add boost on mac

macx64/include/boost/math/quadrature/trapezoidal.hpp

@@ -0,0 +1,125 @@
+/*
+ * Copyright Nick Thompson, 2017
+ * Use, modification and distribution are subject to the
+ * Boost Software License, Version 1.0. (See accompanying file
+ * LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
+ *
+ * Use the adaptive trapezoidal rule to estimate the integral of periodic functions over a period,
+ * or to integrate a function whose derivative vanishes at the endpoints.
+ *
+ * If your function does not satisfy these conditions, and instead is simply continuous and bounded
+ * over the whole interval, then this routine will still converge, albeit slowly. However, there
+ * are much more efficient methods in this case, including Romberg, Simpson, and double exponential quadrature.
+ */
+
+#ifndef BOOST_MATH_QUADRATURE_TRAPEZOIDAL_HPP
+#define BOOST_MATH_QUADRATURE_TRAPEZOIDAL_HPP
+
+#include <cmath>
+#include <limits>
+#include <stdexcept>
+#include <boost/math/constants/constants.hpp>
+#include <boost/math/special_functions/fpclassify.hpp>
+#include <boost/math/policies/error_handling.hpp>
+
+namespace boost{ namespace math{ namespace quadrature {
+
+template<class F, class Real, class Policy>
+auto trapezoidal(F f, Real a, Real b, Real tol, std::size_t max_refinements, Real* error_estimate, Real* L1, const Policy& pol)->decltype(std::declval<F>()(std::declval<Real>()))
+{
+    static const char* function = "boost::math::quadrature::trapezoidal<%1%>(F, %1%, %1%, %1%)";
+    using std::abs;
+    using boost::math::constants::half;
+    // In many math texts, K represents the field of real or complex numbers.
+    // Too bad we can't put blackboard bold into C++ source!
+    typedef decltype(f(a)) K;
+    if(a >= b)
+    {
+       return static_cast<K>(boost::math::policies::raise_domain_error(function, "a < b for integration over the region [a, b] is required, but got a = %1%.\n", a, pol));
+    }
+    if (!(boost::math::isfinite)(a))
+    {
+       return static_cast<K>(boost::math::policies::raise_domain_error(function, "Left endpoint of integration must be finite for adaptive trapezoidal integration but got a = %1%.\n", a, pol));
+    }
+    if (!(boost::math::isfinite)(b))
+    {
+       return static_cast<K>(boost::math::policies::raise_domain_error(function, "Right endpoint of integration must be finite for adaptive trapedzoidal integration but got b = %1%.\n", b, pol));
+    }
+
+
+    K ya = f(a);
+    K yb = f(b);
+    Real h = (b - a)*half<Real>();
+    K I0 = (ya + yb)*h;
+    Real IL0 = (abs(ya) + abs(yb))*h;
+
+    K yh = f(a + h);
+    K I1;
+    I1 = I0*half<Real>() + yh*h;
+    Real IL1 = IL0*half<Real>() + abs(yh)*h;
+
+    // The recursion is:
+    // I_k = 1/2 I_{k-1} + 1/2^k \sum_{j=1; j odd, j < 2^k} f(a + j(b-a)/2^k)
+    std::size_t k = 2;
+    // We want to go through at least 4 levels so we have sampled the function at least 10 times.
+    // Otherwise, we could terminate prematurely and miss essential features.
+    // This is of course possible anyway, but 10 samples seems to be a reasonable compromise.
+    Real error = abs(I0 - I1);
+    while (k < 4 || (k < max_refinements && error > tol*IL1) )
+    {
+        I0 = I1;
+        IL0 = IL1;
+
+        I1 = I0*half<Real>();
+        IL1 = IL0*half<Real>();
+        std::size_t p = static_cast<std::size_t>(1u) << k;
+        h *= half<Real>();
+        K sum = 0;
+        Real absum = 0;
+
+        for(std::size_t j = 1; j < p; j += 2)
+        {
+            K y = f(a + j*h);
+            sum += y;
+            absum += abs(y);
+        }
+
+        I1 += sum*h;
+        IL1 += absum*h;
+        ++k;
+        error = abs(I0 - I1);
+    }
+
+    if (error_estimate)
+    {
+        *error_estimate = error;
+    }
+
+    if (L1)
+    {
+        *L1 = IL1;
+    }
+
+    return static_cast<K>(I1);
+}
+#if BOOST_WORKAROUND(BOOST_MSVC, < 1800)
+// Template argument dedcution failure otherwise:
+template<class F, class Real>
+auto trapezoidal(F f, Real a, Real b, Real tol = 0, std::size_t max_refinements = 12, Real* error_estimate = 0, Real* L1 = 0)->decltype(std::declval<F>()(std::declval<Real>()))
+#elif !defined(BOOST_NO_CXX11_NULLPTR)
+template<class F, class Real>
+auto trapezoidal(F f, Real a, Real b, Real tol = boost::math::tools::root_epsilon<Real>(), std::size_t max_refinements = 12, Real* error_estimate = nullptr, Real* L1 = nullptr)->decltype(std::declval<F>()(std::declval<Real>()))
+#else
+template<class F, class Real>
+auto trapezoidal(F f, Real a, Real b, Real tol = boost::math::tools::root_epsilon<Real>(), std::size_t max_refinements = 12, Real* error_estimate = 0, Real* L1 = 0)->decltype(std::declval<F>()(std::declval<Real>()))
+#endif
+{
+#if BOOST_WORKAROUND(BOOST_MSVC, <= 1600)
+   if (tol == 0)
+      tol = boost::math::tools::root_epsilon<Real>();
+#endif
+   return trapezoidal(f, a, b, tol, max_refinements, error_estimate, L1, boost::math::policies::policy<>());
+}
+
+}}}
+#endif