enbrown bugfixes - resolved conflicts, double-checked changes

Author: mitzimorris

src/stan-users-guide/algebraic-equations.Rmd

@@ -17,10 +17,9 @@ to solve nonlinear equations.
 
 As an illustrative example, we consider the following nonlinear system of two equations
 with two unknowns:
-
 \begin{align*}
-  z_1 &= y_1 - \theta_1 \\
-  z_2 &= y_1 y_2 + \theta_2
+z_1 &= y_1 - \theta_1 \\
+z_2 &= y_1 y_2 + \theta_2
 \end{align*}
 
 Our goal is to simultaneously solve all equations for
@@ -60,8 +59,8 @@ The body of the system function here could also be coded using a row
 vector constructor and transposition,
 
 ```
-  return [ y[1] - theta[1],
-           y[1] * y[2] - theta[2] ]';
+return [ y[1] - theta[1],
+         y[1] * y[2] - theta[2] ]';
 ```
 
 As systems get more complicated, naming the intermediate expressions
@@ -120,9 +119,9 @@ must only involve parameters. Note there are no restrictions on the initial gues
 The Jacobian of the solution with respect to the parameters is computed
 using the implicit function theorem, which imposes certain restrictions. In particular,
 the Jacobian of the algebraic function $f$ with respect to the unknowns $x$ must
-be invertible. This requires the Jacobian to be square, meaning \textit{$f(y)$ and
-$y$ have the same length} or, in other words \textit{the number of equations in
-the system is the same as the number of unknowns.}
+be invertible. This requires the Jacobian to be square, meaning $f(y)$ and
+$y$ have the same length or, in other words *the number of equations in
+the system is the same as the number of unknowns.*
 
 ### Pathological Solutions {-}
 Certain systems may be degenerate, meaning they have multiple solutions. The
@@ -142,15 +141,15 @@ allows three additional parameters, all of which must be supplied if any of them
 supplied.
 
 ```
-  y = algebra_solver(system, y_guess, theta, x_r, x_i,
-                     rel_tol, f_tol, max_steps);
+y = algebra_solver(system, y_guess, theta, x_r, x_i,
+                   rel_tol, f_tol, max_steps);
 ```
 
 The three control arguments are relative tolerance, function tolerance, and maximum
 number of steps. Both tolerances need to be satisfied. If one of them is not met, the
 metropolis proposal gets rejected with a warning message explaining which criterion
 was not satisfied. The default values for the control arguments are respectively
-`1e-10` ($10^{-10}$), `1e-6` ($10^{-6}$), and `1e3` ($10^3$).
+`rel_tol = 1e-10` ($10^{-10}$), `f_tol = 1e-6` ($10^{-6}$), and `max_steps = 1e3` ($10^3$).
 
 ### Tolerance {-}