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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 | ||
92f5a8d4 TL |
35 | Real double_prime(Real x) const; |
36 | ||
b32b8144 FG |
37 | private: |
38 | std::vector<Real> m_beta; | |
39 | Real m_h_inv; | |
40 | Real m_a; | |
41 | Real m_avg; | |
42 | }; | |
43 | ||
44 | ||
45 | ||
46 | template <class Real> | |
47 | Real b3_spline(Real x) | |
48 | { | |
49 | using std::abs; | |
50 | Real absx = abs(x); | |
51 | if (absx < 1) | |
52 | { | |
53 | Real y = 2 - absx; | |
54 | Real z = 1 - absx; | |
55 | return boost::math::constants::sixth<Real>()*(y*y*y - 4*z*z*z); | |
56 | } | |
57 | if (absx < 2) | |
58 | { | |
59 | Real y = 2 - absx; | |
60 | return boost::math::constants::sixth<Real>()*y*y*y; | |
61 | } | |
62 | return (Real) 0; | |
63 | } | |
64 | ||
65 | template<class Real> | |
66 | Real b3_spline_prime(Real x) | |
67 | { | |
68 | if (x < 0) | |
69 | { | |
70 | return -b3_spline_prime(-x); | |
71 | } | |
72 | ||
73 | if (x < 1) | |
74 | { | |
75 | return x*(3*boost::math::constants::half<Real>()*x - 2); | |
76 | } | |
77 | if (x < 2) | |
78 | { | |
79 | return -boost::math::constants::half<Real>()*(2 - x)*(2 - x); | |
80 | } | |
81 | return (Real) 0; | |
82 | } | |
83 | ||
92f5a8d4 TL |
84 | template<class Real> |
85 | Real b3_spline_double_prime(Real x) | |
86 | { | |
87 | if (x < 0) | |
88 | { | |
89 | return b3_spline_double_prime(-x); | |
90 | } | |
91 | ||
92 | if (x < 1) | |
93 | { | |
94 | return 3*x - 2; | |
95 | } | |
96 | if (x < 2) | |
97 | { | |
98 | return (2 - x); | |
99 | } | |
100 | return (Real) 0; | |
101 | } | |
102 | ||
b32b8144 FG |
103 | |
104 | template <class Real> | |
105 | template <class BidiIterator> | |
106 | cubic_b_spline_imp<Real>::cubic_b_spline_imp(BidiIterator f, BidiIterator end_p, Real left_endpoint, Real step_size, | |
107 | Real left_endpoint_derivative, Real right_endpoint_derivative) : m_a(left_endpoint), m_avg(0) | |
108 | { | |
109 | using boost::math::constants::third; | |
110 | ||
111 | std::size_t length = end_p - f; | |
112 | ||
113 | if (length < 5) | |
114 | { | |
115 | if (boost::math::isnan(left_endpoint_derivative) || boost::math::isnan(right_endpoint_derivative)) | |
116 | { | |
117 | throw std::logic_error("Interpolation using a cubic b spline with derivatives estimated at the endpoints requires at least 5 points.\n"); | |
118 | } | |
119 | if (length < 3) | |
120 | { | |
121 | throw std::logic_error("Interpolation using a cubic b spline requires at least 3 points.\n"); | |
122 | } | |
123 | } | |
124 | ||
125 | if (boost::math::isnan(left_endpoint)) | |
126 | { | |
127 | throw std::logic_error("Left endpoint is NAN; this is disallowed.\n"); | |
128 | } | |
129 | if (left_endpoint + length*step_size >= (std::numeric_limits<Real>::max)()) | |
130 | { | |
131 | throw std::logic_error("Right endpoint overflows the maximum representable number of the specified precision.\n"); | |
132 | } | |
133 | if (step_size <= 0) | |
134 | { | |
135 | throw std::logic_error("The step size must be strictly > 0.\n"); | |
136 | } | |
137 | // Storing the inverse of the stepsize does provide a measurable speedup. | |
138 | // It's not huge, but nonetheless worthwhile. | |
139 | m_h_inv = 1/step_size; | |
140 | ||
141 | // Following Kress's notation, s'(a) = a1, s'(b) = b1 | |
142 | Real a1 = left_endpoint_derivative; | |
143 | // See the finite-difference table on Wikipedia for reference on how | |
144 | // to construct high-order estimates for one-sided derivatives: | |
145 | // https://en.wikipedia.org/wiki/Finite_difference_coefficient#Forward_and_backward_finite_difference | |
146 | // Here, we estimate then to O(h^4), as that is the maximum accuracy we could obtain from this method. | |
147 | if (boost::math::isnan(a1)) | |
148 | { | |
149 | // For simple functions (linear, quadratic, so on) | |
150 | // almost all the error comes from derivative estimation. | |
151 | // This does pairwise summation which gives us another digit of accuracy over naive summation. | |
152 | Real t0 = 4*(f[1] + third<Real>()*f[3]); | |
153 | Real t1 = -(25*third<Real>()*f[0] + f[4])/4 - 3*f[2]; | |
154 | a1 = m_h_inv*(t0 + t1); | |
155 | } | |
156 | ||
157 | Real b1 = right_endpoint_derivative; | |
158 | if (boost::math::isnan(b1)) | |
159 | { | |
160 | size_t n = length - 1; | |
161 | Real t0 = 4*(f[n-3] + third<Real>()*f[n - 1]); | |
162 | Real t1 = -(25*third<Real>()*f[n - 4] + f[n])/4 - 3*f[n - 2]; | |
163 | ||
164 | b1 = m_h_inv*(t0 + t1); | |
165 | } | |
166 | ||
167 | // s(x) = \sum \alpha_i B_{3}( (x- x_i - a)/h ) | |
168 | // Of course we must reindex from Kress's notation, since he uses negative indices which make C++ unhappy. | |
169 | m_beta.resize(length + 2, std::numeric_limits<Real>::quiet_NaN()); | |
170 | ||
171 | // Since the splines have compact support, they decay to zero very fast outside the endpoints. | |
172 | // This is often very annoying; we'd like to evaluate the interpolant a little bit outside the | |
173 | // boundary [a,b] without massive error. | |
174 | // A simple way to deal with this is just to subtract the DC component off the signal, so we need the average. | |
175 | // This algorithm for computing the average is recommended in | |
176 | // http://www.heikohoffmann.de/htmlthesis/node134.html | |
177 | Real t = 1; | |
178 | for (size_t i = 0; i < length; ++i) | |
179 | { | |
180 | if (boost::math::isnan(f[i])) | |
181 | { | |
182 | std::string err = "This function you are trying to interpolate is a nan at index " + std::to_string(i) + "\n"; | |
183 | throw std::logic_error(err); | |
184 | } | |
185 | m_avg += (f[i] - m_avg) / t; | |
186 | t += 1; | |
187 | } | |
188 | ||
189 | ||
190 | // Now we must solve an almost-tridiagonal system, which requires O(N) operations. | |
191 | // There are, in fact 5 diagonals, but they only differ from zero on the first and last row, | |
192 | // so we can patch up the tridiagonal row reduction algorithm to deal with two special rows. | |
193 | // See Kress, equations 8.41 | |
194 | // The the "tridiagonal" matrix is: | |
195 | // 1 0 -1 | |
196 | // 1 4 1 | |
197 | // 1 4 1 | |
198 | // 1 4 1 | |
199 | // .... | |
200 | // 1 4 1 | |
201 | // 1 0 -1 | |
202 | // Numerical estimate indicate that as N->Infinity, cond(A) -> 6.9, so this matrix is good. | |
203 | std::vector<Real> rhs(length + 2, std::numeric_limits<Real>::quiet_NaN()); | |
204 | std::vector<Real> super_diagonal(length + 2, std::numeric_limits<Real>::quiet_NaN()); | |
205 | ||
206 | rhs[0] = -2*step_size*a1; | |
207 | rhs[rhs.size() - 1] = -2*step_size*b1; | |
208 | ||
209 | super_diagonal[0] = 0; | |
210 | ||
211 | for(size_t i = 1; i < rhs.size() - 1; ++i) | |
212 | { | |
213 | rhs[i] = 6*(f[i - 1] - m_avg); | |
214 | super_diagonal[i] = 1; | |
215 | } | |
216 | ||
217 | ||
218 | // One step of row reduction on the first row to patch up the 5-diagonal problem: | |
219 | // 1 0 -1 | r0 | |
220 | // 1 4 1 | r1 | |
221 | // mapsto: | |
222 | // 1 0 -1 | r0 | |
223 | // 0 4 2 | r1 - r0 | |
224 | // mapsto | |
225 | // 1 0 -1 | r0 | |
226 | // 0 1 1/2| (r1 - r0)/4 | |
227 | super_diagonal[1] = 0.5; | |
228 | rhs[1] = (rhs[1] - rhs[0])/4; | |
229 | ||
230 | // Now do a tridiagonal row reduction the standard way, until just before the last row: | |
231 | for (size_t i = 2; i < rhs.size() - 1; ++i) | |
232 | { | |
233 | Real diagonal = 4 - super_diagonal[i - 1]; | |
234 | rhs[i] = (rhs[i] - rhs[i - 1])/diagonal; | |
235 | super_diagonal[i] /= diagonal; | |
236 | } | |
237 | ||
238 | // Now the last row, which is in the form | |
239 | // 1 sd[n-3] 0 | rhs[n-3] | |
240 | // 0 1 sd[n-2] | rhs[n-2] | |
241 | // 1 0 -1 | rhs[n-1] | |
242 | Real final_subdiag = -super_diagonal[rhs.size() - 3]; | |
243 | rhs[rhs.size() - 1] = (rhs[rhs.size() - 1] - rhs[rhs.size() - 3])/final_subdiag; | |
244 | Real final_diag = -1/final_subdiag; | |
245 | // Now we're here: | |
246 | // 1 sd[n-3] 0 | rhs[n-3] | |
247 | // 0 1 sd[n-2] | rhs[n-2] | |
248 | // 0 1 final_diag | (rhs[n-1] - rhs[n-3])/diag | |
249 | ||
250 | final_diag = final_diag - super_diagonal[rhs.size() - 2]; | |
251 | rhs[rhs.size() - 1] = rhs[rhs.size() - 1] - rhs[rhs.size() - 2]; | |
252 | ||
253 | ||
254 | // Back substitutions: | |
255 | m_beta[rhs.size() - 1] = rhs[rhs.size() - 1]/final_diag; | |
256 | for(size_t i = rhs.size() - 2; i > 0; --i) | |
257 | { | |
258 | m_beta[i] = rhs[i] - super_diagonal[i]*m_beta[i + 1]; | |
259 | } | |
260 | m_beta[0] = m_beta[2] + rhs[0]; | |
261 | } | |
262 | ||
263 | template<class Real> | |
264 | Real cubic_b_spline_imp<Real>::operator()(Real x) const | |
265 | { | |
266 | // See Kress, 8.40: Since B3 has compact support, we don't have to sum over all terms, | |
267 | // just the (at most 5) whose support overlaps the argument. | |
268 | Real z = m_avg; | |
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) (max)((min)(static_cast<long>(m_beta.size() - 1), boost::math::ltrunc(floor(t + 2))), (long) 0); | |
278 | for (size_t k = k_min; k <= k_max; ++k) | |
279 | { | |
280 | z += m_beta[k]*b3_spline(t - k); | |
281 | } | |
282 | ||
283 | return z; | |
284 | } | |
285 | ||
286 | template<class Real> | |
287 | Real cubic_b_spline_imp<Real>::prime(Real x) const | |
288 | { | |
289 | Real z = 0; | |
290 | Real t = m_h_inv*(x - m_a) + 1; | |
291 | ||
292 | using std::max; | |
293 | using std::min; | |
294 | using std::ceil; | |
295 | using std::floor; | |
296 | ||
297 | size_t k_min = (size_t) (max)(static_cast<long>(0), boost::math::ltrunc(ceil(t - 2))); | |
298 | size_t k_max = (size_t) (min)(static_cast<long>(m_beta.size() - 1), boost::math::ltrunc(floor(t + 2))); | |
299 | ||
300 | for (size_t k = k_min; k <= k_max; ++k) | |
301 | { | |
302 | z += m_beta[k]*b3_spline_prime(t - k); | |
303 | } | |
304 | return z*m_h_inv; | |
305 | } | |
306 | ||
92f5a8d4 TL |
307 | template<class Real> |
308 | Real cubic_b_spline_imp<Real>::double_prime(Real x) const | |
309 | { | |
310 | Real z = 0; | |
311 | Real t = m_h_inv*(x - m_a) + 1; | |
312 | ||
313 | using std::max; | |
314 | using std::min; | |
315 | using std::ceil; | |
316 | using std::floor; | |
317 | ||
318 | size_t k_min = (size_t) (max)(static_cast<long>(0), boost::math::ltrunc(ceil(t - 2))); | |
319 | size_t k_max = (size_t) (min)(static_cast<long>(m_beta.size() - 1), boost::math::ltrunc(floor(t + 2))); | |
320 | ||
321 | for (size_t k = k_min; k <= k_max; ++k) | |
322 | { | |
323 | z += m_beta[k]*b3_spline_double_prime(t - k); | |
324 | } | |
325 | return z*m_h_inv*m_h_inv; | |
326 | } | |
327 | ||
328 | ||
b32b8144 FG |
329 | }}} |
330 | #endif |