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+// (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_TOOLS_NUMERICAL_DIFFERENTIATION_HPP
+#define BOOST_MATH_TOOLS_NUMERICAL_DIFFERENTIATION_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>
+
+// 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;
+ }
+}
+
+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