lab5.cpp

using namespace std;

#include <math.h>
#include <iomanip>
#include <iostream>

double Function(double x) {
    return exp(pow(x - 15, 0.25)) * sin(1.2 * x);
}

double Legendre(int n, double x) {
    double Ln_next, Ln = x, Ln_first = 1;

    if (n == 0) return Ln_first;

    for (int i = 1; i < n; i++) {
        Ln_next = (1.0 * (2 * i + 1) / (i + 1)) * x * Ln - (1.0 * i / (i + 1)) * Ln_first;
        Ln_first = Ln;
        Ln = Ln_next;
    }
    return Ln;
}

double func(int N, double t, int k1, int k2, double a, double b) {
    double x = (2 * t - a - b) / (b - a);
    if (k2 == N) return Function(t) * Legendre(k1, x);
    else return Legendre(k1, x) * Legendre(k2, x);
}

double Simpson(int k1, int k2, double a, double b, int n, int N) {
    double sig1 = 0, sig2 = 0, y0 = func(N, a, k1, k2, a, b), yn = func(N, b, k1, k2, a, b);

    double h = (b - a) / n;
    for (int i = 1; i < n; i++) {
        if (i % 2 == 0) sig2 += func(N, a + i * h, k1, k2, a, b);
        else sig1 += func(N, a + i * h, k1, k2, a, b);
    }

    return h / 3 * (2 * sig2 + 4 * sig1 + y0 + yn);
}

double Runge(int k1, int k2, double a, double b, int N) {
    double error = 1e-8;

    int n = 2 * ceil((b - a) / pow(error, 0.25) / 2.0);
    double h = (b - a) / n;

    double R = 0, In, I2n = Simpson(k1, k2, a, b, n, N);
    do {
        In = I2n;
        I2n = Simpson(k1, k2, a, b, n *= 2, N);
        R = fabs(In - I2n) / 15;
    } while (R > error);

    return I2n;
}

double *CompletePivoting(double **matrix, int n, int m) {
    int p, q, line_new = 0;
    double M[n], t[n][m];

    double *x = new double[n];

    for (int k = 0; k < n - 1; k++){
        double a = 0;
        for (int i = 0; i < n; i++) {
            for (int j = 0; j < m - 1; j++) {
                if (fabs(matrix[i][j]) > a) {
                    a = fabs(matrix[i][j]);
                    p = i;
                    q = j;
                }
            }
        }

        for (int j = 0; j < m; j++) {
            t[line_new][j] = matrix[p][j];
        }
        line_new++;

        for (int i = 0; i < n; i++) {
            M[i] = -matrix[i][q] / a;
        }

        for (int i = 0; i < n; i++) {
            for (int j = 0; j < m; j++) {
                if (i != p) {
                    matrix[i][j] += matrix[p][j] * M[i];
                }
            }
        }

        for (int j = 0; j < m; j++) {
            matrix[p][j] = 0;
        }

        for (int i = 0; i < n; i++) {
            matrix[i][q] = 0;
        }

    }

    for (int i = 0; i < n; i++) {
        if (matrix[i][m - 1] != 0) {
            p = i;
        }
    }

    for (int j = 0; j < m; j++) {
        t[line_new][j] = matrix[p][j];
    }

    for (int j = 0; j < m-1; j++) {
        x[j] = 0;
    }

    for (int i = n - 1; i >= 0; i--){
        double R = t[i][m - 1];
        for (int j = 0; j < m - 1; j++) {
            if ((t[i][j] != 0) && (x[j] != 0)) {
                R -= t[i][j] * x[j];
            }
        }
        for (int j = 0; j < m - 1; j++) {
            if ((t[i][j] != 0) && (x[j] == 0)) {
                x[j] = R / t[i][j];
            }
        }
    }

    return x;
}

double *MakeA(double a, double b, int N) {
    double ** A = new double*[N];
    for (int i = 0; i < N; i++)
        A[i] = new double[N + 1];

    for (int i = 0; i < N; i++) {
        for (int j = 0; j < N; j++) {
            A[i][j] = A[j][i] = Runge(i, j, a, b, N);
        }
        A[i][N] = Runge(i, N, a, b, N);
    }

    return CompletePivoting(A, N, N + 1);

}

double Pm(double x, double a, double b, double *A, int n) {
    double t = (2 * x - b - a) / (b - a), s = 0;
    for (int j = 0; j < n; j++) {
        s += A[j] * Legendre(j, t);
    }
    return s;
}

int main() {
    double a = 20, b = 30;
    double eps = 1e-3;

    int k = (b - a) * 2;
    double step = (b - a) / (double)k;
    int n = 2;

    while (true) {
        n *= 2;
        double deviation = 0;
        double x = a;
        double *A = MakeA(a, b, n);
        for (int i = 0; i <= k; ++i) {
            double y = Function(x) - Pm(x, a, b, A, n);
            deviation += y * y;
            x += step;
        }

        if (sqrt(deviation / (k + 1)) <= eps) break;
    }

    cout << "N = " << n << endl << endl;

    double *A = MakeA(a, b, n), x = a;
    cout << "X     P(x)" << endl;
    for (int i = 0; i <= k; i++, x += step) {
        printf("%-4f  %.8f\n", x, Pm(x, a, b, A, n));
    }
}