src/math/CubicSpline.cpp

/*
 * Copyright (c) 2005 The University of Notre Dame. All Rights Reserved.
 *
 * The University of Notre Dame grants you ("Licensee") a
 * non-exclusive, royalty free, license to use, modify and
 * redistribute this software in source and binary code form, provided
 * that the following conditions are met:
 *
 * 1. Redistributions of source code must retain the above copyright
 *    notice, this list of conditions and the following disclaimer.
 *
 * 2. Redistributions in binary form must reproduce the above copyright
 *    notice, this list of conditions and the following disclaimer in the
 *    documentation and/or other materials provided with the
 *    distribution.
 *
 * This software is provided "AS IS," without a warranty of any
 * kind. All express or implied conditions, representations and
 * warranties, including any implied warranty of merchantability,
 * fitness for a particular purpose or non-infringement, are hereby
 * excluded.  The University of Notre Dame and its licensors shall not
 * be liable for any damages suffered by licensee as a result of
 * using, modifying or distributing the software or its
 * derivatives. In no event will the University of Notre Dame or its
 * licensors be liable for any lost revenue, profit or data, or for
 * direct, indirect, special, consequential, incidental or punitive
 * damages, however caused and regardless of the theory of liability,
 * arising out of the use of or inability to use software, even if the
 * University of Notre Dame has been advised of the possibility of
 * such damages.
 *
 * SUPPORT OPEN SCIENCE!  If you use OpenMD or its source code in your
 * research, please cite the appropriate papers when you publish your
 * work.  Good starting points are:
 *                                                                      
 * [1]  Meineke, et al., J. Comp. Chem. 26, 252-271 (2005).             
 * [2]  Fennell & Gezelter, J. Chem. Phys. 124, 234104 (2006).          
 * [3]  Sun, Lin & Gezelter, J. Chem. Phys. 128, 234107 (2008).          
 * [4]  Kuang & Gezelter,  J. Chem. Phys. 133, 164101 (2010).
 * [5]  Vardeman, Stocker & Gezelter, J. Chem. Theory Comput. 7, 834 (2011).
 */
 
#include "math/CubicSpline.hpp"
#include <cmath>
#include <cassert>
#include <cstdio>
#include <algorithm>

using namespace OpenMD;
using namespace std;

CubicSpline::CubicSpline() : generated(false), isUniform(true) {
  data_.clear();
}

void CubicSpline::addPoint(const RealType xp, const RealType yp) {
  data_.push_back(make_pair(xp, yp));
}

void CubicSpline::addPoints(const vector<RealType>& xps, 
                            const vector<RealType>& yps) {
  
  assert(xps.size() == yps.size());
  
  for (unsigned int i = 0; i < xps.size(); i++) 
    data_.push_back(make_pair(xps[i], yps[i]));
}

void CubicSpline::generate() { 
  // Calculate coefficients defining a smooth cubic interpolatory spline.
  //
  // class values constructed:
  //   n   = number of data_ points.
  //   x   = vector of independent variable values 
  //   y   = vector of dependent variable values
  //   b   = vector of S'(x[i]) values.
  //   c   = vector of S"(x[i])/2 values.
  //   d   = vector of S'''(x[i]+)/6 values (i < n).
  // Local variables:   
 
  RealType fp1, fpn, h, p;
  
  // make sure the sizes match
  
  n = data_.size();  
  b.resize(n);
  c.resize(n);
  d.resize(n);
  
  // make sure we are monotonically increasing in x:
  
  bool sorted = true;
  
  for (int i = 1; i < n; i++) {
    if ( (data_[i].first - data_[i-1].first ) <= 0.0 ) sorted = false;
  }
  
  // sort if necessary
  
  if (!sorted) sort(data_.begin(), data_.end());  
  
  // Calculate coefficients for the tridiagonal system: store
  // sub-diagonal in B, diagonal in D, difference quotient in C.  
  
  b[0] = data_[1].first - data_[0].first;
  c[0] = (data_[1].second - data_[0].second) / b[0];
  
  if (n == 2) {

    // Assume the derivatives at both endpoints are zero. Another
    // assumption could be made to have a linear interpolant between
    // the two points.  In that case, the b coefficients below would be
    // (data_[1].second - data_[0].second) / (data_[1].first - data_[0].first)
    // and the c and d coefficients would both be zero.
    b[0] = 0.0;
    c[0] = -3.0 * pow((data_[1].second - data_[0].second) /
                      (data_[1].first-data_[0].first), 2);
    d[0] = -2.0 * pow((data_[1].second - data_[0].second) / 
                      (data_[1].first-data_[0].first), 3);
    b[1] = b[0];
    c[1] = 0.0;
    d[1] = 0.0;
    dx = 1.0 / (data_[1].first - data_[0].first);
    isUniform = true;
    generated = true;
    return;
  }
  
  d[0] = 2.0 * b[0];
  
  for (int i = 1; i < n-1; i++) {
    b[i] = data_[i+1].first - data_[i].first;
    if ( fabs( b[i] - b[0] ) / b[0] > 1.0e-5) isUniform = false;
    c[i] = (data_[i+1].second - data_[i].second) / b[i];
    d[i] = 2.0 * (b[i] + b[i-1]);
  }
  
  d[n-1] = 2.0 * b[n-2];
  
  // Calculate estimates for the end slopes using polynomials
  // that interpolate the data_ nearest the end.
  
  fp1 = c[0] - b[0]*(c[1] - c[0])/(b[0] + b[1]);
  if (n > 3) fp1 = fp1 + b[0]*((b[0] + b[1]) * (c[2] - c[1]) / 
                               (b[1] + b[2]) - 
                               c[1] + c[0]) / (data_[3].first - data_[0].first);
  
  fpn = c[n-2] + b[n-2]*(c[n-2] - c[n-3])/(b[n-3] + b[n-2]);
  
  if (n > 3)  fpn = fpn + b[n-2] * 
                (c[n-2] - c[n-3] - (b[n-3] + b[n-2]) * 
                 (c[n-3] - c[n-4])/(b[n-3] + b[n-4])) /
                (data_[n-1].first - data_[n-4].first);
  
  // Calculate the right hand side and store it in C.
  
  c[n-1] = 3.0 * (fpn - c[n-2]);
  for (int i = n-2; i > 0; i--) 
    c[i] = 3.0 * (c[i] - c[i-1]);  
  c[0] = 3.0 * (c[0] - fp1);
  
  // Solve the tridiagonal system.
  
  for (int k = 1; k < n; k++) {
    p = b[k-1] / d[k-1];
    d[k] = d[k] - p*b[k-1];
    c[k] = c[k] - p*c[k-1];
  }
  
  c[n-1] = c[n-1] / d[n-1];
  
  for (int k = n-2; k >= 0; k--) 
    c[k] = (c[k] - b[k] * c[k+1]) / d[k];
  
  // Calculate the coefficients defining the spline.
  
  for (int i = 0; i < n-1; i++) {
    h = data_[i+1].first - data_[i].first;
    d[i] = (c[i+1] - c[i]) / (3.0 * h);
    b[i] = (data_[i+1].second - data_[i].second)/h - h * (c[i] + h * d[i]);
  }
  
  b[n-1] = b[n-2] + h * (2.0 * c[n-2] + h * 3.0 * d[n-2]);
  
  if (isUniform) dx = 1.0 / (data_[1].first - data_[0].first); 
  
  generated = true;
  return;
}

RealType CubicSpline::getValueAt(const RealType& t) {
  // Evaluate the spline at t using coefficients 
  //
  // Input parameters
  //   t = point where spline is to be evaluated.
  // Output:
  //   value of spline at t.
  
  if (!generated) generate();
  
  assert(t >= data_.front().first);
  assert(t <= data_.back().first);

  //  Find the interval ( x[j], x[j+1] ) that contains or is nearest
  //  to t.

  if (isUniform) {    
    
    j = max(0, min(n-1, int((t - data_[0].first) * dx)));   

  } else { 

    j = n-1;
    
    for (int i = 0; i < n; i++) {
      if ( t < data_[i].first ) {
        j = i-1;
        break;
      }      
    }
  }
  
  //  Evaluate the cubic polynomial.
  
  dt = t - data_[j].first;
  return data_[j].second + dt*(b[j] + dt*(c[j] + dt*d[j]));  
}


void CubicSpline::getValueAt(const RealType& t, RealType& v) {
  // Evaluate the spline at t using coefficients 
  //
  // Input parameters
  //   t = point where spline is to be evaluated.
  // Output:
  //   value of spline at t.
  
  if (!generated) generate();
  
  assert(t >= data_.front().first);
  assert(t <= data_.back().first);

  //  Find the interval ( x[j], x[j+1] ) that contains or is nearest
  //  to t.

  if (isUniform) {    
    
    j = max(0, min(n-1, int((t - data_[0].first) * dx)));   

  } else { 

    j = n-1;
    
    for (int i = 0; i < n; i++) {
      if ( t < data_[i].first ) {
        j = i-1;
        break;
      }      
    }
  }
  
  //  Evaluate the cubic polynomial.
  
  dt = t - data_[j].first;
  v = data_[j].second + dt*(b[j] + dt*(c[j] + dt*d[j]));  
}


pair<RealType, RealType> CubicSpline::getValueAndDerivativeAt(const RealType& t){
  // Evaluate the spline and first derivative at t using coefficients 
  //
  // Input parameters
  //   t = point where spline is to be evaluated.
  // Output:
  //   pair containing value of spline at t and first derivative at t

  if (!generated) generate();
  
  assert(t >= data_.front().first);
  assert(t <= data_.back().first);

  //  Find the interval ( x[j], x[j+1] ) that contains or is nearest
  //  to t.

  if (isUniform) {    
    
    j = max(0, min(n-1, int((t - data_[0].first) * dx)));   

  } else { 

    j = n-1;
    
    for (int i = 0; i < n; i++) {
      if ( t < data_[i].first ) {
        j = i-1;
        break;
      }      
    }
  }
  
  //  Evaluate the cubic polynomial.
  
  dt = t - data_[j].first;

  yval = data_[j].second + dt*(b[j] + dt*(c[j] + dt*d[j]));
  dydx = b[j] + dt*(2.0 * c[j] + 3.0 * dt * d[j]); 

  return make_pair(yval, dydx);
}

pair<RealType, RealType> CubicSpline::getLimits(){
  if (!generated) generate();
  return make_pair( data_.front().first, data_.back().first );
}

void CubicSpline::getValueAndDerivativeAt(const RealType& t, RealType& v, 
                                          RealType &dv) {
  // Evaluate the spline and first derivative at t using coefficients 
  //
  // Input parameters
  //   t = point where spline is to be evaluated.

  if (!generated) generate();
  
  assert(t >= data_.front().first);
  assert(t <= data_.back().first);

  //  Find the interval ( x[j], x[j+1] ) that contains or is nearest
  //  to t.

  if (isUniform) {    
    
    j = max(0, min(n-1, int((t - data_[0].first) * dx)));   

  } else { 

    j = n-1;
    
    for (int i = 0; i < n; i++) {
      if ( t < data_[i].first ) {
        j = i-1;
        break;
      }      
    }
  }
  
  //  Evaluate the cubic polynomial.
  
  dt = t - data_[j].first;

  v = data_[j].second + dt*(b[j] + dt*(c[j] + dt*d[j]));
  dv = b[j] + dt*(2.0 * c[j] + 3.0 * dt * d[j]); 
}
Revision:	1928
Committed:	Sat Aug 17 13:03:17 2013 UTC (11 years, 8 months ago) by gezelter
File size:	10044 byte(s)
Log Message:	Fixed a few bugs in EAM and Electrostatic.
#	User	Rev	Content
1	gezelter	1475	/*
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	gezelter	1879	* [3] Sun, Lin & Gezelter, J. Chem. Phys. 128, 234107 (2008).
39	gezelter	1665	* [4] Kuang & Gezelter, J. Chem. Phys. 133, 164101 (2010).
40			* [5] Vardeman, Stocker & Gezelter, J. Chem. Theory Comput. 7, 834 (2011).
41	gezelter	1475	*/
42
43			#include "math/CubicSpline.hpp"
44			#include <cmath>
45	gezelter	1879	#include <cassert>
46	gezelter	1618	#include <cstdio>
47	gezelter	1475	#include <algorithm>
48
49			using namespace OpenMD;
50			using namespace std;
51
52	gezelter	1536	CubicSpline::CubicSpline() : generated(false), isUniform(true) {
53			data_.clear();
54			}
55	gezelter	1475
56	gezelter	1536	void CubicSpline::addPoint(const RealType xp, const RealType yp) {
57			data_.push_back(make_pair(xp, yp));
58	gezelter	1475	}
59
60			void CubicSpline::addPoints(const vector<RealType>& xps,
61			const vector<RealType>& yps) {
62
63	gezelter	1879	assert(xps.size() == yps.size());
64
65	gezelter	1767	for (unsigned int i = 0; i < xps.size(); i++)
66	gezelter	1536	data_.push_back(make_pair(xps[i], yps[i]));
67	gezelter	1475	}
68
69			void CubicSpline::generate() {
70			// Calculate coefficients defining a smooth cubic interpolatory spline.
71			//
72			// class values constructed:
73	gezelter	1536	// n = number of data_ points.
74	gezelter	1475	// x = vector of independent variable values
75			// y = vector of dependent variable values
76			// b = vector of S'(x[i]) values.
77			// c = vector of S"(x[i])/2 values.
78			// d = vector of S'''(x[i]+)/6 values (i < n).
79			// Local variables:
80	gezelter	1536
81	gezelter	1475	RealType fp1, fpn, h, p;
82
83			// make sure the sizes match
84
85	gezelter	1536	n = data_.size();
86	gezelter	1475	b.resize(n);
87			c.resize(n);
88			d.resize(n);
89
90			// make sure we are monotonically increasing in x:
91
92			bool sorted = true;
93
94			for (int i = 1; i < n; i++) {
95	gezelter	1536	if ( (data_[i].first - data_[i-1].first ) <= 0.0 ) sorted = false;
96	gezelter	1475	}
97
98			// sort if necessary
99
100	gezelter	1536	if (!sorted) sort(data_.begin(), data_.end());
101	gezelter	1475
102			// Calculate coefficients for the tridiagonal system: store
103			// sub-diagonal in B, diagonal in D, difference quotient in C.
104
105	gezelter	1536	b[0] = data_[1].first - data_[0].first;
106			c[0] = (data_[1].second - data_[0].second) / b[0];
107	gezelter	1475
108			if (n == 2) {
109
110			// Assume the derivatives at both endpoints are zero. Another
111			// assumption could be made to have a linear interpolant between
112			// the two points. In that case, the b coefficients below would be
113	gezelter	1536	// (data_[1].second - data_[0].second) / (data_[1].first - data_[0].first)
114	gezelter	1475	// and the c and d coefficients would both be zero.
115			b[0] = 0.0;
116	gezelter	1536	c[0] = -3.0 * pow((data_[1].second - data_[0].second) /
117			(data_[1].first-data_[0].first), 2);
118			d[0] = -2.0 * pow((data_[1].second - data_[0].second) /
119			(data_[1].first-data_[0].first), 3);
120	gezelter	1475	b[1] = b[0];
121			c[1] = 0.0;
122			d[1] = 0.0;
123	gezelter	1536	dx = 1.0 / (data_[1].first - data_[0].first);
124	gezelter	1475	isUniform = true;
125			generated = true;
126			return;
127			}
128
129			d[0] = 2.0 * b[0];
130
131			for (int i = 1; i < n-1; i++) {
132	gezelter	1536	b[i] = data_[i+1].first - data_[i].first;
133	gezelter	1475	if ( fabs( b[i] - b[0] ) / b[0] > 1.0e-5) isUniform = false;
134	gezelter	1536	c[i] = (data_[i+1].second - data_[i].second) / b[i];
135	gezelter	1475	d[i] = 2.0 * (b[i] + b[i-1]);
136			}
137
138			d[n-1] = 2.0 * b[n-2];
139
140			// Calculate estimates for the end slopes using polynomials
141	gezelter	1536	// that interpolate the data_ nearest the end.
142	gezelter	1475
143			fp1 = c[0] - b[0]*(c[1] - c[0])/(b[0] + b[1]);
144			if (n > 3) fp1 = fp1 + b[0]((b[0] + b[1]) (c[2] - c[1]) /
145			(b[1] + b[2]) -
146	gezelter	1536	c[1] + c[0]) / (data_[3].first - data_[0].first);
147	gezelter	1475
148			fpn = c[n-2] + b[n-2]*(c[n-2] - c[n-3])/(b[n-3] + b[n-2]);
149	gezelter	1879
150	gezelter	1475	if (n > 3) fpn = fpn + b[n-2] *
151	gezelter	1879	(c[n-2] - c[n-3] - (b[n-3] + b[n-2]) *
152			(c[n-3] - c[n-4])/(b[n-3] + b[n-4])) /
153			(data_[n-1].first - data_[n-4].first);
154	gezelter	1475
155			// Calculate the right hand side and store it in C.
156
157			c[n-1] = 3.0 * (fpn - c[n-2]);
158			for (int i = n-2; i > 0; i--)
159			c[i] = 3.0 * (c[i] - c[i-1]);
160			c[0] = 3.0 * (c[0] - fp1);
161
162			// Solve the tridiagonal system.
163
164			for (int k = 1; k < n; k++) {
165			p = b[k-1] / d[k-1];
166			d[k] = d[k] - p*b[k-1];
167			c[k] = c[k] - p*c[k-1];
168			}
169
170			c[n-1] = c[n-1] / d[n-1];
171
172			for (int k = n-2; k >= 0; k--)
173			c[k] = (c[k] - b[k] * c[k+1]) / d[k];
174
175			// Calculate the coefficients defining the spline.
176
177			for (int i = 0; i < n-1; i++) {
178	gezelter	1536	h = data_[i+1].first - data_[i].first;
179	gezelter	1475	d[i] = (c[i+1] - c[i]) / (3.0 * h);
180	gezelter	1536	b[i] = (data_[i+1].second - data_[i].second)/h - h * (c[i] + h * d[i]);
181	gezelter	1475	}
182
183			b[n-1] = b[n-2] + h * (2.0 * c[n-2] + h * 3.0 * d[n-2]);
184
185	gezelter	1536	if (isUniform) dx = 1.0 / (data_[1].first - data_[0].first);
186	gezelter	1475
187			generated = true;
188			return;
189			}
190
191	gezelter	1879	RealType CubicSpline::getValueAt(const RealType& t) {
192	gezelter	1475	// Evaluate the spline at t using coefficients
193			//
194			// Input parameters
195			// t = point where spline is to be evaluated.
196			// Output:
197			// value of spline at t.
198
199			if (!generated) generate();
200
201	gezelter	1928	assert(t >= data_.front().first);
202			assert(t <= data_.back().first);
203	gezelter	1879
204			// Find the interval ( x[j], x[j+1] ) that contains or is nearest
205			// to t.
206
207			if (isUniform) {
208
209			j = max(0, min(n-1, int((t - data_[0].first) * dx)));
210
211			} else {
212
213			j = n-1;
214
215			for (int i = 0; i < n; i++) {
216			if ( t < data_[i].first ) {
217			j = i-1;
218			break;
219			}
220			}
221	gezelter	1475	}
222	gezelter	1879
223			// Evaluate the cubic polynomial.
224
225			dt = t - data_[j].first;
226			return data_[j].second + dt(b[j] + dt(c[j] + dt*d[j]));
227			}
228	gezelter	1475
229	gezelter	1879
230			void CubicSpline::getValueAt(const RealType& t, RealType& v) {
231			// Evaluate the spline at t using coefficients
232			//
233			// Input parameters
234			// t = point where spline is to be evaluated.
235			// Output:
236			// value of spline at t.
237
238			if (!generated) generate();
239
240	gezelter	1928	assert(t >= data_.front().first);
241			assert(t <= data_.back().first);
242	gezelter	1879
243	gezelter	1475	// Find the interval ( x[j], x[j+1] ) that contains or is nearest
244			// to t.
245
246			if (isUniform) {
247
248	gezelter	1536	j = max(0, min(n-1, int((t - data_[0].first) * dx)));
249	gezelter	1475
250			} else {
251
252			j = n-1;
253
254			for (int i = 0; i < n; i++) {
255	gezelter	1536	if ( t < data_[i].first ) {
256	gezelter	1475	j = i-1;
257			break;
258			}
259			}
260			}
261
262			// Evaluate the cubic polynomial.
263
264	gezelter	1536	dt = t - data_[j].first;
265	gezelter	1879	v = data_[j].second + dt(b[j] + dt(c[j] + dt*d[j]));
266	gezelter	1475	}
267
268
269	gezelter	1879	pair<RealType, RealType> CubicSpline::getValueAndDerivativeAt(const RealType& t){
270	gezelter	1475	// Evaluate the spline and first derivative at t using coefficients
271			//
272			// Input parameters
273			// t = point where spline is to be evaluated.
274			// Output:
275			// pair containing value of spline at t and first derivative at t
276
277			if (!generated) generate();
278
279	gezelter	1928	assert(t >= data_.front().first);
280			assert(t <= data_.back().first);
281	gezelter	1475
282			// Find the interval ( x[j], x[j+1] ) that contains or is nearest
283			// to t.
284
285			if (isUniform) {
286
287	gezelter	1536	j = max(0, min(n-1, int((t - data_[0].first) * dx)));
288	gezelter	1475
289			} else {
290
291			j = n-1;
292
293			for (int i = 0; i < n; i++) {
294	gezelter	1536	if ( t < data_[i].first ) {
295	gezelter	1475	j = i-1;
296			break;
297			}
298			}
299			}
300
301			// Evaluate the cubic polynomial.
302
303	gezelter	1536	dt = t - data_[j].first;
304	gezelter	1475
305	gezelter	1879	yval = data_[j].second + dt(b[j] + dt(c[j] + dt*d[j]));
306			dydx = b[j] + dt(2.0 c[j] + 3.0 * dt * d[j]);
307
308	gezelter	1475	return make_pair(yval, dydx);
309			}
310	gezelter	1879
311	gezelter	1928	pair<RealType, RealType> CubicSpline::getLimits(){
312			if (!generated) generate();
313			return make_pair( data_.front().first, data_.back().first );
314			}
315
316	gezelter	1879	void CubicSpline::getValueAndDerivativeAt(const RealType& t, RealType& v,
317			RealType &dv) {
318			// Evaluate the spline and first derivative at t using coefficients
319			//
320			// Input parameters
321			// t = point where spline is to be evaluated.
322
323			if (!generated) generate();
324
325	gezelter	1928	assert(t >= data_.front().first);
326			assert(t <= data_.back().first);
327	gezelter	1879
328			// Find the interval ( x[j], x[j+1] ) that contains or is nearest
329			// to t.
330
331			if (isUniform) {
332
333			j = max(0, min(n-1, int((t - data_[0].first) * dx)));
334
335			} else {
336
337			j = n-1;
338
339			for (int i = 0; i < n; i++) {
340			if ( t < data_[i].first ) {
341			j = i-1;
342			break;
343			}
344			}
345			}
346
347			// Evaluate the cubic polynomial.
348
349			dt = t - data_[j].first;
350
351			v = data_[j].second + dt(b[j] + dt(c[j] + dt*d[j]));
352			dv = b[j] + dt(2.0 c[j] + 3.0 * dt * d[j]);
353			}