app.cpp

// app.cpp - Modified version with cubic equation solver
#include <opencv2/opencv.hpp>
#include <algorithm>
#include <cmath>
#include <iostream>
#include <iomanip>
#include <numeric>
#include <random>
#include <vector>
#include <limits>
#include <sstream>
#include <string>
#include <fstream>
#include <complex>

// Struct to hold cubic equation roots
struct CubicRoots {
    std::complex<double> root1;
    std::complex<double> root2;
    std::complex<double> root3;
};

// Function to solve cubic equation: az^3 + bz^2 + cz + d = 0
CubicRoots solveCubic(double a, double b, double c, double d) {
    // Handle special case for a == 0 (quadratic)
    if (std::abs(a) < 1e-14) {
        CubicRoots roots;
        // For a quadratic equation: bz^2 + cz + d = 0
        double discriminant = c * c - 4.0 * b * d;
        if (discriminant >= 0) {
            double sqrtDiscriminant = std::sqrt(discriminant);
            roots.root1 = std::complex<double>((-c + sqrtDiscriminant) / (2.0 * b), 0.0);
            roots.root2 = std::complex<double>((-c - sqrtDiscriminant) / (2.0 * b), 0.0);
            roots.root3 = std::complex<double>(1e99, 0.0); // Infinity for third root
        } else {
            double real = -c / (2.0 * b);
            double imag = std::sqrt(-discriminant) / (2.0 * b);
            roots.root1 = std::complex<double>(real, imag);
            roots.root2 = std::complex<double>(real, -imag);
            roots.root3 = std::complex<double>(1e99, 0.0); // Infinity for third root
        }
        return roots;
    }

    // Normalize equation: z^3 + (b/a)z^2 + (c/a)z + (d/a) = 0
    double p = b / a;
    double q = c / a;
    double r = d / a;

    // Substitute z = t - p/3 to get t^3 + pt^2 + qt + r = 0
    double p1 = q - p * p / 3.0;
    double q1 = r - p * q / 3.0 + 2.0 * p * p * p / 27.0;

    // Calculate discriminant
    double D = q1 * q1 / 4.0 + p1 * p1 * p1 / 27.0;

    // Precompute values
    const double two_pi = 2.0 * M_PI;
    const double third = 1.0 / 3.0;
    const double p_over_3 = p / 3.0;

    CubicRoots roots;

    if (D > 1e-10) { // One real root and two complex conjugate roots
        double sqrtD = std::sqrt(D);
        double u = std::cbrt(-q1 / 2.0 + sqrtD);
        double v = std::cbrt(-q1 / 2.0 - sqrtD);
        
        // Real root
        roots.root1 = std::complex<double>(u + v - p_over_3, 0.0);
        
        // Complex conjugate roots
        double real_part = -(u + v) / 2.0 - p_over_3;
        double imag_part = (u - v) * std::sqrt(3.0) / 2.0;
        roots.root2 = std::complex<double>(real_part, imag_part);
        roots.root3 = std::complex<double>(real_part, -imag_part);
    } 
    else if (D < -1e-10) { // Three distinct real roots
        double angle = std::acos(-q1 / 2.0 * std::sqrt(-27.0 / (p1 * p1 * p1)));
        double magnitude = 2.0 * std::sqrt(-p1 / 3.0);
        
        roots.root1 = std::complex<double>(magnitude * std::cos(angle / 3.0) - p_over_3, 0.0);
        roots.root2 = std::complex<double>(magnitude * std::cos((angle + two_pi) / 3.0) - p_over_3, 0.0);
        roots.root3 = std::complex<double>(magnitude * std::cos((angle + 2.0 * two_pi) / 3.0) - p_over_3, 0.0);
    } 
    else { // D ≈ 0, at least two equal roots
        double u = std::cbrt(-q1 / 2.0);
        
        roots.root1 = std::complex<double>(2.0 * u - p_over_3, 0.0);
        roots.root2 = std::complex<double>(-u - p_over_3, 0.0);
        roots.root3 = roots.root2; // Duplicate root
    }

    return roots;
}

// Function to compute the cubic equation for Im(s) vs z
std::vector<std::vector<double>> computeImSVsZ(double a, double y, double beta, int num_points) {
    std::vector<double> z_values(num_points);
    std::vector<double> ims_values1(num_points);
    std::vector<double> ims_values2(num_points);
    std::vector<double> ims_values3(num_points);
    
    // Generate z values from 0 to 10 (or adjust range as needed)
    double z_start = 0.01;  // Avoid z=0 to prevent potential division issues
    double z_end = 10.0;
    double z_step = (z_end - z_start) / (num_points - 1);
    
    for (int i = 0; i < num_points; ++i) {
        double z = z_start + i * z_step;
        z_values[i] = z;
        
        // Coefficients for the cubic equation:
        // zas^3 + [z(a+1)+a(1-y)]s^2 + [z+(a+1)-y-yβ(a-1)]s + 1 = 0
        double coef_a = z * a;
        double coef_b = z * (a + 1) + a * (1 - y);
        double coef_c = z + (a + 1) - y - y * beta * (a - 1);
        double coef_d = 1.0;
        
        // Solve the cubic equation
        CubicRoots roots = solveCubic(coef_a, coef_b, coef_c, coef_d);
        
        // Extract imaginary parts
        ims_values1[i] = std::abs(roots.root1.imag());
        ims_values2[i] = std::abs(roots.root2.imag());
        ims_values3[i] = std::abs(roots.root3.imag());
    }
    
    // Create output vector
    std::vector<std::vector<double>> result = {
        z_values, ims_values1, ims_values2, ims_values3
    };
    
    return result;
}

// Function to save Im(s) vs z data as JSON
void saveImSDataAsJSON(const std::string& filename, 
                      const std::vector<std::vector<double>>& data) {
    std::ofstream outfile(filename);
    
    if (!outfile.is_open()) {
        std::cerr << "Error: Could not open file " << filename << " for writing." << std::endl;
        return;
    }
    
    // Start JSON object
    outfile << "{\n";
    
    // Write z values
    outfile << "  \"z_values\": [";
    for (size_t i = 0; i < data[0].size(); ++i) {
        outfile << data[0][i];
        if (i < data[0].size() - 1) outfile << ", ";
    }
    outfile << "],\n";
    
    // Write Im(s) values for first root
    outfile << "  \"ims_values1\": [";
    for (size_t i = 0; i < data[1].size(); ++i) {
        outfile << data[1][i];
        if (i < data[1].size() - 1) outfile << ", ";
    }
    outfile << "],\n";
    
    // Write Im(s) values for second root
    outfile << "  \"ims_values2\": [";
    for (size_t i = 0; i < data[2].size(); ++i) {
        outfile << data[2][i];
        if (i < data[2].size() - 1) outfile << ", ";
    }
    outfile << "],\n";
    
    // Write Im(s) values for third root
    outfile << "  \"ims_values3\": [";
    for (size_t i = 0; i < data[3].size(); ++i) {
        outfile << data[3][i];
        if (i < data[3].size() - 1) outfile << ", ";
    }
    outfile << "]\n";
    
    // Close JSON object
    outfile << "}\n";
    
    outfile.close();
}

// Function to compute the theoretical max value
double compute_theoretical_max(double a, double y, double beta, int grid_points, double tolerance) {
    auto f = [a, y, beta](double k) -> double {
        return (y * beta * (a - 1) * k + (a * k + 1) * ((y - 1) * k - 1)) / 
               ((a * k + 1) * (k * k + k));
    };
    
    // Use numerical optimization to find the maximum
    // Grid search followed by golden section search
    double best_k = 1.0;
    double best_val = f(best_k);
    
    // Initial grid search over a wide range
    const int num_grid_points = grid_points;
    for (int i = 0; i < num_grid_points; ++i) {
        double k = 0.01 + 100.0 * i / (num_grid_points - 1); // From 0.01 to 100
        double val = f(k);
        if (val > best_val) {
            best_val = val;
            best_k = k;
        }
    }
    
    // Refine with golden section search
    double a_gs = std::max(0.01, best_k / 10.0);
    double b_gs = best_k * 10.0;
    const double golden_ratio = (1.0 + std::sqrt(5.0)) / 2.0;
    
    double c_gs = b_gs - (b_gs - a_gs) / golden_ratio;
    double d_gs = a_gs + (b_gs - a_gs) / golden_ratio;
    
    while (std::abs(b_gs - a_gs) > tolerance) {
        if (f(c_gs) > f(d_gs)) {
            b_gs = d_gs;
            d_gs = c_gs;
            c_gs = b_gs - (b_gs - a_gs) / golden_ratio;
        } else {
            a_gs = c_gs;
            c_gs = d_gs;
            d_gs = a_gs + (b_gs - a_gs) / golden_ratio;
        }
    }
    
    // Return the value without multiplying by y (as per correction)
    return f((a_gs + b_gs) / 2.0);
}

// Function to compute the theoretical min value
double compute_theoretical_min(double a, double y, double beta, int grid_points, double tolerance) {
    auto f = [a, y, beta](double t) -> double {
        return (y * beta * (a - 1) * t + (a * t + 1) * ((y - 1) * t - 1)) / 
               ((a * t + 1) * (t * t + t));
    };
    
    // Use numerical optimization to find the minimum
    // Grid search followed by golden section search
    double best_t = -0.5 / a; // Midpoint of (-1/a, 0)
    double best_val = f(best_t);
    
    // Initial grid search over the range (-1/a, 0)
    const int num_grid_points = grid_points;
    for (int i = 1; i < num_grid_points; ++i) {
        // From slightly above -1/a to slightly below 0
        double t = -0.999/a + 0.998/a * i / (num_grid_points - 1);
        if (t >= 0 || t <= -1.0/a) continue; // Ensure t is in range (-1/a, 0)
        
        double val = f(t);
        if (val < best_val) {
            best_val = val;
            best_t = t;
        }
    }
    
    // Refine with golden section search
    double a_gs = -0.999/a; // Slightly above -1/a
    double b_gs = -0.001/a; // Slightly below 0
    const double golden_ratio = (1.0 + std::sqrt(5.0)) / 2.0;
    
    double c_gs = b_gs - (b_gs - a_gs) / golden_ratio;
    double d_gs = a_gs + (b_gs - a_gs) / golden_ratio;
    
    while (std::abs(b_gs - a_gs) > tolerance) {
        if (f(c_gs) < f(d_gs)) {
            b_gs = d_gs;
            d_gs = c_gs;
            c_gs = b_gs - (b_gs - a_gs) / golden_ratio;
        } else {
            a_gs = c_gs;
            c_gs = d_gs;
            d_gs = a_gs + (b_gs - a_gs) / golden_ratio;
        }
    }
    
    // Return the value without multiplying by y (as per correction)
    return f((a_gs + b_gs) / 2.0);
}

// Function to save data as JSON
void save_as_json(const std::string& filename, 
                 const std::vector<double>& beta_values,
                 const std::vector<double>& max_eigenvalues,
                 const std::vector<double>& min_eigenvalues,
                 const std::vector<double>& theoretical_max_values,
                 const std::vector<double>& theoretical_min_values) {
    
    std::ofstream outfile(filename);
    
    if (!outfile.is_open()) {
        std::cerr << "Error: Could not open file " << filename << " for writing." << std::endl;
        return;
    }
    
    // Start JSON object
    outfile << "{\n";
    
    // Write beta values
    outfile << "  \"beta_values\": [";
    for (size_t i = 0; i < beta_values.size(); ++i) {
        outfile << beta_values[i];
        if (i < beta_values.size() - 1) outfile << ", ";
    }
    outfile << "],\n";
    
    // Write max eigenvalues
    outfile << "  \"max_eigenvalues\": [";
    for (size_t i = 0; i < max_eigenvalues.size(); ++i) {
        outfile << max_eigenvalues[i];
        if (i < max_eigenvalues.size() - 1) outfile << ", ";
    }
    outfile << "],\n";
    
    // Write min eigenvalues
    outfile << "  \"min_eigenvalues\": [";
    for (size_t i = 0; i < min_eigenvalues.size(); ++i) {
        outfile << min_eigenvalues[i];
        if (i < min_eigenvalues.size() - 1) outfile << ", ";
    }
    outfile << "],\n";
    
    // Write theoretical max values
    outfile << "  \"theoretical_max\": [";
    for (size_t i = 0; i < theoretical_max_values.size(); ++i) {
        outfile << theoretical_max_values[i];
        if (i < theoretical_max_values.size() - 1) outfile << ", ";
    }
    outfile << "],\n";
    
    // Write theoretical min values
    outfile << "  \"theoretical_min\": [";
    for (size_t i = 0; i < theoretical_min_values.size(); ++i) {
        outfile << theoretical_min_values[i];
        if (i < theoretical_min_values.size() - 1) outfile << ", ";
    }
    outfile << "]\n";
    
    // Close JSON object
    outfile << "}\n";
    
    outfile.close();
}

int main(int argc, char* argv[]) {
    // Check command mode
    if (argc < 2) {
        std::cerr << "Usage: " << argv[0] << " [eigenvalues|cubic] [parameters...]" << std::endl;
        return 1;
    }
    
    std::string mode = argv[1];
    
    if (mode == "eigenvalues") {
        // ─── Eigenvalue analysis mode ───────────────────────────────────────────
        if (argc != 9) {
            std::cerr << "Usage: " << argv[0] << " eigenvalues <n> <p> <a> <y> <fineness> <theory_grid_points> <theory_tolerance> <output_file>" << std::endl;
            return 1;
        }
        
        int n = std::stoi(argv[2]);
        int p = std::stoi(argv[3]);
        double a = std::stod(argv[4]);
        double y = std::stod(argv[5]);
        int fineness = std::stoi(argv[6]);
        int theory_grid_points = std::stoi(argv[7]);
        double theory_tolerance = std::stod(argv[8]);
        std::string output_file = argv[9];
        const double b = 1.0;
        
        std::cout << "Running eigenvalue analysis with parameters: n = " << n << ", p = " << p 
                << ", a = " << a << ", y = " << y << ", fineness = " << fineness 
                << ", theory_grid_points = " << theory_grid_points
                << ", theory_tolerance = " << theory_tolerance << std::endl;
        std::cout << "Output will be saved to: " << output_file << std::endl;
        
        // ─── Beta range parameters ────────────────────────────────────────
        const int num_beta_points = fineness; // Controlled by fineness parameter
        std::vector<double> beta_values(num_beta_points);
        for (int i = 0; i < num_beta_points; ++i) {
            beta_values[i] = static_cast<double>(i) / (num_beta_points - 1);
        }
        
        // ─── Storage for results ────────────────────────────────────────
        std::vector<double> max_eigenvalues(num_beta_points);
        std::vector<double> min_eigenvalues(num_beta_points);
        std::vector<double> theoretical_max_values(num_beta_points);
        std::vector<double> theoretical_min_values(num_beta_points);
        
        // ─── Random‐Gaussian X and S_n ────────────────────────────────
        std::mt19937_64 rng{std::random_device{}()};
        std::normal_distribution<double> norm(0.0, 1.0);
        
        cv::Mat X(p, n, CV_64F);
        for(int i = 0; i < p; ++i)
            for(int j = 0; j < n; ++j)
                X.at<double>(i,j) = norm(rng);
        
        // ─── Process each beta value ─────────────────────────────────
        for (int beta_idx = 0; beta_idx < num_beta_points; ++beta_idx) {
            double beta = beta_values[beta_idx];
            
            // Compute theoretical values with customizable precision
            theoretical_max_values[beta_idx] = compute_theoretical_max(a, y, beta, theory_grid_points, theory_tolerance);
            theoretical_min_values[beta_idx] = compute_theoretical_min(a, y, beta, theory_grid_points, theory_tolerance);
            
            // ─── Build T_n matrix ──────────────────────────────────
            int k = static_cast<int>(std::floor(beta * p));
            std::vector<double> diags(p);
            std::fill_n(diags.begin(), k, a);
            std::fill_n(diags.begin()+k, p-k, b);
            std::shuffle(diags.begin(), diags.end(), rng);
            
            cv::Mat T_n = cv::Mat::zeros(p, p, CV_64F);
            for(int i = 0; i < p; ++i){
                T_n.at<double>(i,i) = diags[i];
            }
            
            // ─── Form B_n = (1/n) * X * T_n * X^T ────────────
            cv::Mat B = (X.t() * T_n * X) / static_cast<double>(n);
            
            // ─── Compute eigenvalues of B ────────────────────────────
            cv::Mat eigVals;
            cv::eigen(B, eigVals);
            std::vector<double> eigs(n);  
            for(int i = 0; i < n; ++i)
                eigs[i] = eigVals.at<double>(i, 0);
            
            max_eigenvalues[beta_idx] = *std::max_element(eigs.begin(), eigs.end());
            min_eigenvalues[beta_idx] = *std::min_element(eigs.begin(), eigs.end());
            
            // Progress indicator for Streamlit
            double progress = static_cast<double>(beta_idx + 1) / num_beta_points;
            std::cout << "PROGRESS:" << progress << std::endl;
            
            // Less verbose output for Streamlit
            if (beta_idx % 20 == 0 || beta_idx == num_beta_points - 1) {
                std::cout << "Processing beta = " << beta 
                        << " (" << beta_idx+1 << "/" << num_beta_points << ")" << std::endl;
            }
        }
        
        // Save data as JSON for Python to read
        save_as_json(output_file, beta_values, max_eigenvalues, min_eigenvalues, 
                    theoretical_max_values, theoretical_min_values);
        
        std::cout << "Data saved to " << output_file << std::endl;
        
    } else if (mode == "cubic") {
        // ─── Cubic equation analysis mode ───────────────────────────────────────────
        if (argc != 6) {
            std::cerr << "Usage: " << argv[0] << " cubic <a> <y> <beta> <num_points> <output_file>" << std::endl;
            return 1;
        }
        
        double a = std::stod(argv[2]);
        double y = std::stod(argv[3]);
        double beta = std::stod(argv[4]);
        int num_points = std::stoi(argv[5]);
        std::string output_file = argv[6];
        
        std::cout << "Running cubic equation analysis with parameters: a = " << a 
                  << ", y = " << y << ", beta = " << beta << ", num_points = " << num_points << std::endl;
        std::cout << "Output will be saved to: " << output_file << std::endl;
        
        // Compute Im(s) vs z data
        std::vector<std::vector<double>> ims_data = computeImSVsZ(a, y, beta, num_points);
        
        // Save to JSON
        saveImSDataAsJSON(output_file, ims_data);
        
        std::cout << "Cubic equation data saved to " << output_file << std::endl;
        
    } else {
        std::cerr << "Unknown mode: " << mode << std::endl;
        std::cerr << "Use 'eigenvalues' or 'cubic'" << std::endl;
        return 1;
    }
    
    return 0;
}