changing the title and adding roadmaps

Author: HumphreyYang

lectures/newton_method.md

@@ -20,7 +20,7 @@ kernelspec:
 ```{index} single: python
 ```
 
-# Application of Newton's Method
+# Using Newton's Method to Solve Economic Models
 
 ```{contents} Contents
 :depth: 2
@@ -51,6 +51,10 @@ plt.rcParams["figure.figsize"] = (10, 5.7)
 
 ## Fixed Point Computation Using Newton's Method
 
+In this section, we will solve the fixed point of the law of motion for capital under the Solow model.
+
+We will inspect the fixed point visually, solve it by successive approximation, and then apply Newton's method to achieve faster convergence.
+
 (solow)=
 ### The Solow Model
 
@@ -355,6 +359,10 @@ This naturally leads us to use Newton's method to solve multi-dimensional proble
 
 ## Multivariate Newton’s Method
 
+In this section, we will first introduce a two-good problem, present a visualization of the problem, and solve the equilibrium of the two-good market using both a root finder in `SciPy` and Newton's method.
+
+We will then expand the idea to a larger market with 5000 goods and compare the performance of the two methods again to show a significant improvement in performance using Netwon's method.
+
 ### A Two Goods Market Equilibrium
 
 Before moving to higher dimensional settings, let's compute the market equilibrium of a two-good problem.