boost/math/differentiation/finite_difference.hpp
// (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, static_cast<Real>(1) / static_cast<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, static_cast<Real>(1) / static_cast<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, static_cast<Real>(1) / static_cast<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, Real, Real*, const tag&) { // Always fails, but condition is template-arg-dependent so only evaluated if we get instantiated. static_assert(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