From a250691ed190cac0bf261eee92a10fed43fec1f4 Mon Sep 17 00:00:00 2001
From: Nick Thompson <nathompson7@protonmail.com>
Date: Thu, 27 Dec 2018 18:59:44 -0700
Subject: [PATCH 01/10] Begin rearrangement.

---
 .../numerical_differentiation.qbk             |  10 +-
 .../differentiation/finite_difference.hpp     | 266 +++++++++++++++++
 .../math/tools/numerical_differentiation.hpp  | 267 +-----------------
 test/test_numerical_differentiation.cpp       |   6 +-
 4 files changed, 284 insertions(+), 265 deletions(-)
 create mode 100644 include/boost/math/differentiation/finite_difference.hpp

diff --git a/doc/differentiation/numerical_differentiation.qbk b/doc/differentiation/numerical_differentiation.qbk
index 857627bd4..6f7956b8d 100644
--- a/doc/differentiation/numerical_differentiation.qbk
+++ b/doc/differentiation/numerical_differentiation.qbk
@@ -10,10 +10,10 @@ LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
 [heading Synopsis]
 
 ``
-#include <boost/math/tools/numerical_differentiation.hpp>
+#include <boost/math/differentiaton/finite_difference.hpp>
 
 
-namespace boost::math::tools {
+namespace boost { namespace math { namespace differentiation {
 
     template <class F, class Real>
     Real complex_step_derivative(const F f, Real x);
@@ -21,19 +21,19 @@ namespace boost::math::tools {
     template <class F, class Real, size_t order = 6>
     Real finite_difference_derivative(const F f, Real x, Real* error = nullptr);
 
-} // namespaces
+}}} // namespaces
 ``
 
 [heading Description]
 
-The function `finite_difference_derivative` calculates a finite-difference approximation to the derivative of of a function /f/ at point /x/.
+The function `finite_difference_derivative` calculates a finite-difference approximation to the derivative of a function /f/ at point /x/.
 A basic usage is
 
     auto f = [](double x) { return std::exp(x); };
     double x = 1.7;
     double dfdx = finite_difference_derivative(f, x);
 
-Finite differencing is complicated, as finite-difference approximations to the derivative are /infinitely/ ill-conditioned.
+Finite differencing is complicated, as differentiation is /infinitely/ ill-conditioned.
 In addition, for any function implemented in finite-precision arithmetic, the "true" derivative is /zero/ almost everywhere, and undefined at representables.
 However, some tricks allow for reasonable results to be obtained in many cases.
 
diff --git a/include/boost/math/differentiation/finite_difference.hpp b/include/boost/math/differentiation/finite_difference.hpp
new file mode 100644
index 000000000..c7615b6b2
--- /dev/null
+++ b/include/boost/math/differentiation/finite_difference.hpp
@@ -0,0 +1,266 @@
+//  (C) Copyright Nick Thompson 2018.
+//  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)
+
+#ifndef BOOST_MATH_DIFFERENTIATION_FINITE_DIFFERENCE_HPP
+#define BOOST_MATH_DIFFERENTIATION_FINITE_DIFFERENCE_HPP
+
+/*
+ * Performs numerical differentiation by finite-differences.
+ *
+ * All numerical differentiation using finite-differences are ill-conditioned, and these routines are no exception.
+ * A simple argument demonstrates that the error is unbounded as h->0.
+ * Take the one sides finite difference formula f'(x) = (f(x+h)-f(x))/h.
+ * The evaluation of f induces an error as well as the error from the finite-difference approximation, giving
+ * |f'(x) - (f(x+h) -f(x))/h| < h|f''(x)|/2 + (|f(x)|+|f(x+h)|)eps/h =: g(h), where eps is the unit roundoff for the type.
+ * It is reasonable to choose h in a way that minimizes the maximum error bound g(h).
+ * The value of h that minimizes g is h = sqrt(2eps(|f(x)| + |f(x+h)|)/|f''(x)|), and for this value of h the error bound is
+ * sqrt(2eps(|f(x+h) +f(x)||f''(x)|)).
+ * In fact it is not necessary to compute the ratio (|f(x+h)| + |f(x)|)/|f''(x)|; the error bound of ~\sqrt{\epsilon} still holds if we set it to one.
+ *
+ *
+ * For more details on this method of analysis, see
+ *
+ * http://www.uio.no/studier/emner/matnat/math/MAT-INF1100/h08/kompendiet/diffint.pdf
+ * http://web.archive.org/web/20150420195907/http://www.uio.no/studier/emner/matnat/math/MAT-INF1100/h08/kompendiet/diffint.pdf
+ *
+ *
+ * It can be shown on general grounds that when choosing the optimal h, the maximum error in f'(x) is ~(|f(x)|eps)^k/k+1|f^(k-1)(x)|^1/k+1.
+ * From this we can see that full precision can be recovered in the limit k->infinity.
+ *
+ * References:
+ *
+ * 1) Fornberg, Bengt. "Generation of finite difference formulas on arbitrarily spaced grids." Mathematics of computation 51.184 (1988): 699-706.
+ *
+ *
+ * The second algorithm, the complex step derivative, is not ill-conditioned.
+ * However, it requires that your function can be evaluated at complex arguments.
+ * The idea is that f(x+ih) = f(x) +ihf'(x) - h^2f''(x) + ... so f'(x) \approx Im[f(x+ih)]/h.
+ * No subtractive cancellation occurs. The error is ~ eps|f'(x)| + eps^2|f'''(x)|/6; hard to beat that.
+ *
+ * References:
+ *
+ * 1) Squire, William, and George Trapp. "Using complex variables to estimate derivatives of real functions." Siam Review 40.1 (1998): 110-112.
+ */
+
+#include <complex>
+#include <boost/math/special_functions/next.hpp>
+
+namespace boost{ namespace math{ namespace differentiation {
+
+namespace detail {
+    template<class Real>
+    Real make_xph_representable(Real x, Real h)
+    {
+        using std::numeric_limits;
+        // Redefine h so that x + h is representable. Not using this trick leads to large error.
+        // The compiler flag -ffast-math evaporates these operations . . .
+        Real temp = x + h;
+        h = temp - x;
+        // Handle the case x + h == x:
+        if (h == 0)
+        {
+            h = boost::math::nextafter(x, (numeric_limits<Real>::max)()) - x;
+        }
+        return h;
+    }
+}
+
+template<class F, class Real>
+Real complex_step_derivative(const F f, Real x)
+{
+    // Is it really this easy? Yes.
+    // Note that some authors recommend taking the stepsize h to be smaller than epsilon(), some recommending use of the min().
+    // This idea was tested over a few billion test cases and found the make the error *much* worse.
+    // Even 2eps and eps/2 made the error worse, which was surprising.
+    using std::complex;
+    using std::numeric_limits;
+    constexpr const Real step = (numeric_limits<Real>::epsilon)();
+    constexpr const Real inv_step = 1/(numeric_limits<Real>::epsilon)();
+    return f(complex<Real>(x, step)).imag()*inv_step;
+}
+
+namespace detail {
+
+   template <unsigned>
+   struct fd_tag {};
+
+   template<class F, class Real>
+   Real finite_difference_derivative(const F f, Real x, Real* error, const fd_tag<1>&)
+   {
+      using std::sqrt;
+      using std::pow;
+      using std::abs;
+      using std::numeric_limits;
+
+      const Real eps = (numeric_limits<Real>::epsilon)();
+      // Error bound ~eps^1/2
+      // Note that this estimate of h differs from the best estimate by a factor of sqrt((|f(x)| + |f(x+h)|)/|f''(x)|).
+      // Since this factor is invariant under the scaling f -> kf, then we are somewhat justified in approximating it by 1.
+      // This approximation will get better as we move to higher orders of accuracy.
+      Real h = 2 * sqrt(eps);
+      h = detail::make_xph_representable(x, h);
+
+      Real yh = f(x + h);
+      Real y0 = f(x);
+      Real diff = yh - y0;
+      if (error)
+      {
+         Real ym = f(x - h);
+         Real ypph = abs(yh - 2 * y0 + ym) / h;
+         // h*|f''(x)|*0.5 + (|f(x+h)+|f(x)|)*eps/h
+         *error = ypph / 2 + (abs(yh) + abs(y0))*eps / h;
+      }
+      return diff / h;
+   }
+
+   template<class F, class Real>
+   Real finite_difference_derivative(const F f, Real x, Real* error, const fd_tag<2>&)
+   {
+      using std::sqrt;
+      using std::pow;
+      using std::abs;
+      using std::numeric_limits;
+
+      const Real eps = (numeric_limits<Real>::epsilon)();
+      // Error bound ~eps^2/3
+      // See the previous discussion to understand determination of h and the error bound.
+      // Series[(f[x+h] - f[x-h])/(2*h), {h, 0, 4}]
+      Real h = pow(3 * eps, static_cast<Real>(1) / static_cast<Real>(3));
+      h = detail::make_xph_representable(x, h);
+
+      Real yh = f(x + h);
+      Real ymh = f(x - h);
+      Real diff = yh - ymh;
+      if (error)
+      {
+         Real yth = f(x + 2 * h);
+         Real ymth = f(x - 2 * h);
+         *error = eps * (abs(yh) + abs(ymh)) / (2 * h) + abs((yth - ymth) / 2 - diff) / (6 * h);
+      }
+
+      return diff / (2 * h);
+   }
+
+   template<class F, class Real>
+   Real finite_difference_derivative(const F f, Real x, Real* error, const fd_tag<4>&)
+   {
+      using std::sqrt;
+      using std::pow;
+      using std::abs;
+      using std::numeric_limits;
+
+      const Real eps = (numeric_limits<Real>::epsilon)();
+      // Error bound ~eps^4/5
+      Real h = pow(11.25*eps, (Real)1 / (Real)5);
+      h = detail::make_xph_representable(x, h);
+      Real ymth = f(x - 2 * h);
+      Real yth = f(x + 2 * h);
+      Real yh = f(x + h);
+      Real ymh = f(x - h);
+      Real y2 = ymth - yth;
+      Real y1 = yh - ymh;
+      if (error)
+      {
+         // Mathematica code to extract the remainder:
+         // Series[(f[x-2*h]+ 8*f[x+h] - 8*f[x-h] - f[x+2*h])/(12*h), {h, 0, 7}]
+         Real y_three_h = f(x + 3 * h);
+         Real y_m_three_h = f(x - 3 * h);
+         // Error from fifth derivative:
+         *error = abs((y_three_h - y_m_three_h) / 2 + 2 * (ymth - yth) + 5 * (yh - ymh) / 2) / (30 * h);
+         // Error from function evaluation:
+         *error += eps * (abs(yth) + abs(ymth) + 8 * (abs(ymh) + abs(yh))) / (12 * h);
+      }
+      return (y2 + 8 * y1) / (12 * h);
+   }
+
+   template<class F, class Real>
+   Real finite_difference_derivative(const F f, Real x, Real* error, const fd_tag<6>&)
+   {
+      using std::sqrt;
+      using std::pow;
+      using std::abs;
+      using std::numeric_limits;
+
+      const Real eps = (numeric_limits<Real>::epsilon)();
+      // Error bound ~eps^6/7
+      // Error: h^6f^(7)(x)/140 + 5|f(x)|eps/h
+      Real h = pow(eps / 168, (Real)1 / (Real)7);
+      h = detail::make_xph_representable(x, h);
+
+      Real yh = f(x + h);
+      Real ymh = f(x - h);
+      Real y1 = yh - ymh;
+      Real y2 = f(x - 2 * h) - f(x + 2 * h);
+      Real y3 = f(x + 3 * h) - f(x - 3 * h);
+
+      if (error)
+      {
+         // Mathematica code to generate fd scheme for 7th derivative:
+         // Sum[(-1)^i*Binomial[7, i]*(f[x+(3-i)*h] + f[x+(4-i)*h])/2, {i, 0, 7}]
+         // Mathematica to demonstrate that this is a finite difference formula for 7th derivative:
+         // Series[(f[x+4*h]-f[x-4*h] + 6*(f[x-3*h] - f[x+3*h]) + 14*(f[x-h] - f[x+h] + f[x+2*h] - f[x-2*h]))/2, {h, 0, 15}]
+         Real y7 = (f(x + 4 * h) - f(x - 4 * h) - 6 * y3 - 14 * y1 - 14 * y2) / 2;
+         *error = abs(y7) / (140 * h) + 5 * (abs(yh) + abs(ymh))*eps / h;
+      }
+      return (y3 + 9 * y2 + 45 * y1) / (60 * h);
+   }
+
+   template<class F, class Real>
+   Real finite_difference_derivative(const F f, Real x, Real* error, const fd_tag<8>&)
+   {
+      using std::sqrt;
+      using std::pow;
+      using std::abs;
+      using std::numeric_limits;
+
+      const Real eps = (numeric_limits<Real>::epsilon)();
+      // Error bound ~eps^8/9.
+      // In double precision, we only expect to lose two digits of precision while using this formula, at the cost of 8 function evaluations.
+      // Error: h^8|f^(9)(x)|/630 + 7|f(x)|eps/h assuming 7 unstabilized additions.
+      // Mathematica code to get the error:
+      // Series[(f[x+h]-f[x-h])*(4/5) + (1/5)*(f[x-2*h] - f[x+2*h]) + (4/105)*(f[x+3*h] - f[x-3*h]) + (1/280)*(f[x-4*h] - f[x+4*h]), {h, 0, 9}]
+      // If we used Kahan summation, we could get the max error down to h^8|f^(9)(x)|/630 + |f(x)|eps/h.
+      Real h = pow(551.25*eps, (Real)1 / (Real)9);
+      h = detail::make_xph_representable(x, h);
+
+      Real yh = f(x + h);
+      Real ymh = f(x - h);
+      Real y1 = yh - ymh;
+      Real y2 = f(x - 2 * h) - f(x + 2 * h);
+      Real y3 = f(x + 3 * h) - f(x - 3 * h);
+      Real y4 = f(x - 4 * h) - f(x + 4 * h);
+
+      Real tmp1 = 3 * y4 / 8 + 4 * y3;
+      Real tmp2 = 21 * y2 + 84 * y1;
+
+      if (error)
+      {
+         // Mathematica code to generate fd scheme for 7th derivative:
+         // Sum[(-1)^i*Binomial[9, i]*(f[x+(4-i)*h] + f[x+(5-i)*h])/2, {i, 0, 9}]
+         // Mathematica to demonstrate that this is a finite difference formula for 7th derivative:
+         // Series[(f[x+5*h]-f[x- 5*h])/2 + 4*(f[x-4*h] - f[x+4*h]) + 27*(f[x+3*h] - f[x-3*h])/2 + 24*(f[x-2*h]  - f[x+2*h]) + 21*(f[x+h] - f[x-h]), {h, 0, 15}]
+         Real f9 = (f(x + 5 * h) - f(x - 5 * h)) / 2 + 4 * y4 + 27 * y3 / 2 + 24 * y2 + 21 * y1;
+         *error = abs(f9) / (630 * h) + 7 * (abs(yh) + abs(ymh))*eps / h;
+      }
+      return (tmp1 + tmp2) / (105 * h);
+   }
+
+   template<class F, class Real, class tag>
+   Real finite_difference_derivative(const F f, Real x, Real* error, const tag&)
+   {
+      // Always fails, but condition is template-arg-dependent so only evaluated if we get instantiated.
+      BOOST_STATIC_ASSERT_MSG(sizeof(Real) == 0, "Finite difference not implemented for this order: try 1, 2, 4, 6 or 8");
+   }
+
+}
+
+template<class F, class Real, size_t order=6>
+inline Real finite_difference_derivative(const F f, Real x, Real* error = nullptr)
+{
+   return detail::finite_difference_derivative(f, x, error, detail::fd_tag<order>());
+}
+
+}}}  // namespaces
+#endif
diff --git a/include/boost/math/tools/numerical_differentiation.hpp b/include/boost/math/tools/numerical_differentiation.hpp
index 34fef0db8..eb4bdcb44 100644
--- a/include/boost/math/tools/numerical_differentiation.hpp
+++ b/include/boost/math/tools/numerical_differentiation.hpp
@@ -2,266 +2,19 @@
 //  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)
-
 #ifndef BOOST_MATH_TOOLS_NUMERICAL_DIFFERENTIATION_HPP
 #define BOOST_MATH_TOOLS_NUMERICAL_DIFFERENTIATION_HPP
+#include <boost/math/differentiation/finite_difference.hpp>
+#include <boost/config/header_deprecated.hpp>
 
-/*
- * Performs numerical differentiation by finite-differences.
- *
- * All numerical differentiation using finite-differences are ill-conditioned, and these routines are no exception.
- * A simple argument demonstrates that the error is unbounded as h->0.
- * Take the one sides finite difference formula f'(x) = (f(x+h)-f(x))/h.
- * The evaluation of f induces an error as well as the error from the finite-difference approximation, giving
- * |f'(x) - (f(x+h) -f(x))/h| < h|f''(x)|/2 + (|f(x)|+|f(x+h)|)eps/h =: g(h), where eps is the unit roundoff for the type.
- * It is reasonable to choose h in a way that minimizes the maximum error bound g(h).
- * The value of h that minimizes g is h = sqrt(2eps(|f(x)| + |f(x+h)|)/|f''(x)|), and for this value of h the error bound is
- * sqrt(2eps(|f(x+h) +f(x)||f''(x)|)).
- * In fact it is not necessary to compute the ratio (|f(x+h)| + |f(x)|)/|f''(x)|; the error bound of ~\sqrt{\epsilon} still holds if we set it to one.
- *
- *
- * For more details on this method of analysis, see
- *
- * http://www.uio.no/studier/emner/matnat/math/MAT-INF1100/h08/kompendiet/diffint.pdf
- * http://web.archive.org/web/20150420195907/http://www.uio.no/studier/emner/matnat/math/MAT-INF1100/h08/kompendiet/diffint.pdf
- *
- *
- * It can be shown on general grounds that when choosing the optimal h, the maximum error in f'(x) is ~(|f(x)|eps)^k/k+1|f^(k-1)(x)|^1/k+1.
- * From this we can see that full precision can be recovered in the limit k->infinity.
- *
- * References:
- *
- * 1) Fornberg, Bengt. "Generation of finite difference formulas on arbitrarily spaced grids." Mathematics of computation 51.184 (1988): 699-706.
- *
- *
- * The second algorithm, the complex step derivative, is not ill-conditioned.
- * However, it requires that your function can be evaluated at complex arguments.
- * The idea is that f(x+ih) = f(x) +ihf'(x) - h^2f''(x) + ... so f'(x) \approx Im[f(x+ih)]/h.
- * No subtractive cancellation occurs. The error is ~ eps|f'(x)| + eps^2|f'''(x)|/6; hard to beat that.
- *
- * References:
- *
- * 1) Squire, William, and George Trapp. "Using complex variables to estimate derivatives of real functions." Siam Review 40.1 (1998): 110-112.
- */
+BOOST_HEADER_DEPRECATED("<boost/math/differentiation/finite_difference.hpp>");
 
-#include <complex>
-#include <boost/math/special_functions/next.hpp>
-
-// Make sure everyone is informed that C++17 is required:
-namespace boost{ namespace math{ namespace tools {
-
-namespace detail {
-    template<class Real>
-    Real make_xph_representable(Real x, Real h)
-    {
-        using std::numeric_limits;
-        // Redefine h so that x + h is representable. Not using this trick leads to large error.
-        // The compiler flag -ffast-math evaporates these operations . . .
-        Real temp = x + h;
-        h = temp - x;
-        // Handle the case x + h == x:
-        if (h == 0)
-        {
-            h = boost::math::nextafter(x, (numeric_limits<Real>::max)()) - x;
-        }
-        return h;
-    }
+namespace boost {
+   namespace math {
+       namespace differentiation {
+           using finite_difference_derivative;
+           using complex_step_derivative;
+       }
+   }
 }
-
-template<class F, class Real>
-Real complex_step_derivative(const F f, Real x)
-{
-    // Is it really this easy? Yes.
-    // Note that some authors recommend taking the stepsize h to be smaller than epsilon(), some recommending use of the min().
-    // This idea was tested over a few billion test cases and found the make the error *much* worse.
-    // Even 2eps and eps/2 made the error worse, which was surprising.
-    using std::complex;
-    using std::numeric_limits;
-    constexpr const Real step = (numeric_limits<Real>::epsilon)();
-    constexpr const Real inv_step = 1/(numeric_limits<Real>::epsilon)();
-    return f(complex<Real>(x, step)).imag()*inv_step;
-}
-
-namespace detail {
-
-   template <unsigned>
-   struct fd_tag {};
-
-   template<class F, class Real>
-   Real finite_difference_derivative(const F f, Real x, Real* error, const fd_tag<1>&)
-   {
-      using std::sqrt;
-      using std::pow;
-      using std::abs;
-      using std::numeric_limits;
-
-      const Real eps = (numeric_limits<Real>::epsilon)();
-      // Error bound ~eps^1/2
-      // Note that this estimate of h differs from the best estimate by a factor of sqrt((|f(x)| + |f(x+h)|)/|f''(x)|).
-      // Since this factor is invariant under the scaling f -> kf, then we are somewhat justified in approximating it by 1.
-      // This approximation will get better as we move to higher orders of accuracy.
-      Real h = 2 * sqrt(eps);
-      h = detail::make_xph_representable(x, h);
-
-      Real yh = f(x + h);
-      Real y0 = f(x);
-      Real diff = yh - y0;
-      if (error)
-      {
-         Real ym = f(x - h);
-         Real ypph = abs(yh - 2 * y0 + ym) / h;
-         // h*|f''(x)|*0.5 + (|f(x+h)+|f(x)|)*eps/h
-         *error = ypph / 2 + (abs(yh) + abs(y0))*eps / h;
-      }
-      return diff / h;
-   }
-
-   template<class F, class Real>
-   Real finite_difference_derivative(const F f, Real x, Real* error, const fd_tag<2>&)
-   {
-      using std::sqrt;
-      using std::pow;
-      using std::abs;
-      using std::numeric_limits;
-
-      const Real eps = (numeric_limits<Real>::epsilon)();
-      // Error bound ~eps^2/3
-      // See the previous discussion to understand determination of h and the error bound.
-      // Series[(f[x+h] - f[x-h])/(2*h), {h, 0, 4}]
-      Real h = pow(3 * eps, static_cast<Real>(1) / static_cast<Real>(3));
-      h = detail::make_xph_representable(x, h);
-
-      Real yh = f(x + h);
-      Real ymh = f(x - h);
-      Real diff = yh - ymh;
-      if (error)
-      {
-         Real yth = f(x + 2 * h);
-         Real ymth = f(x - 2 * h);
-         *error = eps * (abs(yh) + abs(ymh)) / (2 * h) + abs((yth - ymth) / 2 - diff) / (6 * h);
-      }
-
-      return diff / (2 * h);
-   }
-
-   template<class F, class Real>
-   Real finite_difference_derivative(const F f, Real x, Real* error, const fd_tag<4>&)
-   {
-      using std::sqrt;
-      using std::pow;
-      using std::abs;
-      using std::numeric_limits;
-
-      const Real eps = (numeric_limits<Real>::epsilon)();
-      // Error bound ~eps^4/5
-      Real h = pow(11.25*eps, (Real)1 / (Real)5);
-      h = detail::make_xph_representable(x, h);
-      Real ymth = f(x - 2 * h);
-      Real yth = f(x + 2 * h);
-      Real yh = f(x + h);
-      Real ymh = f(x - h);
-      Real y2 = ymth - yth;
-      Real y1 = yh - ymh;
-      if (error)
-      {
-         // Mathematica code to extract the remainder:
-         // Series[(f[x-2*h]+ 8*f[x+h] - 8*f[x-h] - f[x+2*h])/(12*h), {h, 0, 7}]
-         Real y_three_h = f(x + 3 * h);
-         Real y_m_three_h = f(x - 3 * h);
-         // Error from fifth derivative:
-         *error = abs((y_three_h - y_m_three_h) / 2 + 2 * (ymth - yth) + 5 * (yh - ymh) / 2) / (30 * h);
-         // Error from function evaluation:
-         *error += eps * (abs(yth) + abs(ymth) + 8 * (abs(ymh) + abs(yh))) / (12 * h);
-      }
-      return (y2 + 8 * y1) / (12 * h);
-   }
-
-   template<class F, class Real>
-   Real finite_difference_derivative(const F f, Real x, Real* error, const fd_tag<6>&)
-   {
-      using std::sqrt;
-      using std::pow;
-      using std::abs;
-      using std::numeric_limits;
-
-      const Real eps = (numeric_limits<Real>::epsilon)();
-      // Error bound ~eps^6/7
-      // Error: h^6f^(7)(x)/140 + 5|f(x)|eps/h
-      Real h = pow(eps / 168, (Real)1 / (Real)7);
-      h = detail::make_xph_representable(x, h);
-
-      Real yh = f(x + h);
-      Real ymh = f(x - h);
-      Real y1 = yh - ymh;
-      Real y2 = f(x - 2 * h) - f(x + 2 * h);
-      Real y3 = f(x + 3 * h) - f(x - 3 * h);
-
-      if (error)
-      {
-         // Mathematica code to generate fd scheme for 7th derivative:
-         // Sum[(-1)^i*Binomial[7, i]*(f[x+(3-i)*h] + f[x+(4-i)*h])/2, {i, 0, 7}]
-         // Mathematica to demonstrate that this is a finite difference formula for 7th derivative:
-         // Series[(f[x+4*h]-f[x-4*h] + 6*(f[x-3*h] - f[x+3*h]) + 14*(f[x-h] - f[x+h] + f[x+2*h] - f[x-2*h]))/2, {h, 0, 15}]
-         Real y7 = (f(x + 4 * h) - f(x - 4 * h) - 6 * y3 - 14 * y1 - 14 * y2) / 2;
-         *error = abs(y7) / (140 * h) + 5 * (abs(yh) + abs(ymh))*eps / h;
-      }
-      return (y3 + 9 * y2 + 45 * y1) / (60 * h);
-   }
-
-   template<class F, class Real>
-   Real finite_difference_derivative(const F f, Real x, Real* error, const fd_tag<8>&)
-   {
-      using std::sqrt;
-      using std::pow;
-      using std::abs;
-      using std::numeric_limits;
-
-      const Real eps = (numeric_limits<Real>::epsilon)();
-      // Error bound ~eps^8/9.
-      // In double precision, we only expect to lose two digits of precision while using this formula, at the cost of 8 function evaluations.
-      // Error: h^8|f^(9)(x)|/630 + 7|f(x)|eps/h assuming 7 unstabilized additions.
-      // Mathematica code to get the error:
-      // Series[(f[x+h]-f[x-h])*(4/5) + (1/5)*(f[x-2*h] - f[x+2*h]) + (4/105)*(f[x+3*h] - f[x-3*h]) + (1/280)*(f[x-4*h] - f[x+4*h]), {h, 0, 9}]
-      // If we used Kahan summation, we could get the max error down to h^8|f^(9)(x)|/630 + |f(x)|eps/h.
-      Real h = pow(551.25*eps, (Real)1 / (Real)9);
-      h = detail::make_xph_representable(x, h);
-
-      Real yh = f(x + h);
-      Real ymh = f(x - h);
-      Real y1 = yh - ymh;
-      Real y2 = f(x - 2 * h) - f(x + 2 * h);
-      Real y3 = f(x + 3 * h) - f(x - 3 * h);
-      Real y4 = f(x - 4 * h) - f(x + 4 * h);
-
-      Real tmp1 = 3 * y4 / 8 + 4 * y3;
-      Real tmp2 = 21 * y2 + 84 * y1;
-
-      if (error)
-      {
-         // Mathematica code to generate fd scheme for 7th derivative:
-         // Sum[(-1)^i*Binomial[9, i]*(f[x+(4-i)*h] + f[x+(5-i)*h])/2, {i, 0, 9}]
-         // Mathematica to demonstrate that this is a finite difference formula for 7th derivative:
-         // Series[(f[x+5*h]-f[x- 5*h])/2 + 4*(f[x-4*h] - f[x+4*h]) + 27*(f[x+3*h] - f[x-3*h])/2 + 24*(f[x-2*h]  - f[x+2*h]) + 21*(f[x+h] - f[x-h]), {h, 0, 15}]
-         Real f9 = (f(x + 5 * h) - f(x - 5 * h)) / 2 + 4 * y4 + 27 * y3 / 2 + 24 * y2 + 21 * y1;
-         *error = abs(f9) / (630 * h) + 7 * (abs(yh) + abs(ymh))*eps / h;
-      }
-      return (tmp1 + tmp2) / (105 * h);
-   }
-
-   template<class F, class Real, class tag>
-   Real finite_difference_derivative(const F f, Real x, Real* error, const tag&)
-   {
-      // Always fails, but condition is template-arg-dependent so only evaluated if we get instantiated.
-      BOOST_STATIC_ASSERT_MSG(sizeof(Real) == 0, "Finite difference not implemented for this order: try 1, 2, 4, 6 or 8");
-   }
-
-}
-
-template<class F, class Real, size_t order=6>
-inline Real finite_difference_derivative(const F f, Real x, Real* error = nullptr)
-{
-   return detail::finite_difference_derivative(f, x, error, detail::fd_tag<order>());
-}
-
-}}}  // namespaces
 #endif
diff --git a/test/test_numerical_differentiation.cpp b/test/test_numerical_differentiation.cpp
index 01f42e16e..d49a75daa 100644
--- a/test/test_numerical_differentiation.cpp
+++ b/test/test_numerical_differentiation.cpp
@@ -16,12 +16,12 @@
 #include <boost/math/special_functions/bessel.hpp>
 #include <boost/math/special_functions/bessel_prime.hpp>
 #include <boost/math/special_functions/next.hpp>
-#include <boost/math/tools/numerical_differentiation.hpp>
+#include <boost/math/differentiation/finite_difference.hpp>
 
 using std::abs;
 using std::pow;
-using boost::math::tools::finite_difference_derivative;
-using boost::math::tools::complex_step_derivative;
+using boost::math::differentiation::finite_difference_derivative;
+using boost::math::differentiation::complex_step_derivative;
 using boost::math::cyl_bessel_j;
 using boost::math::cyl_bessel_j_prime;
 using boost::math::constants::half;

From a27c4bc96cfd757bfdb6371ce2be7c1c8483fcaa Mon Sep 17 00:00:00 2001
From: Nick Thompson <nathompson7@protonmail.com>
Date: Thu, 27 Dec 2018 20:38:27 -0700
Subject: [PATCH 02/10] Also change concept and include test [CI SKIP]

---
 include/boost/math/tools/numerical_differentiation.hpp    | 8 --------
 .../numerical_differentiation_concept_test.cpp            | 4 ++--
 test/compile_test/numerical_differentiation_incl_test.cpp | 6 +++---
 3 files changed, 5 insertions(+), 13 deletions(-)

diff --git a/include/boost/math/tools/numerical_differentiation.hpp b/include/boost/math/tools/numerical_differentiation.hpp
index eb4bdcb44..52d278889 100644
--- a/include/boost/math/tools/numerical_differentiation.hpp
+++ b/include/boost/math/tools/numerical_differentiation.hpp
@@ -9,12 +9,4 @@
 
 BOOST_HEADER_DEPRECATED("<boost/math/differentiation/finite_difference.hpp>");
 
-namespace boost {
-   namespace math {
-       namespace differentiation {
-           using finite_difference_derivative;
-           using complex_step_derivative;
-       }
-   }
-}
 #endif
diff --git a/test/compile_test/numerical_differentiation_concept_test.cpp b/test/compile_test/numerical_differentiation_concept_test.cpp
index edf1dfa3d..7fcb898b8 100644
--- a/test/compile_test/numerical_differentiation_concept_test.cpp
+++ b/test/compile_test/numerical_differentiation_concept_test.cpp
@@ -4,12 +4,12 @@
 //  LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
 //
 #include <boost/math/concepts/std_real_concept.hpp>
-#include <boost/math/tools/numerical_differentiation.hpp>
+#include <boost/math/differentiation/finite_difference.hpp>
 
 
 void compile_and_link_test()
 {
    boost::math::concepts::std_real_concept x = 0;
    auto f = [](boost::math::concepts::std_real_concept x) { return x; };
-   boost::math::tools::finite_difference_derivative(f, x);
+   boost::math::differentiation::finite_difference_derivative(f, x);
 }
diff --git a/test/compile_test/numerical_differentiation_incl_test.cpp b/test/compile_test/numerical_differentiation_incl_test.cpp
index 56d2ea607..ab7c32cc6 100644
--- a/test/compile_test/numerical_differentiation_incl_test.cpp
+++ b/test/compile_test/numerical_differentiation_incl_test.cpp
@@ -6,7 +6,7 @@
 // Basic sanity check that header <boost/math/special_functions/gamma.hpp>
 // #includes all the files that it needs to.
 //
-#include <boost/math/tools/numerical_differentiation.hpp>
+#include <boost/math/differentiation/numerical_differentiation.hpp>
 //
 // Note this header includes no other headers, this is
 // important if this test is to be meaningful:
@@ -17,8 +17,8 @@ void compile_and_link_test()
 {
     auto f = [](double x) { return x; };
     double x = 0;
-    check_result<double>(boost::math::tools::finite_difference_derivative(f, x));
+    check_result<double>(boost::math::differentiation::finite_difference_derivative(f, x));
 
     auto g = [](std::complex<double> x){ return x;};
-    check_result<double>(boost::math::tools::complex_step_derivative(g, x));
+    check_result<double>(boost::math::differentiation::complex_step_derivative(g, x));
 }

From bf7b29f13c3fb7b90b258941d9213ac50d2e267c Mon Sep 17 00:00:00 2001
From: Nick Thompson <nathompson7@protonmail.com>
Date: Sat, 29 Dec 2018 18:41:38 -0700
Subject: [PATCH 03/10] Fix typo.

---
 test/compile_test/numerical_differentiation_incl_test.cpp | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/test/compile_test/numerical_differentiation_incl_test.cpp b/test/compile_test/numerical_differentiation_incl_test.cpp
index ab7c32cc6..7286949a1 100644
--- a/test/compile_test/numerical_differentiation_incl_test.cpp
+++ b/test/compile_test/numerical_differentiation_incl_test.cpp
@@ -6,7 +6,7 @@
 // Basic sanity check that header <boost/math/special_functions/gamma.hpp>
 // #includes all the files that it needs to.
 //
-#include <boost/math/differentiation/numerical_differentiation.hpp>
+#include <boost/math/differentiation/finite_difference.hpp>
 //
 // Note this header includes no other headers, this is
 // important if this test is to be meaningful:

From 7bbf05e8ba3dafc9a433347c8fef59af4093eb37 Mon Sep 17 00:00:00 2001
From: Nick Thompson <nathompson7@protonmail.com>
Date: Mon, 31 Dec 2018 20:11:25 -0700
Subject: [PATCH 04/10] Lanczos smoothing differentiators.

---
 doc/differentiation/lanczos_smoothing.qbk     | 110 ++++
 doc/math.qbk                                  |   1 +
 .../differentiation/lanczos_smoothing.hpp     | 297 +++++++++++
 test/Jamfile.v2                               |   1 +
 test/lanczos_smoothing_test.cpp               | 469 ++++++++++++++++++
 5 files changed, 878 insertions(+)
 create mode 100644 doc/differentiation/lanczos_smoothing.qbk
 create mode 100644 include/boost/math/differentiation/lanczos_smoothing.hpp
 create mode 100644 test/lanczos_smoothing_test.cpp

diff --git a/doc/differentiation/lanczos_smoothing.qbk b/doc/differentiation/lanczos_smoothing.qbk
new file mode 100644
index 000000000..213fc964e
--- /dev/null
+++ b/doc/differentiation/lanczos_smoothing.qbk
@@ -0,0 +1,110 @@
+[/
+Copyright (c) 2019 Nick Thompson
+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)
+]
+
+[section:diff Lanczos Smoothing Derivatives]
+
+[heading Synopsis]
+
+``
+#include <boost/math/differentiation/lanczos_smoothing.hpp>
+
+namespace boost { namespace math { namespace differentiation {
+
+    template <class RandomAccessContainer>
+    class lanczos_derivative {
+    public:
+        lanczos_smoothing_derivative(RandomAccessContainer const & v,
+                                     typename RandomAccessContainer::value_type spacing = 1,
+                                     size_t n = 18,
+                                     size_t approximation_order = 3);
+
+        typename RandomAccessContainer::value_type operator[](size_t i) const;
+
+        void reset_data(RandomAccessContainer const &v);
+
+        void reset_spacing(Real spacing);
+    }
+
+}}} // namespaces
+``
+
+[heading Description]
+
+The function `lanczos_derivative` calculates a finite-difference approximation to the derivative of a noisy sequence of equally-spaced values /v/ at an index /i/.
+A basic usage is
+
+    std::vector<double> v(500);
+    // fill v with noisy data.
+    double spacing = 0.001;
+    using boost::math::differentiation::lanczos_derivative;
+    auto lanczos = lanczos_derivative(v, spacing);
+    double dvdt = lanczos[30];
+
+If the data has variance \u03C3[super 2], then the variance of the computed derivative is roughly \u03C3[super 2]/p/[super 3] /n/[super -3] \u0394 /t/[super -2],
+i.e., it increases cubically with the approximation order /p/, linearly with the data variance, and decreases at the cube of the filter length /n/.
+In addition, we must not forget the discretization error which is /O/(\u0394 /t/[super /p/]).
+You can play around with the approximation order /p/ and the filter length /n/:
+
+    size_t n = 12;
+    size_t p = 2;
+    auto lanczos = lanczos_derivative(v, spacing, n, p);
+    double dvdt = lanczos[24];
+
+If /p=2n/, then the Lanczos smoothing derivative is not smoothing:
+It reduces to the standard /2n+1/-point finite-difference formula.
+For /p>2n/, an assertion is hit as the filter is undefined.
+
+In our tests with AWGN, we have found the error decreases monotonically with /n/, as is expected from the theory discussed above.
+So the choice of /n/ is simple:
+As high as possible given your speed requirements (larger /n/ implies a longer filter and hence more compute),
+balanced against the danger of overfitting and averaging over non-stationarity.
+
+The choice of approximation order /p/ for a given /n/ is more difficult.
+If your signal is believed to be a polynomial,
+it does not make sense to set /p/ to larger than the polynomial degree-
+though it may be sensible to take /p/ less than the polynomial degree.
+
+For a sinusoidal signal contaminated with AWGN, we ran a few tests showing that for SNR = 1, p = n/8 gave the best results,
+for SNR = 10, p = n/7 was the best, and for SNR = 100, p = n/6 was the most reasonable choice.
+For SNR = 0.1, the method appears to be useless.
+The user is urged to use these results with caution-they have no theoretical backing and are extrapolated from a single case.
+
+The filters are (regrettably) computed at runtime-the vast number of combinations of approximation order and filter length makes the number of filters that must be stored excessive for compile-time data.
+Hence the constructor call computes the filters.
+Since each filter has length /2n+1/ and there are /n/ filters, whose element each consist of /p/ summands,
+the complexity of the constructor call is O(/n/[super 2]/p/).e
+This is not cheap-though for most cases small /p/ and /n/ not too large (< 20) is desired.
+However, for concreteness, on the author's 2.7GHz Intel laptop CPU, the /n=16/, /p=3/ filter takes 9 microseconds to compute.
+This is far from negligible, and as such we provide API calls which allow the filters to be used with multiple data:
+
+
+    std::vector<double> v(500);
+    // fill v with noisy data.
+    auto lanczos = lanczos_derivative(v, spacing);
+    // use lanczos with v . . .
+    std::vector<double> w(500);
+    lanczos.reset_data(w);
+    // use lanczos with w . . .
+    // need to use a different spacing?
+    lanczos.reset_spacing(0.02);
+
+
+The implementation follows [@https://doi.org/10.1080/00207160.2012.666348 McDevitt, 2012],
+who vastly expanded the ideas of Lanczos to create a very general framework for numerically differentiating noisy equispaced data.
+
+
+
+[heading References]
+
+* Corless, Robert M., and Nicolas Fillion. ['A graduate introduction to numerical methods.] AMC 10 (2013): 12.
+
+* Lanczos, Cornelius. ['Applied analysis.] Courier Corporation, 1988.
+
+* Timothy J. McDevitt (2012): ['Discrete Lanczos derivatives of noisy data], International Journal of Computer Mathematics, 89:7, 916-931
+
+
+[endsect]
diff --git a/doc/math.qbk b/doc/math.qbk
index 711e88f8c..0090e4ea0 100644
--- a/doc/math.qbk
+++ b/doc/math.qbk
@@ -661,6 +661,7 @@ and as a CD ISBN 0-9504833-2-X  978-0-9504833-2-0, Classification 519.2-dc22.
 [include quadrature/double_exponential.qbk]
 [include quadrature/naive_monte_carlo.qbk]
 [include differentiation/numerical_differentiation.qbk]
+[include differentiation/lanczos_smoothing.qbk]
 [endmathpart]
 
 [include complex/complex-tr1.qbk]
diff --git a/include/boost/math/differentiation/lanczos_smoothing.hpp b/include/boost/math/differentiation/lanczos_smoothing.hpp
new file mode 100644
index 000000000..5696fc454
--- /dev/null
+++ b/include/boost/math/differentiation/lanczos_smoothing.hpp
@@ -0,0 +1,297 @@
+//  (C) Copyright Nick Thompson 2018.
+//  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)
+
+#ifndef BOOST_MATH_DIFFERENTIATION_LANCZOS_SMOOTHING_HPP
+#define BOOST_MATH_DIFFERENTIATION_LANCZOS_SMOOTHING_HPP
+#include <vector>
+#include <boost/assert.hpp>
+
+namespace boost {
+namespace math {
+namespace differentiation {
+
+namespace detail {
+template <typename Real>
+class discrete_legendre {
+  public:
+    discrete_legendre(int n) : m_n{n}
+    {
+        // The integer n indexes a family of discrete Legendre polynomials indexed by k <= 2*n
+    }
+
+    Real norm_sq(int r)
+    {
+        Real prod = Real(2) / Real(2 * r + 1);
+        for (int k = -r; k <= r; ++k) {
+            prod *= Real(2 * m_n + 1 + k) / Real(2 * m_n);
+        }
+        return prod;
+    }
+
+    void initialize_recursion(Real x)
+    {
+        m_qrm2 = 1;
+        m_qrm1 = x;
+        m_qrm2p = 0;
+        m_qrm1p = 1;
+
+        m_r = 2;
+        m_x = x;
+    }
+
+    Real next()
+    {
+        Real N = 2 * m_n + 1;
+        Real num = (m_r - 1) * (N * N - (m_r - 1) * (m_r - 1)) * m_qrm2;
+        Real tmp = (2 * m_r - 1) * m_x * m_qrm1 - num / Real(4 * m_n * m_n);
+        m_qrm2 = m_qrm1;
+        m_qrm1 = tmp / m_r;
+        ++m_r;
+        return m_qrm1;
+    }
+
+    Real next_prime()
+    {
+        Real N = 2 * m_n + 1;
+        Real s =
+            (m_r - 1) * (N * N - (m_r - 1) * (m_r - 1)) / Real(4 * m_n * m_n);
+        Real tmp1 = ((2 * m_r - 1) * m_x * m_qrm1 - s * m_qrm2) / m_r;
+        Real tmp2 = ((2 * m_r - 1) * (m_qrm1 + m_x * m_qrm1p) - s * m_qrm2p) / m_r;
+        m_qrm2 = m_qrm1;
+        m_qrm1 = tmp1;
+        m_qrm2p = m_qrm1p;
+        m_qrm1p = tmp2;
+        ++m_r;
+        return m_qrm1p;
+    }
+
+
+    Real operator()(Real x, int k)
+    {
+        BOOST_ASSERT_MSG(k <= 2 * m_n, "r <= 2n is required.");
+        if (k == 0)
+        {
+            return 1;
+        }
+        if (k == 1)
+        {
+            return x;
+        }
+        Real qrm2 = 1;
+        Real qrm1 = x;
+        Real N = 2 * m_n + 1;
+        for (int r = 2; r <= k; ++r) {
+            Real num = (r - 1) * (N * N - (r - 1) * (r - 1)) * qrm2;
+            Real tmp = (2 * r - 1) * x * qrm1 - num / Real(4 * m_n * m_n);
+            qrm2 = qrm1;
+            qrm1 = tmp / r;
+        }
+        return qrm1;
+    }
+
+    Real prime(Real x, int k) {
+        BOOST_ASSERT_MSG(k <= 2 * m_n, "r <= 2n is required.");
+        if (k == 0) {
+            return 0;
+        }
+        if (k == 1) {
+            return 1;
+        }
+        Real qrm2 = 1;
+        Real qrm1 = x;
+        Real qrm2p = 0;
+        Real qrm1p = 1;
+        Real N = 2 * m_n + 1;
+        for (int r = 2; r <= k; ++r) {
+            Real s =
+                (r - 1) * (N * N - (r - 1) * (r - 1)) / Real(4 * m_n * m_n);
+            Real tmp1 = ((2 * r - 1) * x * qrm1 - s * qrm2) / r;
+            Real tmp2 = ((2 * r - 1) * (qrm1 + x * qrm1p) - s * qrm2p) / r;
+            qrm2 = qrm1;
+            qrm1 = tmp1;
+            qrm2p = qrm1p;
+            qrm1p = tmp2;
+        }
+        return qrm1p;
+    }
+
+  private:
+    int m_n;
+    Real m_qrm2;
+    Real m_qrm1;
+    Real m_qrm2p;
+    Real m_qrm1p;
+    int m_r;
+    Real m_x;
+};
+
+template <class Real>
+std::vector<Real> interior_filter(int n, int p) {
+    // We could make the filter length n, as f[0] = 0,
+    // but that'd make the indexing awkward when applying the filter.
+    std::vector<Real> f(n + 1, 0);
+    auto dlp = discrete_legendre<Real>(n);
+    std::vector<Real> coeffs(p+1, std::numeric_limits<Real>::quiet_NaN());
+    dlp.initialize_recursion(0);
+    coeffs[1] = 1/dlp.norm_sq(1);
+    for (int l = 3; l < p + 1; l += 2)
+    {
+        dlp.next_prime();
+        coeffs[l] = dlp.next_prime()/ dlp.norm_sq(l);
+    }
+
+    for (size_t j = 1; j < f.size(); ++j)
+    {
+        Real arg = Real(j) / Real(n);
+        dlp.initialize_recursion(arg);
+        f[j] = coeffs[1]*arg;
+        for (int l = 3; l <= p; l += 2)
+        {
+            dlp.next();
+            f[j] += coeffs[l]*dlp.next();
+        }
+        f[j] /= (n * n);
+    }
+    return f;
+}
+
+template <class Real>
+std::vector<Real> boundary_filter(int n, int p, int s) {
+    std::vector<Real> f(2 * n + 1, 0);
+    auto dlp = discrete_legendre<Real>(n);
+    Real sn = Real(s) / Real(n);
+    std::vector<Real> coeffs(p+1, std::numeric_limits<Real>::quiet_NaN());
+    dlp.initialize_recursion(sn);
+    coeffs[0] = 0;
+    coeffs[1] = 1/dlp.norm_sq(1);
+    for (int l = 2; l < p + 1; ++l)
+    {
+        // Calculation of the norms is common to all filters,
+        // so it seems like an obvious optimization target.
+        // I tried this: The spent in computing the norms time is not negligible,
+        // but still a small fraction of the total compute time.
+        // Hence I'm not refactoring out these norm calculations.
+        coeffs[l] = dlp.next_prime()/ dlp.norm_sq(l);
+    }
+
+    for (int k = 0; k < f.size(); ++k)
+    {
+        int j = k - n;
+        f[k] = 0;
+        Real arg = Real(j) / Real(n);
+        dlp.initialize_recursion(arg);
+        f[k] = coeffs[1]*arg;
+        for (int l = 2; l <= p; ++l)
+        {
+            f[k] += coeffs[l]*dlp.next();
+        }
+        f[k] /= (n * n);
+    }
+    return f;
+}
+
+} // namespace detail
+
+template <class RandomAccessContainer>
+class lanczos_derivative {
+public:
+    using Real = typename RandomAccessContainer::value_type;
+    lanczos_derivative(RandomAccessContainer const &v,
+                       Real spacing = 1,
+                       int filter_length = 18,
+                       int approximation_order = 3)
+        : m_v{v}, dt{spacing}
+    {
+        BOOST_ASSERT_MSG(approximation_order <= 2 * filter_length,
+                         "The approximation order must be <= 2n");
+        BOOST_ASSERT_MSG(approximation_order >= 2,
+                         "The approximation order must be >= 2");
+        BOOST_ASSERT_MSG(spacing > 0, "Spacing between samples must be > 0.");
+        using std::size;
+        BOOST_ASSERT_MSG(size(v) >= filter_length,
+            "Vector must be at least as long as the filter length");
+        m_f = detail::interior_filter<Real>(filter_length, approximation_order);
+
+        boundary_filters.resize(filter_length);
+        for (size_t i = 0; i < filter_length; ++i)
+        {
+            // s = i - n;
+            boundary_filters[i] = detail::boundary_filter<Real>(
+                filter_length, approximation_order, i - filter_length);
+        }
+    }
+
+    void reset_data(RandomAccessContainer const &v)
+    {
+        using std::size;
+        BOOST_ASSERT_MSG(size(v) >= m_f.size(), "Vector must be at least as long as the filter length");
+        m_v = v;
+    }
+
+    void reset_spacing(Real spacing)
+    {
+        BOOST_ASSERT_MSG(spacing > 0, "Spacing between samples must be > 0.");
+        dt = spacing;
+    }
+
+    Real spacing() const
+    {
+        return dt;
+    }
+
+    Real operator[](size_t i) const
+    {
+        using std::size;
+        if (i >= m_f.size() - 1 && i <= size(m_v) - m_f.size())
+        {
+            Real dv = 0;
+            for (size_t j = 1; j < m_f.size(); ++j)
+            {
+                dv += m_f[j] * (m_v[i + j] - m_v[i - j]);
+            }
+            return dv / dt;
+        }
+
+        // m_f.size() = N+1
+        if (i < m_f.size() - 1)
+        {
+            auto &f = boundary_filters[i];
+            Real dv = 0;
+            for (size_t j = 0; j < f.size(); ++j) {
+                dv += f[j] * m_v[j];
+            }
+            return dv / dt;
+        }
+
+        if (i > size(m_v) - m_f.size() && i < size(m_v))
+        {
+            int k = size(m_v) - 1 - i;
+            auto &f = boundary_filters[k];
+            Real dv = 0;
+            for (size_t j = 0; j < f.size(); ++j)
+            {
+                dv += f[j] * m_v[m_v.size() - 1 - j];
+            }
+            return -dv / dt;
+        }
+
+        // OOB access:
+        BOOST_ASSERT_MSG(false, "Out of bounds access in Lanczos derivative");
+        return std::numeric_limits<Real>::quiet_NaN();
+    }
+
+private:
+    const RandomAccessContainer &m_v;
+    std::vector<Real> m_f;
+    std::vector<std::vector<Real>> boundary_filters;
+    Real dt;
+};
+
+// We can also implement lanczos_acceleration, but let's get the API for lanczos_derivative nailed down before doing so.
+
+} // namespace differentiation
+} // namespace math
+} // namespace boost
+#endif
diff --git a/test/Jamfile.v2 b/test/Jamfile.v2
index 911572d66..ac7926265 100644
--- a/test/Jamfile.v2
+++ b/test/Jamfile.v2
@@ -902,6 +902,7 @@ test-suite misc :
    [ run test_constant_generate.cpp : : : release <define>USE_CPP_FLOAT=1 <exception-handling>off:<build>no  ]
    [ run test_cubic_b_spline.cpp ../../test/build//boost_unit_test_framework : : :  [ requires cxx11_smart_ptr cxx11_defaulted_functions ] <debug-symbols>off <toolset>msvc:<cxxflags>/bigobj release ]
    [ run catmull_rom_test.cpp ../../test/build//boost_unit_test_framework : : :  [ requires cxx17_if_constexpr ] ] # does not in fact require C++17 constexpr; requires C++17 std::size.
+   [ run lanczos_smoothing_test.cpp ../../test/build//boost_unit_test_framework : : : [ requires cxx17_if_constexpr ] ]
    [ run test_real_concept.cpp ../../test/build//boost_unit_test_framework  ]
    [ run test_remez.cpp pch ../../test/build//boost_unit_test_framework  ]
    [ run test_roots.cpp pch ../../test/build//boost_unit_test_framework  ]
diff --git a/test/lanczos_smoothing_test.cpp b/test/lanczos_smoothing_test.cpp
new file mode 100644
index 000000000..fd7315e88
--- /dev/null
+++ b/test/lanczos_smoothing_test.cpp
@@ -0,0 +1,469 @@
+/*
+ * 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)
+ */
+#define BOOST_TEST_MODULE lanczos_smoothing_test
+
+#include <random>
+#include <boost/accumulators/statistics/sum_kahan.hpp>
+#include <boost/test/included/unit_test.hpp>
+#include <boost/test/floating_point_comparison.hpp>
+#include <boost/math/differentiation/lanczos_smoothing.hpp>
+#include <boost/multiprecision/cpp_bin_float.hpp>
+
+using std::abs;
+using std::pow;
+using std::sqrt;
+using boost::multiprecision::cpp_bin_float_50;
+using boost::multiprecision::cpp_bin_float_100;
+using boost::math::differentiation::lanczos_derivative;
+using boost::math::differentiation::detail::discrete_legendre;
+using boost::math::differentiation::detail::interior_filter;
+using boost::math::differentiation::detail::boundary_filter;
+
+template<class Real>
+void test_dlp_norms()
+{
+    std::cout << "Testing Discrete Legendre Polynomial norms on type " << typeid(Real).name() << "\n";
+    Real tol = std::numeric_limits<Real>::epsilon();
+    auto dlp = discrete_legendre<Real>(1);
+    BOOST_CHECK_CLOSE_FRACTION(dlp.norm_sq(0), 3, tol);
+    BOOST_CHECK_CLOSE_FRACTION(dlp.norm_sq(1), 2, tol);
+    dlp = discrete_legendre<Real>(2);
+    BOOST_CHECK_CLOSE_FRACTION(dlp.norm_sq(0), Real(5)/Real(2), tol);
+    BOOST_CHECK_CLOSE_FRACTION(dlp.norm_sq(1), Real(5)/Real(4), tol);
+    BOOST_CHECK_CLOSE_FRACTION(dlp.norm_sq(2), Real(3*3*7)/Real(pow(2,6)), 2*tol);
+    dlp = discrete_legendre<Real>(200);
+    for(size_t r = 0; r < 10; ++r)
+    {
+        Real calc = dlp.norm_sq(r);
+        Real expected = Real(2)/Real(2*r+1);
+        // As long as r << n, ||q_r||^2 -> 2/(2r+1) as n->infty
+        BOOST_CHECK_CLOSE_FRACTION(calc, expected, 0.05);
+    }
+
+}
+
+template<class Real>
+void test_dlp_evaluation()
+{
+    std::cout << "Testing evaluation of Discrete Legendre polynomials on type " << typeid(Real).name() << "\n";
+    Real tol = std::numeric_limits<Real>::epsilon();
+    size_t n = 25;
+    auto dlp = discrete_legendre<Real>(n);
+    Real x = 0.72;
+    Real q0 = dlp(x, 0);
+    BOOST_TEST(q0 == 1);
+    Real q1 = dlp(x, 1);
+    BOOST_TEST(q1 == x);
+    Real q2 = dlp(x, 2);
+    int N = 2*n+1;
+    Real expected = 0.5*(3*x*x - Real(N*N - 1)/Real(4*n*n));
+    BOOST_CHECK_CLOSE_FRACTION(q2, expected, tol);
+    Real q3 = dlp(x, 3);
+    expected = (x/3)*(5*expected - (Real(N*N - 4))/(2*n*n));
+    BOOST_CHECK_CLOSE_FRACTION(q3, expected, 2*tol);
+
+    // q_r(x) is even for even r, and odd for odd r:
+    for (size_t n = 8; n < 22; ++n)
+    {
+        dlp = discrete_legendre<Real>(n);
+        for(size_t r = 2; r <= n; ++r)
+        {
+            if (r & 1)
+            {
+                Real q1 = dlp(x, r);
+                Real q2 = -dlp(-x, r);
+                BOOST_CHECK_CLOSE_FRACTION(q1, q2, tol);
+            }
+            else
+            {
+                Real q1 = dlp(x, r);
+                Real q2 = dlp(-x, r);
+                BOOST_CHECK_CLOSE_FRACTION(q1, q2, tol);
+            }
+
+            Real l2_sq = 0;
+            for (int j = -(int)n; j <= (int) n; ++j)
+            {
+                Real y = Real(j)/Real(n);
+                Real term = dlp(y, r);
+                l2_sq += term*term;
+            }
+            l2_sq /= n;
+            Real l2_sq_expected = dlp.norm_sq(r);
+            BOOST_CHECK_CLOSE_FRACTION(l2_sq, l2_sq_expected, 20*tol);
+        }
+    }
+}
+
+template<class Real>
+void test_dlp_next()
+{
+    std::cout << "Testing Discrete Legendre polynomial 'next' function on type " << typeid(Real).name() << "\n";
+    Real tol = std::numeric_limits<Real>::epsilon();
+
+    for(size_t n = 2; n < 20; ++n)
+    {
+        auto dlp = discrete_legendre<Real>(n);
+        for(Real x = -1; x <= 1; x += 0.1)
+        {
+            dlp.initialize_recursion(x);
+            for (size_t k = 2; k < n; ++k)
+            {
+                BOOST_CHECK_CLOSE(dlp.next(), dlp(x, k), tol);
+            }
+
+            dlp.initialize_recursion(x);
+            for (size_t k = 2; k < n; ++k)
+            {
+                BOOST_CHECK_CLOSE(dlp.next_prime(), dlp.prime(x, k), tol);
+            }
+        }
+    }
+}
+
+
+template<class Real>
+void test_dlp_derivatives()
+{
+    std::cout << "Testing Discrete Legendre polynomial derivatives on type " << typeid(Real).name() << "\n";
+    Real tol = 10*std::numeric_limits<Real>::epsilon();
+    int n = 25;
+    auto dlp = discrete_legendre<Real>(n);
+    Real x = 0.72;
+    Real q0p = dlp.prime(x, 0);
+    BOOST_TEST(q0p == 0);
+    Real q1p = dlp.prime(x, 1);
+    BOOST_TEST(q1p == 1);
+    Real q2p = dlp.prime(x, 2);
+    Real expected = 3*x;
+    BOOST_CHECK_CLOSE_FRACTION(q2p, expected, tol);
+}
+
+template<class Real>
+void test_interior_filter()
+{
+    std::cout << "Testing interior filter on type " << typeid(Real).name() << "\n";
+    Real tol = std::numeric_limits<Real>::epsilon();
+    for(int n = 1; n < 10; ++n)
+    {
+        for (int p = 1; p < n; p += 2)
+        {
+            auto f = interior_filter<Real>(n,p);
+            // Since we only store half the filter coefficients,
+            // we need to reindex the moment sums:
+            Real sum = 0;
+            for (int j = 0; j < f.size(); ++j)
+            {
+                sum += j*f[j];
+            }
+            BOOST_CHECK_CLOSE_FRACTION(2*sum, 1, 1000*tol);
+
+            for (int l = 3; l <= p; l += 2)
+            {
+                sum = 0;
+                for (int j = 0; j < f.size(); ++j)
+                {
+                    // The condition number of this sum is infinite!
+                    // No need to get to worked up about the tolerance.
+                    sum += pow(Real(j), l)*f[j];
+                }
+                BOOST_CHECK_SMALL(sum, sqrt(tol)/100);
+            }
+            //std::cout << "(n,p) = (" << n  << "," << p << ") = {";
+            //for (auto & x : f)
+            //{
+            //    std::cout << x << ", ";
+            //}
+            //std::cout << "}\n";
+        }
+    }
+}
+
+template<class Real>
+void test_interior_lanczos()
+{
+    std::cout << "Testing interior Lanczos on type " << typeid(Real).name() << "\n";
+    Real tol = std::numeric_limits<Real>::epsilon();
+    std::vector<Real> v(500);
+    for (auto & x : v)
+    {
+        x = 7;
+    }
+    for (size_t n = 1; n < 10; ++n)
+    {
+        for (size_t p = 2; p < 2*n; p += 2)
+        {
+            auto lsd = lanczos_derivative(v, 0.1, n, p);
+            for (size_t m = n; m < v.size() - n; ++m)
+            {
+                Real dvdt = lsd[m];
+                BOOST_CHECK_SMALL(dvdt, tol);
+            }
+        }
+    }
+
+
+    for(size_t i = 0; i < v.size(); ++i)
+    {
+        v[i] = 7*i+8;
+    }
+
+    for (size_t n = 1; n < 10; ++n)
+    {
+        for (size_t p = 2; p < 2*n; p += 2)
+        {
+            auto lsd = lanczos_derivative(v, Real(1), n, p);
+            for (size_t m = n; m < v.size() - n; ++m)
+            {
+                Real dvdt = lsd[m];
+                BOOST_CHECK_CLOSE_FRACTION(dvdt, 7, 2000*tol);
+            }
+        }
+    }
+
+    //std::random_device rd{};
+    //auto seed = rd();
+    //std::cout << "Seed = " << seed << "\n";
+    std::mt19937 gen(4172378669);
+    std::normal_distribution<> dis{0, 0.01};
+    std::cout << std::fixed;
+    for (size_t i = 0; i < v.size(); ++i)
+    {
+        v[i] = 7*i+8 + dis(gen);
+    }
+
+    for (size_t n = 1; n < 10; ++n)
+    {
+        for (size_t p = 2; p < 2*n; p += 2)
+        {
+            auto lsd = lanczos_derivative(v, Real(1), n, p);
+            for (size_t m = n; m < v.size() - n; ++m)
+            {
+                BOOST_CHECK_CLOSE_FRACTION(lsd[m], Real(7), Real(0.0042));
+            }
+        }
+    }
+
+
+    for (size_t i = 0; i < v.size(); ++i)
+    {
+        v[i] = 15*i*i + 7*i+8 + dis(gen);
+    }
+
+    for (size_t n = 1; n < 10; ++n)
+    {
+        for (size_t p = 2; p < 2*n; p += 2)
+        {
+            auto lsd = lanczos_derivative(v, Real(1), n, p);
+            for (size_t m = n; m < v.size() - n; ++m)
+            {
+                BOOST_CHECK_CLOSE_FRACTION(lsd[m], Real(30*m + 7), Real(0.00008));
+            }
+        }
+    }
+
+    std::normal_distribution<> dis1{0, 0.0001};
+    Real omega = Real(1)/Real(16);
+    for (size_t i = 0; i < v.size(); ++i)
+    {
+        v[i] = sin(i*omega) + dis1(gen);
+    }
+
+    for (size_t n = 10; n < 20; ++n)
+    {
+        for (size_t p = 3; p < 100 && p < n/2; p += 2)
+        {
+            auto lsd = lanczos_derivative(v, Real(1), n, p);
+
+            for (size_t m = n; m < v.size() - n && m < n + 10; ++m)
+            {
+                BOOST_CHECK_CLOSE_FRACTION(lsd[m], omega*cos(omega*m), Real(0.03));
+            }
+        }
+    }
+}
+
+template<class Real>
+void test_boundary_filters()
+{
+    std::cout << "Testing boundary filters on type " << typeid(Real).name() << "\n";
+    Real tol = std::numeric_limits<Real>::epsilon();
+    for(int n = 1; n < 5; ++n)
+    {
+        for (int p = 1; p < 2*n+1; ++p)
+        {
+            for (int s = -n; s <= n; ++s)
+            {
+                auto f = boundary_filter<Real>(n, p, s);
+                // Sum is zero:
+                Real sum = 0;
+                Real c = 0;
+                for (auto & x : f)
+                {
+                    Real y = x - c;
+                    Real t = sum + y;
+                    c = (t-sum) -y;
+                    sum = t;
+                }
+                BOOST_CHECK_SMALL(sum, 200*tol);
+
+                sum = 0;
+                c = 0;
+                for (int k = 0; k < f.size(); ++k)
+                {
+                    int j = k - n;
+                    // note the shifted index here:
+                    Real x = (j-s)*f[k];
+                    Real y = x - c;
+                    Real t = sum + y;
+                    c = (t-sum) -y;
+                    sum = t;
+                }
+                BOOST_CHECK_CLOSE_FRACTION(sum, 1, 350*tol);
+
+
+                for (int l = 2; l <= p; ++l)
+                {
+                    sum = 0;
+                    c = 0;
+                    for (int k = 0; k < f.size(); ++k)
+                    {
+                        int j = k - n;
+                        // The condition number of this sum is infinite!
+                        // No need to get to worked up about the tolerance.
+                        Real x = pow(Real(j-s), l)*f[k];
+                        Real y = x - c;
+                        Real t = sum + y;
+                        c = (t-sum) -y;
+                        sum = t;
+                    }
+                    BOOST_CHECK_SMALL(sum, sqrt(tol)/10);
+                }
+
+                //std::cout << "(n,p,s) = ("<< n << ", " << p << "," << s << ") = {";
+                //for (auto & x : f)
+                //{
+                //    std::cout << x << ", ";
+                //}
+                //std::cout << "}\n";*/
+            }
+        }
+    }
+}
+
+template<class Real>
+void test_boundary_lanczos()
+{
+    std::cout << "Testing Lanczos boundary on type " << typeid(Real).name() << "\n";
+    Real tol = std::numeric_limits<Real>::epsilon();
+    std::vector<Real> v(500);
+    for (auto & x : v)
+    {
+        x = 7;
+    }
+    for (size_t n = 1; n < 10; ++n)
+    {
+        for (size_t p = 2; p < 2*n; ++p)
+        {
+            auto lsd = lanczos_derivative(v, 0.0125, n, p);
+            for (size_t m = 0; m < n; ++m)
+            {
+                Real dvdt = lsd[m];
+                BOOST_CHECK_SMALL(dvdt, 4*sqrt(tol));
+            }
+            for (size_t m = v.size() - n; m < v.size(); ++m)
+            {
+                Real dvdt = lsd[m];
+                BOOST_CHECK_SMALL(dvdt, 4*sqrt(tol));
+            }
+        }
+    }
+
+    for(size_t i = 0; i < v.size(); ++i)
+    {
+        v[i] = 7*i+8;
+    }
+
+    for (size_t n = 3; n < 10; ++n)
+    {
+        for (size_t p = 2; p < 2*n; ++p)
+        {
+            auto lsd = lanczos_derivative(v, Real(1), n, p);
+            for (size_t m = 0; m < n; ++m)
+            {
+                Real dvdt = lsd[m];
+                BOOST_CHECK_CLOSE_FRACTION(dvdt, 7, sqrt(tol));
+            }
+
+            for (size_t m = v.size() - n; m < v.size(); ++m)
+            {
+                Real dvdt = lsd[m];
+                BOOST_CHECK_CLOSE_FRACTION(dvdt, 7, 4*sqrt(tol));
+            }
+        }
+    }
+
+    for (size_t i = 0; i < v.size(); ++i)
+    {
+        v[i] = 15*i*i + 7*i+8;
+    }
+
+    for (size_t n = 1; n < 10; ++n)
+    {
+        for (size_t p = 2; p < 2*n; ++p)
+        {
+            auto lsd = lanczos_derivative(v, Real(1), n, p);
+            for (size_t m = 0; m < v.size(); ++m)
+            {
+                BOOST_CHECK_CLOSE_FRACTION(lsd[m], 30*m+7, 30*sqrt(tol));
+            }
+        }
+    }
+
+    // Demonstrate that the boundary filters are also denoising:
+    //std::random_device rd{};
+    //auto seed = rd();
+    //std::cout << "seed = " << seed << "\n";
+    std::mt19937 gen(311354333);
+    std::normal_distribution<> dis{0, 0.01};
+    for (size_t i = 0; i < v.size(); ++i)
+    {
+        v[i] += dis(gen);
+    }
+
+    for (size_t n = 1; n < 10; ++n)
+    {
+        for (size_t p = 2; p < n; ++p)
+        {
+            auto lsd = lanczos_derivative(v, Real(1), n, p);
+            for (size_t m = 0; m < v.size(); ++m)
+            {
+                BOOST_CHECK_CLOSE_FRACTION(lsd[m], 30*m+7, 0.005);
+            }
+        }
+    }
+
+}
+
+BOOST_AUTO_TEST_CASE(lanczos_smoothing_test)
+{
+    test_dlp_norms<double>();
+    test_dlp_evaluation<double>();
+    test_dlp_derivatives<double>();
+    test_dlp_next<double>();
+    test_dlp_norms<cpp_bin_float_50>();
+    test_boundary_filters<double>();
+    test_boundary_filters<long double>();
+    test_boundary_filters<cpp_bin_float_50>();
+    test_boundary_lanczos<double>();
+    test_boundary_lanczos<long double>();
+    test_boundary_lanczos<cpp_bin_float_50>();
+
+    test_interior_filter<double>();
+    test_interior_filter<long double>();
+    test_interior_lanczos<double>();
+}

From 1cc2ec907daa8e2d9af51ddb72b239fa92a3acfa Mon Sep 17 00:00:00 2001
From: Nick Thompson <nathompson7@protonmail.com>
Date: Tue, 1 Jan 2019 21:14:50 -0700
Subject: [PATCH 05/10] Add example of differentiating the LIGO data [CI SKIP]

---
 doc/differentiation/lanczos_smoothing.qbk |   12 +-
 doc/graphs/ligo_derivative.svg            | 2146 +++++++++++++++++++++
 2 files changed, 2157 insertions(+), 1 deletion(-)
 create mode 100644 doc/graphs/ligo_derivative.svg

diff --git a/doc/differentiation/lanczos_smoothing.qbk b/doc/differentiation/lanczos_smoothing.qbk
index 213fc964e..38162d508 100644
--- a/doc/differentiation/lanczos_smoothing.qbk
+++ b/doc/differentiation/lanczos_smoothing.qbk
@@ -17,7 +17,7 @@ namespace boost { namespace math { namespace differentiation {
     template <class RandomAccessContainer>
     class lanczos_derivative {
     public:
-        lanczos_smoothing_derivative(RandomAccessContainer const & v,
+        lanczos_derivative(RandomAccessContainer const & v,
                                      typename RandomAccessContainer::value_type spacing = 1,
                                      size_t n = 18,
                                      size_t approximation_order = 3);
@@ -96,6 +96,16 @@ This is far from negligible, and as such we provide API calls which allow the fi
 The implementation follows [@https://doi.org/10.1080/00207160.2012.666348 McDevitt, 2012],
 who vastly expanded the ideas of Lanczos to create a very general framework for numerically differentiating noisy equispaced data.
 
+[heading Example]
+
+We have extracted some data from the [@https://www.gw-openscience.org/data/ LIGO signal] and differentiated it
+using the (/n/, /p/) = (60, 4) Lanczos smoothing derivative, as well as using the (/n/, /p/) = (4, 8) (nonsmoothing) derivative.
+
+[graph ligo_derivative]
+
+The original data is in orange, the smoothing derivative in blue, and the non-smoothing standard finite difference formula is in gray.
+(Each time series has been rescaled to fit in the same graph.)
+We can see that the smoothing derivative tracking the increase and decrease in the trend well, whereas the standard finite difference formula produces nonsense.
 
 
 [heading References]
diff --git a/doc/graphs/ligo_derivative.svg b/doc/graphs/ligo_derivative.svg
new file mode 100644
index 000000000..fd1c68201
--- /dev/null
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From b2c0f9eac27f75dcb53f69a45c931da4270105dd Mon Sep 17 00:00:00 2001
From: Nick Thompson <nathompson7@protonmail.com>
Date: Tue, 1 Jan 2019 21:31:56 -0700
Subject: [PATCH 06/10] Remove grammar errors and reduce point radius. [CI
 SKIP]

---
 doc/differentiation/lanczos_smoothing.qbk |    2 +-
 doc/graphs/ligo_derivative.svg            | 4200 ++++++++++-----------
 2 files changed, 2101 insertions(+), 2101 deletions(-)

diff --git a/doc/differentiation/lanczos_smoothing.qbk b/doc/differentiation/lanczos_smoothing.qbk
index 38162d508..af3437676 100644
--- a/doc/differentiation/lanczos_smoothing.qbk
+++ b/doc/differentiation/lanczos_smoothing.qbk
@@ -105,7 +105,7 @@ using the (/n/, /p/) = (60, 4) Lanczos smoothing derivative, as well as using th
 
 The original data is in orange, the smoothing derivative in blue, and the non-smoothing standard finite difference formula is in gray.
 (Each time series has been rescaled to fit in the same graph.)
-We can see that the smoothing derivative tracking the increase and decrease in the trend well, whereas the standard finite difference formula produces nonsense.
+We can see that the smoothing derivative tracks the increase and decrease in the trend well, whereas the standard finite difference formula produces nonsense and amplifies noise.
 
 
 [heading References]
diff --git a/doc/graphs/ligo_derivative.svg b/doc/graphs/ligo_derivative.svg
index fd1c68201..0fd967eca 100644
--- a/doc/graphs/ligo_derivative.svg
+++ b/doc/graphs/ligo_derivative.svg
@@ -42,2105 +42,2105 @@
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From e070ed17e79e850e4100c9e8dcf65520662c1e4d Mon Sep 17 00:00:00 2001
From: Nick Thompson <nathompson7@protonmail.com>
Date: Wed, 2 Jan 2019 12:38:58 -0700
Subject: [PATCH 07/10] Remove sign-compare warnings. Take advice of cppcheck.
 Grammar in documentation [CI SKIP]

---
 doc/differentiation/lanczos_smoothing.qbk     | 36 +++++++-------
 .../differentiation/lanczos_smoothing.hpp     | 37 ++++++++-------
 test/lanczos_smoothing_test.cpp               | 47 +++++++++----------
 3 files changed, 60 insertions(+), 60 deletions(-)

diff --git a/doc/differentiation/lanczos_smoothing.qbk b/doc/differentiation/lanczos_smoothing.qbk
index af3437676..95218e82b 100644
--- a/doc/differentiation/lanczos_smoothing.qbk
+++ b/doc/differentiation/lanczos_smoothing.qbk
@@ -15,37 +15,39 @@ LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
 namespace boost { namespace math { namespace differentiation {
 
     template <class RandomAccessContainer>
-    class lanczos_derivative {
+    class discrete_lanczos_derivative {
     public:
-        lanczos_derivative(RandomAccessContainer const & v,
-                                     typename RandomAccessContainer::value_type spacing = 1,
-                                     size_t n = 18,
-                                     size_t approximation_order = 3);
+        discrete_lanczos_derivative(RandomAccessContainer const & v,
+                           typename RandomAccessContainer::value_type spacing = 1,
+                           size_t n = 18,
+                           size_t approximation_order = 3);
 
         typename RandomAccessContainer::value_type operator[](size_t i) const;
 
         void reset_data(RandomAccessContainer const &v);
 
         void reset_spacing(Real spacing);
-    }
+    };
 
 }}} // namespaces
 ``
 
 [heading Description]
 
-The function `lanczos_derivative` calculates a finite-difference approximation to the derivative of a noisy sequence of equally-spaced values /v/ at an index /i/.
+The `discrete_lanczos_derivative` class calculates a finite-difference approximation to the derivative of a noisy sequence of equally-spaced values /v/ at an index /i/.
 A basic usage is
 
     std::vector<double> v(500);
     // fill v with noisy data.
     double spacing = 0.001;
-    using boost::math::differentiation::lanczos_derivative;
-    auto lanczos = lanczos_derivative(v, spacing);
+    using boost::math::differentiation::discrete_lanczos_derivative;
+    auto lanczos = discrete_lanczos_derivative(v, spacing);
     double dvdt = lanczos[30];
 
-If the data has variance \u03C3[super 2], then the variance of the computed derivative is roughly \u03C3[super 2]/p/[super 3] /n/[super -3] \u0394 /t/[super -2],
-i.e., it increases cubically with the approximation order /p/, linearly with the data variance, and decreases at the cube of the filter length /n/.
+If the data has variance \u03C3[super 2],
+then the variance of the computed derivative is roughly \u03C3[super 2]/p/[super 3] /n/[super -3] \u0394 /t/[super -2],
+i.e., it increases cubically with the approximation order /p/, linearly with the data variance,
+and decreases at the cube of the filter length /n/.
 In addition, we must not forget the discretization error which is /O/(\u0394 /t/[super /p/]).
 You can play around with the approximation order /p/ and the filter length /n/:
 
@@ -54,11 +56,12 @@ You can play around with the approximation order /p/ and the filter length /n/:
     auto lanczos = lanczos_derivative(v, spacing, n, p);
     double dvdt = lanczos[24];
 
-If /p=2n/, then the Lanczos smoothing derivative is not smoothing:
+If /p=2n/, then the discrete Lanczos derivative is not smoothing:
 It reduces to the standard /2n+1/-point finite-difference formula.
 For /p>2n/, an assertion is hit as the filter is undefined.
 
-In our tests with AWGN, we have found the error decreases monotonically with /n/, as is expected from the theory discussed above.
+In our tests with AWGN, we have found the error decreases monotonically with /n/,
+as is expected from the theory discussed above.
 So the choice of /n/ is simple:
 As high as possible given your speed requirements (larger /n/ implies a longer filter and hence more compute),
 balanced against the danger of overfitting and averaging over non-stationarity.
@@ -68,15 +71,16 @@ If your signal is believed to be a polynomial,
 it does not make sense to set /p/ to larger than the polynomial degree-
 though it may be sensible to take /p/ less than the polynomial degree.
 
-For a sinusoidal signal contaminated with AWGN, we ran a few tests showing that for SNR = 1, p = n/8 gave the best results,
+For a sinusoidal signal contaminated with AWGN, we ran a few tests showing that for SNR = 1,
+p = n/8 gave the best results,
 for SNR = 10, p = n/7 was the best, and for SNR = 100, p = n/6 was the most reasonable choice.
 For SNR = 0.1, the method appears to be useless.
 The user is urged to use these results with caution-they have no theoretical backing and are extrapolated from a single case.
 
 The filters are (regrettably) computed at runtime-the vast number of combinations of approximation order and filter length makes the number of filters that must be stored excessive for compile-time data.
-Hence the constructor call computes the filters.
+The constructor call computes the filters.
 Since each filter has length /2n+1/ and there are /n/ filters, whose element each consist of /p/ summands,
-the complexity of the constructor call is O(/n/[super 2]/p/).e
+the complexity of the constructor call is O(/n/[super 2]/p/).
 This is not cheap-though for most cases small /p/ and /n/ not too large (< 20) is desired.
 However, for concreteness, on the author's 2.7GHz Intel laptop CPU, the /n=16/, /p=3/ filter takes 9 microseconds to compute.
 This is far from negligible, and as such we provide API calls which allow the filters to be used with multiple data:
diff --git a/include/boost/math/differentiation/lanczos_smoothing.hpp b/include/boost/math/differentiation/lanczos_smoothing.hpp
index 5696fc454..11f346c41 100644
--- a/include/boost/math/differentiation/lanczos_smoothing.hpp
+++ b/include/boost/math/differentiation/lanczos_smoothing.hpp
@@ -16,7 +16,7 @@ namespace detail {
 template <typename Real>
 class discrete_legendre {
   public:
-    discrete_legendre(int n) : m_n{n}
+    explicit discrete_legendre(int n) : m_n{n}
     {
         // The integer n indexes a family of discrete Legendre polynomials indexed by k <= 2*n
     }
@@ -128,7 +128,7 @@ class discrete_legendre {
 };
 
 template <class Real>
-std::vector<Real> interior_filter(int n, int p) {
+std::vector<Real> interior_filter(size_t n, size_t p) {
     // We could make the filter length n, as f[0] = 0,
     // but that'd make the indexing awkward when applying the filter.
     std::vector<Real> f(n + 1, 0);
@@ -136,7 +136,7 @@ std::vector<Real> interior_filter(int n, int p) {
     std::vector<Real> coeffs(p+1, std::numeric_limits<Real>::quiet_NaN());
     dlp.initialize_recursion(0);
     coeffs[1] = 1/dlp.norm_sq(1);
-    for (int l = 3; l < p + 1; l += 2)
+    for (size_t l = 3; l < p + 1; l += 2)
     {
         dlp.next_prime();
         coeffs[l] = dlp.next_prime()/ dlp.norm_sq(l);
@@ -147,7 +147,7 @@ std::vector<Real> interior_filter(int n, int p) {
         Real arg = Real(j) / Real(n);
         dlp.initialize_recursion(arg);
         f[j] = coeffs[1]*arg;
-        for (int l = 3; l <= p; l += 2)
+        for (size_t l = 3; l <= p; l += 2)
         {
             dlp.next();
             f[j] += coeffs[l]*dlp.next();
@@ -158,7 +158,8 @@ std::vector<Real> interior_filter(int n, int p) {
 }
 
 template <class Real>
-std::vector<Real> boundary_filter(int n, int p, int s) {
+std::vector<Real> boundary_filter(size_t n, size_t p, int64_t s)
+{
     std::vector<Real> f(2 * n + 1, 0);
     auto dlp = discrete_legendre<Real>(n);
     Real sn = Real(s) / Real(n);
@@ -166,7 +167,7 @@ std::vector<Real> boundary_filter(int n, int p, int s) {
     dlp.initialize_recursion(sn);
     coeffs[0] = 0;
     coeffs[1] = 1/dlp.norm_sq(1);
-    for (int l = 2; l < p + 1; ++l)
+    for (size_t l = 2; l < p + 1; ++l)
     {
         // Calculation of the norms is common to all filters,
         // so it seems like an obvious optimization target.
@@ -176,14 +177,14 @@ std::vector<Real> boundary_filter(int n, int p, int s) {
         coeffs[l] = dlp.next_prime()/ dlp.norm_sq(l);
     }
 
-    for (int k = 0; k < f.size(); ++k)
+    for (size_t k = 0; k < f.size(); ++k)
     {
-        int j = k - n;
+        Real j = Real(k) - Real(n);
         f[k] = 0;
-        Real arg = Real(j) / Real(n);
+        Real arg = j/Real(n);
         dlp.initialize_recursion(arg);
         f[k] = coeffs[1]*arg;
-        for (int l = 2; l <= p; ++l)
+        for (size_t l = 2; l <= p; ++l)
         {
             f[k] += coeffs[l]*dlp.next();
         }
@@ -195,13 +196,13 @@ std::vector<Real> boundary_filter(int n, int p, int s) {
 } // namespace detail
 
 template <class RandomAccessContainer>
-class lanczos_derivative {
+class discrete_lanczos_derivative {
 public:
     using Real = typename RandomAccessContainer::value_type;
-    lanczos_derivative(RandomAccessContainer const &v,
-                       Real spacing = 1,
-                       int filter_length = 18,
-                       int approximation_order = 3)
+    discrete_lanczos_derivative(RandomAccessContainer const &v,
+                       Real const & spacing = 1,
+                       size_t filter_length = 18,
+                       size_t approximation_order = 3)
         : m_v{v}, dt{spacing}
     {
         BOOST_ASSERT_MSG(approximation_order <= 2 * filter_length,
@@ -218,8 +219,9 @@ public:
         for (size_t i = 0; i < filter_length; ++i)
         {
             // s = i - n;
+            int64_t s = static_cast<int64_t>(i) - static_cast<int64_t>(filter_length);
             boundary_filters[i] = detail::boundary_filter<Real>(
-                filter_length, approximation_order, i - filter_length);
+                filter_length, approximation_order, s);
         }
     }
 
@@ -230,7 +232,7 @@ public:
         m_v = v;
     }
 
-    void reset_spacing(Real spacing)
+    void reset_spacing(Real const & spacing)
     {
         BOOST_ASSERT_MSG(spacing > 0, "Spacing between samples must be > 0.");
         dt = spacing;
@@ -289,7 +291,6 @@ private:
     Real dt;
 };
 
-// We can also implement lanczos_acceleration, but let's get the API for lanczos_derivative nailed down before doing so.
 
 } // namespace differentiation
 } // namespace math
diff --git a/test/lanczos_smoothing_test.cpp b/test/lanczos_smoothing_test.cpp
index fd7315e88..aef9d94b2 100644
--- a/test/lanczos_smoothing_test.cpp
+++ b/test/lanczos_smoothing_test.cpp
@@ -18,7 +18,7 @@ using std::pow;
 using std::sqrt;
 using boost::multiprecision::cpp_bin_float_50;
 using boost::multiprecision::cpp_bin_float_100;
-using boost::math::differentiation::lanczos_derivative;
+using boost::math::differentiation::discrete_lanczos_derivative;
 using boost::math::differentiation::detail::discrete_legendre;
 using boost::math::differentiation::detail::interior_filter;
 using boost::math::differentiation::detail::boundary_filter;
@@ -156,7 +156,7 @@ void test_interior_filter()
             // Since we only store half the filter coefficients,
             // we need to reindex the moment sums:
             Real sum = 0;
-            for (int j = 0; j < f.size(); ++j)
+            for (size_t j = 0; j < f.size(); ++j)
             {
                 sum += j*f[j];
             }
@@ -165,7 +165,7 @@ void test_interior_filter()
             for (int l = 3; l <= p; l += 2)
             {
                 sum = 0;
-                for (int j = 0; j < f.size(); ++j)
+                for (size_t j = 0; j < f.size(); ++j)
                 {
                     // The condition number of this sum is infinite!
                     // No need to get to worked up about the tolerance.
@@ -189,15 +189,13 @@ void test_interior_lanczos()
     std::cout << "Testing interior Lanczos on type " << typeid(Real).name() << "\n";
     Real tol = std::numeric_limits<Real>::epsilon();
     std::vector<Real> v(500);
-    for (auto & x : v)
-    {
-        x = 7;
-    }
+    std::fill(v.begin(), v.end(), 7);
+
     for (size_t n = 1; n < 10; ++n)
     {
         for (size_t p = 2; p < 2*n; p += 2)
         {
-            auto lsd = lanczos_derivative(v, 0.1, n, p);
+            auto lsd = discrete_lanczos_derivative(v, 0.1, n, p);
             for (size_t m = n; m < v.size() - n; ++m)
             {
                 Real dvdt = lsd[m];
@@ -216,7 +214,7 @@ void test_interior_lanczos()
     {
         for (size_t p = 2; p < 2*n; p += 2)
         {
-            auto lsd = lanczos_derivative(v, Real(1), n, p);
+            auto lsd = discrete_lanczos_derivative(v, Real(1), n, p);
             for (size_t m = n; m < v.size() - n; ++m)
             {
                 Real dvdt = lsd[m];
@@ -240,7 +238,7 @@ void test_interior_lanczos()
     {
         for (size_t p = 2; p < 2*n; p += 2)
         {
-            auto lsd = lanczos_derivative(v, Real(1), n, p);
+            auto lsd = discrete_lanczos_derivative(v, Real(1), n, p);
             for (size_t m = n; m < v.size() - n; ++m)
             {
                 BOOST_CHECK_CLOSE_FRACTION(lsd[m], Real(7), Real(0.0042));
@@ -258,7 +256,7 @@ void test_interior_lanczos()
     {
         for (size_t p = 2; p < 2*n; p += 2)
         {
-            auto lsd = lanczos_derivative(v, Real(1), n, p);
+            auto lsd = discrete_lanczos_derivative(v, Real(1), n, p);
             for (size_t m = n; m < v.size() - n; ++m)
             {
                 BOOST_CHECK_CLOSE_FRACTION(lsd[m], Real(30*m + 7), Real(0.00008));
@@ -277,7 +275,7 @@ void test_interior_lanczos()
     {
         for (size_t p = 3; p < 100 && p < n/2; p += 2)
         {
-            auto lsd = lanczos_derivative(v, Real(1), n, p);
+            auto lsd = discrete_lanczos_derivative(v, Real(1), n, p);
 
             for (size_t m = n; m < v.size() - n && m < n + 10; ++m)
             {
@@ -313,9 +311,9 @@ void test_boundary_filters()
 
                 sum = 0;
                 c = 0;
-                for (int k = 0; k < f.size(); ++k)
+                for (size_t k = 0; k < f.size(); ++k)
                 {
-                    int j = k - n;
+                    Real j = Real(k) - Real(n);
                     // note the shifted index here:
                     Real x = (j-s)*f[k];
                     Real y = x - c;
@@ -330,12 +328,12 @@ void test_boundary_filters()
                 {
                     sum = 0;
                     c = 0;
-                    for (int k = 0; k < f.size(); ++k)
+                    for (size_t k = 0; k < f.size(); ++k)
                     {
-                        int j = k - n;
+                        Real j = Real(k) - Real(n);
                         // The condition number of this sum is infinite!
                         // No need to get to worked up about the tolerance.
-                        Real x = pow(Real(j-s), l)*f[k];
+                        Real x = pow(j-s, l)*f[k];
                         Real y = x - c;
                         Real t = sum + y;
                         c = (t-sum) -y;
@@ -360,16 +358,13 @@ void test_boundary_lanczos()
 {
     std::cout << "Testing Lanczos boundary on type " << typeid(Real).name() << "\n";
     Real tol = std::numeric_limits<Real>::epsilon();
-    std::vector<Real> v(500);
-    for (auto & x : v)
-    {
-        x = 7;
-    }
+    std::vector<Real> v(500, 7);
+
     for (size_t n = 1; n < 10; ++n)
     {
         for (size_t p = 2; p < 2*n; ++p)
         {
-            auto lsd = lanczos_derivative(v, 0.0125, n, p);
+            auto lsd = discrete_lanczos_derivative(v, 0.0125, n, p);
             for (size_t m = 0; m < n; ++m)
             {
                 Real dvdt = lsd[m];
@@ -392,7 +387,7 @@ void test_boundary_lanczos()
     {
         for (size_t p = 2; p < 2*n; ++p)
         {
-            auto lsd = lanczos_derivative(v, Real(1), n, p);
+            auto lsd = discrete_lanczos_derivative(v, Real(1), n, p);
             for (size_t m = 0; m < n; ++m)
             {
                 Real dvdt = lsd[m];
@@ -416,7 +411,7 @@ void test_boundary_lanczos()
     {
         for (size_t p = 2; p < 2*n; ++p)
         {
-            auto lsd = lanczos_derivative(v, Real(1), n, p);
+            auto lsd = discrete_lanczos_derivative(v, Real(1), n, p);
             for (size_t m = 0; m < v.size(); ++m)
             {
                 BOOST_CHECK_CLOSE_FRACTION(lsd[m], 30*m+7, 30*sqrt(tol));
@@ -439,7 +434,7 @@ void test_boundary_lanczos()
     {
         for (size_t p = 2; p < n; ++p)
         {
-            auto lsd = lanczos_derivative(v, Real(1), n, p);
+            auto lsd = discrete_lanczos_derivative(v, Real(1), n, p);
             for (size_t m = 0; m < v.size(); ++m)
             {
                 BOOST_CHECK_CLOSE_FRACTION(lsd[m], 30*m+7, 0.005);

From d76d49533ab0d917579a471f1d542c02612ed421 Mon Sep 17 00:00:00 2001
From: Nick Thompson <nathompson7@protonmail.com>
Date: Wed, 2 Jan 2019 12:53:04 -0700
Subject: [PATCH 08/10] Cleanup [CI SKIP]

---
 doc/differentiation/lanczos_smoothing.qbk     | 13 ++++----
 .../differentiation/lanczos_smoothing.hpp     | 30 +++++++++----------
 2 files changed, 22 insertions(+), 21 deletions(-)

diff --git a/doc/differentiation/lanczos_smoothing.qbk b/doc/differentiation/lanczos_smoothing.qbk
index 95218e82b..f75be257d 100644
--- a/doc/differentiation/lanczos_smoothing.qbk
+++ b/doc/differentiation/lanczos_smoothing.qbk
@@ -12,24 +12,25 @@ LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
 ``
 #include <boost/math/differentiation/lanczos_smoothing.hpp>
 
-namespace boost { namespace math { namespace differentiation {
+namespace boost::math::differentiation {
 
     template <class RandomAccessContainer>
     class discrete_lanczos_derivative {
     public:
+        using Real = typename RandomAccessContainer::value_type;
         discrete_lanczos_derivative(RandomAccessContainer const & v,
-                           typename RandomAccessContainer::value_type spacing = 1,
-                           size_t n = 18,
-                           size_t approximation_order = 3);
+                                    Real spacing = 1,
+                                    size_t n = 18,
+                                    size_t approximation_order = 3);
 
-        typename RandomAccessContainer::value_type operator[](size_t i) const;
+        Real operator[](size_t i) const;
 
         void reset_data(RandomAccessContainer const &v);
 
         void reset_spacing(Real spacing);
     };
 
-}}} // namespaces
+} // namespaces
 ``
 
 [heading Description]
diff --git a/include/boost/math/differentiation/lanczos_smoothing.hpp b/include/boost/math/differentiation/lanczos_smoothing.hpp
index 11f346c41..1a57f2020 100644
--- a/include/boost/math/differentiation/lanczos_smoothing.hpp
+++ b/include/boost/math/differentiation/lanczos_smoothing.hpp
@@ -8,15 +8,18 @@
 #include <vector>
 #include <boost/assert.hpp>
 
-namespace boost {
-namespace math {
-namespace differentiation {
+namespace boost::math::differentiation {
 
 namespace detail {
 template <typename Real>
 class discrete_legendre {
   public:
-    explicit discrete_legendre(int n) : m_n{n}
+    explicit discrete_legendre(size_t n) : m_n{n}, m_r{2},
+                                           m_x{std::numeric_limits<Real>::quiet_NaN()},
+                                           m_qrm2{std::numeric_limits<Real>::quiet_NaN()},
+                                           m_qrm1{std::numeric_limits<Real>::quiet_NaN()},
+                                           m_qrm2p{std::numeric_limits<Real>::quiet_NaN()},
+                                           m_qrm1p{std::numeric_limits<Real>::quiet_NaN()}
     {
         // The integer n indexes a family of discrete Legendre polynomials indexed by k <= 2*n
     }
@@ -68,7 +71,7 @@ class discrete_legendre {
     }
 
 
-    Real operator()(Real x, int k)
+    Real operator()(Real x, size_t k)
     {
         BOOST_ASSERT_MSG(k <= 2 * m_n, "r <= 2n is required.");
         if (k == 0)
@@ -82,7 +85,7 @@ class discrete_legendre {
         Real qrm2 = 1;
         Real qrm1 = x;
         Real N = 2 * m_n + 1;
-        for (int r = 2; r <= k; ++r) {
+        for (size_t r = 2; r <= k; ++r) {
             Real num = (r - 1) * (N * N - (r - 1) * (r - 1)) * qrm2;
             Real tmp = (2 * r - 1) * x * qrm1 - num / Real(4 * m_n * m_n);
             qrm2 = qrm1;
@@ -91,7 +94,7 @@ class discrete_legendre {
         return qrm1;
     }
 
-    Real prime(Real x, int k) {
+    Real prime(Real x, size_t k) {
         BOOST_ASSERT_MSG(k <= 2 * m_n, "r <= 2n is required.");
         if (k == 0) {
             return 0;
@@ -104,7 +107,7 @@ class discrete_legendre {
         Real qrm2p = 0;
         Real qrm1p = 1;
         Real N = 2 * m_n + 1;
-        for (int r = 2; r <= k; ++r) {
+        for (size_t r = 2; r <= k; ++r) {
             Real s =
                 (r - 1) * (N * N - (r - 1) * (r - 1)) / Real(4 * m_n * m_n);
             Real tmp1 = ((2 * r - 1) * x * qrm1 - s * qrm2) / r;
@@ -118,13 +121,13 @@ class discrete_legendre {
     }
 
   private:
-    int m_n;
+    size_t m_n;
+    size_t m_r;
+    Real m_x;
     Real m_qrm2;
     Real m_qrm1;
     Real m_qrm2p;
     Real m_qrm1p;
-    int m_r;
-    Real m_x;
 };
 
 template <class Real>
@@ -291,8 +294,5 @@ private:
     Real dt;
 };
 
-
-} // namespace differentiation
-} // namespace math
-} // namespace boost
+} // namespaces
 #endif

From 95f993c9bcffa05cd1e7ef741589b3b525c92b33 Mon Sep 17 00:00:00 2001
From: Nick Thompson <nathompson7@protonmail.com>
Date: Thu, 3 Jan 2019 11:55:29 -0700
Subject: [PATCH 09/10] Add denoising second derivative.

---
 doc/differentiation/lanczos_smoothing.qbk     |  11 +-
 .../differentiation/lanczos_smoothing.hpp     | 207 ++++++++++++++----
 test/lanczos_smoothing_test.cpp               |  89 ++++++++
 3 files changed, 263 insertions(+), 44 deletions(-)

diff --git a/doc/differentiation/lanczos_smoothing.qbk b/doc/differentiation/lanczos_smoothing.qbk
index f75be257d..169836f9f 100644
--- a/doc/differentiation/lanczos_smoothing.qbk
+++ b/doc/differentiation/lanczos_smoothing.qbk
@@ -14,7 +14,7 @@ LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
 
 namespace boost::math::differentiation {
 
-    template <class RandomAccessContainer>
+    template <class RandomAccessContainer, size_t order=1>
     class discrete_lanczos_derivative {
     public:
         using Real = typename RandomAccessContainer::value_type;
@@ -45,6 +45,15 @@ A basic usage is
     auto lanczos = discrete_lanczos_derivative(v, spacing);
     double dvdt = lanczos[30];
 
+Noise-suppressing second derivatives can also be computed.
+The syntax is as follows:
+
+    std::vector<double> v(500);
+    // fill v with noisy data.
+    auto lanczos = lanczos_derivative<decltype(v), 2>(v, spacing);
+    // evaluate:
+    double dvdt = lanczos[25];
+
 If the data has variance \u03C3[super 2],
 then the variance of the computed derivative is roughly \u03C3[super 2]/p/[super 3] /n/[super -3] \u0394 /t/[super -2],
 i.e., it increases cubically with the approximation order /p/, linearly with the data variance,
diff --git a/include/boost/math/differentiation/lanczos_smoothing.hpp b/include/boost/math/differentiation/lanczos_smoothing.hpp
index 1a57f2020..c3b78f0a4 100644
--- a/include/boost/math/differentiation/lanczos_smoothing.hpp
+++ b/include/boost/math/differentiation/lanczos_smoothing.hpp
@@ -19,7 +19,9 @@ class discrete_legendre {
                                            m_qrm2{std::numeric_limits<Real>::quiet_NaN()},
                                            m_qrm1{std::numeric_limits<Real>::quiet_NaN()},
                                            m_qrm2p{std::numeric_limits<Real>::quiet_NaN()},
-                                           m_qrm1p{std::numeric_limits<Real>::quiet_NaN()}
+                                           m_qrm1p{std::numeric_limits<Real>::quiet_NaN()},
+                                           m_qrm2pp{std::numeric_limits<Real>::quiet_NaN()},
+                                           m_qrm1pp{std::numeric_limits<Real>::quiet_NaN()}
     {
         // The integer n indexes a family of discrete Legendre polynomials indexed by k <= 2*n
     }
@@ -35,10 +37,16 @@ class discrete_legendre {
 
     void initialize_recursion(Real x)
     {
+        using std::abs;
+        BOOST_ASSERT_MSG(abs(x) <= 1, "Three term recurrence is stable only for |x| <=1.");
         m_qrm2 = 1;
         m_qrm1 = x;
+        // Derivatives:
         m_qrm2p = 0;
         m_qrm1p = 1;
+        // Second derivatives:
+        m_qrm2pp = 0;
+        m_qrm1pp = 0;
 
         m_r = 2;
         m_x = x;
@@ -58,8 +66,7 @@ class discrete_legendre {
     Real next_prime()
     {
         Real N = 2 * m_n + 1;
-        Real s =
-            (m_r - 1) * (N * N - (m_r - 1) * (m_r - 1)) / Real(4 * m_n * m_n);
+        Real s = (m_r - 1) * (N * N - (m_r - 1) * (m_r - 1)) / Real(4 * m_n * m_n);
         Real tmp1 = ((2 * m_r - 1) * m_x * m_qrm1 - s * m_qrm2) / m_r;
         Real tmp2 = ((2 * m_r - 1) * (m_qrm1 + m_x * m_qrm1p) - s * m_qrm2p) / m_r;
         m_qrm2 = m_qrm1;
@@ -70,6 +77,23 @@ class discrete_legendre {
         return m_qrm1p;
     }
 
+    Real next_dbl_prime()
+    {
+        Real N = 2*m_n + 1;
+        Real trm1 = 2*m_r - 1;
+        Real s = (m_r - 1) * (N * N - (m_r - 1) * (m_r - 1)) / Real(4 * m_n * m_n);
+        Real rqrpp = 2*trm1*m_qrm1p + trm1*m_x*m_qrm1pp - s*m_qrm2pp;
+        Real tmp1 = ((2 * m_r - 1) * m_x * m_qrm1 - s * m_qrm2) / m_r;
+        Real tmp2 = ((2 * m_r - 1) * (m_qrm1 + m_x * m_qrm1p) - s * m_qrm2p) / m_r;
+        m_qrm2 = m_qrm1;
+        m_qrm1 = tmp1;
+        m_qrm2p = m_qrm1p;
+        m_qrm1p = tmp2;
+        m_qrm2pp = m_qrm1pp;
+        m_qrm1pp = rqrpp/m_r;
+        ++m_r;
+        return m_qrm1pp;
+    }
 
     Real operator()(Real x, size_t k)
     {
@@ -128,6 +152,8 @@ class discrete_legendre {
     Real m_qrm1;
     Real m_qrm2p;
     Real m_qrm1p;
+    Real m_qrm2pp;
+    Real m_qrm1pp;
 };
 
 template <class Real>
@@ -196,35 +222,91 @@ std::vector<Real> boundary_filter(size_t n, size_t p, int64_t s)
     return f;
 }
 
+template <class Real>
+std::vector<Real> acceleration_boundary_filter(size_t n, size_t p, int64_t s)
+{
+    BOOST_ASSERT_MSG(p <= 2*n, "Approximation order must be <= 2*n");
+    BOOST_ASSERT_MSG(p > 2, "Approximation order must be > 2");
+    std::vector<Real> f(2 * n + 1, 0);
+    auto dlp = discrete_legendre<Real>(n);
+    Real sn = Real(s) / Real(n);
+    std::vector<Real> coeffs(p+2, std::numeric_limits<Real>::quiet_NaN());
+    dlp.initialize_recursion(sn);
+    coeffs[0] = 0;
+    coeffs[1] = 0;
+    for (size_t l = 2; l < p + 2; ++l)
+    {
+        coeffs[l] = dlp.next_dbl_prime()/ dlp.norm_sq(l);
+    }
+
+    for (size_t k = 0; k < f.size(); ++k)
+    {
+        Real j = Real(k) - Real(n);
+        f[k] = 0;
+        Real arg = j/Real(n);
+        dlp.initialize_recursion(arg);
+        f[k] = coeffs[1]*arg;
+        for (size_t l = 2; l <= p; ++l)
+        {
+            f[k] += coeffs[l]*dlp.next();
+        }
+        f[k] /= (n * n * n);
+    }
+    return f;
+}
+
+
 } // namespace detail
 
-template <class RandomAccessContainer>
+template <class RandomAccessContainer, size_t order = 1>
 class discrete_lanczos_derivative {
 public:
     using Real = typename RandomAccessContainer::value_type;
     discrete_lanczos_derivative(RandomAccessContainer const &v,
                        Real const & spacing = 1,
-                       size_t filter_length = 18,
+                       size_t n = 18,
                        size_t approximation_order = 3)
         : m_v{v}, dt{spacing}
     {
-        BOOST_ASSERT_MSG(approximation_order <= 2 * filter_length,
-                         "The approximation order must be <= 2n");
-        BOOST_ASSERT_MSG(approximation_order >= 2,
-                         "The approximation order must be >= 2");
         BOOST_ASSERT_MSG(spacing > 0, "Spacing between samples must be > 0.");
         using std::size;
-        BOOST_ASSERT_MSG(size(v) >= filter_length,
+        BOOST_ASSERT_MSG(size(v) >= 2*n+1,
             "Vector must be at least as long as the filter length");
-        m_f = detail::interior_filter<Real>(filter_length, approximation_order);
 
-        boundary_filters.resize(filter_length);
-        for (size_t i = 0; i < filter_length; ++i)
+        if constexpr (order == 1)
         {
-            // s = i - n;
-            int64_t s = static_cast<int64_t>(i) - static_cast<int64_t>(filter_length);
-            boundary_filters[i] = detail::boundary_filter<Real>(
-                filter_length, approximation_order, s);
+            BOOST_ASSERT_MSG(approximation_order <= 2 * n,
+                             "The approximation order must be <= 2n");
+            BOOST_ASSERT_MSG(approximation_order >= 2,
+                             "The approximation order must be >= 2");
+            m_f = detail::interior_filter<Real>(n, approximation_order);
+
+            boundary_filters.resize(n);
+            for (size_t i = 0; i < n; ++i)
+            {
+                // s = i - n;
+                int64_t s = static_cast<int64_t>(i) - static_cast<int64_t>(n);
+                boundary_filters[i] = detail::boundary_filter<Real>(n, approximation_order, s);
+            }
+        }
+        else if constexpr (order == 2)
+        {
+            auto f = detail::acceleration_boundary_filter<Real>(n, approximation_order, 0);
+            m_f.resize(n+1);
+            for (size_t i = 0; i < m_f.size(); ++i)
+            {
+                m_f[i] = f[i+n];
+            }
+            boundary_filters.resize(n);
+            for (size_t i = 0; i < n; ++i)
+            {
+                int64_t s = static_cast<int64_t>(i) - static_cast<int64_t>(n);
+                boundary_filters[i] = detail::acceleration_boundary_filter<Real>(n, approximation_order, s);
+            }
+        }
+        else
+        {
+            BOOST_ASSERT_MSG(false, "Derivatives of order 3 and higher are not implemented.");
         }
     }
 
@@ -248,38 +330,77 @@ public:
 
     Real operator[](size_t i) const
     {
-        using std::size;
-        if (i >= m_f.size() - 1 && i <= size(m_v) - m_f.size())
+        if constexpr (order==1)
         {
-            Real dv = 0;
-            for (size_t j = 1; j < m_f.size(); ++j)
+            using std::size;
+            if (i >= m_f.size() - 1 && i <= size(m_v) - m_f.size())
             {
-                dv += m_f[j] * (m_v[i + j] - m_v[i - j]);
+                Real dv = 0;
+                for (size_t j = 1; j < m_f.size(); ++j)
+                {
+                    dv += m_f[j] * (m_v[i + j] - m_v[i - j]);
+                }
+                return dv / dt;
             }
-            return dv / dt;
-        }
 
-        // m_f.size() = N+1
-        if (i < m_f.size() - 1)
-        {
-            auto &f = boundary_filters[i];
-            Real dv = 0;
-            for (size_t j = 0; j < f.size(); ++j) {
-                dv += f[j] * m_v[j];
-            }
-            return dv / dt;
-        }
-
-        if (i > size(m_v) - m_f.size() && i < size(m_v))
-        {
-            int k = size(m_v) - 1 - i;
-            auto &f = boundary_filters[k];
-            Real dv = 0;
-            for (size_t j = 0; j < f.size(); ++j)
+            // m_f.size() = N+1
+            if (i < m_f.size() - 1)
             {
-                dv += f[j] * m_v[m_v.size() - 1 - j];
+                auto &f = boundary_filters[i];
+                Real dv = 0;
+                for (size_t j = 0; j < f.size(); ++j) {
+                    dv += f[j] * m_v[j];
+                }
+                return dv / dt;
+            }
+
+            if (i > size(m_v) - m_f.size() && i < size(m_v))
+            {
+                int k = size(m_v) - 1 - i;
+                auto &f = boundary_filters[k];
+                Real dv = 0;
+                for (size_t j = 0; j < f.size(); ++j)
+                {
+                    dv += f[j] * m_v[m_v.size() - 1 - j];
+                }
+                return -dv / dt;
+            }
+        }
+        else if constexpr (order==2)
+        {
+            using std::size;
+            if (i >= m_f.size() - 1 && i <= size(m_v) - m_f.size())
+            {
+                Real d2v = m_f[0]*m_v[i];
+                for (size_t j = 1; j < m_f.size(); ++j)
+                {
+                    d2v += m_f[j] * (m_v[i + j] + m_v[i - j]);
+                }
+                return d2v / (dt*dt);
+            }
+
+            // m_f.size() = N+1
+            if (i < m_f.size() - 1)
+            {
+                auto &f = boundary_filters[i];
+                Real d2v = 0;
+                for (size_t j = 0; j < f.size(); ++j) {
+                    d2v += f[j] * m_v[j];
+                }
+                return d2v / (dt*dt);
+            }
+
+            if (i > size(m_v) - m_f.size() && i < size(m_v))
+            {
+                int k = size(m_v) - 1 - i;
+                auto &f = boundary_filters[k];
+                Real d2v = 0;
+                for (size_t j = 0; j < f.size(); ++j)
+                {
+                    d2v += f[j] * m_v[m_v.size() - 1 - j];
+                }
+                return d2v / dt;
             }
-            return -dv / dt;
         }
 
         // OOB access:
diff --git a/test/lanczos_smoothing_test.cpp b/test/lanczos_smoothing_test.cpp
index aef9d94b2..bebc2d562 100644
--- a/test/lanczos_smoothing_test.cpp
+++ b/test/lanczos_smoothing_test.cpp
@@ -143,6 +143,19 @@ void test_dlp_derivatives()
     BOOST_CHECK_CLOSE_FRACTION(q2p, expected, tol);
 }
 
+template<class Real>
+void test_dlp_second_derivative()
+{
+    std::cout << "Testing Discrete Legendre polynomial derivatives on type " << typeid(Real).name() << "\n";
+    int n = 25;
+    auto dlp = discrete_legendre<Real>(n);
+    Real x = Real(1)/Real(3);
+    dlp.initialize_recursion(x);
+    Real q2pp = dlp.next_dbl_prime();
+    BOOST_TEST(q2pp == 3);
+}
+
+
 template<class Real>
 void test_interior_filter()
 {
@@ -444,8 +457,82 @@ void test_boundary_lanczos()
 
 }
 
+template<class Real>
+void test_acceleration_filters()
+{
+    Real eps = std::numeric_limits<Real>::epsilon();
+    for (size_t n = 1; n < 8; ++n)
+    {
+        for(size_t p = 3; p <= 2*n; ++p)
+        {
+            for(int64_t s = -int64_t(n); s <= 0 /*int64_t(n)*/; ++s)
+            {
+                auto f = boost::math::differentiation::detail::acceleration_boundary_filter<Real>(n,p,s);
+                Real sum = 0;
+                for (auto & x : f)
+                {
+                    sum += x;
+                }
+                BOOST_CHECK_SMALL(abs(sum), sqrt(eps));
+
+                sum = 0;
+                for (size_t k = 0; k < f.size(); ++k)
+                {
+                    Real j = Real(k) - Real(n);
+                    sum += (j-s)*f[k];
+                }
+                BOOST_CHECK_SMALL(sum, sqrt(eps));
+
+                sum = 0;
+                for (size_t k = 0; k < f.size(); ++k)
+                {
+                    Real j = Real(k) - Real(n);
+                    sum += (j-s)*(j-s)*f[k];
+                }
+                BOOST_CHECK_CLOSE_FRACTION(sum, 2, 4*sqrt(eps));
+                // More moments vanish, but the moment sums become increasingly poorly conditioned.
+                // We can check them later if we wish.
+            }
+        }
+    }
+}
+
+template<class Real>
+void test_lanczos_acceleration()
+{
+    Real eps = std::numeric_limits<Real>::epsilon();
+    std::vector<Real> v(100, 7);
+    auto lanczos = discrete_lanczos_derivative<decltype(v), 2>(v, Real(1), 4, 3);
+    for (size_t i = 0; i < v.size(); ++i)
+    {
+        BOOST_CHECK_SMALL(lanczos[i], eps);
+    }
+
+    for(size_t i = 0; i < v.size(); ++i)
+    {
+        v[i] = 7*i + 6;
+    }
+    for (size_t i = 0; i < v.size(); ++i)
+    {
+        BOOST_CHECK_SMALL(lanczos[i], 200*eps);
+    }
+
+    for(size_t i = 0; i < v.size(); ++i)
+    {
+        v[i] = 7*i*i + 9*i + 6;
+    }
+    for (size_t i = 0; i < v.size(); ++i)
+    {
+        BOOST_CHECK_CLOSE_FRACTION(lanczos[i], 14, 1000*eps);
+    }
+
+
+}
+
 BOOST_AUTO_TEST_CASE(lanczos_smoothing_test)
 {
+    test_acceleration_filters<double>();
+    test_dlp_second_derivative<double>();
     test_dlp_norms<double>();
     test_dlp_evaluation<double>();
     test_dlp_derivatives<double>();
@@ -461,4 +548,6 @@ BOOST_AUTO_TEST_CASE(lanczos_smoothing_test)
     test_interior_filter<double>();
     test_interior_filter<long double>();
     test_interior_lanczos<double>();
+
+    test_lanczos_acceleration<double>();
 }

From bfa7619954c317b0c37da1c9ef096e33b684a536 Mon Sep 17 00:00:00 2001
From: Nick Thompson <nathompson7@protonmail.com>
Date: Fri, 4 Jan 2019 12:50:58 -0700
Subject: [PATCH 10/10] Refactor so as to not store a reference member, make
 call threadsafe, compute entire vector in one go.

---
 doc/differentiation/lanczos_smoothing.qbk     |  52 ++---
 .../differentiation/lanczos_smoothing.hpp     | 194 +++++++++++++-----
 test/lanczos_smoothing_test.cpp               |  65 +++---
 3 files changed, 205 insertions(+), 106 deletions(-)

diff --git a/doc/differentiation/lanczos_smoothing.qbk b/doc/differentiation/lanczos_smoothing.qbk
index 169836f9f..fb3ba10eb 100644
--- a/doc/differentiation/lanczos_smoothing.qbk
+++ b/doc/differentiation/lanczos_smoothing.qbk
@@ -14,18 +14,18 @@ LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
 
 namespace boost::math::differentiation {
 
-    template <class RandomAccessContainer, size_t order=1>
+    template <class Real, size_t order=1>
     class discrete_lanczos_derivative {
     public:
-        using Real = typename RandomAccessContainer::value_type;
-        discrete_lanczos_derivative(RandomAccessContainer const & v,
-                                    Real spacing = 1,
+        discrete_lanczos_derivative(Real spacing,
                                     size_t n = 18,
                                     size_t approximation_order = 3);
 
-        Real operator[](size_t i) const;
+        template<class RandomAccessContainer>
+        Real operator()(RandomAccessContainer const & v, size_t i) const;
 
-        void reset_data(RandomAccessContainer const &v);
+        template<class RandomAccessContainer>
+        RandomAccessContainer operator()(RandomAccessContainer const & v) const;
 
         void reset_spacing(Real spacing);
     };
@@ -35,24 +35,29 @@ namespace boost::math::differentiation {
 
 [heading Description]
 
-The `discrete_lanczos_derivative` class calculates a finite-difference approximation to the derivative of a noisy sequence of equally-spaced values /v/ at an index /i/.
+The `discrete_lanczos_derivative` class calculates a finite-difference approximation to the derivative of a noisy sequence of equally-spaced values /v/.
 A basic usage is
 
     std::vector<double> v(500);
     // fill v with noisy data.
     double spacing = 0.001;
     using boost::math::differentiation::discrete_lanczos_derivative;
-    auto lanczos = discrete_lanczos_derivative(v, spacing);
-    double dvdt = lanczos[30];
+    auto lanczos = discrete_lanczos_derivative(spacing);
+    // Compute dvdt at index 30:
+    double dvdt30 = lanczos(v, 30);
+    // Compute derivative of entire vector:
+    std::vector<double> dvdt = lanczos(v);
 
 Noise-suppressing second derivatives can also be computed.
 The syntax is as follows:
 
     std::vector<double> v(500);
     // fill v with noisy data.
-    auto lanczos = lanczos_derivative<decltype(v), 2>(v, spacing);
-    // evaluate:
-    double dvdt = lanczos[25];
+    auto lanczos = lanczos_derivative<double, 2>(spacing);
+    // evaluate second derivative at a point:
+    double d2vdt2 = lanczos(v, 18);
+    // evaluate second derivative of entire vector:
+    std::vector<double> d2vdt2 = lanczos(v);
 
 If the data has variance \u03C3[super 2],
 then the variance of the computed derivative is roughly \u03C3[super 2]/p/[super 3] /n/[super -3] \u0394 /t/[super -2],
@@ -63,8 +68,8 @@ You can play around with the approximation order /p/ and the filter length /n/:
 
     size_t n = 12;
     size_t p = 2;
-    auto lanczos = lanczos_derivative(v, spacing, n, p);
-    double dvdt = lanczos[24];
+    auto lanczos = lanczos_derivative(spacing, n, p);
+    double dvdt = lanczos(v, i);
 
 If /p=2n/, then the discrete Lanczos derivative is not smoothing:
 It reduces to the standard /2n+1/-point finite-difference formula.
@@ -79,7 +84,7 @@ balanced against the danger of overfitting and averaging over non-stationarity.
 The choice of approximation order /p/ for a given /n/ is more difficult.
 If your signal is believed to be a polynomial,
 it does not make sense to set /p/ to larger than the polynomial degree-
-though it may be sensible to take /p/ less than the polynomial degree.
+though it may be sensible to take /p/ less than this.
 
 For a sinusoidal signal contaminated with AWGN, we ran a few tests showing that for SNR = 1,
 p = n/8 gave the best results,
@@ -93,18 +98,17 @@ Since each filter has length /2n+1/ and there are /n/ filters, whose element eac
 the complexity of the constructor call is O(/n/[super 2]/p/).
 This is not cheap-though for most cases small /p/ and /n/ not too large (< 20) is desired.
 However, for concreteness, on the author's 2.7GHz Intel laptop CPU, the /n=16/, /p=3/ filter takes 9 microseconds to compute.
-This is far from negligible, and as such we provide API calls which allow the filters to be used with multiple data:
+This is far from negligible, and as such the filters may be used with multiple data, and even shared between threads:
 
 
-    std::vector<double> v(500);
-    // fill v with noisy data.
-    auto lanczos = lanczos_derivative(v, spacing);
-    // use lanczos with v . . .
-    std::vector<double> w(500);
-    lanczos.reset_data(w);
-    // use lanczos with w . . .
+    std::vector<double> v1(500);
+    std::vector<double> v2(500);
+    // fill v1, v2 with noisy data.
+    auto lanczos = lanczos_derivative(spacing);
+    std::vector<double> dv1dt = lanczos(v1); // threadsafe
+    std::vector<double> dv2dt = lanczos(v2); // threadsafe
     // need to use a different spacing?
-    lanczos.reset_spacing(0.02);
+    lanczos.reset_spacing(0.02); // not threadsafe
 
 
 The implementation follows [@https://doi.org/10.1080/00207160.2012.666348 McDevitt, 2012],
diff --git a/include/boost/math/differentiation/lanczos_smoothing.hpp b/include/boost/math/differentiation/lanczos_smoothing.hpp
index c3b78f0a4..6c1349cdf 100644
--- a/include/boost/math/differentiation/lanczos_smoothing.hpp
+++ b/include/boost/math/differentiation/lanczos_smoothing.hpp
@@ -258,20 +258,16 @@ std::vector<Real> acceleration_boundary_filter(size_t n, size_t p, int64_t s)
 
 } // namespace detail
 
-template <class RandomAccessContainer, size_t order = 1>
+template <typename Real, size_t order = 1>
 class discrete_lanczos_derivative {
 public:
-    using Real = typename RandomAccessContainer::value_type;
-    discrete_lanczos_derivative(RandomAccessContainer const &v,
-                       Real const & spacing = 1,
-                       size_t n = 18,
-                       size_t approximation_order = 3)
-        : m_v{v}, dt{spacing}
+    discrete_lanczos_derivative(Real const & spacing,
+                                size_t n = 18,
+                                size_t approximation_order = 3)
+        : m_dt{spacing}
     {
+        static_assert(!std::is_integral_v<Real>, "Spacing must be a floating point type.");
         BOOST_ASSERT_MSG(spacing > 0, "Spacing between samples must be > 0.");
-        using std::size;
-        BOOST_ASSERT_MSG(size(v) >= 2*n+1,
-            "Vector must be at least as long as the filter length");
 
         if constexpr (order == 1)
         {
@@ -281,12 +277,12 @@ public:
                              "The approximation order must be >= 2");
             m_f = detail::interior_filter<Real>(n, approximation_order);
 
-            boundary_filters.resize(n);
+            m_boundary_filters.resize(n);
             for (size_t i = 0; i < n; ++i)
             {
                 // s = i - n;
                 int64_t s = static_cast<int64_t>(i) - static_cast<int64_t>(n);
-                boundary_filters[i] = detail::boundary_filter<Real>(n, approximation_order, s);
+                m_boundary_filters[i] = detail::boundary_filter<Real>(n, approximation_order, s);
             }
         }
         else if constexpr (order == 2)
@@ -297,11 +293,11 @@ public:
             {
                 m_f[i] = f[i+n];
             }
-            boundary_filters.resize(n);
+            m_boundary_filters.resize(n);
             for (size_t i = 0; i < n; ++i)
             {
                 int64_t s = static_cast<int64_t>(i) - static_cast<int64_t>(n);
-                boundary_filters[i] = detail::acceleration_boundary_filter<Real>(n, approximation_order, s);
+                m_boundary_filters[i] = detail::acceleration_boundary_filter<Real>(n, approximation_order, s);
             }
         }
         else
@@ -310,96 +306,96 @@ public:
         }
     }
 
-    void reset_data(RandomAccessContainer const &v)
-    {
-        using std::size;
-        BOOST_ASSERT_MSG(size(v) >= m_f.size(), "Vector must be at least as long as the filter length");
-        m_v = v;
-    }
-
     void reset_spacing(Real const & spacing)
     {
         BOOST_ASSERT_MSG(spacing > 0, "Spacing between samples must be > 0.");
-        dt = spacing;
+        m_dt = spacing;
     }
 
     Real spacing() const
     {
-        return dt;
+        return m_dt;
     }
 
-    Real operator[](size_t i) const
+    template<class RandomAccessContainer>
+    Real operator()(RandomAccessContainer const & v, size_t i) const
     {
+        static_assert(std::is_same_v<typename RandomAccessContainer::value_type, Real>,
+                      "The type of the values in the vector provided does not match the type in the filters.");
+        using std::size;
+        BOOST_ASSERT_MSG(size(v) >= m_boundary_filters[0].size(),
+            "Vector must be at least as long as the filter length");
+
         if constexpr (order==1)
         {
-            using std::size;
-            if (i >= m_f.size() - 1 && i <= size(m_v) - m_f.size())
+            if (i >= m_f.size() - 1 && i <= size(v) - m_f.size())
             {
                 Real dv = 0;
                 for (size_t j = 1; j < m_f.size(); ++j)
                 {
-                    dv += m_f[j] * (m_v[i + j] - m_v[i - j]);
+                    dv += m_f[j] * (v[i + j] - v[i - j]);
                 }
-                return dv / dt;
+                return dv / m_dt;
             }
 
             // m_f.size() = N+1
             if (i < m_f.size() - 1)
             {
-                auto &f = boundary_filters[i];
+                auto &bf = m_boundary_filters[i];
                 Real dv = 0;
-                for (size_t j = 0; j < f.size(); ++j) {
-                    dv += f[j] * m_v[j];
+                for (size_t j = 0; j < bf.size(); ++j)
+                {
+                    dv += bf[j] * v[j];
                 }
-                return dv / dt;
+                return dv / m_dt;
             }
 
-            if (i > size(m_v) - m_f.size() && i < size(m_v))
+            if (i > size(v) - m_f.size() && i < size(v))
             {
-                int k = size(m_v) - 1 - i;
-                auto &f = boundary_filters[k];
+                int k = size(v) - 1 - i;
+                auto &bf = m_boundary_filters[k];
                 Real dv = 0;
-                for (size_t j = 0; j < f.size(); ++j)
+                for (size_t j = 0; j < bf.size(); ++j)
                 {
-                    dv += f[j] * m_v[m_v.size() - 1 - j];
+                    dv += bf[j] * v[size(v) - 1 - j];
                 }
-                return -dv / dt;
+                return -dv / m_dt;
             }
         }
         else if constexpr (order==2)
         {
-            using std::size;
-            if (i >= m_f.size() - 1 && i <= size(m_v) - m_f.size())
+            if (i >= m_f.size() - 1 && i <= size(v) - m_f.size())
             {
-                Real d2v = m_f[0]*m_v[i];
+                Real d2v = m_f[0]*v[i];
                 for (size_t j = 1; j < m_f.size(); ++j)
                 {
-                    d2v += m_f[j] * (m_v[i + j] + m_v[i - j]);
+                    d2v += m_f[j] * (v[i + j] + v[i - j]);
                 }
-                return d2v / (dt*dt);
+                return d2v / (m_dt*m_dt);
             }
 
             // m_f.size() = N+1
             if (i < m_f.size() - 1)
             {
-                auto &f = boundary_filters[i];
+                auto &bf = m_boundary_filters[i];
                 Real d2v = 0;
-                for (size_t j = 0; j < f.size(); ++j) {
-                    d2v += f[j] * m_v[j];
+                for (size_t j = 0; j < bf.size(); ++j)
+                {
+                    d2v += bf[j] * v[j];
                 }
-                return d2v / (dt*dt);
+                return d2v / (m_dt*m_dt);
             }
 
-            if (i > size(m_v) - m_f.size() && i < size(m_v))
+            if (i > size(v) - m_f.size() && i < size(v))
             {
-                int k = size(m_v) - 1 - i;
-                auto &f = boundary_filters[k];
+                int k = size(v) - 1 - i;
+                auto &bf = m_boundary_filters[k];
                 Real d2v = 0;
-                for (size_t j = 0; j < f.size(); ++j)
+                for (size_t j = 0; j < bf.size(); ++j)
                 {
-                    d2v += f[j] * m_v[m_v.size() - 1 - j];
+                    d2v += bf[j] * v[size(v) - 1 - j];
                 }
-                return d2v / dt;
+                return d2v / (m_dt*m_dt);
             }
         }
 
@@ -408,11 +404,97 @@ public:
         return std::numeric_limits<Real>::quiet_NaN();
     }
 
+    template<class RandomAccessContainer>
+    RandomAccessContainer operator()(RandomAccessContainer const & v) const
+    {
+        static_assert(std::is_same_v<typename RandomAccessContainer::value_type, Real>,
+                      "The type of the values in the vector provided does not match the type in the filters.");
+        using std::size;
+        BOOST_ASSERT_MSG(size(v) >= m_boundary_filters[0].size(),
+            "Vector must be at least as long as the filter length");
+
+        RandomAccessContainer w(size(v));
+        if constexpr (order==1)
+        {
+            for (size_t i = 0; i < m_f.size() - 1; ++i)
+            {
+                auto &bf = m_boundary_filters[i];
+                Real dv = 0;
+                for (size_t j = 0; j < bf.size(); ++j)
+                {
+                    dv += bf[j] * v[j];
+                }
+                w[i] = dv / m_dt;
+            }
+
+            for(size_t i = m_f.size() - 1; i <= size(v) - m_f.size(); ++i)
+            {
+                Real dv = 0;
+                for (size_t j = 1; j < m_f.size(); ++j)
+                {
+                    dv += m_f[j] * (v[i + j] - v[i - j]);
+                }
+                w[i] = dv / m_dt;
+            }
+
+
+            for(size_t i = size(v) - m_f.size() + 1; i < size(v); ++i)
+            {
+                int k = size(v) - 1 - i;
+                auto &f = m_boundary_filters[k];
+                Real dv = 0;
+                for (size_t j = 0; j < f.size(); ++j)
+                {
+                    dv += f[j] * v[size(v) - 1 - j];
+                }
+                w[i] = -dv / m_dt;
+            }
+        }
+        else if constexpr (order==2)
+        {
+            // m_f.size() = N+1
+            for (size_t i = 0; i < m_f.size() - 1; ++i)
+            {
+                auto &bf = m_boundary_filters[i];
+                Real d2v = 0;
+                for (size_t j = 0; j < bf.size(); ++j)
+                {
+                    d2v += bf[j] * v[j];
+                }
+                w[i] = d2v / (m_dt*m_dt);
+            }
+
+            for (size_t i = m_f.size() - 1; i <= size(v) - m_f.size(); ++i)
+            {
+                Real d2v = m_f[0]*v[i];
+                for (size_t j = 1; j < m_f.size(); ++j)
+                {
+                    d2v += m_f[j] * (v[i + j] + v[i - j]);
+                }
+                w[i] = d2v / (m_dt*m_dt);
+            }
+
+
+            for (size_t i = size(v) - m_f.size() + 1; i < size(v); ++i)
+            {
+                int k = size(v) - 1 - i;
+                auto &bf = m_boundary_filters[k];
+                Real d2v = 0;
+                for (size_t j = 0; j < bf.size(); ++j)
+                {
+                    d2v += bf[j] * v[size(v) - 1 - j];
+                }
+                w[i] = d2v / (m_dt*m_dt);
+            }
+        }
+
+        return w;
+    }
+
 private:
-    const RandomAccessContainer &m_v;
     std::vector<Real> m_f;
-    std::vector<std::vector<Real>> boundary_filters;
-    Real dt;
+    std::vector<std::vector<Real>> m_boundary_filters;
+    Real m_dt;
 };
 
 } // namespaces
diff --git a/test/lanczos_smoothing_test.cpp b/test/lanczos_smoothing_test.cpp
index bebc2d562..cf2d958af 100644
--- a/test/lanczos_smoothing_test.cpp
+++ b/test/lanczos_smoothing_test.cpp
@@ -208,12 +208,17 @@ void test_interior_lanczos()
     {
         for (size_t p = 2; p < 2*n; p += 2)
         {
-            auto lsd = discrete_lanczos_derivative(v, 0.1, n, p);
+            auto dld = discrete_lanczos_derivative(Real(0.1), n, p);
             for (size_t m = n; m < v.size() - n; ++m)
             {
-                Real dvdt = lsd[m];
+                Real dvdt = dld(v, m);
                 BOOST_CHECK_SMALL(dvdt, tol);
             }
+            auto dvdt = dld(v);
+            for (size_t m = n; m < v.size() - n; ++m)
+            {
+                BOOST_CHECK_SMALL(dvdt[m], tol);
+            }
         }
     }
 
@@ -227,12 +232,17 @@ void test_interior_lanczos()
     {
         for (size_t p = 2; p < 2*n; p += 2)
         {
-            auto lsd = discrete_lanczos_derivative(v, Real(1), n, p);
+            auto dld = discrete_lanczos_derivative(Real(1), n, p);
             for (size_t m = n; m < v.size() - n; ++m)
             {
-                Real dvdt = lsd[m];
+                Real dvdt = dld(v, m);
                 BOOST_CHECK_CLOSE_FRACTION(dvdt, 7, 2000*tol);
             }
+            auto dvdt = dld(v);
+            for (size_t m = n; m < v.size() - n; ++m)
+            {
+                BOOST_CHECK_CLOSE_FRACTION(dvdt[m], 7, 2000*tol);
+            }
         }
     }
 
@@ -241,7 +251,6 @@ void test_interior_lanczos()
     //std::cout << "Seed = " << seed << "\n";
     std::mt19937 gen(4172378669);
     std::normal_distribution<> dis{0, 0.01};
-    std::cout << std::fixed;
     for (size_t i = 0; i < v.size(); ++i)
     {
         v[i] = 7*i+8 + dis(gen);
@@ -251,10 +260,10 @@ void test_interior_lanczos()
     {
         for (size_t p = 2; p < 2*n; p += 2)
         {
-            auto lsd = discrete_lanczos_derivative(v, Real(1), n, p);
+            auto dld = discrete_lanczos_derivative(Real(1), n, p);
             for (size_t m = n; m < v.size() - n; ++m)
             {
-                BOOST_CHECK_CLOSE_FRACTION(lsd[m], Real(7), Real(0.0042));
+                BOOST_CHECK_CLOSE_FRACTION(dld(v, m), Real(7), Real(0.0042));
             }
         }
     }
@@ -269,10 +278,10 @@ void test_interior_lanczos()
     {
         for (size_t p = 2; p < 2*n; p += 2)
         {
-            auto lsd = discrete_lanczos_derivative(v, Real(1), n, p);
+            auto dld = discrete_lanczos_derivative(Real(1), n, p);
             for (size_t m = n; m < v.size() - n; ++m)
             {
-                BOOST_CHECK_CLOSE_FRACTION(lsd[m], Real(30*m + 7), Real(0.00008));
+                BOOST_CHECK_CLOSE_FRACTION(dld(v,m), Real(30*m + 7), Real(0.00008));
             }
         }
     }
@@ -288,11 +297,11 @@ void test_interior_lanczos()
     {
         for (size_t p = 3; p < 100 && p < n/2; p += 2)
         {
-            auto lsd = discrete_lanczos_derivative(v, Real(1), n, p);
+            auto dld = discrete_lanczos_derivative(Real(1), n, p);
 
             for (size_t m = n; m < v.size() - n && m < n + 10; ++m)
             {
-                BOOST_CHECK_CLOSE_FRACTION(lsd[m], omega*cos(omega*m), Real(0.03));
+                BOOST_CHECK_CLOSE_FRACTION(dld(v,m), omega*cos(omega*m), Real(0.03));
             }
         }
     }
@@ -377,15 +386,15 @@ void test_boundary_lanczos()
     {
         for (size_t p = 2; p < 2*n; ++p)
         {
-            auto lsd = discrete_lanczos_derivative(v, 0.0125, n, p);
+            auto lsd = discrete_lanczos_derivative(Real(0.0125), n, p);
             for (size_t m = 0; m < n; ++m)
             {
-                Real dvdt = lsd[m];
+                Real dvdt = lsd(v,m);
                 BOOST_CHECK_SMALL(dvdt, 4*sqrt(tol));
             }
             for (size_t m = v.size() - n; m < v.size(); ++m)
             {
-                Real dvdt = lsd[m];
+                Real dvdt = lsd(v,m);
                 BOOST_CHECK_SMALL(dvdt, 4*sqrt(tol));
             }
         }
@@ -400,16 +409,16 @@ void test_boundary_lanczos()
     {
         for (size_t p = 2; p < 2*n; ++p)
         {
-            auto lsd = discrete_lanczos_derivative(v, Real(1), n, p);
+            auto lsd = discrete_lanczos_derivative(Real(1), n, p);
             for (size_t m = 0; m < n; ++m)
             {
-                Real dvdt = lsd[m];
+                Real dvdt = lsd(v,m);
                 BOOST_CHECK_CLOSE_FRACTION(dvdt, 7, sqrt(tol));
             }
 
             for (size_t m = v.size() - n; m < v.size(); ++m)
             {
-                Real dvdt = lsd[m];
+                Real dvdt = lsd(v,m);
                 BOOST_CHECK_CLOSE_FRACTION(dvdt, 7, 4*sqrt(tol));
             }
         }
@@ -424,10 +433,10 @@ void test_boundary_lanczos()
     {
         for (size_t p = 2; p < 2*n; ++p)
         {
-            auto lsd = discrete_lanczos_derivative(v, Real(1), n, p);
+            auto lsd = discrete_lanczos_derivative(Real(1), n, p);
             for (size_t m = 0; m < v.size(); ++m)
             {
-                BOOST_CHECK_CLOSE_FRACTION(lsd[m], 30*m+7, 30*sqrt(tol));
+                BOOST_CHECK_CLOSE_FRACTION(lsd(v,m), 30*m+7, 30*sqrt(tol));
             }
         }
     }
@@ -447,14 +456,18 @@ void test_boundary_lanczos()
     {
         for (size_t p = 2; p < n; ++p)
         {
-            auto lsd = discrete_lanczos_derivative(v, Real(1), n, p);
+            auto lsd = discrete_lanczos_derivative(Real(1), n, p);
             for (size_t m = 0; m < v.size(); ++m)
             {
-                BOOST_CHECK_CLOSE_FRACTION(lsd[m], 30*m+7, 0.005);
+                BOOST_CHECK_CLOSE_FRACTION(lsd(v,m), 30*m+7, 0.005);
+            }
+            auto dvdt = lsd(v);
+            for (size_t m = 0; m < v.size(); ++m)
+            {
+                BOOST_CHECK_CLOSE_FRACTION(dvdt[m], 30*m+7, 0.005);
             }
         }
     }
-
 }
 
 template<class Real>
@@ -502,10 +515,10 @@ void test_lanczos_acceleration()
 {
     Real eps = std::numeric_limits<Real>::epsilon();
     std::vector<Real> v(100, 7);
-    auto lanczos = discrete_lanczos_derivative<decltype(v), 2>(v, Real(1), 4, 3);
+    auto lanczos = discrete_lanczos_derivative<Real, 2>(Real(1), 4, 3);
     for (size_t i = 0; i < v.size(); ++i)
     {
-        BOOST_CHECK_SMALL(lanczos[i], eps);
+        BOOST_CHECK_SMALL(lanczos(v, i), eps);
     }
 
     for(size_t i = 0; i < v.size(); ++i)
@@ -514,7 +527,7 @@ void test_lanczos_acceleration()
     }
     for (size_t i = 0; i < v.size(); ++i)
     {
-        BOOST_CHECK_SMALL(lanczos[i], 200*eps);
+        BOOST_CHECK_SMALL(lanczos(v,i), 200*eps);
     }
 
     for(size_t i = 0; i < v.size(); ++i)
@@ -523,7 +536,7 @@ void test_lanczos_acceleration()
     }
     for (size_t i = 0; i < v.size(); ++i)
     {
-        BOOST_CHECK_CLOSE_FRACTION(lanczos[i], 14, 1000*eps);
+        BOOST_CHECK_CLOSE_FRACTION(lanczos(v, i), 14, 1000*eps);
     }