Lab

// Newton-Raphson:
double fun(double x) {
    return 3 * x * x + 2 * x - 4;
}
double dfun(double x) {
    return 6 * x + 2;
}
int main() {
    int i = 0; double x = 0;
    do {
        double x1 = x;
        x = x1 - fun(x1) / dfun(x1);
        cout << "At iteration-" << ++i << " , Root = " << x << '\n';
    } while (fabs(fun(x)) > 0.0001);
}

// Secant Method:
double fun(double x) {
    return 2 * x * x + 3 * x - 7;
}
int main() {
    int i = 1; double x1 = 5, x2 = 1;
    double x3 = (fun(x2) * x1 - fun(x1) * x2) / (fun(x2) - fun(x1));
    cout << fixed << "At iteration-" << i << " :\n";
    cout << "Root = " << x3 << "\nFunction value = " << fun(x3) << '\n';
    while (fabs((x3 - x2) / x3) > 0.0001) {
        x1 = x2; x2 = x3;
        x3 = (fun(x2) * x1 - fun(x1) * x2) / (fun(x2) - fun(x1));
        cout << "At iteration-" << ++i << " :\n";
        cout << "Root = " << x3 << "\nFunction value = " << fun(x3) << '\n';
    }
}

// Guess Eliminaton:
float a[10][10], x[10], ratio;
int i, j, k, n;
cout << setprecision(3) << fixed;
cout << "Enter number of unknowns:";
cin >> n;
cout << "Enter co of augmented matrix:" << '\n';
for (i = 1;i <= n;++i) {
    for (j = 1;j <= n + 1;++j) {
        cout << "a[" << i << "]" << j << "] =";
        cin >> a[i][j];
    }
}
for (k = 1;k <= n - 1;k++) {
    for (i = k + 1;i <= n;++i) {
        ratio = a[i][k] / a[k][k];
        for (j = k;j <= n + 1;++j) {
            a[i][j] -= (ratio * a[k][j]);
        }
    }
}
cout << "after gauess:" << '\n';
for (i = 1;i <= n;++i) {
    for (j = 1;j <= n + 1;++j) {
        cout << a[i][j] << " ";
    }
    cout << '\n';
}
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];
}
cout << '\n' << "Solution: " << '\n';
for (i = 1;i <= n;++i)
{
    cout << "x[" << i << "] = " << x[i] << '\n';
}
return 0;
}

// Jecobi Method:
int n = 3;
double a[n][n + 1] = { {2,1,1,5},{3,5,2,15},{2,1,4,8} }, xn[3], xo[3] = { 0,0,0 };
int c = 0;
while (true) {
    int f = 0;
    for (int i = 0;i < n;i++) {
        double sum = 0;
        for (int j = 0;j < n;j++) {
            if (j != i) sum += (a[i][j] * xo[j]);
        }
        xn[i] = (a[i][n] - sum) / a[i][i];
    }
    for (int k = 0;k < n;k++) {
        if (abs((xn[k] - xo[k])) > 0.0001) {
            xo[k] = xn[k];
            f++;
        }
    }
    c++;
    if (f == 0)break;
}
for (int i = 0;i < n;i++) {
    cout << xn[i] << " " << endl;
}

// Gauss Seidal
int n = 3;
double a[n][n + 1] = { {2,1,1,5},{3,5,2,15},{2,1,4,8} }, xo[3] = { 0,0,0 };
int c = 0;
while (true) {
    int f = 0;
    double xn;
    for (int i = 0;i < n;++i) {
        double sum = 0;
        for (int j = 0;j < n;++j) {
            if (j != i) sum += (a[i][j] * xo[j]);
        }
        double xn = (a[i][n] - sum) / a[i][i];
        if (abs((xn - xo[i]) / xn) > 0.0001) {
            xo[i] = xn;
            f++;
        }
    }
    c++;
    if (f == 0)break;
}
for (int i = 0;i < n;++i) {
    cout << xo[i] << " " << '\n';
}
return 0;
}


// Least Square
#define f(x) 1/(1+pow(x,2))
int i, n;
double x, y, a, b;
double sx = 0, sxx = 0, sy = 0, sxy = 0;
cout << "Enter the number of values for n : ";
cin >> n;
for (int i = 0;i < n;++i) {
    cout << "Enter the value of x and y : ";
    cin >> x >> y;
    sx += x;
    sy += y;
    sxx += (x * x);
    sxy += (x * y);
}
double xb = sx / n;
double yb = sy / n;
b = (sxy - yb * sx) / (sxx - xb * sx);
a = (sy - b * sx) / n;
cout << "Values of a and b are : " << a << " " << b << '\n';


// Simpson 3/8
#define f(x) 1/(1+pow(x,2))
double lower, upper, i = 0.0, h, k;
int i, n;
cout << "Enter lower limit of integration: ";
cin >> lower;
cout << "Enter upper limit of integration: ";
cin >> upper;
cout << "Enter number of sub intervals: ";
cin >> n;
h = (upper - lower) / n;
i = f(lower) + f(upper);
for (i = 1; i <= n - 1; ++i)
{
    k = lower + i * h;
    if (i % 3 == 0) i = i + 2 * (f(k));
    else i = i + 3 * (f(k));
}

i = i * h * 3.0 / 8.0;

cout << '\n' << "Area is: " << i;

// Simpson 1/3
#define f(x) 1/(1+pow(x,2))
double lower, upper, i = 0.0, h, k;
int i, n;
cout << "Enter lower limit of integration: ";
cin >> lower;
cout << "Enter upper limit of integration: ";
cin >> upper;
cout << "Enter number of sub intervals: ";
cin >> n;
h = (upper - lower) / n;

i = f(lower) + f(upper);

for (i = 1; i <= n - 1; ++i)
{
    k = lower + i * h;
    if (i % 2 == 0) i = i + 2 * (f(k));
    else i = i + 4 * (f(k));
}

i = i * h / 3;

cout << '\n' << "Area is: " << i;

/// Trapizoidal
#define f(x) 1/(1+pow(x,2))
double lower, upper, i = 0.0, h, k;
int i, n;
cout << "Enter lower limit of integration: ";
cin >> lower;
cout << "Enter upper limit of integration: ";
cin >> upper;
cout << "Enter number of sub intervals: ";
cin >> n;
h = (upper - lower) / n;

i = f(lower) + f(upper);

for (i = 1; i <= n - 1; ++i)
{
    k = lower + i * h;
    i = i + 2 * (f(k));
}

i = i * h / 2;

cout << '\n' << "Area is: " << i;

// Lagrange
double x[100], y[100], xp, yp = 0, p;
int i, j, n;
cout << "Enter number of data: ";
cin >> n;
cout << "Enter data:" << '\n';
for (i = 1;i <= n;++i)
{
    cout << "x[" << i << "] = ";
    cin >> x[i];
    cout << "y[" << i << "] = ";
    cin >> y[i];
}
cout << "Enter interpolation point: ";
cin >> xp;
for (i = 1;i <= n;++i)
{
    p = 1;
    for (j = 1;j <= n;++j)
    {
        if (i != j)
        {
            p = p * (xp - x[j]) / (x[i] - x[j]);
        }
    }
    yp = yp + p * y[i];
}
cout << '\n' << "Interpolated value at " << xp << " is " << yp;