ceph/src/boost/boost/math/differentiation/finite_difference.hpp

   1 //  (C) Copyright Nick Thompson 2018.
   2 //  Use, modification and distribution are subject to the
   3 //  Boost Software License, Version 1.0. (See accompanying file
   4 //  LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
   5
   6 #ifndef BOOST_MATH_DIFFERENTIATION_FINITE_DIFFERENCE_HPP
   7 #define BOOST_MATH_DIFFERENTIATION_FINITE_DIFFERENCE_HPP
   8
   9 /*
  10  * Performs numerical differentiation by finite-differences.
  11  *
  12  * All numerical differentiation using finite-differences are ill-conditioned, and these routines are no exception.
  13  * A simple argument demonstrates that the error is unbounded as h->0.
  14  * Take the one sides finite difference formula f'(x) = (f(x+h)-f(x))/h.
  15  * The evaluation of f induces an error as well as the error from the finite-difference approximation, giving
  16  * |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.
  17  * It is reasonable to choose h in a way that minimizes the maximum error bound g(h).
  18  * 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
  19  * sqrt(2eps(|f(x+h) +f(x)||f''(x)|)).
  20  * 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.
  21  *
  22  *
  23  * For more details on this method of analysis, see
  24  *
  25  * http://www.uio.no/studier/emner/matnat/math/MAT-INF1100/h08/kompendiet/diffint.pdf
  26  * http://web.archive.org/web/20150420195907/http://www.uio.no/studier/emner/matnat/math/MAT-INF1100/h08/kompendiet/diffint.pdf
  27  *
  28  *
  29  * 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.
  30  * From this we can see that full precision can be recovered in the limit k->infinity.
  31  *
  32  * References:
  33  *
  34  * 1) Fornberg, Bengt. "Generation of finite difference formulas on arbitrarily spaced grids." Mathematics of computation 51.184 (1988): 699-706.
  35  *
  36  *
  37  * The second algorithm, the complex step derivative, is not ill-conditioned.
  38  * However, it requires that your function can be evaluated at complex arguments.
  39  * The idea is that f(x+ih) = f(x) +ihf'(x) - h^2f''(x) + ... so f'(x) \approx Im[f(x+ih)]/h.
  40  * No subtractive cancellation occurs. The error is ~ eps|f'(x)| + eps^2|f'''(x)|/6; hard to beat that.
  41  *
  42  * References:
  43  *
  44  * 1) Squire, William, and George Trapp. "Using complex variables to estimate derivatives of real functions." Siam Review 40.1 (1998): 110-112.
  45  */
  46
  47 #include <complex>
  48 #include <boost/math/special_functions/next.hpp>
  49
  50 namespace boost{ namespace math{ namespace differentiation {
  51
  52 namespace detail {
  53     template<class Real>
  54     Real make_xph_representable(Real x, Real h)
  55     {
  56         using std::numeric_limits;
  57         // Redefine h so that x + h is representable. Not using this trick leads to large error.
  58         // The compiler flag -ffast-math evaporates these operations . . .
  59         Real temp = x + h;
  60         h = temp - x;
  61         // Handle the case x + h == x:
  62         if (h == 0)
  63         {
  64             h = boost::math::nextafter(x, (numeric_limits<Real>::max)()) - x;
  65         }
  66         return h;
  67     }
  68 }
  69
  70 template<class F, class Real>
  71 Real complex_step_derivative(const F f, Real x)
  72 {
  73     // Is it really this easy? Yes.
  74     // Note that some authors recommend taking the stepsize h to be smaller than epsilon(), some recommending use of the min().
  75     // This idea was tested over a few billion test cases and found the make the error *much* worse.
  76     // Even 2eps and eps/2 made the error worse, which was surprising.
  77     using std::complex;
  78     using std::numeric_limits;
  79     constexpr const Real step = (numeric_limits<Real>::epsilon)();
  80     constexpr const Real inv_step = 1/(numeric_limits<Real>::epsilon)();
  81     return f(complex<Real>(x, step)).imag()*inv_step;
  82 }
  83
  84 namespace detail {
  85
  86    template <unsigned>
  87    struct fd_tag {};
  88
  89    template<class F, class Real>
  90    Real finite_difference_derivative(const F f, Real x, Real* error, const fd_tag<1>&)
  91    {
  92       using std::sqrt;
  93       using std::pow;
  94       using std::abs;
  95       using std::numeric_limits;
  96
  97       const Real eps = (numeric_limits<Real>::epsilon)();
  98       // Error bound ~eps^1/2
  99       // Note that this estimate of h differs from the best estimate by a factor of sqrt((|f(x)| + |f(x+h)|)/|f''(x)|).
 100       // Since this factor is invariant under the scaling f -> kf, then we are somewhat justified in approximating it by 1.
 101       // This approximation will get better as we move to higher orders of accuracy.
 102       Real h = 2 * sqrt(eps);
 103       h = detail::make_xph_representable(x, h);
 104
 105       Real yh = f(x + h);
 106       Real y0 = f(x);
 107       Real diff = yh - y0;
 108       if (error)
 109       {
 110          Real ym = f(x - h);
 111          Real ypph = abs(yh - 2 * y0 + ym) / h;
 112          // h*|f''(x)|*0.5 + (|f(x+h)+|f(x)|)*eps/h
 113          *error = ypph / 2 + (abs(yh) + abs(y0))*eps / h;
 114       }
 115       return diff / h;
 116    }
 117
 118    template<class F, class Real>
 119    Real finite_difference_derivative(const F f, Real x, Real* error, const fd_tag<2>&)
 120    {
 121       using std::sqrt;
 122       using std::pow;
 123       using std::abs;
 124       using std::numeric_limits;
 125
 126       const Real eps = (numeric_limits<Real>::epsilon)();
 127       // Error bound ~eps^2/3
 128       // See the previous discussion to understand determination of h and the error bound.
 129       // Series[(f[x+h] - f[x-h])/(2*h), {h, 0, 4}]
 130       Real h = pow(3 * eps, static_cast<Real>(1) / static_cast<Real>(3));
 131       h = detail::make_xph_representable(x, h);
 132
 133       Real yh = f(x + h);
 134       Real ymh = f(x - h);
 135       Real diff = yh - ymh;
 136       if (error)
 137       {
 138          Real yth = f(x + 2 * h);
 139          Real ymth = f(x - 2 * h);
 140          *error = eps * (abs(yh) + abs(ymh)) / (2 * h) + abs((yth - ymth) / 2 - diff) / (6 * h);
 141       }
 142
 143       return diff / (2 * h);
 144    }
 145
 146    template<class F, class Real>
 147    Real finite_difference_derivative(const F f, Real x, Real* error, const fd_tag<4>&)
 148    {
 149       using std::sqrt;
 150       using std::pow;
 151       using std::abs;
 152       using std::numeric_limits;
 153
 154       const Real eps = (numeric_limits<Real>::epsilon)();
 155       // Error bound ~eps^4/5
 156       Real h = pow(11.25*eps, (Real)1 / (Real)5);
 157       h = detail::make_xph_representable(x, h);
 158       Real ymth = f(x - 2 * h);
 159       Real yth = f(x + 2 * h);
 160       Real yh = f(x + h);
 161       Real ymh = f(x - h);
 162       Real y2 = ymth - yth;
 163       Real y1 = yh - ymh;
 164       if (error)
 165       {
 166          // Mathematica code to extract the remainder:
 167          // Series[(f[x-2*h]+ 8*f[x+h] - 8*f[x-h] - f[x+2*h])/(12*h), {h, 0, 7}]
 168          Real y_three_h = f(x + 3 * h);
 169          Real y_m_three_h = f(x - 3 * h);
 170          // Error from fifth derivative:
 171          *error = abs((y_three_h - y_m_three_h) / 2 + 2 * (ymth - yth) + 5 * (yh - ymh) / 2) / (30 * h);
 172          // Error from function evaluation:
 173          *error += eps * (abs(yth) + abs(ymth) + 8 * (abs(ymh) + abs(yh))) / (12 * h);
 174       }
 175       return (y2 + 8 * y1) / (12 * h);
 176    }
 177
 178    template<class F, class Real>
 179    Real finite_difference_derivative(const F f, Real x, Real* error, const fd_tag<6>&)
 180    {
 181       using std::sqrt;
 182       using std::pow;
 183       using std::abs;
 184       using std::numeric_limits;
 185
 186       const Real eps = (numeric_limits<Real>::epsilon)();
 187       // Error bound ~eps^6/7
 188       // Error: h^6f^(7)(x)/140 + 5|f(x)|eps/h
 189       Real h = pow(eps / 168, (Real)1 / (Real)7);
 190       h = detail::make_xph_representable(x, h);
 191
 192       Real yh = f(x + h);
 193       Real ymh = f(x - h);
 194       Real y1 = yh - ymh;
 195       Real y2 = f(x - 2 * h) - f(x + 2 * h);
 196       Real y3 = f(x + 3 * h) - f(x - 3 * h);
 197
 198       if (error)
 199       {
 200          // Mathematica code to generate fd scheme for 7th derivative:
 201          // Sum[(-1)^i*Binomial[7, i]*(f[x+(3-i)*h] + f[x+(4-i)*h])/2, {i, 0, 7}]
 202          // Mathematica to demonstrate that this is a finite difference formula for 7th derivative:
 203          // 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}]
 204          Real y7 = (f(x + 4 * h) - f(x - 4 * h) - 6 * y3 - 14 * y1 - 14 * y2) / 2;
 205          *error = abs(y7) / (140 * h) + 5 * (abs(yh) + abs(ymh))*eps / h;
 206       }
 207       return (y3 + 9 * y2 + 45 * y1) / (60 * h);
 208    }
 209
 210    template<class F, class Real>
 211    Real finite_difference_derivative(const F f, Real x, Real* error, const fd_tag<8>&)
 212    {
 213       using std::sqrt;
 214       using std::pow;
 215       using std::abs;
 216       using std::numeric_limits;
 217
 218       const Real eps = (numeric_limits<Real>::epsilon)();
 219       // Error bound ~eps^8/9.
 220       // In double precision, we only expect to lose two digits of precision while using this formula, at the cost of 8 function evaluations.
 221       // Error: h^8|f^(9)(x)|/630 + 7|f(x)|eps/h assuming 7 unstabilized additions.
 222       // Mathematica code to get the error:
 223       // 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}]
 224       // If we used Kahan summation, we could get the max error down to h^8|f^(9)(x)|/630 + |f(x)|eps/h.
 225       Real h = pow(551.25*eps, (Real)1 / (Real)9);
 226       h = detail::make_xph_representable(x, h);
 227
 228       Real yh = f(x + h);
 229       Real ymh = f(x - h);
 230       Real y1 = yh - ymh;
 231       Real y2 = f(x - 2 * h) - f(x + 2 * h);
 232       Real y3 = f(x + 3 * h) - f(x - 3 * h);
 233       Real y4 = f(x - 4 * h) - f(x + 4 * h);
 234
 235       Real tmp1 = 3 * y4 / 8 + 4 * y3;
 236       Real tmp2 = 21 * y2 + 84 * y1;
 237
 238       if (error)
 239       {
 240          // Mathematica code to generate fd scheme for 7th derivative:
 241          // Sum[(-1)^i*Binomial[9, i]*(f[x+(4-i)*h] + f[x+(5-i)*h])/2, {i, 0, 9}]
 242          // Mathematica to demonstrate that this is a finite difference formula for 7th derivative:
 243          // 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}]
 244          Real f9 = (f(x + 5 * h) - f(x - 5 * h)) / 2 + 4 * y4 + 27 * y3 / 2 + 24 * y2 + 21 * y1;
 245          *error = abs(f9) / (630 * h) + 7 * (abs(yh) + abs(ymh))*eps / h;
 246       }
 247       return (tmp1 + tmp2) / (105 * h);
 248    }
 249
 250    template<class F, class Real, class tag>
 251    Real finite_difference_derivative(const F, Real, Real*, const tag&)
 252    {
 253       // Always fails, but condition is template-arg-dependent so only evaluated if we get instantiated.
 254       BOOST_STATIC_ASSERT_MSG(sizeof(Real) == 0, "Finite difference not implemented for this order: try 1, 2, 4, 6 or 8");
 255    }
 256
 257 }
 258
 259 template<class F, class Real, size_t order=6>
 260 inline Real finite_difference_derivative(const F f, Real x, Real* error = nullptr)
 261 {
 262    return detail::finite_difference_derivative(f, x, error, detail::fd_tag<order>());
 263 }
 264
 265 }}}  // namespaces
 266 #endif