Update README.md

README.md

@@ -15,10 +15,9 @@
   </a>
 </p>
 
-This module overloads `factorize` and `\` to work with hyperdual-valued arrays.
-
-It uses the HyperDualNumbers.jl package.
+This package provides an overloaded `factorize` and `\` that work with hyperdual-valued arrays.
 
+It uses of the hyper dual type defined by the [HyperDualNumbers.jl](https://github.com/JuliaDiff/HyperDualNumbers.jl) package.
 The idea is that for a hyperdual-valued matrix
 
 <a href="https://www.codecogs.com/eqnedit.php?latex=\fn_phv&space;M&space;=&space;A&space;&plus;&space;\varepsilon_1&space;B&space;&plus;&space;\varepsilon_2&space;C&space;&plus;&space;\varepsilon_1&space;\varepsilon_2&space;D" target="_blank"><img src="https://latex.codecogs.com/svg.latex?\fn_phv&space;M&space;=&space;A&space;&plus;&space;\varepsilon_1&space;B&space;&plus;&space;\varepsilon_2&space;C&space;&plus;&space;\varepsilon_1&space;\varepsilon_2&space;D" title="M = A + \varepsilon_1 B + \varepsilon_2 C + \varepsilon_1 \varepsilon_2 D" /></a>,
@@ -30,12 +29,15 @@ its inverse is given by
 Therefore, only the inverse of <a href="https://www.codecogs.com/eqnedit.php?latex=\fn_phv&space;A" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\fn_phv&space;A" title="A" /></a> is required to evaluate the inverse of <a href="https://www.codecogs.com/eqnedit.php?latex=\fn_phv&space;M" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\fn_phv&space;M" title="M" /></a>.
 This package makes available a `HyperDualFactors` type which containts the factors of <a href="https://www.codecogs.com/eqnedit.php?latex=\fn_phv&space;A" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\fn_phv&space;A" title="A" /></a> and the non-real parts of <a href="https://www.codecogs.com/eqnedit.php?latex=\fn_phv&space;M" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\fn_phv&space;M" title="M" /></a>, and overloads `factorize` to create an instance of `HyperDualFactors`, which can then be called with `\` to efficiently solve hyperdual-valued linear systems. 
 
+This package should be useful for evaluation of second derivatives of functions that use `\` (e.g., with iterative solvers).
+
 ## Usage
 
 - Create your hyperdual-valued matrix `M`:
     ```julia
     julia> M = A + ε₁ * B + ε₂ * C + ε₁ε₂ * D
     ```
+    (The `ε₁`, `ε₂`, and `ε₁ε₂` constants are defined for convenience here.)
 
 - Factorize `M`:
     ```julia