Numerical Analysis

//e^(-3x/2)

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

int main(){
	
	double ans=0;
	for(double i=0.1; i<=2; i+=0.1){
		ans += exp((-1)*i*i*i/2);
	}	
	
	printf("%lf", ans);
}

//5x^3 + e^(-2x)

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

int main(){
	
	double ans=0;
	for(double i=0.1; i<=2; i+=0.1){
		ans += 5*i*i*i + exp((-1)*2*i);
	}	
	
	printf("%lf", ans);
}

//e^x
#include<stdio.h>
#include<math.h>

int main(){
	
	double ans=1;
	double x;
	scanf("%lf", &x);
	
	double factorial=1;

	for(double i=2; i<=6; i++){
		ans += pow(x, i-1)/factorial;
		factorial*=i;
	}	
	
	printf("%lf", ans);
}

//ln(1+x)
#include<stdio.h>
#include<math.h>

int main(){
	
	double ans=0;
	double x;
	scanf("%lf", &x);
	
	double factorial=1;

	for(int i=1; i<=6; i++){
		if(i%2==0)
			ans -= pow(x, i)/factorial;
		else 	
			ans += pow(x, i)/factorial;
			
		factorial*=i;
	}	
	
	printf("%lf", ans);
}

// ln (1-x)
#include<stdio.h>
#include<math.h>

int main(){
	
	double ans=0;
	double x;
	scanf("%lf", &x);
	
	double factorial=1;

	for(int i=1; i<=6; i++){	
		ans += pow(x, i)/factorial;	
		factorial*=i;
	}	
	ans *= (-1);
	printf("%lf", ans);
}

//Bisection Method

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

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

int main(){
	double a,b;
	scanf("%lf", &a);
	scanf("%lf", &b);
	
	while(true){
		double c = (a+b)/2;
		if(f(a)*f(c) < 0) b=c;
		else a=c;
		if(fabs((a)-(b)) <= 0.0001) break;
	}
	printf("%lf", a);
}


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

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

int main(){
	double a,b;
	scanf("%lf", &a);
	scanf("%lf", &b);
	
	while(true){
		double c = (a+b)/2;
		if(f(a)*f(c) < 0) b=c;
		else a=c;
		if(fabs((a)-(b)) <= 0.0001) break;
	}
	printf("%lf", a);
}


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

#define f(x) (x*x*x-6*x+4)

int main(){
	double a,b;
	scanf("%lf", &a);
	scanf("%lf", &b);
	
	while(true){
		double c = (a+b)/2;
		if(f(a)*f(c) < 0) b=c;
		else a=c;
		if(fabs((a)-(b)) <= 0.0001) break;
	}
	printf("%lf", a);
}

// False Position

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

#define f(x) (x*x*x-6*x+4)

int main(){
	double a,b;
	scanf("%lf", &a);
	scanf("%lf", &b);
	double ans=a;
	while(true){
		double c = ((a*f(b) - b*f(a))/(f(b)-f(a)));
		if(f(a)*f(c) < 0) b=c;
		else a=c;
		ans=c;
		if(fabs(f(a)-f(b)) <= 0.0001) break;
	}
	printf("%lf", ans);
}

// Newton Raphson

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

#define f(x) (x*x*x-6*x+4)
#define diff(x) (3*x*x - 6)
int main(){
	double x0;
	scanf("%lf", &x0);
	double x=x0;
	while(true){
		x = x0 - (f(x)/diff(x));
		if(fabs(f(x)-f(x0)) <= 0.0001) break;
		x0=x;
	}
	printf("%lf", x);
}

// Upper-triangular method

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

#define ll int

int main(){
	int n,m;
	scanf("%d %d", &n, &m);
	int a[n][m];
	
	for(ll i=0; i<n; i++)
		for(ll j=0; j<m; j++)
			scanf("%d", &a[i][j]);
			
	for(ll i=0; i<n; i++){
		for(ll j=0; j<i; j++) printf("0 ");
		for(ll j=i; j<m; j++){
			printf("%d ", a[i][j]);
		}
		printf("\n");
	}
}

// lower triangular
#include<stdio.h>
#include<math.h>

#define ll int

int main(){
	int n,m;
	scanf("%d %d", &n, &m);
	int a[n][m];
	
	for(ll i=0; i<n; i++)
		for(ll j=0; j<m; j++)
			scanf("%d", &a[i][j]);
			
	for(ll i=0; i<n; i++){
		for(ll j=0; j<=i; j++){
			printf("%d ", a[i][j]);
		}
		for(ll j=i+1; j<m; j++) printf("0 ");
		printf("\n");
	}
}

//Diagonal Matrix
#include<stdio.h>
#include<math.h>

#define ll int

int main(){
	int n,m;
	scanf("%d %d", &n, &m);
	int a[n][m];
	
	for(ll i=0; i<n; i++)
		for(ll j=0; j<m; j++)
			scanf("%d", &a[i][j]);
			
	for(ll i=0; i<n; i++){
		for(ll j=0; j<m; j++){
			if(i==j) printf("%d ", a[i][j]);
			else printf("0 ");
		}
		printf("\n");
	}
}

// Deteminant

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

#define ll int

int main(){
	int n,m;
	scanf("%d %d", &n, &m);
	int a[n][m];
	
	for(ll i=0; i<n; i++)
		for(ll j=0; j<m; j++)
			scanf("%d", &a[i][j]);
			
	double determinant=0;
	for(ll i=0; i<n; i++)
		determinant += a[0][i]*(a[1][(i+1)%n]*a[2][(i+2)%n] - a[1][(i+2)%n]*a[2][(i+1)%n]);
			
	printf("%lf\n", determinant);					
}

//Transpose Matrix
#include<stdio.h>
#include<math.h>

#define ll int

int main(){
	int n,m;
	scanf("%d %d", &n, &m);
	int a[n][m];
	
	for(ll i=0; i<n; i++)
		for(ll j=0; j<m; j++)
			scanf("%d", &a[i][j]);
				
	for(ll i=0; i<n; i++){
		for(ll j=0; j<n; j++){
			printf("%d ", a[j][i]);
		}
		printf("\n");
	}													
}

//Adjoint Matrix
#include<stdio.h>
#include<math.h>

#define ll int

int main(){
	int n,m;
	scanf("%d %d", &n, &m);
	int a[n][m];
	
	for(ll i=0; i<n; i++)
		for(ll j=0; j<m; j++)
			scanf("%d", &a[i][j]);
				
	
	int adj[n][m];
	for(ll i=0; i<n; i++){
		for(ll j=0; j<n; j++){
			adj[i][j] = a[(i+1)%n][(j+1)%n]*a[(i+2)%n][(j+2)%n]
						- a[(i+1)%n][(j+2)%n]*a[(i+2)%n][(j+1)%n];			
		}
	}
	
	for(ll i=0; i<n; i++)
	{
		for(ll j=0; j<m; j++){
			printf("%d ", adj[j][i]);
		}
		printf("\n");
	}
													
}

//Inverse Matrix

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

#define ll int

int main(){
	int n,m;
	scanf("%d %d", &n, &m);
	int a[n][m];
	
	for(ll i=0; i<n; i++)
		for(ll j=0; j<m; j++)
			scanf("%d", &a[i][j]);
				
	
	double determinant = 0;
	for(ll i=0; i<n; i++)
		determinant += a[0][i]*(a[1][(i+1)%n]*a[2][(i+2)%n]- a[1][(i+2)%n]*a[2][(i+1)%n]);
	
	double adj[n][m];
	for(ll i=0; i<n; i++)
		for(ll j=0; j<m; j++)
			adj[i][j] = a[(i+1)%n][(j+1)%n]*a[(i+2)%n][(j+2)%n]
									- a[(i+1)%n][(j+2)%n]*a[(i+2)%n][(j+1)%n];
	
	
	for(ll i=0; i<n; i++){
		for(ll j=0; j<m; j++){
			printf("%0.4lf ", adj[j][i]/determinant);
		}	
		printf("\n");
	}

}

// Gauss Elimination

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

#define ll int

int main(){
	int n;
	scanf("%d", &n);
	double a[n+1][n+2];
	
	for(ll i=1; i<=n; i++){
		for(ll j=1; j<=n; j++){
			scanf("%lf", &a[i][j]);
		}
	}
	
	for(ll i=1; i<=n; i++) 
		scanf("%lf", &a[i][n+1]);
	
	double c;
	for(ll j=1; j<=n; j++){
		for(ll i=1; i<=n; i++){
			if(i > j){
				c=a[i][j]/a[j][j];
				for(ll k=1; k<=n+1; k++){
					a[i][k]=a[i][k]-c*a[j][k];
				}	
			}
		}
	}
	
	double x[n+1];
	x[n]=a[n][n+1]/a[n][n];
	for(ll i=n-1; i>=1; i--){
		double sum=0;
		for(ll j=i+1; j<=n; j++){
			sum += a[i][j]*x[j];
		}
		x[i]=(a[i][n+1]-sum)/a[i][i]; 
	}
	
	for(ll i=1; i<=n; i++) printf("%lf\n", x[i]);

}

// Matrix Inversion Method

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

#define ll int

int main(){
	int n;
	scanf("%d", &n);
	double a[n][n], b[n];
	
	for(ll i=0; i<n; i++){
		for(ll j=0; j<n; j++){
			scanf("%lf", &a[i][j]);
		}
	}
	
	for(ll i=0; i<n; i++) 
		scanf("%lf", &b[i]);
		
	double determinant=0;
	for(ll i=0; i<n; i++)
		determinant += a[0][i]*(a[1][(i+1)%n]*a[2][(i+2)%n]
					    - a[1][(i+2)%n]*a[2][(i+1)%n]);
	
	double adj[n][n];
	for(ll i=0; i<n; i++)
		for(ll j=0; j<n; j++)
			adj[i][j] = a[(i+1)%n][(j+1)%n]*a[(i+2)%n][(j+2)%n]
						- a[(i+1)%n][(j+2)%n]*a[(i+2)%n][(j+1)%n];

	double adjoint[n][n];
	for(ll i=0; i<n; i++)
		for(ll j=0; j<n; j++)
			adjoint[j][i]=(adj[i][j]);
			
	for(ll i=0; i<n; i++){
		double sum=0;
		for(ll j=0; j<n; j++){
			sum += b[j]*adjoint[i][j];
		}
		printf("%0.4lf\n", (sum/determinant));
	}
}

// Trapezoidal Rule from X and Y Values

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

#define ll int

int main(){
	ll n;	scanf("%d", &n);
	double a[n], b[n];
	
	for(ll i=0; i<n; i++)
		scanf("%lf %lf", &a[i], &b[i]);
	double xi, xf, h;
	scanf("%lf %lf %lf", &xi, &xf, &h);
	
	double ans=(b[0]+b[n-1])/2;
	for(ll i=1; i<n-1; i++) ans += b[i];
	ans *= h;
	
	printf("%lf", ans);
	
}

// Trapezoidal Rule from f(x), limit, and h

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

#define f(x) (1/ (1+pow(x, 2)))


int main(){
	float lb, ub, h;
	scanf("%f %f %f", &lb, &ub, &h );
	
	int n = (ub-lb)/h + 1;
	
	float y[n];
	int j=0;
	for(int i=0; i<=ub; i+=h){
		y[j] = f(i);
		j++;
	}
	
	float ans = y[0]+y[n-1];
	for(int i=1; i<n-1; i++) ans += 2*y[i];
	
	
	ans *= (h/2);
	
	printf("%f", ans);
	
}

// Simpson one-third rule

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

#define ll int

int main(){
	ll n;	scanf("%d", &n);
	double a[n], b[n];
	
	for(ll i=0; i<n; i++)
		scanf("%lf %lf", &a[i], &b[i]);
	double xi, xf, h;
	scanf("%lf %lf %lf", &xi, &xf, &h);
	
	double ans=(b[0]+b[n-1]);
	for(ll i=1; i<n-1; i+=2) ans += 4*b[i];
	for(ll i=2; i<n-1; i+=2) ans += 2*b[i];
	ans *= (h/3);
	
	printf("%lf", ans);
}

//Newton Forward Interpolation

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

int main()
{
	int n;
	printf("The number of x and f(x)");
	scanf("%d", &n);
	double x[n], y[n], xi;
	printf("Write the number of x:");
	for (int i = 0; i < n; i++)
	{
		scanf("%lf", &x[i]);
	}
	printf("Write the number of f(x):");
	for (int i = 0; i < n; i++)
	{
		scanf("%lf", &y[i]);
	}

	printf("X = ");
	scanf("%lf", &xi);

	double da[n][n];
	for (int j = 0; j < n; j++)
	{
		da[j][0] = y[j];
	}

	for (int i = 1; i < n; i++)
	{

		for (int j = 0; j < n - i; j++)
		{

			da[j][i] = (da[j + 1][i - 1] - da[j][i - 1]);
		}
	}

	double p = (xi - x[0]) / (x[1] - x[0]);
	double ans = y[0];
	double fact=1;
	for (int i = 1; i < n; i++)
	{
		double localP = p * 1.00;
		fact *= 1.00;

		for (int j = 1; j < i; j++)
		{
			localP *= (p - j);
		}
		ans += (localP / fact) * da[0][i];
	}

	printf(" %lf ", ans);

	return 0;
}

//Newton Backward

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

int main()
{
	int n;
	printf("The number of x and f(x)");
	scanf("%d", &n);
	double x[n], y[n], xi;
	printf("Write the number of x:");
	for (int i = 0; i < n; i++)
	{
		scanf("%lf", &x[i]);
	}
	printf("Write the number of f(x):");
	for (int i = 0; i < n; i++)
	{
		scanf("%lf", &y[i]);
	}

	printf("X = ");
	scanf("%lf", &xi);

	double da[n][n];
	for (int j = 0; j < n; j++)
	{
		da[j][0] = y[j];
	}

	for (int i = 1; i < n; i++)
	{

		for (int j = 0; j < n - i; j++)
		{

			// printf(" %lf  %lf ", da[j + 1][i - 1], da[j][i - 1]);
			da[j][i] = (da[j + 1][i - 1] - da[j][i - 1]);
			// printf(" %lf \n ", da[j][i]);
		}
	}

	double p = (xi - x[n - 1]) / (x[1] - x[0]);
	double ans = y[n - 1];
	for (int i = n - 1; i >= 1; i--)
	{
		double localP = p * 1.00;
		double fact = 1.00;

		for (int j = 1; j <= n - i - 1; j++)
		{
			fact = fact * j;
		}
		for (int j = 1; j < n - i - 1; j++)
		{
			localP *= (p - j);
		}
		ans += (localP / fact) * da[n - i - 1][i];
	}

	printf("%lf ", ans);

	return 0;
}

//Newton Devided Difference

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


int main()
{
	int n;
	//taking the total number of x
	printf("Enter number of x and y or f(x)\n");
	scanf("%d", &n);
	double x[n], y[n], X;
	
	
	// taking the value of X
	printf("jot down the value of x\n");
	for (int i = 0; i < n; i++)
		scanf("%lf", &x[i]);
	
	// taking the value of y/f(x)
	printf("write down the number of y or f(x):\n");
	for (int i = 0; i < n; i++)
		scanf("%lf", &y[i]);

	//taking the value of question
	printf("The X you want to get = ");
	scanf("%lf", &X);

	double table[n][n];
	for (int j = 0; j < n; j++)
		table[j][0] = y[j];
	

	for (int i = 1; i < n; i++)
		for (int k = 0; k < n - i; k++)
		table[k][i] = (table[k + 1][i - 1] - table[k][i - 1]) / (x[i + k] - x[k]);
	
	double ans = y[0];
	for (int i = 1; i < n-1; i++)
	{
		double ansa = 1.00;
		for (int k = 0; k <= i-1; k++)
			ansa *= (X - x[k]);
		ans += (ansa)*table[0][i];
	}


	//printing ans
	printf("Ans: %lf ", ans);
	return 0;
}

// Version 1: Numerical Differentiation
// Warning : Bhuuul acheee

#include <stdio.h>
#include <stdlib.h>
#include <math.h>
int n;
int main()
{
	//taking the total number of x
	printf("Enter number of x and y or f(x)\n");
	scanf("%d", &n);
	double x[n], y[n], X;
	
	
	// taking the value of X
	printf("jot down the value of x\n");
	for (int i = 0; i < n; i++)
		scanf("%lf", &x[i]);
	
	// taking the value of y/f(x)
	printf("write down the number of y or f(x):\n");
	for (int i = 0; i < n; i++)
		scanf("%lf", &y[i]);

	//taking the value of question
	printf("The X you want to get = ");
	scanf("%lf", &X);
	
	int Xindex;
	
	for(int i=0; i<n; i++){
		if(X == x[i]) Xindex=i;
	}

	double table[n][n];
	for (int j = 0; j < n; j++)
	{
		table[j][0] = y[j];
	}

	for (int i = 1; i < n; i++)
		for (int j = 0; j < n - i; j++)
			table[j][i] = (table[j + 1][i - 1] - table[j][i - 1]);

	double ans = 0.000000;
			

	for (int i = 0; i < n; i++)
	{
		if(i%2 == 1) 
			ans-=(1/(i+1))*(table[Xindex-i][i]);
		
		else  
			ans+=(1/(i+1))*(table[Xindex-i][i]);
	}

	printf("Ans for the 3rd derivative  :  %lf ", ans);

	return 0;
}

/* 
Version 02 : Numerical Differentiation 
From : Aminul Imam Robi (MUH2101038M)
*/

#include <stdio.h>
#include <math.h>
int main()
{
    int n;
    printf("Enter number of data points : ");
    scanf("%d", &n);
    double x[n], y[n];

    printf("Enter values of x : ");
    for (int i = 0; i < n; i++)
        scanf("%lf", &x[i]);

    printf("Enter values of y : ");
    for (int i = 0; i < n; i++)
        scanf("%lf", &y[i]);

    double value;
    printf("Enter a value of x : ");
    scanf("%lf", &value);

    int index;
    for (int i = 0; i < n; i++)
    {
        if (value == x[i])
        {
            index = i;
            break;
        }
    }

    // Making table
    double table[n][n];
    for (int i = 0; i < n; i++)
        table[i][0] = y[i];
    for (int i = 1; i < n; i++)
    {
        for (int j = 0; j < n - i; j++)
        {
            table[j][i] = table[j + 1][i - 1] - table[j][i - 1];
        }
    }

    // Result calculation
    double h = x[1] - x[0];
    double p = (value - x[0]) / (x[1] - x[0]);
    double a = (2 * p - 1) / 2;
    double b = (3 * p * p - 6 * p + 2) / 6;
    double answer = 0;
    for (int i = 1; i < 4; i++)
    {
        if (i == 2)
        {
            answer += (a * table[0][i]);
        }
        if (i == 3)
        {
            answer += (b * table[0][i]);
        }
        else
        {
            answer += table[0][i];
        }
    }
    answer *= (1 / h);
    printf("Answer : %.4lf\n", answer);
}

//Lagrange Interpolation

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

int main()
{
    int n;
    printf("The number of x and f(x)");
    scanf("%d", &n);
    double x[n], y[n], xi;
    printf("Write the number of x:");
    for (int i = 0; i < n; i++)
    {
        scanf("%lf", &x[i]);
    }
    printf("Write the number of f(x):");
    for (int i = 0; i < n; i++)
    {
        scanf("%lf", &y[i]);
    }

    printf("X = ");
    scanf("%lf", &xi);

    double da[n][n];
    for (int j = 0; j < n; j++)
    {
        da[j][0] = y[j];
    }

    double ans = 0;
    for (int i = 0; i < n; i++)
    {
        double localAns = 1;
        for (int j = 0; j < n; j++)
        {
            if (i == j)
                continue;
            localAns *= (xi - x[j]);
            localAns /= (x[i] - x[j]);
        }
        ans += localAns * y[i];
    }
    printf(" %lf ", ans);

    return 0;
}

//Iterative Method

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

#define f(x) (x * x * x + x * x - 1)
#define phi(x) 1 / sqrt(1 + x)
#define diffPhi(x) (0.5 / sqrt(1 + x))

int main()
{
	double a = 0, b = 0;
	double x1 = 0, x2 = 0;

	// Step 1 : finding the value of x1 and x2
	while (1)
	{
		/*
			ekhane dui bhabe check kortechi,
			dhoren,
			1) (0 1), (1,2) , ...
			2) (0,-1), (-1,-2), ....
			ig, yk why.
		*/
		if (f(a) * f((a - (double)1.0000)) < 0)
		{
			printf("%lf %lf", f(a), f((a - (double)1.0000)));
			x1 = a;
			x2 = a - 1;

			break;
		}
		if (f(b) * f((b + (double)1.0000)) < 0)
		{
			x1 = b;
			x2 = b + 1;
			break;
		}
		a--;
		b++;
	}
	// Step 2 & 3 : finding the root
	// we'll just jot down the steps which are written on the blog
	double ans = 1e9; // Just assume a number which can never be the answer
	double x0 = (x1 + x2) / 2;

	if (abs(diffPhi(x0)) < 1)
	{
		while (1)
		{
			double xn = phi(x0);
			if (fabs(xn - x0) <= 0.001)
			{
				ans = xn;
				break;
			}
			x0 = xn;
		}
	}
	else
	{
		printf("NOT FOUND");
		return 0;
	}

	// Printing the answer
	printf("%lf", ans);
	return 0;
}

//Secant Method

#include <stdio.h>
#include <math.h>
#define f(x) (x * x * x - 2 * x - 5)

int main()
{
	double a = 0, b = 0;
	double x1 = 0, x2 = 0;

	// Step 1 : finding the value of x1 and x2
	while (1)
	{
		/*
			ekhane dui bhabe check kortechi,
			dhoren,
			1) (0 1), (1,2) , ...
			2) (0,-1), (-1,-2), ....
			ig, yk why.
		*/
		if (f(a) * f((a - (double)1.0000)) < 0)
		{
			printf("%lf %lf", f(a), f((a - (double)1.0000)));
			x1 = a;
			x2 = a - 1;

			break;
		}
		if (f(b) * f((b + (double)1.0000)) < 0)
		{
			x1 = b;
			x2 = b + 1;
			break;
		}
		a--;
		b++;
	}
	// Step 2 & 3 : finding the root

	// we'll just jot down the steps which are written on the blog
	double ans = 1e9; // Just assume a number which can never be the answer
	while (1)
	{
		double prevAns = ans;
		double x = ((x1 * f(x2) - x2 * f(x1)) / (f(x2) - f(x1)));
		ans = x;
		x1 = x2;
		x2 = x;
		// Calculating the the  literation of last two answer
		if (fabs((prevAns) - (ans)) <= 0.001)
		{
			break;
		}
	}

	// Printing the answer
	printf("%lf", ans);
	return 0;
}

//Seidel
/*
	From : Saiful
*/

#include<stdio.h>

#define f1(x,y,z) (95-11*y+4*z)/83
#define f2(x,y,z) (104 - 7*x - 13*z)/52
#define f3(x,y,z)   (71-3*x-8*y)/29

int main(){
    double x=0, y=0, z=0;
    double ex=100, ey=100, ez=100;
    double check = 0.00001;
    while(fabs(x-ex) > check && fabs(y-ey)> check && fabs(y-ez) > check){
        ex=x, ey=y, ez=z;
        x=f1(x,y,z);
        y=f2(x,y,z);
        z=f3(x,y,z);
    }
    printf("%lf %lf %lf", x,y,z);


    return 0;
}

// Euler Method from 15th Batch

#include<stdio.h>
float fun(float x,float y)
{
    float f;
    f=x+y;
    return f;
}
main()
{
    float a,b,x,y,h,t,k;
    printf("\nEnter x0,y0,h,xn: ");
    scanf("%f%f%f%f",&a,&b,&h,&t);
    x=a;
    y=b;
    printf("\n  x\t  y\n");
    while(x<=t)
    {
        k=h*fun(x,y);
        y=y+k;
        x=x+h;
        printf("%0.3f\t%0.3f\n",x,y);
    }
}