misc

lectures/opt_invest.md

@@ -245,26 +245,26 @@ $$ v_\sigma(y, z) = r_\sigma(y, z) + \beta \sum_{z'} v_\sigma(\sigma(y, z), z')
 
 for $v$.
 
-Suppose we define the linear operator $R_\sigma$ by
+Suppose we define the linear operator $L_\sigma$ by
 
-$$ (R_\sigma v)(y, z) = v(y, z) - \beta \sum_{z'} v(\sigma(y, z), z') Q(z, z) $$
+$$ (L_\sigma v)(y, z) = v(y, z) - \beta \sum_{z'} v(\sigma(y, z), z') Q(z, z) $$
 
 With this notation, the problem is to solve for $v$ via
 
 $$
-    (R_{\sigma} v)(y, z) = r_\sigma(y, z)
+    (L_{\sigma} v)(y, z) = r_\sigma(y, z)
 $$
 
-In vector for this is $R_\sigma v = r_\sigma$, which tells us that the function
+In vector for this is $L_\sigma v = r_\sigma$, which tells us that the function
 we seek is
 
-$$ v_\sigma = R_\sigma^{-1} r_\sigma $$
+$$ v_\sigma = L_\sigma^{-1} r_\sigma $$
 
 JAX allows us to solve linear systems defined in terms of operators; the first
-step is to define the function $R_{\sigma}$.
+step is to define the function $L_{\sigma}$.
 
 ```{code-cell} ipython3
-def R_σ(v, σ, constants, sizes, arrays):
+def L_σ(v, σ, constants, sizes, arrays):
 
     β, a_0, a_1, γ, c = constants
     y_size, z_size = sizes
@@ -282,7 +282,7 @@ def R_σ(v, σ, constants, sizes, arrays):
     # Compute and return v[i, j] - β Σ_jp v[σ[i, j], jp] * Q[j, jp]
     return v - β * jnp.sum(V * Q, axis=2)
 
-R_σ = jax.jit(R_σ, static_argnums=(3,))
+L_σ = jax.jit(L_σ, static_argnums=(3,))
 ```
 
 Now we can define a function to compute $v_{\sigma}$ 
@@ -298,10 +298,10 @@ def get_value(σ, constants, sizes, arrays):
 
     r_σ = compute_r_σ(σ, constants, sizes, arrays)
 
-    # Reduce R_σ to a function in v
-    partial_R_σ = lambda v: R_σ(v, σ, constants, sizes, arrays)
+    # Reduce L_σ to a function in v
+    partial_L_σ = lambda v: L_σ(v, σ, constants, sizes, arrays)
 
-    return jax.scipy.sparse.linalg.bicgstab(partial_R_σ, r_σ)[0]
+    return jax.scipy.sparse.linalg.bicgstab(partial_L_σ, r_σ)[0]
 
 get_value = jax.jit(get_value, static_argnums=(2,))
 ```