| 1 |
/* |
| 2 |
* Copyright (c) 2005 The University of Notre Dame. All Rights Reserved. |
| 3 |
* |
| 4 |
* The University of Notre Dame grants you ("Licensee") a |
| 5 |
* non-exclusive, royalty free, license to use, modify and |
| 6 |
* redistribute this software in source and binary code form, provided |
| 7 |
* that the following conditions are met: |
| 8 |
* |
| 9 |
* 1. Redistributions of source code must retain the above copyright |
| 10 |
* notice, this list of conditions and the following disclaimer. |
| 11 |
* |
| 12 |
* 2. Redistributions in binary form must reproduce the above copyright |
| 13 |
* notice, this list of conditions and the following disclaimer in the |
| 14 |
* documentation and/or other materials provided with the |
| 15 |
* distribution. |
| 16 |
* |
| 17 |
* This software is provided "AS IS," without a warranty of any |
| 18 |
* kind. All express or implied conditions, representations and |
| 19 |
* warranties, including any implied warranty of merchantability, |
| 20 |
* fitness for a particular purpose or non-infringement, are hereby |
| 21 |
* excluded. The University of Notre Dame and its licensors shall not |
| 22 |
* be liable for any damages suffered by licensee as a result of |
| 23 |
* using, modifying or distributing the software or its |
| 24 |
* derivatives. In no event will the University of Notre Dame or its |
| 25 |
* licensors be liable for any lost revenue, profit or data, or for |
| 26 |
* direct, indirect, special, consequential, incidental or punitive |
| 27 |
* damages, however caused and regardless of the theory of liability, |
| 28 |
* arising out of the use of or inability to use software, even if the |
| 29 |
* University of Notre Dame has been advised of the possibility of |
| 30 |
* such damages. |
| 31 |
* |
| 32 |
* SUPPORT OPEN SCIENCE! If you use OpenMD or its source code in your |
| 33 |
* research, please cite the appropriate papers when you publish your |
| 34 |
* work. Good starting points are: |
| 35 |
* |
| 36 |
* [1] Meineke, et al., J. Comp. Chem. 26, 252-271 (2005). |
| 37 |
* [2] Fennell & Gezelter, J. Chem. Phys. 124, 234104 (2006). |
| 38 |
* [3] Sun, Lin & Gezelter, J. Chem. Phys. 128, 234107 (2008). |
| 39 |
* [4] Kuang & Gezelter, J. Chem. Phys. 133, 164101 (2010). |
| 40 |
* [5] Vardeman, Stocker & Gezelter, J. Chem. Theory Comput. 7, 834 (2011). |
| 41 |
*/ |
| 42 |
|
| 43 |
#include "math/CubicSpline.hpp" |
| 44 |
#include <cmath> |
| 45 |
#include <cassert> |
| 46 |
#include <cstdio> |
| 47 |
#include <algorithm> |
| 48 |
#include <numeric> |
| 49 |
|
| 50 |
using namespace OpenMD; |
| 51 |
using namespace std; |
| 52 |
|
| 53 |
CubicSpline::CubicSpline() : isUniform(true), generated(false) { |
| 54 |
x_.clear(); |
| 55 |
y_.clear(); |
| 56 |
} |
| 57 |
|
| 58 |
void CubicSpline::addPoint(const RealType xp, const RealType yp) { |
| 59 |
x_.push_back(xp); |
| 60 |
y_.push_back(yp); |
| 61 |
} |
| 62 |
|
| 63 |
void CubicSpline::addPoints(const vector<RealType>& xps, |
| 64 |
const vector<RealType>& yps) { |
| 65 |
|
| 66 |
assert(xps.size() == yps.size()); |
| 67 |
|
| 68 |
for (unsigned int i = 0; i < xps.size(); i++){ |
| 69 |
x_.push_back(xps[i]); |
| 70 |
y_.push_back(yps[i]); |
| 71 |
} |
| 72 |
} |
| 73 |
|
| 74 |
void CubicSpline::generate() { |
| 75 |
// Calculate coefficients defining a smooth cubic interpolatory spline. |
| 76 |
// |
| 77 |
// class values constructed: |
| 78 |
// n = number of data_ points. |
| 79 |
// x_ = vector of independent variable values |
| 80 |
// y_ = vector of dependent variable values |
| 81 |
// b = vector of S'(x_[i]) values. |
| 82 |
// c = vector of S"(x_[i])/2 values. |
| 83 |
// d = vector of S'''(x_[i]+)/6 values (i < n). |
| 84 |
// Local variables: |
| 85 |
|
| 86 |
RealType fp1, fpn, p; |
| 87 |
RealType h(0.0); |
| 88 |
|
| 89 |
// make sure the sizes match |
| 90 |
|
| 91 |
n = x_.size(); |
| 92 |
b.resize(n); |
| 93 |
c.resize(n); |
| 94 |
d.resize(n); |
| 95 |
|
| 96 |
// make sure we are monotonically increasing in x: |
| 97 |
|
| 98 |
bool sorted = true; |
| 99 |
|
| 100 |
for (int i = 1; i < n; i++) { |
| 101 |
if ( (x_[i] - x_[i-1] ) <= 0.0 ) sorted = false; |
| 102 |
} |
| 103 |
|
| 104 |
// sort if necessary |
| 105 |
|
| 106 |
if (!sorted) { |
| 107 |
vector<int> p = sort_permutation(x_); |
| 108 |
x_ = apply_permutation(x_, p); |
| 109 |
y_ = apply_permutation(y_, p); |
| 110 |
} |
| 111 |
|
| 112 |
|
| 113 |
// Calculate coefficients for the tridiagonal system: store |
| 114 |
// sub-diagonal in B, diagonal in D, difference quotient in C. |
| 115 |
|
| 116 |
b[0] = x_[1] - x_[0]; |
| 117 |
c[0] = (y_[1] - y_[0]) / b[0]; |
| 118 |
|
| 119 |
if (n == 2) { |
| 120 |
|
| 121 |
// Assume the derivatives at both endpoints are zero. Another |
| 122 |
// assumption could be made to have a linear interpolant between |
| 123 |
// the two points. In that case, the b coefficients below would be |
| 124 |
// (y_[1] - y_[0]) / (x_[1] - x_[0]) |
| 125 |
// and the c and d coefficients would both be zero. |
| 126 |
b[0] = 0.0; |
| 127 |
c[0] = -3.0 * pow((y_[1] - y_[0]) / (x_[1] - x_[0]), 2); |
| 128 |
d[0] = -2.0 * pow((y_[1] - y_[0]) / (x_[1] - x_[0]), 3); |
| 129 |
b[1] = b[0]; |
| 130 |
c[1] = 0.0; |
| 131 |
d[1] = 0.0; |
| 132 |
dx = 1.0 / (x_[1] - x_[0]); |
| 133 |
isUniform = true; |
| 134 |
generated = true; |
| 135 |
return; |
| 136 |
} |
| 137 |
|
| 138 |
d[0] = 2.0 * b[0]; |
| 139 |
|
| 140 |
for (int i = 1; i < n-1; i++) { |
| 141 |
b[i] = x_[i+1] - x_[i]; |
| 142 |
if ( fabs( b[i] - b[0] ) / b[0] > 1.0e-5) isUniform = false; |
| 143 |
c[i] = (y_[i+1] - y_[i]) / b[i]; |
| 144 |
d[i] = 2.0 * (b[i] + b[i-1]); |
| 145 |
} |
| 146 |
|
| 147 |
d[n-1] = 2.0 * b[n-2]; |
| 148 |
|
| 149 |
// Calculate estimates for the end slopes using polynomials |
| 150 |
// that interpolate the data_ nearest the end. |
| 151 |
|
| 152 |
fp1 = c[0] - b[0]*(c[1] - c[0])/(b[0] + b[1]); |
| 153 |
if (n > 3) fp1 = fp1 + b[0]*((b[0] + b[1]) * (c[2] - c[1]) / |
| 154 |
(b[1] + b[2]) - |
| 155 |
c[1] + c[0]) / (x_[3] - x_[0]); |
| 156 |
|
| 157 |
fpn = c[n-2] + b[n-2]*(c[n-2] - c[n-3])/(b[n-3] + b[n-2]); |
| 158 |
|
| 159 |
if (n > 3) fpn = fpn + b[n-2] * |
| 160 |
(c[n-2] - c[n-3] - (b[n-3] + b[n-2]) * |
| 161 |
(c[n-3] - c[n-4])/(b[n-3] + b[n-4])) / |
| 162 |
(x_[n-1] - x_[n-4]); |
| 163 |
|
| 164 |
// Calculate the right hand side and store it in C. |
| 165 |
|
| 166 |
c[n-1] = 3.0 * (fpn - c[n-2]); |
| 167 |
for (int i = n-2; i > 0; i--) |
| 168 |
c[i] = 3.0 * (c[i] - c[i-1]); |
| 169 |
c[0] = 3.0 * (c[0] - fp1); |
| 170 |
|
| 171 |
// Solve the tridiagonal system. |
| 172 |
|
| 173 |
for (int k = 1; k < n; k++) { |
| 174 |
p = b[k-1] / d[k-1]; |
| 175 |
d[k] = d[k] - p*b[k-1]; |
| 176 |
c[k] = c[k] - p*c[k-1]; |
| 177 |
} |
| 178 |
|
| 179 |
c[n-1] = c[n-1] / d[n-1]; |
| 180 |
|
| 181 |
for (int k = n-2; k >= 0; k--) |
| 182 |
c[k] = (c[k] - b[k] * c[k+1]) / d[k]; |
| 183 |
|
| 184 |
// Calculate the coefficients defining the spline. |
| 185 |
|
| 186 |
for (int i = 0; i < n-1; i++) { |
| 187 |
h = x_[i+1] - x_[i]; |
| 188 |
d[i] = (c[i+1] - c[i]) / (3.0 * h); |
| 189 |
b[i] = (y_[i+1] - y_[i])/h - h * (c[i] + h * d[i]); |
| 190 |
} |
| 191 |
|
| 192 |
b[n-1] = b[n-2] + h * (2.0 * c[n-2] + h * 3.0 * d[n-2]); |
| 193 |
|
| 194 |
if (isUniform) dx = 1.0 / (x_[1] - x_[0]); |
| 195 |
|
| 196 |
generated = true; |
| 197 |
return; |
| 198 |
} |
| 199 |
|
| 200 |
RealType CubicSpline::getValueAt(const RealType& t) { |
| 201 |
// Evaluate the spline at t using coefficients |
| 202 |
// |
| 203 |
// Input parameters |
| 204 |
// t = point where spline is to be evaluated. |
| 205 |
// Output: |
| 206 |
// value of spline at t. |
| 207 |
|
| 208 |
if (!generated) generate(); |
| 209 |
|
| 210 |
assert(t >= x_.front()); |
| 211 |
assert(t <= x_.back()); |
| 212 |
|
| 213 |
// Find the interval ( x[j], x[j+1] ) that contains or is nearest |
| 214 |
// to t. |
| 215 |
|
| 216 |
if (isUniform) { |
| 217 |
|
| 218 |
j = max(0, min(n-1, int((t - x_[0]) * dx))); |
| 219 |
|
| 220 |
} else { |
| 221 |
|
| 222 |
j = n-1; |
| 223 |
|
| 224 |
for (int i = 0; i < n; i++) { |
| 225 |
if ( t < x_[i] ) { |
| 226 |
j = i-1; |
| 227 |
break; |
| 228 |
} |
| 229 |
} |
| 230 |
} |
| 231 |
|
| 232 |
// Evaluate the cubic polynomial. |
| 233 |
|
| 234 |
dt = t - x_[j]; |
| 235 |
return y_[j] + dt*(b[j] + dt*(c[j] + dt*d[j])); |
| 236 |
} |
| 237 |
|
| 238 |
|
| 239 |
void CubicSpline::getValueAt(const RealType& t, RealType& v) { |
| 240 |
// Evaluate the spline at t using coefficients |
| 241 |
// |
| 242 |
// Input parameters |
| 243 |
// t = point where spline is to be evaluated. |
| 244 |
// Output: |
| 245 |
// value of spline at t. |
| 246 |
|
| 247 |
if (!generated) generate(); |
| 248 |
|
| 249 |
assert(t >= x_.front()); |
| 250 |
assert(t <= x_.back()); |
| 251 |
|
| 252 |
// Find the interval ( x[j], x[j+1] ) that contains or is nearest |
| 253 |
// to t. |
| 254 |
|
| 255 |
if (isUniform) { |
| 256 |
|
| 257 |
j = max(0, min(n-1, int((t - x_[0]) * dx))); |
| 258 |
|
| 259 |
} else { |
| 260 |
|
| 261 |
j = n-1; |
| 262 |
|
| 263 |
for (int i = 0; i < n; i++) { |
| 264 |
if ( t < x_[i] ) { |
| 265 |
j = i-1; |
| 266 |
break; |
| 267 |
} |
| 268 |
} |
| 269 |
} |
| 270 |
|
| 271 |
// Evaluate the cubic polynomial. |
| 272 |
|
| 273 |
dt = t - x_[j]; |
| 274 |
v = y_[j] + dt*(b[j] + dt*(c[j] + dt*d[j])); |
| 275 |
} |
| 276 |
|
| 277 |
pair<RealType, RealType> CubicSpline::getLimits(){ |
| 278 |
if (!generated) generate(); |
| 279 |
return make_pair( x_.front(), x_.back() ); |
| 280 |
} |
| 281 |
|
| 282 |
void CubicSpline::getValueAndDerivativeAt(const RealType& t, RealType& v, |
| 283 |
RealType &dv) { |
| 284 |
// Evaluate the spline and first derivative at t using coefficients |
| 285 |
// |
| 286 |
// Input parameters |
| 287 |
// t = point where spline is to be evaluated. |
| 288 |
|
| 289 |
if (!generated) generate(); |
| 290 |
|
| 291 |
assert(t >= x_.front()); |
| 292 |
assert(t <= x_.back()); |
| 293 |
|
| 294 |
// Find the interval ( x[j], x[j+1] ) that contains or is nearest |
| 295 |
// to t. |
| 296 |
|
| 297 |
if (isUniform) { |
| 298 |
|
| 299 |
j = max(0, min(n-1, int((t - x_[0]) * dx))); |
| 300 |
|
| 301 |
} else { |
| 302 |
|
| 303 |
j = n-1; |
| 304 |
|
| 305 |
for (int i = 0; i < n; i++) { |
| 306 |
if ( t < x_[i] ) { |
| 307 |
j = i-1; |
| 308 |
break; |
| 309 |
} |
| 310 |
} |
| 311 |
} |
| 312 |
|
| 313 |
// Evaluate the cubic polynomial. |
| 314 |
|
| 315 |
dt = t - x_[j]; |
| 316 |
|
| 317 |
v = y_[j] + dt*(b[j] + dt*(c[j] + dt*d[j])); |
| 318 |
dv = b[j] + dt*(2.0 * c[j] + 3.0 * dt * d[j]); |
| 319 |
} |
| 320 |
|
| 321 |
std::vector<int> CubicSpline::sort_permutation(std::vector<RealType>& v) { |
| 322 |
std::vector<int> p(v.size()); |
| 323 |
|
| 324 |
// 6 lines to replace std::iota(p.begin(), p.end(), 0); |
| 325 |
int value = 0; |
| 326 |
std::vector<int>::iterator i; |
| 327 |
for (i = p.begin(); i != p.end(); ++i) { |
| 328 |
(*i) = value; |
| 329 |
++value; |
| 330 |
} |
| 331 |
|
| 332 |
std::sort(p.begin(), p.end(), OpenMD::Comparator(v) ); |
| 333 |
return p; |
| 334 |
} |
| 335 |
|
| 336 |
std::vector<RealType> CubicSpline::apply_permutation(std::vector<RealType> const& v, |
| 337 |
std::vector<int> const& p) { |
| 338 |
int n = p.size(); |
| 339 |
std::vector<double> sorted_vec(n); |
| 340 |
for (int i = 0; i < n; ++i) { |
| 341 |
sorted_vec[i] = v[p[i]]; |
| 342 |
} |
| 343 |
return sorted_vec; |
| 344 |
} |
