Implemented 5th order gaussian quadrature.

Change-Id: Id7023b4f175c60b6930b5cb8aa829636604b2d25
diff --git a/frc971/control_loops/fixed_quadrature.h b/frc971/control_loops/fixed_quadrature.h
new file mode 100644
index 0000000..572c235
--- /dev/null
+++ b/frc971/control_loops/fixed_quadrature.h
@@ -0,0 +1,37 @@
+#ifndef FRC971_CONTROL_LOOPS_FIXED_QUADRATURE_H_
+#define FRC971_CONTROL_LOOPS_FIXED_QUADRATURE_H_
+
+#include <array>
+
+namespace frc971 {
+namespace control_loops {
+
+// Implements Gaussian Quadrature integration (5th order).  fn is the function to
+// integrate.  It must take 1 argument of type T.  The integration is between a
+// and b.
+template <typename F, typename T>
+T GaussianQuadrature5(const F &fn, T a, T b) {
+  // Pulled from Python.
+  // numpy.set_printoptions(precision=20)
+  // scipy.special.p_roots(5)
+  const ::std::array<double, 5> x{{
+      -9.06179845938663630633e-01, -5.38469310105682885670e-01,
+      3.24607628916367383789e-17, 5.38469310105683218737e-01,
+      9.06179845938663408589e-01}};
+
+  const ::std::array<double, 5> w{{
+      0.23692688505618844652, 0.4786286704993669705, 0.56888888888888811124,
+      0.47862867049936674846, 0.23692688505618875183}};
+
+  double answer = 0.0;
+  for (int i = 0; i < 5; ++i) {
+    const double y = (b - a) * (x[i] + 1) / 2.0 + a;
+    answer += (b - a) / 2.0 * w[i] * fn(y);
+  }
+  return answer;
+}
+
+}  // namespace control_loops
+}  // namespace frc971
+
+#endif  // FRC971_CONTROL_LOOPS_FIXED_QUADRATURE_H_