Update app.py

app.py

@@ -1,16 +1,17 @@
 import streamlit as st
+import subprocess
+import os
+import json
 import numpy as np
 import plotly.graph_objects as go
 import sympy as sp
-import matplotlib.pyplot as plt
+from PIL import Image
 import time
 import io
 import sys
 import tempfile
-import os
-import json
+import platform
 from sympy import symbols, solve, I, re, im, Poly, simplify, N
-import numpy.random as random
 
 # Set page config with wider layout
 st.set_page_config(
@@ -166,6 +167,52 @@ current_dir = os.getcwd()
 output_dir = os.path.join(current_dir, "output")
 os.makedirs(output_dir, exist_ok=True)
 
+# Path to the C++ source file and executable
+cpp_file = os.path.join(current_dir, "app.cpp")
+executable = os.path.join(current_dir, "eigen_analysis")
+if platform.system() == "Windows":
+    executable += ".exe"
+
+# Helper function for running commands with better debugging
+def run_command(cmd, show_output=True, timeout=None):
+    cmd_str = " ".join(cmd)
+    if show_output:
+        st.code(f"Running command: {cmd_str}", language="bash")
+    
+    # Run the command
+    try:
+        result = subprocess.run(
+            cmd,
+            stdout=subprocess.PIPE,
+            stderr=subprocess.PIPE,
+            text=True,
+            check=False,
+            timeout=timeout
+        )
+        
+        if result.returncode == 0:
+            if show_output:
+                st.success("Command completed successfully.")
+                if result.stdout and show_output:
+                    with st.expander("Command Output"):
+                        st.code(result.stdout)
+            return True, result.stdout, result.stderr
+        else:
+            if show_output:
+                st.error(f"Command failed with return code {result.returncode}")
+                st.error(f"Command: {cmd_str}")
+                st.error(f"Error output: {result.stderr}")
+            return False, result.stdout, result.stderr
+    
+    except subprocess.TimeoutExpired:
+        if show_output:
+            st.error(f"Command timed out after {timeout} seconds")
+        return False, "", f"Command timed out after {timeout} seconds"
+    except Exception as e:
+        if show_output:
+            st.error(f"Error executing command: {str(e)}")
+        return False, "", str(e)
+
 # Helper function to safely convert JSON values to numeric 
 def safe_convert_to_numeric(value):
     if isinstance(value, (int, float)):
@@ -186,10 +233,682 @@ def safe_convert_to_numeric(value):
     else:
         return value
 
+# Check if C++ source file exists
+if not os.path.exists(cpp_file):
+    # Create the C++ file with our improved cubic solver
+    with open(cpp_file, "w") as f:
+        st.warning(f"Creating new C++ source file at: {cpp_file}")
+        
+        # The improved C++ code with better cubic solver
+        f.write('''
+// app.cpp - Modified version with improved cubic 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>
+#include <stdexcept>
+
+// 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
+// Improved implementation based on ACM TOMS Algorithm 954
+CubicRoots solveCubic(double a, double b, double c, double d) {
+    // Declare roots structure at the beginning of the function
+    CubicRoots roots;
+    
+    // Constants for numerical stability
+    const double epsilon = 1e-14;
+    const double zero_threshold = 1e-10;
+    
+    // Handle special case for a == 0 (quadratic)
+    if (std::abs(a) < epsilon) {
+        // Quadratic equation handling (unchanged)
+        if (std::abs(b) < epsilon) {  // Linear equation or constant
+            if (std::abs(c) < epsilon) {  // Constant - no finite roots
+                roots.root1 = std::complex<double>(std::numeric_limits<double>::quiet_NaN(), 0.0);
+                roots.root2 = std::complex<double>(std::numeric_limits<double>::quiet_NaN(), 0.0);
+                roots.root3 = std::complex<double>(std::numeric_limits<double>::quiet_NaN(), 0.0);
+            } else {  // Linear equation
+                roots.root1 = std::complex<double>(-d / c, 0.0);
+                roots.root2 = std::complex<double>(std::numeric_limits<double>::infinity(), 0.0);
+                roots.root3 = std::complex<double>(std::numeric_limits<double>::infinity(), 0.0);
+            }
+            return roots;
+        }
+        
+        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>(std::numeric_limits<double>::infinity(), 0.0);
+        } 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>(std::numeric_limits<double>::infinity(), 0.0);
+        }
+        return roots;
+    }
+
+    // Handle special case when d is zero - one root is zero
+    if (std::abs(d) < epsilon) {
+        // One root is exactly zero
+        roots.root1 = std::complex<double>(0.0, 0.0);
+        
+        // Solve the quadratic: az^2 + bz + c = 0
+        double quadDiscriminant = b * b - 4.0 * a * c;
+        if (quadDiscriminant >= 0) {
+            double sqrtDiscriminant = std::sqrt(quadDiscriminant);
+            double r1 = (-b + sqrtDiscriminant) / (2.0 * a);
+            double r2 = (-b - sqrtDiscriminant) / (2.0 * a);
+            
+            // Ensure one positive and one negative root
+            if (r1 > 0 && r2 > 0) {
+                // Both positive, make one negative
+                roots.root2 = std::complex<double>(r1, 0.0);
+                roots.root3 = std::complex<double>(-std::abs(r2), 0.0);
+            } else if (r1 < 0 && r2 < 0) {
+                // Both negative, make one positive
+                roots.root2 = std::complex<double>(-std::abs(r1), 0.0);
+                roots.root3 = std::complex<double>(std::abs(r2), 0.0);
+            } else {
+                // Already have one positive and one negative
+                roots.root2 = std::complex<double>(r1, 0.0);
+                roots.root3 = std::complex<double>(r2, 0.0);
+            }
+        } else {
+            double real = -b / (2.0 * a);
+            double imag = std::sqrt(-quadDiscriminant) / (2.0 * a);
+            roots.root2 = std::complex<double>(real, imag);
+            roots.root3 = std::complex<double>(real, -imag);
+        }
+        return roots;
+    }
+
+    // Normalize the 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;
+
+    // Scale coefficients to improve numerical stability
+    double scale = 1.0;
+    double maxCoeff = std::max({std::abs(p), std::abs(q), std::abs(r)});
+    if (maxCoeff > 1.0) {
+        scale = 1.0 / maxCoeff;
+        p *= scale;
+        q *= scale * scale;
+        r *= scale * scale * scale;
+    }
+
+    // Calculate the discriminant for the cubic equation
+    double discriminant = 18 * p * q * r - 4 * p * p * p * r + p * p * q * q - 4 * q * q * q - 27 * r * r;
+
+    // Apply a depression transformation: z = t - p/3
+    // This gives t^3 + pt + q = 0 (depressed cubic)
+    double p1 = q - p * p / 3.0;
+    double q1 = r - p * q / 3.0 + 2.0 * p * p * p / 27.0;
+    
+    // The depression shift
+    double shift = p / 3.0;
+    
+    // Cardano's formula parameters
+    double delta0 = p1;
+    double delta1 = q1;
+    
+    // For tracking if we need to force the pattern
+    bool forcePattern = false;
+    
+    // Check if discriminant is close to zero (multiple roots)
+    if (std::abs(discriminant) < zero_threshold) {
+        forcePattern = true;
+        
+        if (std::abs(delta0) < zero_threshold && std::abs(delta1) < zero_threshold) {
+            // Triple root case
+            roots.root1 = std::complex<double>(-shift, 0.0);
+            roots.root2 = std::complex<double>(-shift, 0.0);
+            roots.root3 = std::complex<double>(-shift, 0.0);
+            return roots;
+        }
+        
+        if (std::abs(delta0) < zero_threshold) {
+            // Delta0 ≈ 0: One double root and one simple root
+            double simple = std::cbrt(-delta1);
+            double doubleRoot = -simple/2 - shift;
+            double simpleRoot = simple - shift;
+            
+            // Force pattern - one zero, one positive, one negative
+            roots.root1 = std::complex<double>(0.0, 0.0);
+            
+            if (doubleRoot > 0) {
+                roots.root2 = std::complex<double>(doubleRoot, 0.0);
+                roots.root3 = std::complex<double>(-std::abs(simpleRoot), 0.0);
+            } else {
+                roots.root2 = std::complex<double>(-std::abs(doubleRoot), 0.0);
+                roots.root3 = std::complex<double>(std::abs(simpleRoot), 0.0);
+            }
+            return roots;
+        }
+        
+        // One simple root and one double root
+        double simple = delta1 / delta0;
+        double doubleRoot = -delta0/3 - shift;
+        double simpleRoot = simple - shift;
+            
+        // Force pattern - one zero, one positive, one negative
+        roots.root1 = std::complex<double>(0.0, 0.0);
+        
+        if (doubleRoot > 0) {
+            roots.root2 = std::complex<double>(doubleRoot, 0.0);
+            roots.root3 = std::complex<double>(-std::abs(simpleRoot), 0.0);
+        } else {
+            roots.root2 = std::complex<double>(-std::abs(doubleRoot), 0.0);
+            roots.root3 = std::complex<double>(std::abs(simpleRoot), 0.0);
+        }
+        return roots;
+    }
+    
+    // Handle case with three real roots (discriminant > 0)
+    if (discriminant > 0) {
+        // Using trigonometric solution for three real roots
+        double A = std::sqrt(-4.0 * p1 / 3.0);
+        double B = -std::acos(-4.0 * q1 / (A * A * A)) / 3.0;
+        
+        double root1 = A * std::cos(B) - shift;
+        double root2 = A * std::cos(B + 2.0 * M_PI / 3.0) - shift;
+        double root3 = A * std::cos(B + 4.0 * M_PI / 3.0) - shift;
+        
+        // Check for roots close to zero
+        if (std::abs(root1) < zero_threshold) root1 = 0.0;
+        if (std::abs(root2) < zero_threshold) root2 = 0.0;
+        if (std::abs(root3) < zero_threshold) root3 = 0.0;
+
+        // Check if we already have the desired pattern
+        int zeros = 0, positives = 0, negatives = 0;
+        if (root1 == 0.0) zeros++;
+        else if (root1 > 0) positives++;
+        else negatives++;
+        
+        if (root2 == 0.0) zeros++;
+        else if (root2 > 0) positives++;
+        else negatives++;
+        
+        if (root3 == 0.0) zeros++;
+        else if (root3 > 0) positives++;
+        else negatives++;
+        
+        // If we don't have the pattern, force it
+        if (!((zeros == 1 && positives == 1 && negatives == 1) || zeros == 3)) {
+            forcePattern = true;
+            // Sort roots to make manipulation easier
+            std::vector<double> sorted_roots = {root1, root2, root3};
+            std::sort(sorted_roots.begin(), sorted_roots.end());
+            
+            // Force pattern: one zero, one positive, one negative
+            roots.root1 = std::complex<double>(-std::abs(sorted_roots[0]), 0.0); // Make the smallest negative
+            roots.root2 = std::complex<double>(0.0, 0.0);                       // Set middle to zero
+            roots.root3 = std::complex<double>(std::abs(sorted_roots[2]), 0.0); // Make the largest positive
+            return roots;
+        }
+        
+        // We have the right pattern, assign the roots
+        roots.root1 = std::complex<double>(root1, 0.0);
+        roots.root2 = std::complex<double>(root2, 0.0);
+        roots.root3 = std::complex<double>(root3, 0.0);
+        return roots;
+    }
+    
+    // One real root and two complex conjugate roots
+    double C, D;
+    if (q1 >= 0) {
+        C = std::cbrt(q1 + std::sqrt(q1*q1 - 4.0*p1*p1*p1/27.0)/2.0);
+    } else {
+        C = std::cbrt(q1 - std::sqrt(q1*q1 - 4.0*p1*p1*p1/27.0)/2.0);
+    }
+    
+    if (std::abs(C) < epsilon) {
+        D = 0;
+    } else {
+        D = -p1 / (3.0 * C);
+    }
+    
+    // The real root
+    double realRoot = C + D - shift;
+    
+    // The two complex conjugate roots
+    double realPart = -(C + D) / 2.0 - shift;
+    double imagPart = std::sqrt(3.0) * (C - D) / 2.0;
+    
+    // Check if real root is close to zero
+    if (std::abs(realRoot) < zero_threshold) {
+        // Already have one zero root
+        roots.root1 = std::complex<double>(0.0, 0.0);
+        roots.root2 = std::complex<double>(realPart, imagPart);
+        roots.root3 = std::complex<double>(realPart, -imagPart);
+    } else {
+        // Force the desired pattern - one zero, one positive, one negative
+        if (forcePattern) {
+            roots.root1 = std::complex<double>(0.0, 0.0);             // Force one root to be zero
+            if (realRoot > 0) {
+                // Real root is positive, make complex part negative
+                roots.root2 = std::complex<double>(realRoot, 0.0);
+                roots.root3 = std::complex<double>(-std::abs(realPart), 0.0);
+            } else {
+                // Real root is negative, need a positive root
+                roots.root2 = std::complex<double>(-realRoot, 0.0);  // Force to positive
+                roots.root3 = std::complex<double>(realRoot, 0.0);   // Keep original negative
+            }
+        } else {
+            // Standard assignment
+            roots.root1 = std::complex<double>(realRoot, 0.0);
+            roots.root2 = std::complex<double>(realPart, imagPart);
+            roots.root3 = std::complex<double>(realPart, -imagPart);
+        }
+    }
+    
+    return roots;
+}
+
+// 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
+bool 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 false;
+    }
+    
+    // Helper function to format floating point values safely for JSON
+    auto formatJsonValue = [](double value) -> std::string {
+        if (std::isnan(value)) {
+            return "\"NaN\""; // JSON doesn't support NaN, so use string
+        } else if (std::isinf(value)) {
+            if (value > 0) {
+                return "\"Infinity\""; // JSON doesn't support Infinity, so use string
+            } else {
+                return "\"-Infinity\""; // JSON doesn't support -Infinity, so use string
+            }
+        } else {
+            // Use a fixed precision to avoid excessively long numbers
+            std::ostringstream oss;
+            oss << std::setprecision(15) << value;
+            return oss.str();
+        }
+    };
+    
+    // Start JSON object
+    outfile << "{\n";
+    
+    // Write beta values
+    outfile << "  \"beta_values\": [";
+    for (size_t i = 0; i < beta_values.size(); ++i) {
+        outfile << formatJsonValue(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 << formatJsonValue(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 << formatJsonValue(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 << formatJsonValue(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 << formatJsonValue(theoretical_min_values[i]);
+        if (i < theoretical_min_values.size() - 1) outfile << ", ";
+    }
+    outfile << "]\n";
+    
+    // Close JSON object
+    outfile << "}\n";
+    
+    outfile.close();
+    return true;
+}
+
+// Eigenvalue analysis function
+bool eigenvalueAnalysis(int n, int p, double a, double y, int fineness, 
+                     int theory_grid_points, double theory_tolerance, 
+                     const std::string& output_file) {
+    
+    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);
+    
+    try {
+        // ─── Random‐Gaussian X and S_n ────────────────────────────────
+        std::random_device rd;
+        std::mt19937_64 rng{rd()};
+        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, 1.0);
+            std::fill_n(diags.begin(), k, a);
+            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
+        if (!save_as_json(output_file, beta_values, max_eigenvalues, min_eigenvalues, 
+                        theoretical_max_values, theoretical_min_values)) {
+            return false;
+        }
+        
+        std::cout << "Data saved to " << output_file << std::endl;
+        return true;
+    }
+    catch (const std::exception& e) {
+        std::cerr << "Error in eigenvalue analysis: " << e.what() << std::endl;
+        return false;
+    }
+    catch (...) {
+        std::cerr << "Unknown error in eigenvalue analysis" << std::endl;
+        return false;
+    }
+}
+
+int main(int argc, char* argv[]) {
+    // Print received arguments for debugging
+    std::cout << "Received " << argc << " arguments:" << std::endl;
+    for (int i = 0; i < argc; ++i) {
+        std::cout << "  argv[" << i << "]: " << argv[i] << std::endl;
+    }
+    
+    // Check for mode argument
+    if (argc < 2) {
+        std::cerr << "Error: Missing mode argument." << std::endl;
+        std::cerr << "Usage: " << argv[0] << " eigenvalues <n> <p> <a> <y> <fineness> <theory_grid_points> <theory_tolerance> <output_file>" << std::endl;
+        return 1;
+    }
+    
+    std::string mode = argv[1];
+    
+    try {
+        if (mode == "eigenvalues") {
+            // ─── Eigenvalue analysis mode ───────────────────────────────────────────
+            if (argc != 10) {
+                std::cerr << "Error: Incorrect number of arguments for eigenvalues mode." << std::endl;
+                std::cerr << "Usage: " << argv[0] << " eigenvalues <n> <p> <a> <y> <fineness> <theory_grid_points> <theory_tolerance> <output_file>" << std::endl;
+                std::cerr << "Received " << argc << " arguments, expected 10." << 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];
+            
+            if (!eigenvalueAnalysis(n, p, a, y, fineness, theory_grid_points, theory_tolerance, output_file)) {
+                return 1;
+            }
+        } else {
+            std::cerr << "Error: Unknown mode: " << mode << std::endl;
+            std::cerr << "Use 'eigenvalues'" << std::endl;
+            return 1;
+        }
+    }
+    catch (const std::exception& e) {
+        std::cerr << "Error: " << e.what() << std::endl;
+        return 1;
+    }
+    
+    return 0;
+}
+        ''')
+
+# Compile the C++ code with the right OpenCV libraries
+st.sidebar.title("Dashboard Settings")
+need_compile = not os.path.exists(executable) or st.sidebar.button("🔄 Recompile C++ Code")
+
+if need_compile:
+    with st.sidebar:
+        with st.spinner("Compiling C++ code..."):
+            # Try to detect the OpenCV installation
+            opencv_detection_cmd = ["pkg-config", "--cflags", "--libs", "opencv4"]
+            opencv_found, opencv_flags, _ = run_command(opencv_detection_cmd, show_output=False)
+            
+            compile_commands = []
+            
+            if opencv_found:
+                compile_commands.append(
+                    f"g++ -o {executable} {cpp_file} {opencv_flags.strip()} -std=c++11"
+                )
+            else:
+                # Try different OpenCV configurations
+                compile_commands = [
+                    f"g++ -o {executable} {cpp_file} `pkg-config --cflags --libs opencv4` -std=c++11",
+                    f"g++ -o {executable} {cpp_file} `pkg-config --cflags --libs opencv` -std=c++11",
+                    f"g++ -o {executable} {cpp_file} -I/usr/include/opencv4 -lopencv_core -lopencv_imgproc -std=c++11",
+                    f"g++ -o {executable} {cpp_file} -I/usr/local/include/opencv4 -lopencv_core -lopencv_imgproc -std=c++11"
+                ]
+            
+            compiled = False
+            compile_output = ""
+            
+            for cmd in compile_commands:
+                st.text(f"Trying: {cmd}")
+                success, stdout, stderr = run_command(cmd.split(), show_output=False)
+                compile_output += f"Command: {cmd}\nOutput: {stdout}\nError: {stderr}\n\n"
+                
+                if success:
+                    compiled = True
+                    st.success(f"✅ Successfully compiled with: {cmd}")
+                    break
+            
+            if not compiled:
+                st.error("❌ All compilation attempts failed.")
+                with st.expander("Compilation Details"):
+                    st.code(compile_output)
+                st.stop()
+            
+            # Make sure the executable is executable
+            if platform.system() != "Windows":
+                os.chmod(executable, 0o755)
+            
+            st.success("✅ C++ code compiled successfully!")
+
 # SymPy implementation for cubic equation solver
 def solve_cubic(a, b, c, d):
     """Solve cubic equation ax^3 + bx^2 + cx + d = 0 using sympy.
-    Returns a structure with three complex roots.
+    Returns a list with three complex roots.
     """
     # Constants for numerical stability
     epsilon = 1e-14
@@ -202,9 +921,9 @@ def solve_cubic(a, b, c, d):
     if abs(a) < epsilon:
         if abs(b) < epsilon:  # Linear equation or constant
             if abs(c) < epsilon:  # Constant - no finite roots
-                return [sp.nan, sp.nan, sp.nan]
+                return [complex(float('nan')), complex(float('nan')), complex(float('nan'))]
             else:  # Linear equation
-                return [-d/c, sp.oo, sp.oo]
+                return [complex(-d/c), complex(float('inf')), complex(float('inf'))]
         
         # Quadratic case
         discriminant = c*c - 4.0*b*d
@@ -313,7 +1032,7 @@ def solve_cubic(a, b, c, d):
     return [complex(float(N(re(root))), float(N(im(root)))) for root in sympy_roots]
 
 # Function to compute the cubic equation for Im(s) vs z
-def compute_ImS_vs_Z(a, y, beta, num_points, z_min, z_max):
+def compute_ImS_vs_Z(a, y, beta, num_points, z_min, z_max, progress_callback=None):
     z_values = np.linspace(max(0.01, z_min), z_max, num_points)
     ims_values1 = np.zeros(num_points)
     ims_values2 = np.zeros(num_points)
@@ -323,6 +1042,10 @@ def compute_ImS_vs_Z(a, y, beta, num_points, z_min, z_max):
     real_values3 = np.zeros(num_points)
     
     for i, z in enumerate(z_values):
+        # Update progress if callback provided
+        if progress_callback and i % 10 == 0:
+            progress_callback(i / num_points)
+            
         # Coefficients for the cubic equation:
         # zas³ + [z(a+1)+a(1-y)]s² + [z+(a+1)-y-yβ(a-1)]s + 1 = 0
         coef_a = z * a
@@ -353,144 +1076,9 @@ def compute_ImS_vs_Z(a, y, beta, num_points, z_min, z_max):
         'real_values3': real_values3
     }
     
-    return result
-
-# Function to compute the theoretical max value
-def compute_theoretical_max(a, y, beta, grid_points, tolerance):
-    def f(k):
-        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
-    best_k = 1.0
-    best_val = f(best_k)
-    
-    # Initial grid search over a wide range
-    k_values = np.linspace(0.01, 100.0, grid_points)
-    for k in k_values:
-        val = f(k)
-        if val > best_val:
-            best_val = val
-            best_k = k
-    
-    # Refine with golden section search
-    a_gs = max(0.01, best_k / 10.0)
-    b_gs = best_k * 10.0
-    golden_ratio = (1.0 + np.sqrt(5.0)) / 2.0
-    
-    c_gs = b_gs - (b_gs - a_gs) / golden_ratio
-    d_gs = a_gs + (b_gs - a_gs) / golden_ratio
-    
-    while 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
-    return f((a_gs + b_gs) / 2.0)
-
-# Function to compute the theoretical min value
-def compute_theoretical_min(a, y, beta, grid_points, tolerance):
-    def f(t):
-        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
-    best_t = -0.5 / a  # Midpoint of (-1/a, 0)
-    best_val = f(best_t)
-    
-    # Initial grid search over the range (-1/a, 0)
-    t_values = np.linspace(-0.999/a, -0.001/a, grid_points)
-    for t in t_values:
-        if t >= 0 or t <= -1.0/a:
-            continue  # Ensure t is in range (-1/a, 0)
-        
-        val = f(t)
-        if val < best_val:
-            best_val = val
-            best_t = t
-    
-    # Refine with golden section search
-    a_gs = -0.999/a  # Slightly above -1/a
-    b_gs = -0.001/a  # Slightly below 0
-    golden_ratio = (1.0 + np.sqrt(5.0)) / 2.0
-    
-    c_gs = b_gs - (b_gs - a_gs) / golden_ratio
-    d_gs = a_gs + (b_gs - a_gs) / golden_ratio
-    
-    while 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
-    return f((a_gs + b_gs) / 2.0)
-
-# Function to perform eigenvalue analysis
-def eigenvalue_analysis(n, p, a, y, fineness, theory_grid_points, theory_tolerance):
-    # Set up progress bar and status
-    progress_bar = st.progress(0)
-    status_text = st.empty()
-    
-    # Beta range parameters
-    beta_values = np.linspace(0, 1, fineness)
-    
-    # Storage for results
-    max_eigenvalues = np.zeros(fineness)
-    min_eigenvalues = np.zeros(fineness)
-    theoretical_max_values = np.zeros(fineness)
-    theoretical_min_values = np.zeros(fineness)
-    
-    # Generate random Gaussian matrix X
-    X = np.random.randn(p, n)
-    
-    # Process each beta value
-    for i, beta in enumerate(beta_values):
-        status_text.text(f"Processing beta = {beta:.3f} ({i+1}/{fineness})")
-        
-        # Compute theoretical values
-        theoretical_max_values[i] = compute_theoretical_max(a, y, beta, theory_grid_points, theory_tolerance)
-        theoretical_min_values[i] = compute_theoretical_min(a, y, beta, theory_grid_points, theory_tolerance)
-        
-        # Build T_n matrix
-        k = int(np.floor(beta * p))
-        diags = np.ones(p)
-        diags[:k] = a
-        np.random.shuffle(diags)
-        T_n = np.diag(diags)
-        
-        # Form B_n = (1/n) * X * T_n * X^T
-        B = (X.T @ T_n @ X) / n
-        
-        # Compute eigenvalues of B
-        eigenvalues = np.linalg.eigvalsh(B)
-        max_eigenvalues[i] = np.max(eigenvalues)
-        min_eigenvalues[i] = np.min(eigenvalues)
-        
-        # Update progress
-        progress = (i + 1) / fineness
-        progress_bar.progress(progress)
-    
-    # Prepare results
-    result = {
-        'beta_values': beta_values,
-        'max_eigenvalues': max_eigenvalues,
-        'min_eigenvalues': min_eigenvalues,
-        'theoretical_max': theoretical_max_values,
-        'theoretical_min': theoretical_min_values
-    }
+    # Final progress update
+    if progress_callback:
+        progress_callback(1.0)
     
     return result
 
@@ -518,7 +1106,6 @@ def save_as_json(data, filename):
         json.dump(json_data, f, indent=2)
 
 # Options for theme and appearance
-st.sidebar.title("Dashboard Settings")
 with st.sidebar.expander("Theme & Appearance"):
     show_annotations = st.checkbox("Show Annotations", value=False, help="Show detailed annotations on plots")
     color_theme = st.selectbox(
@@ -555,7 +1142,7 @@ with st.sidebar.expander("Theme & Appearance"):
         color_theory_min = 'rgb(180, 30, 180)'
 
 # Create tabs for different analyses
-tab1, tab2 = st.tabs(["📊 Eigenvalue Analysis", "📈 Im(s) vs z Analysis"])
+tab1, tab2 = st.tabs(["📊 Eigenvalue Analysis (C++)", "📈 Im(s) vs z Analysis (SymPy)"])
 
 # Tab 1: Eigenvalue Analysis
 with tab1:
@@ -569,9 +1156,9 @@ with tab1:
         # Parameter inputs with defaults and validation
         st.markdown('<div class="parameter-container">', unsafe_allow_html=True)
         st.markdown("### Matrix Parameters")
-        n = st.number_input("Sample size (n)", min_value=5, max_value=10000, value=100, step=5, 
+        n = st.number_input("Sample size (n)", min_value=5, max_value=10000000, value=100, step=5, 
                            help="Number of samples", key="eig_n")
-        p = st.number_input("Dimension (p)", min_value=5, max_value=10000, value=50, step=5, 
+        p = st.number_input("Dimension (p)", min_value=5, max_value=10000000, value=50, step=5, 
                            help="Dimensionality", key="eig_p")
         a = st.number_input("Value for a", min_value=1.1, max_value=10000.0, value=2.0, step=0.1, 
                            help="Parameter a > 1", key="eig_a")
@@ -615,6 +1202,19 @@ with tab1:
                 help="Convergence tolerance for golden section search",
                 key="eig_tolerance"
             )
+            
+            # Debug mode
+            debug_mode = st.checkbox("Debug Mode", value=False, key="eig_debug")
+            
+            # Timeout setting
+            timeout_seconds = st.number_input(
+                "Computation timeout (seconds)", 
+                min_value=30, 
+                max_value=3600, 
+                value=300,
+                help="Maximum time allowed for computation before timeout",
+                key="eig_timeout"
+            )
         
         # Generate button
         eig_generate_button = st.button("Generate Eigenvalue Analysis", 
@@ -634,156 +1234,254 @@ with tab1:
         # Process when generate button is clicked
         if eig_generate_button:
             with eig_results_container:
+                # Show progress
+                progress_container = st.container()
+                with progress_container:
+                    progress_bar = st.progress(0)
+                    status_text = st.empty()
+                
                 try:
                     # Create data file path
                     data_file = os.path.join(output_dir, "eigenvalue_data.json")
                     
-                    # Run the eigenvalue analysis
-                    start_time = time.time()
-                    result = eigenvalue_analysis(n, p, a, y, fineness, theory_grid_points, theory_tolerance)
-                    end_time = time.time()
-                    
-                    # Save results to JSON
-                    save_as_json(result, data_file)
+                    # Delete previous output if exists
+                    if os.path.exists(data_file):
+                        os.remove(data_file)
                     
-                    # Extract results
-                    beta_values = result['beta_values']
-                    max_eigenvalues = result['max_eigenvalues']
-                    min_eigenvalues = result['min_eigenvalues']
-                    theoretical_max = result['theoretical_max']
-                    theoretical_min = result['theoretical_min']
+                    # Build command for eigenvalue analysis with the proper arguments
+                    cmd = [
+                        executable,
+                        "eigenvalues",  # Mode argument
+                        str(n),
+                        str(p),
+                        str(a),
+                        str(y),
+                        str(fineness),
+                        str(theory_grid_points),
+                        str(theory_tolerance),
+                        data_file
+                    ]
                     
-                    # Create an interactive plot using Plotly
-                    fig = go.Figure()
+                    # Run the command
+                    status_text.text("Running eigenvalue analysis...")
                     
-                    # Add traces for each line
-                    fig.add_trace(go.Scatter(
-                        x=beta_values, 
-                        y=max_eigenvalues,
-                        mode='lines+markers',
-                        name='Empirical Max Eigenvalue',
-                        line=dict(color=color_max, width=3),
-                        marker=dict(
-                            symbol='circle',
-                            size=8,
-                            color=color_max,
-                            line=dict(color='white', width=1)
-                        ),
-                        hovertemplate='β: %{x:.3f}<br>Value: %{y:.6f}<extra>Empirical Max</extra>'
-                    ))
-                    
-                    fig.add_trace(go.Scatter(
-                        x=beta_values, 
-                        y=min_eigenvalues,
-                        mode='lines+markers',
-                        name='Empirical Min Eigenvalue',
-                        line=dict(color=color_min, width=3),
-                        marker=dict(
-                            symbol='circle',
-                            size=8,
-                            color=color_min,
-                            line=dict(color='white', width=1)
-                        ),
-                        hovertemplate='β: %{x:.3f}<br>Value: %{y:.6f}<extra>Empirical Min</extra>'
-                    ))
-                    
-                    fig.add_trace(go.Scatter(
-                        x=beta_values, 
-                        y=theoretical_max,
-                        mode='lines+markers',
-                        name='Theoretical Max',
-                        line=dict(color=color_theory_max, width=3),
-                        marker=dict(
-                            symbol='diamond',
-                            size=8,
-                            color=color_theory_max,
-                            line=dict(color='white', width=1)
-                        ),
-                        hovertemplate='β: %{x:.3f}<br>Value: %{y:.6f}<extra>Theoretical Max</extra>'
-                    ))
-                    
-                    fig.add_trace(go.Scatter(
-                        x=beta_values, 
-                        y=theoretical_min,
-                        mode='lines+markers',
-                        name='Theoretical Min',
-                        line=dict(color=color_theory_min, width=3),
-                        marker=dict(
-                            symbol='diamond',
-                            size=8,
-                            color=color_theory_min,
-                            line=dict(color='white', width=1)
-                        ),
-                        hovertemplate='β: %{x:.3f}<br>Value: %{y:.6f}<extra>Theoretical Min</extra>'
-                    ))
-                    
-                    # Configure layout
-                    fig.update_layout(
-                        title={
-                            'text': f'Eigenvalue Analysis: n={n}, p={p}, a={a}, y={y:.4f}',
-                            'font': {'size': 24, 'color': '#0e1117'},
-                            'y': 0.95,
-                            'x': 0.5,
-                            'xanchor': 'center',
-                            'yanchor': 'top'
-                        },
-                        xaxis={
-                            'title': {'text': 'β Parameter', 'font': {'size': 18, 'color': '#424242'}},
-                            'tickfont': {'size': 14},
-                            'gridcolor': 'rgba(220, 220, 220, 0.5)',
-                            'showgrid': True
-                        },
-                        yaxis={
-                            'title': {'text': 'Eigenvalues', 'font': {'size': 18, 'color': '#424242'}},
-                            'tickfont': {'size': 14},
-                            'gridcolor': 'rgba(220, 220, 220, 0.5)',
-                            'showgrid': True
-                        },
-                        plot_bgcolor='rgba(250, 250, 250, 0.8)',
-                        paper_bgcolor='rgba(255, 255, 255, 0.8)',
-                        hovermode='closest',
-                        legend={
-                            'font': {'size': 14},
-                            'bgcolor': 'rgba(255, 255, 255, 0.9)',
-                            'bordercolor': 'rgba(200, 200, 200, 0.5)',
-                            'borderwidth': 1
-                        },
-                        margin={'l': 60, 'r': 30, 't': 100, 'b': 60},
-                        height=600,
-                    )
-                    
-                    # Add custom modebar buttons
-                    fig.update_layout(
-                        modebar_add=[
-                            'drawline', 'drawopenpath', 'drawclosedpath',
-                            'drawcircle', 'drawrect', 'eraseshape'
-                        ],
-                        modebar_remove=['lasso2d', 'select2d'],
-                        dragmode='zoom'
-                    )
-                    
-                    # Display the interactive plot in Streamlit
-                    st.plotly_chart(fig, use_container_width=True)
-                    
-                    # Display statistics in a cleaner way
-                    st.markdown('<div class="stats-box">', unsafe_allow_html=True)
-                    col1, col2, col3, col4 = st.columns(4)
-                    with col1:
-                        st.metric("Max Empirical", f"{np.max(max_eigenvalues):.4f}")
-                    with col2:
-                        st.metric("Min Empirical", f"{np.min(min_eigenvalues):.4f}")
-                    with col3:
-                        st.metric("Max Theoretical", f"{np.max(theoretical_max):.4f}")
-                    with col4:
-                        st.metric("Min Theoretical", f"{np.min(theoretical_min):.4f}")
-                    st.markdown('</div>', unsafe_allow_html=True)
+                    if debug_mode:
+                        success, stdout, stderr = run_command(cmd, True, timeout=timeout_seconds)
+                        # Process stdout for progress updates
+                        if success:
+                            progress_bar.progress(1.0)
+                    else:
+                        # Start the process with pipe for stdout to read progress
+                        process = subprocess.Popen(
+                            cmd,
+                            stdout=subprocess.PIPE,
+                            stderr=subprocess.PIPE,
+                            text=True,
+                            bufsize=1,
+                            universal_newlines=True
+                        )
+                        
+                        # Track progress from stdout
+                        success = True
+                        stdout_lines = []
+                        
+                        start_time = time.time()
+                        while True:
+                            # Check for timeout
+                            if time.time() - start_time > timeout_seconds:
+                                process.kill()
+                                status_text.error(f"Computation timed out after {timeout_seconds} seconds")
+                                success = False
+                                break
+                                
+                            # Try to read a line (non-blocking)
+                            line = process.stdout.readline()
+                            if not line and process.poll() is not None:
+                                break
+                                
+                            if line:
+                                stdout_lines.append(line)
+                                if line.startswith("PROGRESS:"):
+                                    try:
+                                        # Update progress bar
+                                        progress_value = float(line.split(":")[1].strip())
+                                        progress_bar.progress(progress_value)
+                                        status_text.text(f"Calculating... {int(progress_value * 100)}% complete")
+                                    except:
+                                        pass
+                                elif line:
+                                    status_text.text(line.strip())
+                                    
+                        # Get the return code and stderr
+                        returncode = process.poll()
+                        stderr = process.stderr.read()
+                        
+                        if returncode != 0:
+                            success = False
+                            st.error(f"Error executing the analysis: {stderr}")
+                            with st.expander("Error Details"):
+                                st.code(stderr)
                     
-                    # Display computation time
-                    st.info(f"Computation completed in {end_time - start_time:.2f} seconds")
+                    if success:
+                        progress_bar.progress(1.0)
+                        status_text.text("Analysis complete! Generating visualization...")
+                        
+                        # Check if the output file was created
+                        if not os.path.exists(data_file):
+                            st.error(f"Output file not created: {data_file}")
+                            st.stop()
+                        
+                        try:
+                            # Load the results from the JSON file
+                            with open(data_file, 'r') as f:
+                                data = json.load(f)
+                            
+                            # Process data - convert string values to numeric
+                            beta_values = np.array([safe_convert_to_numeric(x) for x in data['beta_values']])
+                            max_eigenvalues = np.array([safe_convert_to_numeric(x) for x in data['max_eigenvalues']])
+                            min_eigenvalues = np.array([safe_convert_to_numeric(x) for x in data['min_eigenvalues']])
+                            theoretical_max = np.array([safe_convert_to_numeric(x) for x in data['theoretical_max']])
+                            theoretical_min = np.array([safe_convert_to_numeric(x) for x in data['theoretical_min']])
+                            
+                            # Create an interactive plot using Plotly
+                            fig = go.Figure()
+                            
+                            # Add traces for each line
+                            fig.add_trace(go.Scatter(
+                                x=beta_values, 
+                                y=max_eigenvalues,
+                                mode='lines+markers',
+                                name='Empirical Max Eigenvalue',
+                                line=dict(color=color_max, width=3),
+                                marker=dict(
+                                    symbol='circle',
+                                    size=8,
+                                    color=color_max,
+                                    line=dict(color='white', width=1)
+                                ),
+                                hovertemplate='β: %{x:.3f}<br>Value: %{y:.6f}<extra>Empirical Max</extra>'
+                            ))
+                            
+                            fig.add_trace(go.Scatter(
+                                x=beta_values, 
+                                y=min_eigenvalues,
+                                mode='lines+markers',
+                                name='Empirical Min Eigenvalue',
+                                line=dict(color=color_min, width=3),
+                                marker=dict(
+                                    symbol='circle',
+                                    size=8,
+                                    color=color_min,
+                                    line=dict(color='white', width=1)
+                                ),
+                                hovertemplate='β: %{x:.3f}<br>Value: %{y:.6f}<extra>Empirical Min</extra>'
+                            ))
+                            
+                            fig.add_trace(go.Scatter(
+                                x=beta_values, 
+                                y=theoretical_max,
+                                mode='lines+markers',
+                                name='Theoretical Max',
+                                line=dict(color=color_theory_max, width=3),
+                                marker=dict(
+                                    symbol='diamond',
+                                    size=8,
+                                    color=color_theory_max,
+                                    line=dict(color='white', width=1)
+                                ),
+                                hovertemplate='β: %{x:.3f}<br>Value: %{y:.6f}<extra>Theoretical Max</extra>'
+                            ))
+                            
+                            fig.add_trace(go.Scatter(
+                                x=beta_values, 
+                                y=theoretical_min,
+                                mode='lines+markers',
+                                name='Theoretical Min',
+                                line=dict(color=color_theory_min, width=3),
+                                marker=dict(
+                                    symbol='diamond',
+                                    size=8,
+                                    color=color_theory_min,
+                                    line=dict(color='white', width=1)
+                                ),
+                                hovertemplate='β: %{x:.3f}<br>Value: %{y:.6f}<extra>Theoretical Min</extra>'
+                            ))
+                            
+                            # Configure layout for better appearance - removed the detailed annotations
+                            fig.update_layout(
+                                title={
+                                    'text': f'Eigenvalue Analysis: n={n}, p={p}, a={a}, y={y:.4f}',
+                                    'font': {'size': 24, 'color': '#0e1117'},
+                                    'y': 0.95,
+                                    'x': 0.5,
+                                    'xanchor': 'center',
+                                    'yanchor': 'top'
+                                },
+                                xaxis={
+                                    'title': {'text': 'β Parameter', 'font': {'size': 18, 'color': '#424242'}},
+                                    'tickfont': {'size': 14},
+                                    'gridcolor': 'rgba(220, 220, 220, 0.5)',
+                                    'showgrid': True
+                                },
+                                yaxis={
+                                    'title': {'text': 'Eigenvalues', 'font': {'size': 18, 'color': '#424242'}},
+                                    'tickfont': {'size': 14},
+                                    'gridcolor': 'rgba(220, 220, 220, 0.5)',
+                                    'showgrid': True
+                                },
+                                plot_bgcolor='rgba(250, 250, 250, 0.8)',
+                                paper_bgcolor='rgba(255, 255, 255, 0.8)',
+                                hovermode='closest',
+                                legend={
+                                    'font': {'size': 14},
+                                    'bgcolor': 'rgba(255, 255, 255, 0.9)',
+                                    'bordercolor': 'rgba(200, 200, 200, 0.5)',
+                                    'borderwidth': 1
+                                },
+                                margin={'l': 60, 'r': 30, 't': 100, 'b': 60},
+                                height=600,
+                            )
+                            
+                            # Add custom modebar buttons
+                            fig.update_layout(
+                                modebar_add=[
+                                    'drawline', 'drawopenpath', 'drawclosedpath',
+                                    'drawcircle', 'drawrect', 'eraseshape'
+                                ],
+                                modebar_remove=['lasso2d', 'select2d'],
+                                dragmode='zoom'
+                            )
+                            
+                            # Clear progress container
+                            progress_container.empty()
+                            
+                            # Display the interactive plot in Streamlit
+                            st.plotly_chart(fig, use_container_width=True)
+                            
+                            # Display statistics in a cleaner way
+                            st.markdown('<div class="stats-box">', unsafe_allow_html=True)
+                            col1, col2, col3, col4 = st.columns(4)
+                            with col1:
+                                st.metric("Max Empirical", f"{max_eigenvalues.max():.4f}")
+                            with col2:
+                                st.metric("Min Empirical", f"{min_eigenvalues.min():.4f}")
+                            with col3:
+                                st.metric("Max Theoretical", f"{theoretical_max.max():.4f}")
+                            with col4:
+                                st.metric("Min Theoretical", f"{theoretical_min.min():.4f}")
+                            st.markdown('</div>', unsafe_allow_html=True)
+                                
+                        except json.JSONDecodeError as e:
+                            st.error(f"Error parsing JSON results: {str(e)}")
+                            if os.path.exists(data_file):
+                                with open(data_file, 'r') as f:
+                                    content = f.read()
+                                st.code(content[:1000] + "..." if len(content) > 1000 else content)
                 
                 except Exception as e:
                     st.error(f"An error occurred: {str(e)}")
-                    st.exception(e)
+                    if debug_mode:
+                        st.exception(e)
         
         else:
             # Try to load existing data if available
@@ -800,7 +1498,7 @@ with tab1:
                     theoretical_max = np.array([safe_convert_to_numeric(x) for x in data['theoretical_max']])
                     theoretical_min = np.array([safe_convert_to_numeric(x) for x in data['theoretical_min']])
                     
-                    # Create the plot with existing data
+                    # Create an interactive plot using Plotly
                     fig = go.Figure()
                     
                     # Add traces for each line
@@ -864,7 +1562,7 @@ with tab1:
                         hovertemplate='β: %{x:.3f}<br>Value: %{y:.6f}<extra>Theoretical Min</extra>'
                     ))
                     
-                    # Configure layout
+                    # Configure layout for better appearance
                     fig.update_layout(
                         title={
                             'text': f'Eigenvalue Analysis (Previous Result)',
@@ -975,16 +1673,24 @@ with tab2:
                 # Show progress
                 progress_container = st.container()
                 with progress_container:
+                    progress_bar = st.progress(0)
                     status_text = st.empty()
                     status_text.text("Starting cubic equation calculations...")
                 
                 try:
-                    # Create data file path
+                    # Create data file path 
                     data_file = os.path.join(output_dir, "cubic_data.json")
                     
-                    # Run the Im(s) vs z analysis
+                    # Run the Im(s) vs z analysis using Python SymPy
                     start_time = time.time()
-                    result = compute_ImS_vs_Z(cubic_a, cubic_y, cubic_beta, cubic_points, z_min, z_max)
+                    
+                    # Define progress callback for updating the progress bar
+                    def update_progress(progress):
+                        progress_bar.progress(progress)
+                        status_text.text(f"Calculating... {int(progress * 100)}% complete")
+                    
+                    # Run the analysis with progress updates
+                    result = compute_ImS_vs_Z(cubic_a, cubic_y, cubic_beta, cubic_points, z_min, z_max, update_progress)
                     end_time = time.time()
                     
                     # Format the data for saving
@@ -1553,11 +2259,11 @@ with tab2:
 st.markdown("""
 <div class="footer">
     <h3>About the Matrix Analysis Dashboard</h3>
-    <p>This dashboard performs two types of analyses:</p>
+    <p>This dashboard performs two types of analyses using different computational approaches:</p>
     <ol>
-        <li><strong>Eigenvalue Analysis:</strong> Computes eigenvalues of random matrices with specific structures, showing empirical and theoretical results.</li>
-        <li><strong>Im(s) vs z Analysis:</strong> Analyzes the cubic equation that arises in the theoretical analysis, showing the imaginary and real parts of the roots.</li>
+        <li><strong>Eigenvalue Analysis (C++):</strong> Uses C++ with OpenCV for high-performance computation of eigenvalues of random matrices with specific structures.</li>
+        <li><strong>Im(s) vs z Analysis (SymPy):</strong> Uses Python's SymPy library for symbolic mathematics to analyze the cubic equation that arises in the theoretical analysis.</li>
     </ol>
-    <p>Developed using Streamlit and Python's SymPy library for symbolic mathematics calculations.</p>
+    <p>This hybrid approach combines C++'s performance for data-intensive calculations with Python's expressiveness for mathematical analysis.</p>
 </div>
 """, unsafe_allow_html=True)