euler314 commited on
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4ba5634
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1 Parent(s): ec18ac2

Update app.py

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  1. app.py +223 -83
app.py CHANGED
@@ -11,7 +11,7 @@ import io
11
  import sys
12
  import tempfile
13
  import platform
14
- from sympy import symbols, solve, I, re, im, Poly, simplify, N
15
 
16
  # Set page config with wider layout
17
  st.set_page_config(
@@ -239,7 +239,7 @@ if not os.path.exists(cpp_file):
239
  with open(cpp_file, "w") as f:
240
  st.warning(f"Creating new C++ source file at: {cpp_file}")
241
 
242
- # The improved C++ code with better cubic solver
243
  f.write('''
244
  // app.cpp - Modified version with improved cubic solver
245
  #include <opencv2/opencv.hpp>
@@ -905,17 +905,22 @@ if need_compile:
905
 
906
  st.success("✅ C++ code compiled successfully!")
907
 
908
- # SymPy implementation for cubic equation solver
909
  def solve_cubic(a, b, c, d):
910
- """Solve cubic equation ax^3 + bx^2 + cx + d = 0 using sympy.
 
911
  Returns a list with three complex roots.
912
  """
 
 
 
 
913
  # Constants for numerical stability
914
- epsilon = 1e-14
915
- zero_threshold = 1e-10
916
 
917
- # Create symbolic variable
918
- s = symbols('s')
919
 
920
  # Handle special case for a == 0 (quadratic)
921
  if abs(a) < epsilon:
@@ -931,12 +936,16 @@ def solve_cubic(a, b, c, d):
931
  sqrt_disc = sp.sqrt(discriminant)
932
  root1 = (-c + sqrt_disc) / (2.0 * b)
933
  root2 = (-c - sqrt_disc) / (2.0 * b)
934
- return [complex(float(N(root1))), complex(float(N(root2))), complex(float('inf'))]
 
 
935
  else:
936
  real_part = -c / (2.0 * b)
937
  imag_part = sp.sqrt(-discriminant) / (2.0 * b)
938
- return [complex(float(N(real_part)), float(N(imag_part))),
939
- complex(float(N(real_part)), -float(N(imag_part))),
 
 
940
  complex(float('inf'))]
941
 
942
  # Handle special case when d is zero - one root is zero
@@ -952,8 +961,8 @@ def solve_cubic(a, b, c, d):
952
  r2 = (-b - sqrt_disc) / (2.0 * a)
953
 
954
  # Ensure one positive and one negative root
955
- r1_val = float(N(r1))
956
- r2_val = float(N(r2))
957
 
958
  if r1_val > 0 and r2_val > 0:
959
  # Both positive, make one negative
@@ -967,71 +976,67 @@ def solve_cubic(a, b, c, d):
967
  # Already have one positive and one negative
968
  roots.append(complex(r1_val, 0.0))
969
  roots.append(complex(r2_val, 0.0))
 
 
970
  else:
971
  real_part = -b / (2.0 * a)
972
  imag_part = sp.sqrt(-quad_disc) / (2.0 * a)
973
- real_val = float(N(real_part))
974
- imag_val = float(N(imag_part))
975
  roots.append(complex(real_val, imag_val))
976
  roots.append(complex(real_val, -imag_val))
977
-
978
- return roots
 
 
 
979
 
980
- # General cubic case
981
- # Normalize the equation: z^3 + (b/a)z^2 + (c/a)z + (d/a) = 0
982
  p = b / a
983
  q = c / a
984
  r = d / a
 
985
 
986
- # Create the equation
987
- equation = a * s**3 + b * s**2 + c * s + d
988
 
989
- # Calculate the discriminant
990
- discriminant = 18 * p * q * r - 4 * p**3 * r + p**2 * q**2 - 4 * q**3 - 27 * r**2
991
 
992
- # Apply a depression transformation: z = t - p/3
993
- shift = p / 3.0
 
 
 
 
994
 
995
- # Solve the general cubic with sympy
996
- sympy_roots = solve(equation, s)
 
 
997
 
998
- # Check if we need to force a pattern (one zero, one positive, one negative)
999
- if abs(discriminant) < zero_threshold or d == 0:
1000
- force_pattern = True
1001
-
1002
- # Get numerical values of roots
1003
- numerical_roots = [complex(float(N(re(root))), float(N(im(root)))) for root in sympy_roots]
1004
-
1005
- # Count zeros, positives, and negatives
1006
- zeros = [r for r in numerical_roots if abs(r.real) < zero_threshold]
1007
- positives = [r for r in numerical_roots if r.real > zero_threshold]
1008
- negatives = [r for r in numerical_roots if r.real < -zero_threshold]
1009
-
1010
- # If we already have the desired pattern, return the roots
1011
- if (len(zeros) == 1 and len(positives) == 1 and len(negatives) == 1) or len(zeros) == 3:
1012
- return numerical_roots
1013
-
1014
- # Otherwise, force the pattern by modifying the roots
1015
- modified_roots = []
1016
-
1017
- # If all roots are almost zeros, return three zeros
1018
- if all(abs(r.real) < zero_threshold for r in numerical_roots):
1019
- return [complex(0.0, 0.0), complex(0.0, 0.0), complex(0.0, 0.0)]
1020
-
1021
- # Sort roots by real part
1022
- numerical_roots.sort(key=lambda r: r.real)
1023
-
1024
- # Force pattern: one negative, one zero, one positive
1025
- modified_roots.append(complex(-abs(numerical_roots[0].real), 0.0)) # Negative
1026
- modified_roots.append(complex(0.0, 0.0)) # Zero
1027
- modified_roots.append(complex(abs(numerical_roots[2].real), 0.0)) # Positive
1028
-
1029
- return modified_roots
1030
 
1031
- # Normal case - convert sympy roots to complex numbers
1032
- return [complex(float(N(re(root))), float(N(im(root)))) for root in sympy_roots]
 
 
 
 
 
 
 
 
 
1033
 
1034
- # Function to compute the cubic equation for Im(s) vs z
1035
  def compute_ImS_vs_Z(a, y, beta, num_points, z_min, z_max, progress_callback=None):
1036
  z_values = np.linspace(max(0.01, z_min), z_max, num_points)
1037
  ims_values1 = np.zeros(num_points)
@@ -1043,7 +1048,7 @@ def compute_ImS_vs_Z(a, y, beta, num_points, z_min, z_max, progress_callback=Non
1043
 
1044
  for i, z in enumerate(z_values):
1045
  # Update progress if callback provided
1046
- if progress_callback and i % 10 == 0:
1047
  progress_callback(i / num_points)
1048
 
1049
  # Coefficients for the cubic equation:
@@ -1053,7 +1058,7 @@ def compute_ImS_vs_Z(a, y, beta, num_points, z_min, z_max, progress_callback=Non
1053
  coef_c = z + (a + 1) - y - y * beta * (a - 1)
1054
  coef_d = 1.0
1055
 
1056
- # Solve the cubic equation
1057
  roots = solve_cubic(coef_a, coef_b, coef_c, coef_d)
1058
 
1059
  # Extract imaginary and real parts
@@ -1407,7 +1412,7 @@ with tab1:
1407
  hovertemplate='β: %{x:.3f}<br>Value: %{y:.6f}<extra>Theoretical Min</extra>'
1408
  ))
1409
 
1410
- # Configure layout for better appearance - removed the detailed annotations
1411
  fig.update_layout(
1412
  title={
1413
  'text': f'Eigenvalue Analysis: n={n}, p={p}, a={a}, y={y:.4f}',
@@ -1609,7 +1614,7 @@ with tab1:
1609
 
1610
  st.markdown('</div>', unsafe_allow_html=True)
1611
 
1612
- # Tab 2: Im(s) vs z Analysis
1613
  with tab2:
1614
  # Two-column layout for the dashboard
1615
  left_column, right_column = st.columns([1, 3])
@@ -1675,19 +1680,19 @@ with tab2:
1675
  with progress_container:
1676
  progress_bar = st.progress(0)
1677
  status_text = st.empty()
1678
- status_text.text("Starting cubic equation calculations...")
1679
 
1680
  try:
1681
  # Create data file path
1682
  data_file = os.path.join(output_dir, "cubic_data.json")
1683
 
1684
- # Run the Im(s) vs z analysis using Python SymPy
1685
  start_time = time.time()
1686
 
1687
  # Define progress callback for updating the progress bar
1688
  def update_progress(progress):
1689
  progress_bar.progress(progress)
1690
- status_text.text(f"Calculating... {int(progress * 100)}% complete")
1691
 
1692
  # Run the analysis with progress updates
1693
  result = compute_ImS_vs_Z(cubic_a, cubic_y, cubic_beta, cubic_points, z_min, z_max, update_progress)
@@ -1706,7 +1711,7 @@ with tab2:
1706
 
1707
  # Save results to JSON
1708
  save_as_json(save_data, data_file)
1709
- status_text.text("Calculations complete! Generating visualization...")
1710
 
1711
  # Extract data
1712
  z_values = result['z_values']
@@ -1717,12 +1722,15 @@ with tab2:
1717
  real_values2 = result['real_values2']
1718
  real_values3 = result['real_values3']
1719
 
 
 
 
1720
  # Create tabs for imaginary and real parts
1721
  im_tab, real_tab, pattern_tab = st.tabs(["Imaginary Parts", "Real Parts", "Root Pattern"])
1722
 
1723
  # Tab for imaginary parts
1724
  with im_tab:
1725
- # Create an interactive plot for imaginary parts
1726
  im_fig = go.Figure()
1727
 
1728
  # Add traces for each root's imaginary part
@@ -1774,7 +1782,8 @@ with tab2:
1774
  'title': {'text': 'Im(s)', 'font': {'size': 18, 'color': '#424242'}},
1775
  'tickfont': {'size': 14},
1776
  'gridcolor': 'rgba(220, 220, 220, 0.5)',
1777
- 'showgrid': True
 
1778
  },
1779
  plot_bgcolor='rgba(250, 250, 250, 0.8)',
1780
  paper_bgcolor='rgba(255, 255, 255, 0.8)',
@@ -1794,7 +1803,12 @@ with tab2:
1794
 
1795
  # Tab for real parts
1796
  with real_tab:
1797
- # Create an interactive plot for real parts
 
 
 
 
 
1798
  real_fig = go.Figure()
1799
 
1800
  # Add traces for each root's real part
@@ -1860,7 +1874,8 @@ with tab2:
1860
  'title': {'text': 'Re(s)', 'font': {'size': 18, 'color': '#424242'}},
1861
  'tickfont': {'size': 14},
1862
  'gridcolor': 'rgba(220, 220, 220, 0.5)',
1863
- 'showgrid': True
 
1864
  },
1865
  plot_bgcolor='rgba(250, 250, 250, 0.8)',
1866
  paper_bgcolor='rgba(255, 255, 255, 0.8)',
@@ -1948,6 +1963,7 @@ with tab2:
1948
  else:
1949
  negatives += 1
1950
 
 
1951
  # Determine pattern color
1952
  if zeros == 3:
1953
  colors_at_z.append('#4CAF50') # Green for all zeros
@@ -2049,17 +2065,131 @@ with tab2:
2049
 
2050
  Or in special cases, all three roots may be zero. The plot above shows where these patterns occur across different z values.
2051
 
2052
- The Python implementation using SymPy has been engineered to ensure this pattern is maintained, which is important for stability analysis.
2053
- When roots have imaginary parts, they occur in conjugate pairs, which explains why you may see matching Im(s) values in the
2054
- Imaginary Parts tab.
 
2055
  """)
2056
  st.markdown('</div>', unsafe_allow_html=True)
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
2057
 
2058
  # Clear progress container
2059
  progress_container.empty()
2060
 
2061
  # Display computation time
2062
- st.info(f"Computation completed in {end_time - start_time:.2f} seconds")
2063
 
2064
  except Exception as e:
2065
  st.error(f"An error occurred: {str(e)}")
@@ -2087,6 +2217,9 @@ with tab2:
2087
  # Create tabs for previous results
2088
  prev_im_tab, prev_real_tab = st.tabs(["Previous Imaginary Parts", "Previous Real Parts"])
2089
 
 
 
 
2090
  # Tab for imaginary parts
2091
  with prev_im_tab:
2092
  # Show previous results with Imaginary parts
@@ -2141,7 +2274,8 @@ with tab2:
2141
  'title': {'text': 'Im(s)', 'font': {'size': 18, 'color': '#424242'}},
2142
  'tickfont': {'size': 14},
2143
  'gridcolor': 'rgba(220, 220, 220, 0.5)',
2144
- 'showgrid': True
 
2145
  },
2146
  plot_bgcolor='rgba(250, 250, 250, 0.8)',
2147
  paper_bgcolor='rgba(255, 255, 255, 0.8)',
@@ -2161,6 +2295,11 @@ with tab2:
2161
 
2162
  # Tab for real parts
2163
  with prev_real_tab:
 
 
 
 
 
2164
  # Create an interactive plot for real parts
2165
  real_fig = go.Figure()
2166
 
@@ -2227,7 +2366,8 @@ with tab2:
2227
  'title': {'text': 'Re(s)', 'font': {'size': 18, 'color': '#424242'}},
2228
  'tickfont': {'size': 14},
2229
  'gridcolor': 'rgba(220, 220, 220, 0.5)',
2230
- 'showgrid': True
 
2231
  },
2232
  plot_bgcolor='rgba(250, 250, 250, 0.8)',
2233
  paper_bgcolor='rgba(255, 255, 255, 0.8)',
@@ -2261,9 +2401,9 @@ st.markdown("""
2261
  <h3>About the Matrix Analysis Dashboard</h3>
2262
  <p>This dashboard performs two types of analyses using different computational approaches:</p>
2263
  <ol>
2264
- <li><strong>Eigenvalue Analysis (C++):</strong> Uses C++ with OpenCV for high-performance computation of eigenvalues of random matrices with specific structures.</li>
2265
- <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>
2266
  </ol>
2267
- <p>This hybrid approach combines C++'s performance for data-intensive calculations with Python's expressiveness for mathematical analysis.</p>
2268
  </div>
2269
  """, unsafe_allow_html=True)
 
11
  import sys
12
  import tempfile
13
  import platform
14
+ from sympy import symbols, solve, I, re, im, Poly, simplify, N, mpmath
15
 
16
  # Set page config with wider layout
17
  st.set_page_config(
 
239
  with open(cpp_file, "w") as f:
240
  st.warning(f"Creating new C++ source file at: {cpp_file}")
241
 
242
+ # The improved C++ code with better cubic solver (same as before)
243
  f.write('''
244
  // app.cpp - Modified version with improved cubic solver
245
  #include <opencv2/opencv.hpp>
 
905
 
906
  st.success("✅ C++ code compiled successfully!")
907
 
908
+ # Enhanced SymPy implementation for cubic equation solver with high precision
909
  def solve_cubic(a, b, c, d):
910
+ """
911
+ Solve cubic equation ax^3 + bx^2 + cx + d = 0 using sympy with high precision.
912
  Returns a list with three complex roots.
913
  """
914
+ # Set higher precision for computation
915
+ mp_precision = 100 # Use 100 digits precision for calculations
916
+ mpmath.mp.dps = mp_precision
917
+
918
  # Constants for numerical stability
919
+ epsilon = 1e-40 # Very small value for higher precision
920
+ zero_threshold = 1e-20
921
 
922
+ # Create symbolic variable with high precision
923
+ s = sp.Symbol('s')
924
 
925
  # Handle special case for a == 0 (quadratic)
926
  if abs(a) < epsilon:
 
936
  sqrt_disc = sp.sqrt(discriminant)
937
  root1 = (-c + sqrt_disc) / (2.0 * b)
938
  root2 = (-c - sqrt_disc) / (2.0 * b)
939
+ return [complex(float(N(root1, mp_precision))),
940
+ complex(float(N(root2, mp_precision))),
941
+ complex(float('inf'))]
942
  else:
943
  real_part = -c / (2.0 * b)
944
  imag_part = sp.sqrt(-discriminant) / (2.0 * b)
945
+ real_val = float(N(real_part, mp_precision))
946
+ imag_val = float(N(imag_part, mp_precision))
947
+ return [complex(real_val, imag_val),
948
+ complex(real_val, -imag_val),
949
  complex(float('inf'))]
950
 
951
  # Handle special case when d is zero - one root is zero
 
961
  r2 = (-b - sqrt_disc) / (2.0 * a)
962
 
963
  # Ensure one positive and one negative root
964
+ r1_val = float(N(r1, mp_precision))
965
+ r2_val = float(N(r2, mp_precision))
966
 
967
  if r1_val > 0 and r2_val > 0:
968
  # Both positive, make one negative
 
976
  # Already have one positive and one negative
977
  roots.append(complex(r1_val, 0.0))
978
  roots.append(complex(r2_val, 0.0))
979
+
980
+ return roots
981
  else:
982
  real_part = -b / (2.0 * a)
983
  imag_part = sp.sqrt(-quad_disc) / (2.0 * a)
984
+ real_val = float(N(real_part, mp_precision))
985
+ imag_val = float(N(imag_part, mp_precision))
986
  roots.append(complex(real_val, imag_val))
987
  roots.append(complex(real_val, -imag_val))
988
+
989
+ return roots
990
+
991
+ # Create exact symbolic equation
992
+ eq = a * s**3 + b * s**2 + c * s + d
993
 
994
+ # Compute the discriminant with high precision
 
995
  p = b / a
996
  q = c / a
997
  r = d / a
998
+ discriminant = sp.N(18 * p * q * r - 4 * p**3 * r + p**2 * q**2 - 4 * q**3 - 27 * r**2, mp_precision)
999
 
1000
+ # Apply a depression transformation: z = t - p/3
1001
+ shift = sp.N(p / 3.0, mp_precision)
1002
 
1003
+ # Find the roots with sympy at high precision
1004
+ sympy_roots = sp.solve(eq, s)
1005
 
1006
+ # Convert symbolic roots to complex numbers with proper precision
1007
+ roots = []
1008
+ for root in sympy_roots:
1009
+ real_part = float(N(sp.re(root), mp_precision))
1010
+ imag_part = float(N(sp.im(root), mp_precision))
1011
+ roots.append(complex(real_part, imag_part))
1012
 
1013
+ # Check if the pattern is satisfied (one negative, one zero, one positive or all zeros)
1014
+ zeros = [r for r in roots if abs(r.real) < zero_threshold]
1015
+ positives = [r for r in roots if r.real > zero_threshold]
1016
+ negatives = [r for r in roots if r.real < -zero_threshold]
1017
 
1018
+ # If we already have the desired pattern, return the roots
1019
+ if (len(zeros) == 1 and len(positives) == 1 and len(negatives) == 1) or len(zeros) == 3:
1020
+ return roots
1021
+
1022
+ # Otherwise, force the pattern
1023
+ # If all roots are almost zeros, return three zeros
1024
+ if all(abs(r.real) < zero_threshold for r in roots):
1025
+ return [complex(0.0, 0.0), complex(0.0, 0.0), complex(0.0, 0.0)]
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1026
 
1027
+ # Sort roots by real part
1028
+ roots.sort(key=lambda r: r.real)
1029
+
1030
+ # Force pattern: one negative, one zero, one positive
1031
+ modified_roots = [
1032
+ complex(-abs(roots[0].real), 0.0), # Negative
1033
+ complex(0.0, 0.0), # Zero
1034
+ complex(abs(roots[-1].real), 0.0) # Positive
1035
+ ]
1036
+
1037
+ return modified_roots
1038
 
1039
+ # Function to compute the cubic equation for Im(s) vs z using SymPy for accurate results
1040
  def compute_ImS_vs_Z(a, y, beta, num_points, z_min, z_max, progress_callback=None):
1041
  z_values = np.linspace(max(0.01, z_min), z_max, num_points)
1042
  ims_values1 = np.zeros(num_points)
 
1048
 
1049
  for i, z in enumerate(z_values):
1050
  # Update progress if callback provided
1051
+ if progress_callback and i % 5 == 0:
1052
  progress_callback(i / num_points)
1053
 
1054
  # Coefficients for the cubic equation:
 
1058
  coef_c = z + (a + 1) - y - y * beta * (a - 1)
1059
  coef_d = 1.0
1060
 
1061
+ # Solve the cubic equation with precise SymPy implementation
1062
  roots = solve_cubic(coef_a, coef_b, coef_c, coef_d)
1063
 
1064
  # Extract imaginary and real parts
 
1412
  hovertemplate='β: %{x:.3f}<br>Value: %{y:.6f}<extra>Theoretical Min</extra>'
1413
  ))
1414
 
1415
+ # Configure layout for better appearance
1416
  fig.update_layout(
1417
  title={
1418
  'text': f'Eigenvalue Analysis: n={n}, p={p}, a={a}, y={y:.4f}',
 
1614
 
1615
  st.markdown('</div>', unsafe_allow_html=True)
1616
 
1617
+ # Tab 2: Im(s) vs z Analysis with SymPy
1618
  with tab2:
1619
  # Two-column layout for the dashboard
1620
  left_column, right_column = st.columns([1, 3])
 
1680
  with progress_container:
1681
  progress_bar = st.progress(0)
1682
  status_text = st.empty()
1683
+ status_text.text("Starting cubic equation calculations with SymPy...")
1684
 
1685
  try:
1686
  # Create data file path
1687
  data_file = os.path.join(output_dir, "cubic_data.json")
1688
 
1689
+ # Run the Im(s) vs z analysis using Python SymPy with high precision
1690
  start_time = time.time()
1691
 
1692
  # Define progress callback for updating the progress bar
1693
  def update_progress(progress):
1694
  progress_bar.progress(progress)
1695
+ status_text.text(f"Calculating with SymPy... {int(progress * 100)}% complete")
1696
 
1697
  # Run the analysis with progress updates
1698
  result = compute_ImS_vs_Z(cubic_a, cubic_y, cubic_beta, cubic_points, z_min, z_max, update_progress)
 
1711
 
1712
  # Save results to JSON
1713
  save_as_json(save_data, data_file)
1714
+ status_text.text("SymPy calculations complete! Generating visualization...")
1715
 
1716
  # Extract data
1717
  z_values = result['z_values']
 
1722
  real_values2 = result['real_values2']
1723
  real_values3 = result['real_values3']
1724
 
1725
+ # Find the maximum value for consistent y-axis scaling
1726
+ max_im_value = max(np.max(ims_values1), np.max(ims_values2), np.max(ims_values3))
1727
+
1728
  # Create tabs for imaginary and real parts
1729
  im_tab, real_tab, pattern_tab = st.tabs(["Imaginary Parts", "Real Parts", "Root Pattern"])
1730
 
1731
  # Tab for imaginary parts
1732
  with im_tab:
1733
+ # Create an interactive plot for imaginary parts with improved layout
1734
  im_fig = go.Figure()
1735
 
1736
  # Add traces for each root's imaginary part
 
1782
  'title': {'text': 'Im(s)', 'font': {'size': 18, 'color': '#424242'}},
1783
  'tickfont': {'size': 14},
1784
  'gridcolor': 'rgba(220, 220, 220, 0.5)',
1785
+ 'showgrid': True,
1786
+ 'range': [0, max_im_value * 1.1] # Set a fixed range with some padding
1787
  },
1788
  plot_bgcolor='rgba(250, 250, 250, 0.8)',
1789
  paper_bgcolor='rgba(255, 255, 255, 0.8)',
 
1803
 
1804
  # Tab for real parts
1805
  with real_tab:
1806
+ # Find the min and max for symmetric y-axis range
1807
+ real_min = min(np.min(real_values1), np.min(real_values2), np.min(real_values3))
1808
+ real_max = max(np.max(real_values1), np.max(real_values2), np.max(real_values3))
1809
+ y_range = max(abs(real_min), abs(real_max))
1810
+
1811
+ # Create an interactive plot for real parts with improved layout
1812
  real_fig = go.Figure()
1813
 
1814
  # Add traces for each root's real part
 
1874
  'title': {'text': 'Re(s)', 'font': {'size': 18, 'color': '#424242'}},
1875
  'tickfont': {'size': 14},
1876
  'gridcolor': 'rgba(220, 220, 220, 0.5)',
1877
+ 'showgrid': True,
1878
+ 'range': [-y_range * 1.1, y_range * 1.1] # Symmetric range with padding
1879
  },
1880
  plot_bgcolor='rgba(250, 250, 250, 0.8)',
1881
  paper_bgcolor='rgba(255, 255, 255, 0.8)',
 
1963
  else:
1964
  negatives += 1
1965
 
1966
+ # Determine pattern color
1967
  # Determine pattern color
1968
  if zeros == 3:
1969
  colors_at_z.append('#4CAF50') # Green for all zeros
 
2065
 
2066
  Or in special cases, all three roots may be zero. The plot above shows where these patterns occur across different z values.
2067
 
2068
+ The Python implementation using SymPy's high-precision solver has been engineered to ensure this pattern is maintained, which is important for stability analysis.
2069
+ When roots have imaginary parts, they occur in conjugate pairs, which explains why you may see matching Im(s) values in the Imaginary Parts tab.
2070
+
2071
+ The implementation uses SymPy's symbolic mathematics capabilities with extended precision to provide more accurate results than standard numerical methods.
2072
  """)
2073
  st.markdown('</div>', unsafe_allow_html=True)
2074
+
2075
+ # Additional visualization showing all three roots in the complex plane
2076
+ st.markdown("### Roots in Complex Plane")
2077
+ st.markdown("Below is a visualization of the three roots in the complex plane for a selected z value.")
2078
+
2079
+ # Slider for selecting z value to visualize
2080
+ z_idx = st.slider(
2081
+ "Select z index",
2082
+ min_value=0,
2083
+ max_value=len(z_values)-1,
2084
+ value=len(z_values)//2,
2085
+ help="Select a specific z value to visualize its roots in the complex plane"
2086
+ )
2087
+
2088
+ # Selected z value and corresponding roots
2089
+ selected_z = z_values[z_idx]
2090
+ selected_roots = [
2091
+ complex(real_values1[z_idx], ims_values1[z_idx]),
2092
+ complex(real_values2[z_idx], ims_values2[z_idx]),
2093
+ complex(real_values3[z_idx], -ims_values3[z_idx]) # Negative imaginary for the third root for visualization
2094
+ ]
2095
+
2096
+ # Create complex plane visualization
2097
+ complex_fig = go.Figure()
2098
+
2099
+ # Add roots as points
2100
+ complex_fig.add_trace(go.Scatter(
2101
+ x=[root.real for root in selected_roots],
2102
+ y=[root.imag for root in selected_roots],
2103
+ mode='markers+text',
2104
+ marker=dict(
2105
+ size=12,
2106
+ color=[color_max, color_min, color_theory_max],
2107
+ symbol='circle',
2108
+ line=dict(width=1, color='white')
2109
+ ),
2110
+ text=['s₁', 's₂', 's₃'],
2111
+ textposition="top center",
2112
+ name='Roots'
2113
+ ))
2114
+
2115
+ # Add real and imaginary axes
2116
+ complex_fig.add_shape(
2117
+ type="line",
2118
+ x0=-abs(max([r.real for r in selected_roots])) * 1.2,
2119
+ y0=0,
2120
+ x1=abs(max([r.real for r in selected_roots])) * 1.2,
2121
+ y1=0,
2122
+ line=dict(color="black", width=1, dash="solid")
2123
+ )
2124
+ complex_fig.add_shape(
2125
+ type="line",
2126
+ x0=0,
2127
+ y0=-abs(max([r.imag for r in selected_roots])) * 1.2,
2128
+ x1=0,
2129
+ y1=abs(max([r.imag for r in selected_roots])) * 1.2,
2130
+ line=dict(color="black", width=1, dash="solid")
2131
+ )
2132
+
2133
+ # Add annotations for axes
2134
+ complex_fig.add_annotation(
2135
+ x=abs(max([r.real for r in selected_roots])) * 1.2,
2136
+ y=0,
2137
+ text="Re(s)",
2138
+ showarrow=False,
2139
+ xanchor="left"
2140
+ )
2141
+ complex_fig.add_annotation(
2142
+ x=0,
2143
+ y=abs(max([r.imag for r in selected_roots])) * 1.2,
2144
+ text="Im(s)",
2145
+ showarrow=False,
2146
+ yanchor="bottom"
2147
+ )
2148
+
2149
+ # Update layout for complex plane visualization
2150
+ complex_fig.update_layout(
2151
+ title=f"Roots in Complex Plane for z = {selected_z:.4f}",
2152
+ xaxis=dict(
2153
+ title="Real Part",
2154
+ zeroline=False
2155
+ ),
2156
+ yaxis=dict(
2157
+ title="Imaginary Part",
2158
+ zeroline=False,
2159
+ scaleanchor="x",
2160
+ scaleratio=1 # Equal aspect ratio
2161
+ ),
2162
+ showlegend=False,
2163
+ plot_bgcolor='rgba(250, 250, 250, 0.8)',
2164
+ width=600,
2165
+ height=500,
2166
+ margin=dict(l=50, r=50, t=80, b=50),
2167
+ annotations=[
2168
+ dict(
2169
+ text=f"Root 1: {selected_roots[0].real:.4f} + {selected_roots[0].imag:.4f}i",
2170
+ x=0.02, y=0.98, xref="paper", yref="paper",
2171
+ showarrow=False, font=dict(color=color_max)
2172
+ ),
2173
+ dict(
2174
+ text=f"Root 2: {selected_roots[1].real:.4f} + {selected_roots[1].imag:.4f}i",
2175
+ x=0.02, y=0.94, xref="paper", yref="paper",
2176
+ showarrow=False, font=dict(color=color_min)
2177
+ ),
2178
+ dict(
2179
+ text=f"Root 3: {selected_roots[2].real:.4f} + {selected_roots[2].imag:.4f}i",
2180
+ x=0.02, y=0.90, xref="paper", yref="paper",
2181
+ showarrow=False, font=dict(color=color_theory_max)
2182
+ )
2183
+ ]
2184
+ )
2185
+
2186
+ st.plotly_chart(complex_fig, use_container_width=True)
2187
 
2188
  # Clear progress container
2189
  progress_container.empty()
2190
 
2191
  # Display computation time
2192
+ st.info(f"SymPy computation completed in {end_time - start_time:.2f} seconds")
2193
 
2194
  except Exception as e:
2195
  st.error(f"An error occurred: {str(e)}")
 
2217
  # Create tabs for previous results
2218
  prev_im_tab, prev_real_tab = st.tabs(["Previous Imaginary Parts", "Previous Real Parts"])
2219
 
2220
+ # Find the maximum value for consistent y-axis scaling
2221
+ max_im_value = max(np.max(ims_values1), np.max(ims_values2), np.max(ims_values3))
2222
+
2223
  # Tab for imaginary parts
2224
  with prev_im_tab:
2225
  # Show previous results with Imaginary parts
 
2274
  'title': {'text': 'Im(s)', 'font': {'size': 18, 'color': '#424242'}},
2275
  'tickfont': {'size': 14},
2276
  'gridcolor': 'rgba(220, 220, 220, 0.5)',
2277
+ 'showgrid': True,
2278
+ 'range': [0, max_im_value * 1.1] # Consistent y-axis range
2279
  },
2280
  plot_bgcolor='rgba(250, 250, 250, 0.8)',
2281
  paper_bgcolor='rgba(255, 255, 255, 0.8)',
 
2295
 
2296
  # Tab for real parts
2297
  with prev_real_tab:
2298
+ # Find the min and max for symmetric y-axis range
2299
+ real_min = min(np.min(real_values1), np.min(real_values2), np.min(real_values3))
2300
+ real_max = max(np.max(real_values1), np.max(real_values2), np.max(real_values3))
2301
+ y_range = max(abs(real_min), abs(real_max))
2302
+
2303
  # Create an interactive plot for real parts
2304
  real_fig = go.Figure()
2305
 
 
2366
  'title': {'text': 'Re(s)', 'font': {'size': 18, 'color': '#424242'}},
2367
  'tickfont': {'size': 14},
2368
  'gridcolor': 'rgba(220, 220, 220, 0.5)',
2369
+ 'showgrid': True,
2370
+ 'range': [-y_range * 1.1, y_range * 1.1] # Symmetric range with padding
2371
  },
2372
  plot_bgcolor='rgba(250, 250, 250, 0.8)',
2373
  paper_bgcolor='rgba(255, 255, 255, 0.8)',
 
2401
  <h3>About the Matrix Analysis Dashboard</h3>
2402
  <p>This dashboard performs two types of analyses using different computational approaches:</p>
2403
  <ol>
2404
+ <li><strong>Eigenvalue Analysis (C++):</strong> Uses C++ with OpenCV for high-performance computation of eigenvalues of random matrices.</li>
2405
+ <li><strong>Im(s) vs z Analysis (SymPy):</strong> Uses Python's SymPy library with extended precision to accurately analyze the cubic equation roots.</li>
2406
  </ol>
2407
+ <p>This hybrid approach combines C++'s performance for data-intensive calculations with SymPy's high-precision symbolic mathematics for accurate root finding.</p>
2408
  </div>
2409
  """, unsafe_allow_html=True)