Update app.py

app.py

@@ -5,9 +5,9 @@ import plotly.graph_objects as go
 
 # 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 +25,8 @@ 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:
@@ -34,189 +34,219 @@ 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_additional_curve(betas, z_a, y):
-    """
-    Compute the additional curve with proper handling of divide by zero cases
-    """
-    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
-        
-        mask = (denominator == 0) | np.isnan(denominator) | np.isnan(numerator)
-        result = np.zeros_like(denominator)
-        result[~mask] = numerator[~mask] / denominator[~mask]
-        result[mask] = np.nan
-        
-        return result
+def compute_low_y_curve(betas, z_a, y):
+   """
+   Compute the additional curve with proper handling of divide by zero cases
+   """
+   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
+       
+       mask = (denominator == 0) | np.isnan(denominator) | np.isnan(numerator)
+       result = np.zeros_like(denominator)
+       result[~mask] = numerator[~mask] / denominator[~mask]
+       result[mask] = np.nan
+       
+       return result
+
+@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
+   with np.errstate(invalid='ignore', divide='ignore'):
+       numerator = (4*y + 12)*(4 - a) + 16*y*betas*(a - 1)
+       denominator = 3*(4 - a)
+       
+       # Handle division by zero
+       mask = (denominator == 0)
+       result = np.zeros_like(denominator, dtype=float)
+       result[~mask] = numerator[~mask] / denominator[~mask]
+       result[mask] = np.nan
+       
+       return result
 
 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, z_mins, z_maxs = sweep_beta_and_find_z_bounds(z_a, y, z_min, z_max, beta_steps=beta_steps)
-    new_curve = compute_additional_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=new_curve,
-            mode="markers+lines",
-            name="Additional Expression",
-            marker=dict(size=5, color='red'),
-            line=dict(color='red'),
-        )
-    )
-    
-    fig.update_layout(
-        title="Curves vs β: z*(β) boundaries (blue) and Additional Expression (red)",
-        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, 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  # 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
 
 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
 
 # Streamlit UI
 st.title("Cubic Root Analysis")
@@ -224,36 +254,51 @@ st.title("Cubic Root Analysis")
 tab1, tab2 = st.tabs(["z*(β) Curves", "Im{s} vs. z"])
 
 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.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")
-    
-    col1, col2 = st.columns([1, 2])
-    
-    with col1:
-        beta = st.number_input("β", value=0.5, min_value=0.0, max_value=1.0)
-        y_2 = st.number_input("y", value=1.0, key="y_2")
-        z_a_2 = st.number_input("z_a", value=1.0, key="z_a_2")
-        z_min_2 = st.number_input("z_min", value=-10.0, key="z_min_2")
-        z_max_2 = st.number_input("z_max", value=10.0, key="z_max_2")
-        
-        if st.button("Compute Im{s} vs. z"):
-            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.header("Plot Imaginary Parts of Roots vs. z")
+   
+   col1, col2 = st.columns([1, 2])
+   
+   with col1:
+       beta = st.number_input("β", value=0.5, min_value=0.0, max_value=1.0)
+       y_2 = st.number_input("y", value=1.0, key="y_2")
+       z_a_2 = st.number_input("z_a", value=1.0, key="z_a_2")
+       z_min_2 = st.number_input("z_min", value=-10.0, key="z_min_2")
+       z_max_2 = st.number_input("z_max", value=10.0, key="z_max_2")
+       
+       if st.button("Compute Im{s} vs. z"):
+           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)