Update app.py

app.py

@@ -2,12 +2,13 @@ import streamlit as st
 import sympy as sp
 import numpy as np
 import plotly.graph_objects as go
+from scipy.optimize import fsolve
 
 # Configure Streamlit for Hugging Face Spaces
 st.set_page_config(
-    page_title="Cubic Root Analysis",
-    layout="wide",
-    initial_sidebar_state="collapsed"
+   page_title="Cubic Root Analysis",
+   layout="wide",
+   initial_sidebar_state="collapsed"
 )
 
 #############################
@@ -25,8 +26,8 @@ d_sym = 1
 
 # Symbolic expression for the standard cubic discriminant
 Delta_expr = (
-    ( (b_sym*c_sym)/(6*a_sym**2) - (b_sym**3)/(27*a_sym**3) - d_sym/(2*a_sym) )**2
-    + ( c_sym/(3*a_sym) - (b_sym**2)/(9*a_sym**2) )**3
+   ( (b_sym*c_sym)/(6*a_sym**2) - (b_sym**3)/(27*a_sym**3) - d_sym/(2*a_sym) )**2
+   + ( c_sym/(3*a_sym) - (b_sym**2)/(9*a_sym**2) )**3
 )
 
 # Turn that into a fast numeric function:
@@ -34,258 +35,335 @@ discriminant_func = sp.lambdify((z_sym, beta_sym, z_a_sym, y_sym), Delta_expr, "
 
 @st.cache_data
 def find_z_at_discriminant_zero(z_a, y, beta, z_min, z_max, steps=20000):
-    """
-    Numerically scan z in [z_min, z_max] looking for sign changes of
-    Delta(z) = 0. Returns all roots found via bisection.
-    """
-    z_grid = np.linspace(z_min, z_max, steps)
-    disc_vals = discriminant_func(z_grid, beta, z_a, y)
-    
-    roots_found = []
-    
-    # Scan for sign changes
-    for i in range(len(z_grid) - 1):
-        f1, f2 = disc_vals[i], disc_vals[i+1]
-        if np.isnan(f1) or np.isnan(f2):
-            continue
-        
-        if f1 == 0.0:
-            roots_found.append(z_grid[i])
-        elif f2 == 0.0:
-            roots_found.append(z_grid[i+1])
-        elif f1*f2 < 0:
-            zl = z_grid[i]
-            zr = z_grid[i+1]
-            for _ in range(50):
-                mid = 0.5*(zl + zr)
-                fm = discriminant_func(mid, beta, z_a, y)
-                if fm == 0:
-                    zl = zr = mid
-                    break
-                if np.sign(fm) == np.sign(f1):
-                    zl = mid
-                    f1 = fm
-                else:
-                    zr = mid
-                    f2 = fm
-            root_approx = 0.5*(zl + zr)
-            roots_found.append(root_approx)
-    
-    return np.array(roots_found)
+   """
+   Numerically scan z in [z_min, z_max] looking for sign changes of
+   Delta(z) = 0. Returns all roots found via bisection.
+   """
+   z_grid = np.linspace(z_min, z_max, steps)
+   disc_vals = discriminant_func(z_grid, beta, z_a, y)
+   
+   roots_found = []
+   
+   # Scan for sign changes
+   for i in range(len(z_grid) - 1):
+       f1, f2 = disc_vals[i], disc_vals[i+1]
+       if np.isnan(f1) or np.isnan(f2):
+           continue
+       
+       if f1 == 0.0:
+           roots_found.append(z_grid[i])
+       elif f2 == 0.0:
+           roots_found.append(z_grid[i+1])
+       elif f1*f2 < 0:
+           zl = z_grid[i]
+           zr = z_grid[i+1]
+           for _ in range(50):
+               mid = 0.5*(zl + zr)
+               fm = discriminant_func(mid, beta, z_a, y)
+               if fm == 0:
+                   zl = zr = mid
+                   break
+               if np.sign(fm) == np.sign(f1):
+                   zl = mid
+                   f1 = fm
+               else:
+                   zr = mid
+                   f2 = fm
+           root_approx = 0.5*(zl + zr)
+           roots_found.append(root_approx)
+   
+   return np.array(roots_found)
 
 @st.cache_data
 def sweep_beta_and_find_z_bounds(z_a, y, z_min, z_max, beta_steps=51):
-    """
-    For each beta, find both the largest and smallest z where discriminant=0.
-    Returns (betas, z_min_values, z_max_values).
-    """
-    betas = np.linspace(0, 1, beta_steps)
-    z_min_values = []
-    z_max_values = []
-    
-    for b in betas:
-        roots = find_z_at_discriminant_zero(z_a, y, b, z_min, z_max)
-        if len(roots) == 0:
-            z_min_values.append(np.nan)
-            z_max_values.append(np.nan)
-        else:
-            z_min_values.append(np.min(roots))
-            z_max_values.append(np.max(roots))
-    
-    return betas, np.array(z_min_values), np.array(z_max_values)
+   """
+   For each beta, find both the largest and smallest z where discriminant=0.
+   Returns (betas, z_min_values, z_max_values).
+   """
+   betas = np.linspace(0, 1, beta_steps)
+   z_min_values = []
+   z_max_values = []
+   
+   for b in betas:
+       roots = find_z_at_discriminant_zero(z_a, y, b, z_min, z_max)
+       if len(roots) == 0:
+           z_min_values.append(np.nan)
+           z_max_values.append(np.nan)
+       else:
+           z_min_values.append(np.min(roots))
+           z_max_values.append(np.max(roots))
+   
+   return betas, np.array(z_min_values), np.array(z_max_values)
 
 @st.cache_data
 def compute_low_y_curve(betas, z_a, y):
-    """
-    Compute the additional curve with proper handling of divide by zero cases
-    """
-    betas = np.array(betas)
-    with np.errstate(invalid='ignore', divide='ignore'):
-        sqrt_term = y * betas * (z_a - 1)
-        sqrt_term = np.where(sqrt_term < 0, np.nan, np.sqrt(sqrt_term))
-        
-        term = (-1 + sqrt_term)/z_a
-        numerator = (y - 2)*term + y * betas * ((z_a - 1)/z_a) - 1/z_a - 1
-        denominator = term**2 + term
-        
-        # Handle division by zero and invalid values
-        result = np.zeros_like(betas)
-        mask = (denominator != 0) & ~np.isnan(denominator) & ~np.isnan(numerator)
-        result[mask] = numerator[mask] / denominator[mask]
-        result[~mask] = np.nan
-        
-        return result
+   """
+   Compute the additional curve with proper handling of divide by zero cases
+   """
+   betas = np.array(betas)
+   with np.errstate(invalid='ignore', divide='ignore'):
+       sqrt_term = y * betas * (z_a - 1)
+       sqrt_term = np.where(sqrt_term < 0, np.nan, np.sqrt(sqrt_term))
+       
+       term = (-1 + sqrt_term)/z_a
+       numerator = (y - 2)*term + y * betas * ((z_a - 1)/z_a) - 1/z_a - 1
+       denominator = term**2 + term
+       
+       # Handle division by zero and invalid values
+       mask = (denominator != 0) & ~np.isnan(denominator) & ~np.isnan(numerator)
+       return np.where(mask, numerator/denominator, np.nan)
 
 @st.cache_data
 def compute_high_y_curve(betas, z_a, y):
-    """
-    Compute the expression: ((4y + 12)(4 - a) + 16y*β*(a - 1))/(3(4 - a))
-    """
-    a = z_a  # for clarity in the formula
-    betas = np.array(betas)
-    
-    denominator = 3*(4 - a)
-    if denominator == 0:
-        return np.full_like(betas, np.nan)
-        
-    numerator = (4*y + 12)*(4 - a) + 16*y*betas*(a - 1)
-    return numerator/denominator
+   """
+   Compute the expression: ((4y + 12)(4 - a) + 16y*β*(a - 1))/(3(4 - a))
+   """
+   betas = np.array(betas)
+   denominator = 3*(4 - z_a)
+   
+   if denominator == 0:
+       return np.full_like(betas, np.nan)
+       
+   numerator = (4*y + 12)*(4 - z_a) + 16*y*betas*(z_a - 1)
+   return numerator/denominator
 
 def generate_z_vs_beta_plot(z_a, y, z_min, z_max):
-    if z_a <= 0 or y <= 0 or z_min >= z_max:
-        st.error("Invalid input parameters.")
-        return None
-    
-    beta_steps = 101
-    betas = np.linspace(0, 1, beta_steps)
-    
-    betas, z_mins, z_maxs = sweep_beta_and_find_z_bounds(z_a, y, z_min, z_max, beta_steps=beta_steps)
-    low_y_curve = compute_low_y_curve(betas, z_a, y)
-    high_y_curve = compute_high_y_curve(betas, z_a, y)
-    
-    fig = go.Figure()
-    
-    # Upper and lower z*(β) boundaries
-    fig.add_trace(
-        go.Scatter(
-            x=betas,
-            y=z_maxs,
-            mode="markers+lines",
-            name="Upper z*(β)",
-            marker=dict(size=5, color='blue'),
-            line=dict(color='blue'),
-        )
-    )
-    
-    fig.add_trace(
-        go.Scatter(
-            x=betas,
-            y=z_mins,
-            mode="markers+lines",
-            name="Lower z*(β)",
-            marker=dict(size=5, color='lightblue'),
-            line=dict(color='lightblue'),
-        )
-    )
-    
-    # Asymptotic expressions
-    fig.add_trace(
-        go.Scatter(
-            x=betas,
-            y=low_y_curve,
-            mode="markers+lines",
-            name="Low y Expression",
-            marker=dict(size=5, color='red'),
-            line=dict(color='red'),
-        )
-    )
-    
-    fig.add_trace(
-        go.Scatter(
-            x=betas,
-            y=high_y_curve,
-            mode="markers+lines",
-            name="High y Expression",
-            marker=dict(size=5, color='green'),
-            line=dict(color='green'),
-        )
-    )
-    
-    fig.update_layout(
-        title="Curves vs β: z*(β) boundaries and Asymptotic Expressions",
-        xaxis_title="β",
-        yaxis_title="Value",
-        hovermode="x unified",
-    )
-    return fig
+   if z_a <= 0 or y <= 0 or z_min >= z_max:
+       st.error("Invalid input parameters.")
+       return None
+   
+   beta_steps = 101
+   betas = np.linspace(0, 1, beta_steps)
+   
+   betas, z_mins, z_maxs = sweep_beta_and_find_z_bounds(z_a, y, z_min, z_max, beta_steps=beta_steps)
+   low_y_curve = compute_low_y_curve(betas, z_a, y)
+   high_y_curve = compute_high_y_curve(betas, z_a, y)
+   
+   fig = go.Figure()
+   
+   fig.add_trace(
+       go.Scatter(
+           x=betas,
+           y=z_maxs,
+           mode="markers+lines",
+           name="Upper z*(β)",
+           marker=dict(size=5, color='blue'),
+           line=dict(color='blue'),
+       )
+   )
+   
+   fig.add_trace(
+       go.Scatter(
+           x=betas,
+           y=z_mins,
+           mode="markers+lines",
+           name="Lower z*(β)",
+           marker=dict(size=5, color='lightblue'),
+           line=dict(color='lightblue'),
+       )
+   )
+   
+   fig.add_trace(
+       go.Scatter(
+           x=betas,
+           y=low_y_curve,
+           mode="markers+lines",
+           name="Low y Expression",
+           marker=dict(size=5, color='red'),
+           line=dict(color='red'),
+       )
+   )
+   
+   fig.add_trace(
+       go.Scatter(
+           x=betas,
+           y=high_y_curve,
+           mode="markers+lines",
+           name="High y Expression",
+           marker=dict(size=5, color='green'),
+           line=dict(color='green'),
+       )
+   )
+   
+   fig.update_layout(
+       title="Curves vs β: z*(β) boundaries and Asymptotic Expressions",
+       xaxis_title="β",
+       yaxis_title="Value",
+       hovermode="x unified",
+   )
+   return fig
 
-@st.cache_data
 def compute_cubic_roots(z, beta, z_a, y):
-    """
-    Compute the roots of the cubic equation for given parameters.
-    Returns array of complex roots.
-    """
-    a = z * z_a
-    b = z * z_a + z + z_a - z_a*y  # Fixed coefficient b
-    c = z + z_a + 1 - y*(beta*z_a + 1 - beta)
-    d = 1
-    
-    coeffs = [a, b, c, d]
-    roots = np.roots(coeffs)
-    return roots
+   """
+   Compute the roots of the cubic equation for given parameters.
+   Returns array of complex roots.
+   """
+   a = z * z_a
+   b = z * z_a + z + z_a - z_a*y
+   c = z + z_a + 1 - y*(beta*z_a + 1 - beta)
+   d = 1
+   
+   coeffs = [a, b, c, d]
+   roots = np.roots(coeffs)
+   return roots
 
 def generate_ims_vs_z_plot(beta, y, z_a, z_min, z_max):
-    if z_a <= 0 or y <= 0 or z_min >= z_max:
-        st.error("Invalid input parameters.")
-        return None
-    
-    z_points = np.linspace(z_min, z_max, 1000)
-    ims = []
-    
-    for z in z_points:
-        roots = compute_cubic_roots(z, beta, z_a, y)
-        roots = sorted(roots, key=lambda x: abs(x.imag))
-        ims.append([root.imag for root in roots])
-    
-    ims = np.array(ims)
-    
-    fig = go.Figure()
-    
-    for i in range(3):
-        fig.add_trace(
-            go.Scatter(
-                x=z_points,
-                y=ims[:,i],
-                mode="lines",
-                name=f"Im{{s{i+1}}}",
-                line=dict(width=2),
-            )
-        )
-    
-    fig.update_layout(
-        title=f"Im{{s}} vs. z (β={beta:.3f}, y={y:.3f}, z_a={z_a:.3f})",
-        xaxis_title="z",
-        yaxis_title="Im{s}",
-        hovermode="x unified",
-    )
-    return fig
+   if z_a <= 0 or y <= 0 or z_min >= z_max:
+       st.error("Invalid input parameters.")
+       return None
+   
+   z_points = np.linspace(z_min, z_max, 1000)
+   ims = []
+   
+   for z in z_points:
+       roots = compute_cubic_roots(z, beta, z_a, y)
+       roots = sorted(roots, key=lambda x: abs(x.imag))
+       ims.append([root.imag for root in roots])
+   
+   ims = np.array(ims)
+   
+   fig = go.Figure()
+   
+   for i in range(3):
+       fig.add_trace(
+           go.Scatter(
+               x=z_points,
+               y=ims[:,i],
+               mode="lines",
+               name=f"Im{{s{i+1}}}",
+               line=dict(width=2),
+           )
+       )
+   
+   fig.update_layout(
+       title=f"Im{{s}} vs. z (β={beta:.3f}, y={y:.3f}, z_a={z_a:.3f})",
+       xaxis_title="z",
+       yaxis_title="Im{s}",
+       hovermode="x unified",
+   )
+   return fig
+
+def curve1(s, z, y):
+   """First curve: z*s^2 + (z-y+1)*s + 1"""
+   return z*s**2 + (z-y+1)*s + 1
+
+def curve2(s, y, beta, a):
+   """Second curve: y*β*((a-1)*s)/(a*s+1)"""
+   return y*beta*((a-1)*s)/(a*s+1)
+
+def find_intersections(z, y, beta, a, s_range):
+   """Find intersections between the two curves"""
+   def equation(s):
+       return curve1(s, z, y) - curve2(s, y, beta, a)
+   
+   # Create grid of initial guesses
+   s_guesses = np.linspace(s_range[0], s_range[1], 20)
+   intersections = []
+   
+   for s_guess in s_guesses:
+       try:
+           s_sol = fsolve(equation, s_guess)[0]
+           # Check if solution is within range and not already found
+           if (s_range[0] <= s_sol <= s_range[1] and 
+               not any(abs(s_sol - s_prev) < 1e-6 for s_prev in intersections)):
+               intersections.append(s_sol)
+       except:
+           continue
+           
+   return np.array(intersections)
+
+def generate_curves_plot(z, y, beta, a, s_range):
+   s = np.linspace(s_range[0], s_range[1], 1000)
+   
+   # Compute curves
+   y1 = curve1(s, z, y)
+   y2 = curve2(s, y, beta, a)
+   
+   # Find intersections
+   intersections = find_intersections(z, y, beta, a, s_range)
+   
+   fig = go.Figure()
+   
+   # Plot curves
+   fig.add_trace(
+       go.Scatter(
+           x=s, y=y1,
+           mode='lines',
+           name='z*s² + (z-y+1)*s + 1',
+           line=dict(color='blue')
+       )
+   )
+   
+   fig.add_trace(
+       go.Scatter(
+           x=s, y=y2,
+           mode='lines',
+           name='y*β*((a-1)*s)/(a*s+1)',
+           line=dict(color='red')
+       )
+   )
+   
+   # Plot intersections
+   if len(intersections) > 0:
+       fig.add_trace(
+           go.Scatter(
+               x=intersections,
+               y=curve1(intersections, z, y),
+               mode='markers',
+               name='Intersections',
+               marker=dict(
+                   size=10,
+                   color='green',
+                   symbol='x'
+               )
+           )
+       )
+   
+   fig.update_layout(
+       title=f"Curve Intersection Analysis (y={y:.2f}, β={beta:.2f}, a={a:.2f})",
+       xaxis_title="s",
+       yaxis_title="Value",
+       hovermode="closest"
+   )
+   
+   return fig, intersections
 
 # Streamlit UI
 st.title("Cubic Root Analysis")
 
-tab1, tab2 = st.tabs(["z*(β) Curves", "Im{s} vs. z"])
+tab1, tab2, tab3 = st.tabs(["z*(β) Curves", "Im{s} vs. z", "Curve Intersections"])
 
 with tab1:
-    st.header("Find z Values where Cubic Roots Transition Between Real and Complex")
-    
-    col1, col2 = st.columns([1, 2])
-    
-    with col1:
-        z_a_1 = st.number_input("z_a", value=1.0, key="z_a_1")
-        y_1 = st.number_input("y", value=1.0, key="y_1")
-        z_min_1 = st.number_input("z_min", value=-10.0, key="z_min_1")
-        z_max_1 = st.number_input("z_max", value=10.0, key="z_max_1")
-        
-        if st.button("Compute z vs. β Curves"):
-            with col2:
-                fig = generate_z_vs_beta_plot(z_a_1, y_1, z_min_1, z_max_1)
-                if fig is not None:
-                    st.plotly_chart(fig, use_container_width=True)
-                    
-                    st.markdown("### Additional Expressions")
-                    st.markdown("""
-                    **Low y Expression (Red):**
-                    ```
-                    ((y - 2)*(-1 + sqrt(y*β*(a-1)))/a + y*β*((a-1)/a) - 1/a - 1) / 
-                    ((-1 + sqrt(y*β*(a-1)))/a)^2 + (-1 + sqrt(y*β*(a-1)))/a)
-                    ```
-                    
-                    **High y Expression (Green):**
-                    ```
-                    ((4y + 12)(4 - a) + 16y*β*(a - 1))/(3(4 - a))
-                    ```
-                    where a = z_a
-                    """)
+   st.header("Find z Values where Cubic Roots Transition Between Real and Complex")
+   
+   col1, col2 = st.columns([1, 2])
+   
+   with col1:
+       z_a_1 = st.number_input("z_a", value=1.0, key="z_a_1")
+       y_1 = st.number_input("y", value=1.0, key="y_1")
+       z_min_1 = st.number_input("z_min", value=-10.0, key="z_min_1")
+       z_max_1 = st.number_input("z_max", value=10.0, key="z_max_1")
+       
+       if st.button("Compute z vs. β Curves"):
+           with col2:
+               fig = generate_z_vs_beta_plot(z_a_1, y_1, z_min_1, z_max_1)
+               if fig is not None:
+                   st.plotly_chart(fig, use_container_width=True)
+                   
+                   st.markdown("### Additional Expressions")
+                   st.markdown("""
+                   **Low y Expression (Red):**
+                   ```
+                   ((y - 2)*(-1 + sqrt(y*β*(a-1)))/a + y*β*((a-1)/a) - 1/a - 1) / 
+                   ((-1 + sqrt(y*β*(a-1)))/a)^2 + (-1 + sqrt(y*β*(a-1)))/a)
+                   ```
+                   
+                   **High y Expression (Green):**
+                   ```
+                   ((4y + 12)(4 - a) + 16y*β*(a - 1))/(3(4 - a))
+                   ```
+                   where a = z_a
+                   """)
 
 with tab2:
     st.header("Plot Imaginary Parts of Roots vs. z")
@@ -303,4 +381,35 @@ with tab2:
             with col2:
                 fig = generate_ims_vs_z_plot(beta, y_2, z_a_2, z_min_2, z_max_2)
                 if fig is not None:
-                    st.plotly_chart(fig, use_container_width=True)
+                    st.plotly_chart(fig, use_container_width=True)
+
+with tab3:
+    st.header("Curve Intersection Analysis")
+    
+    col1, col2 = st.columns([1, 2])
+    
+    with col1:
+        # Add sliders for parameters
+        z = st.slider("z", min_value=-10.0, max_value=10.0, value=1.0, step=0.1)
+        y_3 = st.slider("y", min_value=0.1, max_value=10.0, value=1.0, step=0.1, key="y_3")
+        beta_3 = st.slider("β", min_value=0.0, max_value=1.0, value=0.5, step=0.01, key="beta_3")
+        a = st.slider("a", min_value=0.1, max_value=10.0, value=1.0, step=0.1)
+        
+        # Add range inputs for s
+        st.subheader("s Range")
+        s_min = st.number_input("s_min", value=-5.0)
+        s_max = st.number_input("s_max", value=5.0)
+        
+        if st.button("Compute Intersections"):
+            with col2:
+                s_range = (s_min, s_max)
+                fig, intersections = generate_curves_plot(z, y_3, beta_3, a, s_range)
+                st.plotly_chart(fig, use_container_width=True)
+                
+                if len(intersections) > 0:
+                    st.subheader("Intersection Points")
+                    for i, s_val in enumerate(intersections):
+                        y_val = curve1(s_val, z, y_3)
+                        st.write(f"Point {i+1}: s = {s_val:.4f}, y = {y_val:.4f}")
+                else:
+                    st.write("No intersections found in the given range.")