ceph/src/boost/boost/math/interpolators/detail/cubic_b_spline_detail.hpp

   1 // Copyright Nick Thompson, 2017
   2 // Use, modification and distribution are subject to the
   3 // Boost Software License, Version 1.0.
   4 // (See accompanying file LICENSE_1_0.txt
   5 // or copy at http://www.boost.org/LICENSE_1_0.txt)
   6
   7 #ifndef CUBIC_B_SPLINE_DETAIL_HPP
   8 #define CUBIC_B_SPLINE_DETAIL_HPP
   9
  10 #include <limits>
  11 #include <cmath>
  12 #include <vector>
  13 #include <memory>
  14 #include <boost/math/constants/constants.hpp>
  15 #include <boost/math/special_functions/fpclassify.hpp>
  16
  17 namespace boost{ namespace math{ namespace detail{
  18
  19
  20 template <class Real>
  21 class cubic_b_spline_imp
  22 {
  23 public:
  24     // If you don't know the value of the derivative at the endpoints, leave them as nans and the routine will estimate them.
  25     // f[0] = f(a), f[length -1] = b, step_size = (b - a)/(length -1).
  26     template <class BidiIterator>
  27     cubic_b_spline_imp(BidiIterator f, BidiIterator end_p, Real left_endpoint, Real step_size,
  28                        Real left_endpoint_derivative = std::numeric_limits<Real>::quiet_NaN(),
  29                        Real right_endpoint_derivative = std::numeric_limits<Real>::quiet_NaN());
  30
  31     Real operator()(Real x) const;
  32
  33     Real prime(Real x) const;
  34
  35 private:
  36     std::vector<Real> m_beta;
  37     Real m_h_inv;
  38     Real m_a;
  39     Real m_avg;
  40 };
  41
  42
  43
  44 template <class Real>
  45 Real b3_spline(Real x)
  46 {
  47     using std::abs;
  48     Real absx = abs(x);
  49     if (absx < 1)
  50     {
  51         Real y = 2 - absx;
  52         Real z = 1 - absx;
  53         return boost::math::constants::sixth<Real>()*(y*y*y - 4*z*z*z);
  54     }
  55     if (absx < 2)
  56     {
  57         Real y = 2 - absx;
  58         return boost::math::constants::sixth<Real>()*y*y*y;
  59     }
  60     return (Real) 0;
  61 }
  62
  63 template<class Real>
  64 Real b3_spline_prime(Real x)
  65 {
  66     if (x < 0)
  67     {
  68         return -b3_spline_prime(-x);
  69     }
  70
  71     if (x < 1)
  72     {
  73         return x*(3*boost::math::constants::half<Real>()*x - 2);
  74     }
  75     if (x < 2)
  76     {
  77         return -boost::math::constants::half<Real>()*(2 - x)*(2 - x);
  78     }
  79     return (Real) 0;
  80 }
  81
  82
  83 template <class Real>
  84 template <class BidiIterator>
  85 cubic_b_spline_imp<Real>::cubic_b_spline_imp(BidiIterator f, BidiIterator end_p, Real left_endpoint, Real step_size,
  86                                              Real left_endpoint_derivative, Real right_endpoint_derivative) : m_a(left_endpoint), m_avg(0)
  87 {
  88     using boost::math::constants::third;
  89
  90     std::size_t length = end_p - f;
  91
  92     if (length < 5)
  93     {
  94         if (boost::math::isnan(left_endpoint_derivative) || boost::math::isnan(right_endpoint_derivative))
  95         {
  96             throw std::logic_error("Interpolation using a cubic b spline with derivatives estimated at the endpoints requires at least 5 points.\n");
  97         }
  98         if (length < 3)
  99         {
 100             throw std::logic_error("Interpolation using a cubic b spline requires at least 3 points.\n");
 101         }
 102     }
 103
 104     if (boost::math::isnan(left_endpoint))
 105     {
 106         throw std::logic_error("Left endpoint is NAN; this is disallowed.\n");
 107     }
 108     if (left_endpoint + length*step_size >= (std::numeric_limits<Real>::max)())
 109     {
 110         throw std::logic_error("Right endpoint overflows the maximum representable number of the specified precision.\n");
 111     }
 112     if (step_size <= 0)
 113     {
 114         throw std::logic_error("The step size must be strictly > 0.\n");
 115     }
 116     // Storing the inverse of the stepsize does provide a measurable speedup.
 117     // It's not huge, but nonetheless worthwhile.
 118     m_h_inv = 1/step_size;
 119
 120     // Following Kress's notation, s'(a) = a1, s'(b) = b1
 121     Real a1 = left_endpoint_derivative;
 122     // See the finite-difference table on Wikipedia for reference on how
 123     // to construct high-order estimates for one-sided derivatives:
 124     // https://en.wikipedia.org/wiki/Finite_difference_coefficient#Forward_and_backward_finite_difference
 125     // Here, we estimate then to O(h^4), as that is the maximum accuracy we could obtain from this method.
 126     if (boost::math::isnan(a1))
 127     {
 128         // For simple functions (linear, quadratic, so on)
 129         // almost all the error comes from derivative estimation.
 130         // This does pairwise summation which gives us another digit of accuracy over naive summation.
 131         Real t0 = 4*(f[1] + third<Real>()*f[3]);
 132         Real t1 = -(25*third<Real>()*f[0] + f[4])/4  - 3*f[2];
 133         a1 = m_h_inv*(t0 + t1);
 134     }
 135
 136     Real b1 = right_endpoint_derivative;
 137     if (boost::math::isnan(b1))
 138     {
 139         size_t n = length - 1;
 140         Real t0 = 4*(f[n-3] + third<Real>()*f[n - 1]);
 141         Real t1 = -(25*third<Real>()*f[n - 4] + f[n])/4  - 3*f[n - 2];
 142
 143         b1 = m_h_inv*(t0 + t1);
 144     }
 145
 146     // s(x) = \sum \alpha_i B_{3}( (x- x_i - a)/h )
 147     // Of course we must reindex from Kress's notation, since he uses negative indices which make C++ unhappy.
 148     m_beta.resize(length + 2, std::numeric_limits<Real>::quiet_NaN());
 149
 150     // Since the splines have compact support, they decay to zero very fast outside the endpoints.
 151     // This is often very annoying; we'd like to evaluate the interpolant a little bit outside the
 152     // boundary [a,b] without massive error.
 153     // A simple way to deal with this is just to subtract the DC component off the signal, so we need the average.
 154     // This algorithm for computing the average is recommended in
 155     // http://www.heikohoffmann.de/htmlthesis/node134.html
 156     Real t = 1;
 157     for (size_t i = 0; i < length; ++i)
 158     {
 159         if (boost::math::isnan(f[i]))
 160         {
 161             std::string err = "This function you are trying to interpolate is a nan at index " + std::to_string(i) + "\n";
 162             throw std::logic_error(err);
 163         }
 164         m_avg += (f[i] - m_avg) / t;
 165         t += 1;
 166     }
 167
 168
 169     // Now we must solve an almost-tridiagonal system, which requires O(N) operations.
 170     // There are, in fact 5 diagonals, but they only differ from zero on the first and last row,
 171     // so we can patch up the tridiagonal row reduction algorithm to deal with two special rows.
 172     // See Kress, equations 8.41
 173     // The the "tridiagonal" matrix is:
 174     // 1  0 -1
 175     // 1  4  1
 176     //    1  4  1
 177     //       1  4  1
 178     //          ....
 179     //          1  4  1
 180     //          1  0 -1
 181     // Numerical estimate indicate that as N->Infinity, cond(A) -> 6.9, so this matrix is good.
 182     std::vector<Real> rhs(length + 2, std::numeric_limits<Real>::quiet_NaN());
 183     std::vector<Real> super_diagonal(length + 2, std::numeric_limits<Real>::quiet_NaN());
 184
 185     rhs[0] = -2*step_size*a1;
 186     rhs[rhs.size() - 1] = -2*step_size*b1;
 187
 188     super_diagonal[0] = 0;
 189
 190     for(size_t i = 1; i < rhs.size() - 1; ++i)
 191     {
 192         rhs[i] = 6*(f[i - 1] - m_avg);
 193         super_diagonal[i] = 1;
 194     }
 195
 196
 197     // One step of row reduction on the first row to patch up the 5-diagonal problem:
 198     // 1 0 -1 | r0
 199     // 1 4 1  | r1
 200     // mapsto:
 201     // 1 0 -1 | r0
 202     // 0 4 2  | r1 - r0
 203     // mapsto
 204     // 1 0 -1 | r0
 205     // 0 1 1/2| (r1 - r0)/4
 206     super_diagonal[1] = 0.5;
 207     rhs[1] = (rhs[1] - rhs[0])/4;
 208
 209     // Now do a tridiagonal row reduction the standard way, until just before the last row:
 210     for (size_t i = 2; i < rhs.size() - 1; ++i)
 211     {
 212         Real diagonal = 4 - super_diagonal[i - 1];
 213         rhs[i] = (rhs[i] - rhs[i - 1])/diagonal;
 214         super_diagonal[i] /= diagonal;
 215     }
 216
 217     // Now the last row, which is in the form
 218     // 1 sd[n-3] 0      | rhs[n-3]
 219     // 0  1     sd[n-2] | rhs[n-2]
 220     // 1  0     -1      | rhs[n-1]
 221     Real final_subdiag = -super_diagonal[rhs.size() - 3];
 222     rhs[rhs.size() - 1] = (rhs[rhs.size() - 1] - rhs[rhs.size() - 3])/final_subdiag;
 223     Real final_diag = -1/final_subdiag;
 224     // Now we're here:
 225     // 1 sd[n-3] 0         | rhs[n-3]
 226     // 0  1     sd[n-2]    | rhs[n-2]
 227     // 0  1     final_diag | (rhs[n-1] - rhs[n-3])/diag
 228
 229     final_diag = final_diag - super_diagonal[rhs.size() - 2];
 230     rhs[rhs.size() - 1] = rhs[rhs.size() - 1] - rhs[rhs.size() - 2];
 231
 232
 233     // Back substitutions:
 234     m_beta[rhs.size() - 1] = rhs[rhs.size() - 1]/final_diag;
 235     for(size_t i = rhs.size() - 2; i > 0; --i)
 236     {
 237         m_beta[i] = rhs[i] - super_diagonal[i]*m_beta[i + 1];
 238     }
 239     m_beta[0] = m_beta[2] + rhs[0];
 240 }
 241
 242 template<class Real>
 243 Real cubic_b_spline_imp<Real>::operator()(Real x) const
 244 {
 245     // See Kress, 8.40: Since B3 has compact support, we don't have to sum over all terms,
 246     // just the (at most 5) whose support overlaps the argument.
 247     Real z = m_avg;
 248     Real t = m_h_inv*(x - m_a) + 1;
 249
 250     using std::max;
 251     using std::min;
 252     using std::ceil;
 253     using std::floor;
 254
 255     size_t k_min = (size_t) (max)(static_cast<long>(0), boost::math::ltrunc(ceil(t - 2)));
 256     size_t k_max = (size_t) (max)((min)(static_cast<long>(m_beta.size() - 1), boost::math::ltrunc(floor(t + 2))), (long) 0);
 257     for (size_t k = k_min; k <= k_max; ++k)
 258     {
 259         z += m_beta[k]*b3_spline(t - k);
 260     }
 261
 262     return z;
 263 }
 264
 265 template<class Real>
 266 Real cubic_b_spline_imp<Real>::prime(Real x) const
 267 {
 268     Real z = 0;
 269     Real t = m_h_inv*(x - m_a) + 1;
 270
 271     using std::max;
 272     using std::min;
 273     using std::ceil;
 274     using std::floor;
 275
 276     size_t k_min = (size_t) (max)(static_cast<long>(0), boost::math::ltrunc(ceil(t - 2)));
 277     size_t k_max = (size_t) (min)(static_cast<long>(m_beta.size() - 1), boost::math::ltrunc(floor(t + 2)));
 278
 279     for (size_t k = k_min; k <= k_max; ++k)
 280     {
 281         z += m_beta[k]*b3_spline_prime(t - k);
 282     }
 283     return z*m_h_inv;
 284 }
 285
 286 }}}
 287 #endif