X2=x0
Step - 10. Else
Step - 11.
X1=x0
Step - 12. Condition:
Step - 13.if | (x1-x2)/x1) | < maximum possible error
or f(x0)=0 Step - 14. Print output
Step - 15. End of program.
PROGRAM:
#include<stdio.h>
#include<math.h>
#include<conio.h>
#include<stdlib.h>
/*Uncomment the function u want to use or else add u r own*/
//#define f(x) ((x*x*x)-(4*x)-9)
//#define f(x) ((x*x*x)-(x)-1)
//#define f(x) ((x)-(cos(x)))
//#define f(x) ((x*sin(x))-1)
//#define f(x) ((exp(x))-(3*x))
//#define f(x) ((x*x*x)-(9*x)+1) //”range to be entered mannually 2,4”
void main()
{
float a=-1,b=-1,c,prv=-1;
int i;
char ch;
system("cls");
printf("Do you want to choose interval (y/n)?: ");
scanf("%c",&ch);
//If interval exists
if (ch=='y'||ch=='Y')
{
printf("Enter value of A and B\n");
scanf("%f%f",&a,&b);
}
//For finding intervals automatically
else
{
for(i=0;a==-1||b==-1;i++)
{
if(f(0)<0)
{
if(f(i)<0)
a=i;
if(f(i)>0)
b=i;
}
else
{
if(f(i)>0)
a=i;
if(f(i)<0)
b=i;
}
}
printf("\nFinding values of A and B...\nA=%.f\nB=%.f\n",a,b);
}
printf(" A\t\tB\t\tRoot\n");
//Iterration Table
while((ceil(10000*prv)/10000)!=(ceil(10000*c)/10000))
{
//Prv checks previous iteration value, loop terminates when root and
//prev are equal
prv=c;
c=((a+b)/2);
printf(" %.4lf\t\t%.4f\t\t%.4f\n",a,b,c);
if(f(a)>f(b)) //f(a)>f(b)
{
if(f(c)>0)
a=c;
else
b=c;
}
else //f(b)>f(a)
{
if(f(c)>0)
b=c;
else
a=c;
}
}
printf("\nThe Aproximate Root is %.4f",c);
getch();
}
INPUT/OUTPUT:
f(x)=((x*x*x)-(x)-1)
CRITERIA MAX.MARKS MARKS OBTAINED
AIM & ALGORITHM 5
EXECUTION & OUTPUT 10
VIVA 5
TOTAL 20
RESULT:
EX.NO:2
ROOTS OF NON – LINEAR EQUATION USING NEWTON’S
METHOD.
DATE:
AIM:
To write a program in “C” Language to implement Newton’s Raphson method.
ALGORITHM:
Step 1. Start the program.
Step 2. Define the function f(x), f’(x)
Step 3. Enter the initial guess of the root, say x0
Step 4. Calculate xn+1 = xn – [f(xn)/ f’(xn)], n = 0, 1, 2, …
Step 5. If |xn+1 – xn| < ϵ, ϵ being prescribed accuracy then
go to step 7. Step 6. Set xn = xn+1 and go to step 4.
Step 7. Print the value
of xn. Step 8. Stop the
program.
PROGRAM:
#include<stdio.h>
#include<stdlib.h>
#include<math.h>
#define f(x) ((3*x)-cos(x)-1) //Function
#define f1(x) (3+sin(x)) //Derivative of the function
void main()
{
float a=-1,b=-1,c,prv=-1;
int i;
char ch;
system("cls");
printf("Do you want to choose interval (y/n)?: ");
scanf("%c",&ch);
if (ch=='y'||ch=='Y')
{
printf("Enter value of A and B\n");
scanf("%f%f",&a,&b);
}
else
{
for(i=0;a==-1||b==-1;i++)
{
if(f(0)<0)
{
if(f(i)<0)
a=i;
if(f(i)>0)
b=i;
}
else
{
if(f(i)>0)
a=i;
if(f(i)<0)
b=i;
}
}
printf("\n Finding values of A and B...\nA=%.f\nB=%.f\n",a,b);
}
printf(" Itt\t\t\tRoot\n");
c=(abs(f(a))>=abs(f(b)))?a:b;
i=0;
EX.NO:3
SOLVE THE SYSTEM OF LINEAR EQUATIONS USING GAUSS -
ELIMINATION METHOD.
DATE:
while((ceil(10000*prv)/10000)!=(ceil(10000*c)/10000))
{
prv=c;
c=(c-(f(c)/f1(c)));
printf(" %d\t\t%.4f\n",i,c);
i++;
}
printf("\nThe Aproximate Root is %.4f",c);
}
INPUT/OUTPUT:
CRITERIA MAX.MARKS MARKS OBTAINED
AIM & ALGORITHM 5
EXECUTION & OUTPUT 10
VIVA 5
TOTAL 20
RESULT:
AIM:
To write a C program to implement the system of linear equations using gauss
- elimination method.
ALGORIHTM:
1. Start
2. Read Number of Unknowns: n
3. Read Augmented Matrix (A) of n by n+1 Size
4. Transform Augmented Matrix (A) to Upper Trainagular Matrix by Row Operations.
5. Obtain Solution by Back Substitution.
6. Display Result.
7. Stop
PROGRAM:
#include <stdio.h>
#include <conio.h>
#include <stdlib.h>
int main()
{
int i, j, k, n;
float a[10][11], x[10], r;
system("cls");
printf("Enter the values:\n\n");
n = 3;
for (i = 1; i <= n; i++)
{
for (j = 1; j <= n + 1; j++)
{
printf("a[%d][%d] = ", i, j);
scanf("%f", &a[i][j]);
}
}
for (i = 1; i <= n - 1; i++)
{
if (a[i][i] == 0.0)
{
printf("Error: Division by zero");
exit(0);
}
for (j = i + 1; j <= n; j++)
{
r = a[j][i] / a[i][i];
for (k = 1; k <= n + 1; k++)
{
a[j][k] = a[j][k] - r * a[i][k];
}
}
}
x[n] = a[n][n + 1] / a[n][n];
for (i = n - 1; i >= 1; i--)
{
x[i] = a[i][n + 1];
for (j = i + 1; j <= n; j++)
{
x[i] = x[i] - a[i][j] * x[j];
}
x[i] = x[i] / a[i][i];
}
printf("\nAnswer:\n");
for (i = 1; i <= n; i++)
{
printf("x%d = %.3f\n", i, x[i]);
}
return 0;
}
INPUT/OUTPUT:
CRITERIA MAX.MARKS MARKS OBTAINED
AIM & ALGORITHM 5
EXECUTION & OUTPUT 10
VIVA 5
TOTAL 20
RESULT:
EX.NO:4
SOLVE THE SYSTEM OF LINEAR EQUATIONS USING GAUSS -
JORDAN METHOD
DATE:
AIM:
To write a C program to implement the system of linear equations using gauss - jordan
method.
ALGORITHM:
1. Start
2. Read Number of Unknowns: n
3. Read Augmented Matrix (A) of n by n+1 Size
4. Transform Augmented Matrix (A) to Diagonal Matrix by Row Operations.
5. Obtain Solution by Making All Diagonal Elements to 1.
6. Display Result.
7. Stop
PROGRAM:
#include <stdio.h>
#include <stdlib.h>
int main() {
int i, j, k, n;
float a[10][11], x[10], r;
system("cls");
printf("Enter the number of equations: ");
scanf("%d", &n);
if (n <= 0 || n > 10)
{
printf("Invalid number of equations. Please enter a value between 1 and 10.\n");
return 1;
}
printf("Enter the augmented matrix (coefficients and constants):\n");
for (i = 1; i <= n; i++)
{
for (j = 1; j <= n + 1; j++)
{
printf("a[%d][%d] = ", i, j);
scanf("%f", &a[i][j]);
}
}
for (i = 1; i <= n - 1; i++)
{
if (a[i][i] == 0.0)
{
printf("Error: Division by zero. The program cannot continue.\n");
return 1;
}
for (j = i + 1; j <= n; j++)
{
r = a[j][i] / a[i][i];
for (k = 1; k <= n + 1; k++)
{
a[j][k] = a[j][k] - r * a[i][k];
}
}
}
x[n] = a[n][n + 1] / a[n][n];
for (i = n - 1; i >= 1; i--) {
x[i] = a[i][n + 1];
for (j = i + 1; j <= n; j++)
{
x[i] = x[i] - a[i][j] * x[j];
}
x[i] = x[i] / a[i][i];
}
printf("\nSolution:\n");
for (i = 1; i <= n; i++)
{
printf("x%d = %.3f\n", i, x[i]);
}
return 0;
INPUT/OUTPUT:
RESULT:
CRITERIA MAX.MARKS MARKS OBTAINED
AIM & ALGORITHM 5
EXECUTION & OUTPUT 10
VIVA 5
TOTAL 20
EX.NO:5
SOLVE THE SYSTEM OF LINEAR EQUATIONS USING GAUSS -
SEIDAL ITERATION METHOD.
DATE:
AI
M: To write a C program to implement the system of linear equations using Gauss -
Seidal iteration method.
ALGORITHM:
1. Start
2. Read Number of Unknowns: n
3. Read Augmented Matrix (A) of n by n+1 Size
4. Transform Augmented Matrix (A) to Diagonal Matrix by Row Operations.
5. Obtain Solution by Making All Diagonal Elements to 1.
6. Display Result.
7. Stop
PROGRAM:
#include <stdio.h>
int main()
{
float a[10][10], x[10], ratio;
int i, j, k, n;
printf("Enter the number of unknowns: ");
scanf("%d", &n);
printf("Enter coefficients of the augmented matrix:\n");
for (i = 1; i <= n; i++)
{
for (j = 1; j <= n + 1; j++)
{
printf("a[%d][%d] = ", i, j);
scanf("%f", &a[i][j]);
}
}
for (i = 1; i <= n; i++)
{
if (a[i][i] == 0.0)
{
printf("Mathematical Error: Division by zero!\n");
return 1; // Exit the program with an error code
}
for (j = 1; j <= n; j++)
{
if (i != j)
{
ratio = a[j][i] / a[i][i];
for (k = 1; k <= n + 1; k++)
{
a[j][k] = a[j][k] - ratio * a[i][k];
}
}
}
}
for (i = 1; i <= n; i++)
{
x[i] = a[i][n + 1] / a[i][i];
}
/* Displaying the Solution */
printf("\n Solution:\n");
for (i = 1; i <= n; i++)
{
printf("x[%d] = %0.3f\n", i, x[i]);
}
return 0;
}
INPUT AND OUTPUT:
CRITERIA MAX.MARKS MARKS OBTAINED
AIM & ALGORITHM 5
EXECUTION & OUTPUT 10
VIVA 5
TOTAL 20
RESULT:
EX.NO:6
FIND AREA BY USING TRAPEZOIDAL RULE
DATE:
AIM:
To write a C program to implement find area by using trapezoidal rule.
ALGORITHM:
1. Start
2. Define function f(x)
3. Read lower limit of integration, upper limit of integration and number of sub interval
4. Calculate: step size = (upper limit - lower limit)/number of sub interval
5. Set: integration value = f(lower limit) + f(upper limit)
6. Set: i = 1
7. If i > number of sub interval then goto
8. Calculate: k = lower limit + i * h
9. Calculate: Integration value = Integration Value + 2* f(k)
10. Increment i by 1 i.e. i = i+1 and go to step 7
11. Calculate: Integration value = Integration value * step size/2
12. Display Integration value as required answer
13. Stop
PROGRAM:
#include <stdio.h>
#include <math.h>
/* Define function here */
#define f(x) (1.0 / (1.0 + pow(x, 2)))
int main() {
float lower, upper, integration = 0.0, stepSize, k;
int i, subInterval;
/* Input */
printf("Enter lower limit of integration: ");
scanf("%f", &lower);
printf("Enter upper limit of integration: ");
scanf("%f", &upper);
printf("Enter number of sub intervals: ");
scanf("%d", &subInterval);
/* Calculation */
/* Finding step size */
stepSize = (upper - lower) / subInterval;
/* Finding Integration Value */
integration = f(lower) + f(upper);
for (i = 1; i <= subInterval - 1; i++)
{
k = lower + i * stepSize;
integration += 2 * f(k);
}
integration = integration * stepSize / 2.0;
printf("\n Required value of integration is: %.3f", integration);
return 0;
}
INPUT AND OUTPUT:
CRITERIA MAX.MARKS MARKS OBTAINED
AIM & ALGORITHM 5
EXECUTION & OUTPUT 10
VIVA 5
TOTAL 20
RESULT:
EX.NO:7
FIND AREA BY USING SIMPSON’S 1/3RD RULE
DATE:
AIM:
To write a C program to implement find area by using Simpson’s 1/3 rule.
ALGORITHM:
1. Start
2. Define function f(x)
3. Read lower limit of integration, upper
limit of integration and number of sub
interval
4. Calculate: step size = (upper limit - lower limit)/number of sub interval
5. Set: integration value = f(lower limit) + f(upper limit)
6. Set: i = 1
7. If i > number of sub interval then goto
8. Calculate: k = lower limit + i * h
9. If i mod 2 =0 then
Integration value = Integration Value +
2* f(k) Otherwise
Integration Value = Integration Value +
4 * f(k) End If
10. Increment i by 1 i.e. i = i+1 and go to step 7
11. Calculate: Integration value = Integration value * step size/3
12. Display Integration value as required answer
13. Stop
PROGRAM:
#include <stdio.h>
#include <math.h>
/* Define function here */
#define f(x) (1.0 / (1.0 + x * x))
int main() {
float lower, upper, integration = 0.0, stepSize, k;
int i, subInterval;
/* Input */
printf("Enter lower limit of integration: ");
scanf("%f", &lower);
printf("Enter upper limit of integration: ");
scanf("%f", &upper);
printf("Enter number of sub intervals (must be even): ");
scanf("%d", &subInterval);
/* Check if the number of sub intervals is even */
if (subInterval % 2 != 0) {
printf("Number of sub intervals must be even.\n");
return 1; // Exit with an error code
}
/* Calculation */
/* Finding step size */
stepSize = (upper - lower) / subInterval;
/* Finding Integration Value */
integration = f(lower) + f(upper);
for (i = 1; i <= subInterval - 1; i++) {
k = lower + i * stepSize;
if (i % 2 == 0) {
integration += 2 * f(k);
} else {
integration += 4 * f(k);
}
}
integration = integration * stepSize / 3.0;
printf("\nRequired value of integration is: %.3f\n", integration);
return 0;
}
INPUT/OUTPUT:
CRITERIA MAX.MARKS MARKS OBTAINED
AIM & ALGORITHM 5
EXECUTION & OUTPUT 10
VIVA 5
TOTAL 20
RESULT:
EX.NO: 8
TO FIND MEAN, MEDIAN AND MODE
DATE:
AIM:
To write a program in C to implement to find Mean, Median and Mode.
PROGRAM:
#include <stdio.h>
#include <stdlib.h>
// Function to calculate the mean
float calculateMean(int arr[], int n)
{
int sum = 0;
for (int i = 0; i < n; i++)
{
sum += arr[i];
}
return (float)sum / n;
}
// Function to calculate the median
float calculateMedian(int arr[], int n)
{
// Sort the array
for (int i = 0; i < n - 1; i++)
{
for (int j = 0; j < n - i - 1; j++)
{
if (arr[j] > arr[j + 1])
{
// Swap arr[j] and arr[j+1]
int temp = arr[j];
arr[j] = arr[j + 1];
arr[j + 1] = temp;
}}}
// Calculate the median
if (n % 2 == 0)
{
// If the number of elements is even, average the middle two elements
return (float)(arr[(n - 1) / 2] + arr[n / 2]) / 2.0;
}
else
{
// If the number of elements is odd, return the middle element
return (float)arr[n / 2];
}}
// Function to calculate the mode
int calculateMode(int arr[], int n)
{
int mode = arr[0]; // Initialize mode to the first element
int maxCount = 1; // Initialize maxCount to 1
int current = arr[0];
int currentCount = 1;
for (int i = 1; i < n; i++)
{
if (arr[i] == current)
{
currentCount++;
}
else
{
if (currentCount > maxCount)
{
maxCount = currentCount;
mode = current;
}
current = arr[i];
currentCount = 1;
}
}
// Check for mode at the end of the array
if (currentCount > maxCount)
{
mode = current;
}
return mode;
}
int main()
{
int n;
printf("Enter the number of elements: ");
scanf("%d", &n);
int *arr = (int *)malloc(n * sizeof(int));
printf("Enter the elements:\n");
for (int i = 0; i < n; i++)
{
scanf("%d", &arr[i]);
}
float mean = calculateMean(arr, n);
float median = calculateMedian(arr, n);
int mode = calculateMode(arr, n);
printf("Mean: %.2f\n", mean);
printf("Median: %.2f\n", median);
printf("Mode: %d\n", mode);
free(arr); // Free dynamically allocated memory
return 0;
}
INPUT/OUTPUT:
CRITERIA MAX.MARKS MARKS OBTAINED
AIM & ALGORITHM 5
EXECUTION & OUTPUT 10
VIVA 5
TOTAL 20
RESULT:
EX.NO: 9
TO FIND QUARTILE DEVIATION
DATE:
AIM:
To write a program in C to implement to find Quartile Deviatrion
PROGRAM:
#include <stdio.h>
#include <stdlib.h>
// Function to compare two integers for sorting
int compare(const void *a, const void *b)
{
return (*(int *)a - *(int *)b);
}
// Function to calculate the quartile deviation
float calculateQuartileDeviation(int arr[], int n)
{
// Sort the array in ascending order
qsort(arr, n, sizeof(int), compare);
// Calculate the median (Q2)
float median;
if (n % 2 == 0)
{
median = (arr[n / 2 - 1] + arr[n / 2]) / 2.0;
}
else
{
median = arr[n / 2];
}
// Calculate the first quartile (Q1)
float q1;
int mid = n / 2;
if (mid % 2 == 0)
{
q1 = (arr[mid / 2 - 1] + arr[mid / 2]) / 2.0;
}
else
{
q1 = arr[mid / 2];
}
// calculate the third quartile (Q3)
float q3;
if (mid % 2 == 0) {
q3 = (arr[mid + mid / 2 - 1] + arr[mid + mid / 2]) / 2.0;
}
else
{
q3 = arr[mid + mid / 2];
}
// Calculate the quartile deviation
return (q3 - q1) / 2.0;
}
int main()
{
int n;
printf("Enter the number of elements: ");
scanf("%d", &n);
int *arr = (int *)malloc(n * sizeof(int));
printf("Enter the elements:\n");
for (int i = 0; i < n; i++)
{
scanf("%d", &arr[i]);
}
float quartileDeviation = calculateQuartileDeviation(arr, n);
printf("Quartile Deviation: %.2f\n", quartileDeviation);
free(arr); // Free dynamically allocated memory
return 0;
}
INPUT \ OUTPUT:
CRITERIA MAX.MARKS MARKS OBTAINED
AIM & ALGORITHM 5
EXECUTION & OUTPUT 10
VIVA 5
TOTAL 20
RESULT:
EX.NO:10
TO FIND PEARSON'S COEFFICIENT OF SKEWNESS
DATE:
AIM:
To write a program in C to implement to find Pearson's coefficient of skewness.
PROGRAM:
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
// Function to compare two integers for sorting
int compare(const void *a, const void *b)
{
return (*(int *)a - *(int *)b);
}
// Function to calculate the mean of an array
float calculateMean(int arr[], int n)
{
int sum = 0;
for (int i = 0; i < n; i++)
{
sum += arr[i];
}
return (float)sum / n;
}
// Function to calculate the median of an array
float calculateMedian(int arr[], int n)
{
// Sort the array
qsort(arr, n, sizeof(int), compare);
// Calculate the median
if (n % 2 == 0)
{
return (arr[n / 2 - 1] + arr[n / 2]) / 2.0;
}
else
{
return arr[n / 2];
}
}
// Function to calculate Pearson's coefficient of skewness
float calculateSkewness(int arr[], int n)
{
float mean = calculateMean(arr, n);
float median = calculateMedian(arr, n);
// Calculate the standard deviation
float sum_squared_diff = 0.0;
for (int i = 0; i < n; i++)
{
float diff = arr[i] - mean;
sum_squared_diff += diff * diff;
}
float standard_deviation = sqrt(sum_squared_diff / n);
// Calculate skewness
return (3.0 * (mean - median)) / standard_deviation;
}
int main()
{
int n;
printf("Enter the number of elements: ");
scanf("%d", &n);
int *arr = (int *)malloc(n * sizeof(int));
printf("Enter the elements:\n");
for (int i = 0; i < n; i++)
{
scanf("%d", &arr[i]);
}
float skewness = calculateSkewness(arr, n);
printf("Pearson's Coefficient of Skewness: %.2f\n", skewness);
free(arr); // Free dynamically allocated memory
return 0;
}
INPUT \ OUTPUT:
CRITERIA MAX.MARKS MARKS OBTAINED
AIM & ALGORITHM 5
EXECUTION & OUTPUT 10
VIVA 5
TOTAL 20
