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/* |
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* Copyright (c) 2005 The University of Notre Dame. All Rights Reserved. |
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* |
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* The University of Notre Dame grants you ("Licensee") a |
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* non-exclusive, royalty free, license to use, modify and |
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* redistribute this software in source and binary code form, provided |
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* that the following conditions are met: |
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* |
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* 1. Redistributions of source code must retain the above copyright |
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* notice, this list of conditions and the following disclaimer. |
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* |
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* 2. Redistributions in binary form must reproduce the above copyright |
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* notice, this list of conditions and the following disclaimer in the |
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* documentation and/or other materials provided with the |
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* distribution. |
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* |
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* This software is provided "AS IS," without a warranty of any |
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* kind. All express or implied conditions, representations and |
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* warranties, including any implied warranty of merchantability, |
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* fitness for a particular purpose or non-infringement, are hereby |
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* excluded. The University of Notre Dame and its licensors shall not |
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* be liable for any damages suffered by licensee as a result of |
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* using, modifying or distributing the software or its |
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* derivatives. In no event will the University of Notre Dame or its |
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* licensors be liable for any lost revenue, profit or data, or for |
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* direct, indirect, special, consequential, incidental or punitive |
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* damages, however caused and regardless of the theory of liability, |
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* arising out of the use of or inability to use software, even if the |
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* University of Notre Dame has been advised of the possibility of |
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* such damages. |
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* |
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* SUPPORT OPEN SCIENCE! If you use OpenMD or its source code in your |
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* research, please cite the appropriate papers when you publish your |
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* work. Good starting points are: |
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* |
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* [1] Meineke, et al., J. Comp. Chem. 26, 252-271 (2005). |
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* [2] Fennell & Gezelter, J. Chem. Phys. 124, 234104 (2006). |
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* [3] Sun, Lin & Gezelter, J. Chem. Phys. 128, 234107 (2008). |
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* [4] Kuang & Gezelter, J. Chem. Phys. 133, 164101 (2010). |
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* [5] Vardeman, Stocker & Gezelter, J. Chem. Theory Comput. 7, 834 (2011). |
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*/ |
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|
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#include "math/CubicSpline.hpp" |
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#include <cmath> |
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#include <cassert> |
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#include <cstdio> |
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#include <algorithm> |
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|
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using namespace OpenMD; |
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using namespace std; |
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|
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CubicSpline::CubicSpline() : generated(false), isUniform(true) { |
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data_.clear(); |
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} |
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|
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void CubicSpline::addPoint(const RealType xp, const RealType yp) { |
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data_.push_back(make_pair(xp, yp)); |
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} |
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|
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void CubicSpline::addPoints(const vector<RealType>& xps, |
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const vector<RealType>& yps) { |
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|
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assert(xps.size() == yps.size()); |
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|
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for (unsigned int i = 0; i < xps.size(); i++) |
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data_.push_back(make_pair(xps[i], yps[i])); |
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} |
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|
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void CubicSpline::generate() { |
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// Calculate coefficients defining a smooth cubic interpolatory spline. |
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// |
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// class values constructed: |
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// n = number of data_ points. |
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// x = vector of independent variable values |
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// y = vector of dependent variable values |
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// b = vector of S'(x[i]) values. |
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// c = vector of S"(x[i])/2 values. |
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// d = vector of S'''(x[i]+)/6 values (i < n). |
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// Local variables: |
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|
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RealType fp1, fpn, h, p; |
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|
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// make sure the sizes match |
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|
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n = data_.size(); |
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b.resize(n); |
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c.resize(n); |
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d.resize(n); |
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|
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// make sure we are monotonically increasing in x: |
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|
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bool sorted = true; |
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|
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for (int i = 1; i < n; i++) { |
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if ( (data_[i].first - data_[i-1].first ) <= 0.0 ) sorted = false; |
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} |
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|
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// sort if necessary |
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|
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if (!sorted) sort(data_.begin(), data_.end()); |
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|
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// Calculate coefficients for the tridiagonal system: store |
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// sub-diagonal in B, diagonal in D, difference quotient in C. |
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|
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b[0] = data_[1].first - data_[0].first; |
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c[0] = (data_[1].second - data_[0].second) / b[0]; |
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|
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if (n == 2) { |
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|
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// Assume the derivatives at both endpoints are zero. Another |
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// assumption could be made to have a linear interpolant between |
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// the two points. In that case, the b coefficients below would be |
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// (data_[1].second - data_[0].second) / (data_[1].first - data_[0].first) |
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// and the c and d coefficients would both be zero. |
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b[0] = 0.0; |
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c[0] = -3.0 * pow((data_[1].second - data_[0].second) / |
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(data_[1].first-data_[0].first), 2); |
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d[0] = -2.0 * pow((data_[1].second - data_[0].second) / |
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(data_[1].first-data_[0].first), 3); |
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b[1] = b[0]; |
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c[1] = 0.0; |
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d[1] = 0.0; |
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dx = 1.0 / (data_[1].first - data_[0].first); |
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isUniform = true; |
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generated = true; |
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return; |
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} |
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|
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d[0] = 2.0 * b[0]; |
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|
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for (int i = 1; i < n-1; i++) { |
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b[i] = data_[i+1].first - data_[i].first; |
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if ( fabs( b[i] - b[0] ) / b[0] > 1.0e-5) isUniform = false; |
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c[i] = (data_[i+1].second - data_[i].second) / b[i]; |
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d[i] = 2.0 * (b[i] + b[i-1]); |
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} |
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|
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d[n-1] = 2.0 * b[n-2]; |
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|
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// Calculate estimates for the end slopes using polynomials |
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// that interpolate the data_ nearest the end. |
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|
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fp1 = c[0] - b[0]*(c[1] - c[0])/(b[0] + b[1]); |
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if (n > 3) fp1 = fp1 + b[0]*((b[0] + b[1]) * (c[2] - c[1]) / |
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(b[1] + b[2]) - |
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c[1] + c[0]) / (data_[3].first - data_[0].first); |
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|
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fpn = c[n-2] + b[n-2]*(c[n-2] - c[n-3])/(b[n-3] + b[n-2]); |
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|
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if (n > 3) fpn = fpn + b[n-2] * |
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(c[n-2] - c[n-3] - (b[n-3] + b[n-2]) * |
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(c[n-3] - c[n-4])/(b[n-3] + b[n-4])) / |
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(data_[n-1].first - data_[n-4].first); |
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|
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// Calculate the right hand side and store it in C. |
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|
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c[n-1] = 3.0 * (fpn - c[n-2]); |
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for (int i = n-2; i > 0; i--) |
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c[i] = 3.0 * (c[i] - c[i-1]); |
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c[0] = 3.0 * (c[0] - fp1); |
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|
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// Solve the tridiagonal system. |
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|
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for (int k = 1; k < n; k++) { |
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p = b[k-1] / d[k-1]; |
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d[k] = d[k] - p*b[k-1]; |
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c[k] = c[k] - p*c[k-1]; |
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} |
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|
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c[n-1] = c[n-1] / d[n-1]; |
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|
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for (int k = n-2; k >= 0; k--) |
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c[k] = (c[k] - b[k] * c[k+1]) / d[k]; |
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|
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// Calculate the coefficients defining the spline. |
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|
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for (int i = 0; i < n-1; i++) { |
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h = data_[i+1].first - data_[i].first; |
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d[i] = (c[i+1] - c[i]) / (3.0 * h); |
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b[i] = (data_[i+1].second - data_[i].second)/h - h * (c[i] + h * d[i]); |
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} |
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|
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b[n-1] = b[n-2] + h * (2.0 * c[n-2] + h * 3.0 * d[n-2]); |
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|
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if (isUniform) dx = 1.0 / (data_[1].first - data_[0].first); |
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|
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generated = true; |
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return; |
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} |
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|
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RealType CubicSpline::getValueAt(RealType t) { |
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// Evaluate the spline at t using coefficients |
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// |
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// Input parameters |
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// t = point where spline is to be evaluated. |
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// Output: |
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// value of spline at t. |
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|
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if (!generated) generate(); |
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|
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assert(t > data_.front().first); |
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assert(t < data_.back().first); |
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|
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// Find the interval ( x[j], x[j+1] ) that contains or is nearest |
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// to t. |
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|
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if (isUniform) { |
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|
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j = max(0, min(n-1, int((t - data_[0].first) * dx))); |
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|
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} else { |
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|
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j = n-1; |
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|
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for (int i = 0; i < n; i++) { |
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if ( t < data_[i].first ) { |
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j = i-1; |
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break; |
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} |
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} |
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} |
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|
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// Evaluate the cubic polynomial. |
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|
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dt = t - data_[j].first; |
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return data_[j].second + dt*(b[j] + dt*(c[j] + dt*d[j])); |
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} |
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|
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|
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pair<RealType, RealType> CubicSpline::getValueAndDerivativeAt(RealType t) { |
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// Evaluate the spline and first derivative at t using coefficients |
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// |
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// Input parameters |
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// t = point where spline is to be evaluated. |
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// Output: |
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// pair containing value of spline at t and first derivative at t |
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|
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if (!generated) generate(); |
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|
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assert(t > data_.front().first); |
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assert(t < data_.back().first); |
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|
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// Find the interval ( x[j], x[j+1] ) that contains or is nearest |
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// to t. |
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|
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if (isUniform) { |
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|
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j = max(0, min(n-1, int((t - data_[0].first) * dx))); |
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|
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} else { |
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|
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j = n-1; |
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|
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for (int i = 0; i < n; i++) { |
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if ( t < data_[i].first ) { |
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j = i-1; |
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break; |
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} |
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} |
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} |
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|
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// Evaluate the cubic polynomial. |
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|
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dt = t - data_[j].first; |
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|
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yval = data_[j].second + dt*(b[j] + dt*(c[j] + dt*d[j])); |
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dydx = b[j] + dt*(2.0 * c[j] + 3.0 * dt * d[j]); |
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|
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return make_pair(yval, dydx); |
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} |
