Edits to derivatives.rst from William Rucklidge
Change-Id: I21991c4d302d6aa569aedf9157d9628c15cc9ed5
diff --git a/docs/source/derivatives.rst b/docs/source/derivatives.rst
index 808c942..3919f86 100644
--- a/docs/source/derivatives.rst
+++ b/docs/source/derivatives.rst
@@ -13,7 +13,7 @@
Introduction
============
-Ceres Solver like all gradient based optimization algorithms, depends
+Ceres Solver, like all gradient based optimization algorithms, depends
on being able to evaluate the objective function and its derivatives
at arbitrary points in its domain. Indeed, defining the objective
function and its `Jacobian
@@ -29,7 +29,7 @@
derivatives herself, by hand or using a tool like
`Maple <https://www.maplesoft.com/products/maple/>`_ or
`Mathematica <https://www.wolfram.com/mathematica/>`_, and
- implements them in a ::class:`CostFunction`.
+ implements them in a :class:`CostFunction`.
2. :ref:`section-numerical_derivatives`: Ceres numerically computes
the derivative using finite differences.
3. :ref:`section-automatic_derivatives`: Ceres automatically computes
@@ -47,7 +47,7 @@
High Level Advice
-----------------
-And for the impatient amongst you, here is some high level advice:
+For the impatient amongst you, here is some high level advice:
1. Use :ref:`section-automatic_derivatives`.
2. In some cases it maybe worth using
@@ -149,7 +149,7 @@
\end{align}
With these derivatives in hand, we can now implement the
-:class:`CostFunction`: as
+:class:`CostFunction` as:
.. code-block:: c++
@@ -186,7 +186,7 @@
const double y_;
};
-This is tedious code, which is hard to read with a lot of
+This is tedious code, hard to read and with a lot of
redundancy. So in practice we will cache some sub-expressions to
improve its efficiency, which would give us something like:
@@ -369,8 +369,8 @@
:class:`NumericDiffCostFunction` implements a generic algorithm to
numerically differentiate a given functor. While the actual
implementation of :class:`NumericDiffCostFunction` is complicated, the
-net result is a ``CostFunction`` that roughly looks something like the
-following:
+net result is a :class:`CostFunction` that roughly looks something
+like the following:
.. code-block:: c++
@@ -508,7 +508,7 @@
One possible approach is to find a method whose error goes down faster
than :math:`O(h^2)`. This can be done by applying `Richardson
Extrapolation
-<https://en.wikipedia.org/wiki/Richardson_extrapolation>_` to the
+<https://en.wikipedia.org/wiki/Richardson_extrapolation>`_ to the
problem of differentiation. This is also known as *Ridders' Method*
[Ridders]_.
@@ -552,7 +552,7 @@
Df(x) = \frac{4 A(1, 2) - A(1,1)}{4 - 1} + O(h^4)
which is an approximation of :math:`Df(x)` with truncation error that
-goes down as :math:`O(h^4)`. But we do not have to stop here, we can
+goes down as :math:`O(h^4)`. But we do not have to stop here. We can
iterate this process to obtain even more accurate estimates as
follows:
@@ -561,7 +561,7 @@
A(n, m) = \begin{cases}
\frac{\displaystyle f(x + h/2^{m-1}) - f(x -
h/2^{m-1})}{\displaystyle 2h/2^{m-1}} & n = 1 \\
- \frac{\displaystyle 4 A(n - 1, m + 1) - A(n - 1, m)}{\displaystyle 4^{n-1} - 1} & n > 1
+ \frac{\displaystyle 4^{n-1} A(n - 1, m + 1) - A(n - 1, m)}{\displaystyle 4^{n-1} - 1} & n > 1
\end{cases}
It is straightforward to show that the approximation error in
@@ -628,7 +628,7 @@
:figwidth: 100%
:align: center
-Using 10 function evaluations that are needed to compute
+Using the 10 function evaluations that are needed to compute
:math:`A(5,1)` we are able to approximate :math:`Df(1.0)` about a 1000
times better than the best central differences estimate. To put these
numbers in perspective, machine epsilon for double precision
@@ -830,7 +830,7 @@
f(x_1,..., x_n) = f(a_1, ..., a_n) + \sum_i D_i f(a_1, ..., a_n) \mathbf{v}_i
So if each :math:`\mathbf{v}_i = e_i` were the :math:`i^{\text{th}}`
-standard basis vector. Then, the above expression would simplify to
+standard basis vector, then, the above expression would simplify to
.. math::
f(x_1,..., x_n) = f(a_1, ..., a_n) + \sum_i D_i f(a_1, ..., a_n) \epsilon_i
@@ -971,7 +971,7 @@
.. rubric:: Footnotes
.. [#f1] The notion of best fit depends on the choice of the objective
- function used to measure the quality of fit. Which in turn
+ function used to measure the quality of fit, which in turn
depends on the underlying noise process which generated the
observations. Minimizing the sum of squared differences is
the right thing to do when the noise is `Gaussian
