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/* -*- Mode: C++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*************************************************************************
*
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* Copyright 2000, 2010 Oracle and/or its affiliates.
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* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU Lesser General Public License version 3 for more details
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* version 3 along with OpenOffice.org. If not, see
* <http://www.openoffice.org/license.html>
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************************************************************************/
// Natural, Clamped, or Periodic Cubic Splines
//
// Input: A list of N+1 points (x_i,a_i), 0 <= i <= N, which are sampled
// from a function, a_i = f(x_i). The function f is unknown. Boundary
// conditions are
// (1) Natural splines: f"(x_0) = f"(x_N) = 0
// (2) Clamped splines: f'(x_0) and f'(x_N) are user-specified.
// (3) Periodic splines: f(x_0) = f(x_N) [in which case a_N = a_0 is
// required in the input], f'(x_0) = f'(x_N), and f"(x_0) = f"(x_N).
//
// Output: b_i, c_i, d_i, 0 <= i <= N-1, which are coefficients for the cubic
// spline S_i(x) = a_i + b_i(x-x_i) + c_i(x-x_i)^2 + d_i(x-x_i)^3 for
// x_i <= x < x_{i+1}.
//
// The natural and clamped algorithms were implemented from
//
// Numerical Analysis, 3rd edition
// Richard L. Burden and J. Douglas Faires
// Prindle, Weber & Schmidt
// Boston, 1985, pp. 122-124.
//
// The algorithm sets up a tridiagonal linear system of equations in the
// c_i values. This can be solved in O(N) time.
//
// The periodic spline algorithm was implemented from my own derivation. The
// linear system of equations is not tridiagonal. For now I use a standard
// linear solver that does not take advantage of the sparseness of the
// matrix. Therefore for very large N, you may have to worry about memory
// usage.
#include "solver.h"
//-----------------------------------------------------------------------------
void NaturalSpline (int N, double* x, double* a, double*& b, double*& c,
double*& d)
{
const double oneThird = 1.0/3.0;
int i;
double* h = new double[N];
double* hdiff = new double[N];
double* alpha = new double[N];
for (i = 0; i < N; i++){
h[i] = x[i+1]-x[i];
}
for (i = 1; i < N; i++)
hdiff[i] = x[i+1]-x[i-1];
for (i = 1; i < N; i++)
{
double numer = 3.0*(a[i+1]*h[i-1]-a[i]*hdiff[i]+a[i-1]*h[i]);
double denom = h[i-1]*h[i];
alpha[i] = numer/denom;
}
double* ell = new double[N+1];
double* mu = new double[N];
double* z = new double[N+1];
double recip;
ell[0] = 1.0;
mu[0] = 0.0;
z[0] = 0.0;
for (i = 1; i < N; i++)
{
ell[i] = 2.0*hdiff[i]-h[i-1]*mu[i-1];
recip = 1.0/ell[i];
mu[i] = recip*h[i];
z[i] = recip*(alpha[i]-h[i-1]*z[i-1]);
}
ell[N] = 1.0;
z[N] = 0.0;
b = new double[N];
c = new double[N+1];
d = new double[N];
c[N] = 0.0;
for (i = N-1; i >= 0; i--)
{
c[i] = z[i]-mu[i]*c[i+1];
recip = 1.0/h[i];
b[i] = recip*(a[i+1]-a[i])-h[i]*(c[i+1]+2.0*c[i])*oneThird;
d[i] = oneThird*recip*(c[i+1]-c[i]);
}
delete[] h;
delete[] hdiff;
delete[] alpha;
delete[] ell;
delete[] mu;
delete[] z;
}
void PeriodicSpline (int N, double* x, double* a, double*& b, double*& c,
double*& d)
{
double* h = new double[N];
int i;
for (i = 0; i < N; i++)
h[i] = x[i+1]-x[i];
mgcLinearSystemD sys;
double** mat = sys.NewMatrix(N+1); // guaranteed to be zeroed memory
c = sys.NewVector(N+1); // guaranteed to be zeroed memory
// c[0] - c[N] = 0
mat[0][0] = +1.0f;
mat[0][N] = -1.0f;
// h[i-1]*c[i-1]+2*(h[i-1]+h[i])*c[i]+h[i]*c[i+1] =
// 3*{(a[i+1]-a[i])/h[i] - (a[i]-a[i-1])/h[i-1]}
for (i = 1; i <= N-1; i++)
{
mat[i][i-1] = h[i-1];
mat[i][i ] = 2.0f*(h[i-1]+h[i]);
mat[i][i+1] = h[i];
c[i] = 3.0f*((a[i+1]-a[i])/h[i] - (a[i]-a[i-1])/h[i-1]);
}
// "wrap around equation" for periodicity
// h[N-1]*c[N-1]+2*(h[N-1]+h[0])*c[0]+h[0]*c[1] =
// 3*{(a[1]-a[0])/h[0] - (a[0]-a[N-1])/h[N-1]}
mat[N][N-1] = h[N-1];
mat[N][0 ] = 2.0f*(h[N-1]+h[0]);
mat[N][1 ] = h[0];
c[N] = 3.0f*((a[1]-a[0])/h[0] - (a[0]-a[N-1])/h[N-1]);
// solve for c[0] through c[N]
sys.Solve(N+1,mat,c);
const double oneThird = 1.0/3.0;
b = new double[N];
d = new double[N];
for (i = 0; i < N; i++)
{
b[i] = (a[i+1]-a[i])/h[i] - oneThird*(c[i+1]+2.0f*c[i])*h[i];
d[i] = oneThird*(c[i+1]-c[i])/h[i];
}
delete[] h;
sys.DeleteMatrix(N+1,mat);
}
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