Get rid of redundant function evaluations in LineSearchMinimizer

1. Replace LineSearch::Summary::optimal_step_size with
   LineSearch:Summary::optimal_point which is a FunctionSample.
2. Add the actual vector position and vector gradient of the
   point in the FunctionSample
3. Use the above two to get rid of an extraneous function evalation
   in LineSearchMinimizer.

Runtime performance is almost 2x improved as a result.

Thanks to @svenpilz for reporting this.

https://github.com/ceres-solver/ceres-solver/issues/296

Change-Id: Iebf2db7acecb2c95c9b1683b73cdc5faab78b02e

internal/ceres/function_sample.h

@@ -32,29 +32,62 @@
 #define CERES_INTERNAL_FUNCTION_SAMPLE_H_
 
 #include <string>
+#include "ceres/internal/eigen.h"
 
 namespace ceres {
 namespace internal {
 
-// Clients can use this struct to communicate the value of the
-// function and or its gradient at a given point x.
+// FunctionSample is used by the line search routines to store and
+// communicate the value and (optionally) the gradient of the function
+// being minimized.
+//
+// Since line search as the name implies happens along a certain
+// line/direction. FunctionSample contains the information in two
+// ways. Information in the ambient space and information along the
+// direction of search.
 struct FunctionSample {
-  FunctionSample()
-      : x(0.0),
-        value(0.0),
-        value_is_valid(false),
-        gradient(0.0),
-        gradient_is_valid(false) {
-  }
+  FunctionSample();
+  FunctionSample(double x, double value);
+  FunctionSample(double x, double value, double gradient);
+
   std::string ToDebugString() const;
 
+  // x is the location of the sample along the search direction.
   double x;
-  double value;      // value = f(x)
+
+  // Let p be a point and d be the search direction then
+  //
+  // vector_x = p + x * d;
+  Vector vector_x;
+  // True if vector_x has been assigned a valid value.
+  bool vector_x_is_valid;
+
+  // value = f(vector_x)
+  double value;
+  // True of the evaluation was successful and value is a finite
+  // number.
   bool value_is_valid;
-  double gradient;   // gradient = f'(x)
+
+  // vector_gradient = Df(vector_position);
+  //
+  // D is the derivative operator.
+  Vector vector_gradient;
+  // True if the vector gradient was evaluated and the evaluation was
+  // successful (the value is a finite number).
+  bool vector_gradient_is_valid;
+
+  // gradient = d.transpose() * vector_gradient
+  //
+  // where d is the search direction.
+  double gradient;
+  // True if the evaluation of the gradient was sucessful and the
+  // value is a finite number.
   bool gradient_is_valid;
 };
 
+
+
+
 }  // namespace internal
 }  // namespace ceres