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/*
* spline.h
*
* simple cubic spline interpolation library without external
* dependencies
*
* ---------------------------------------------------------------------
* Copyright (C) 2011, 2014 Tino Kluge (ttk448 at gmail.com)
*
* This program is free software; you can redistribute it and/or
* modify it under the terms of the GNU General Public License
* as published by the Free Software Foundation; either version 2
* of the License, or (at your option) any later version.
*
* This program is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with this program. If not, see <http://www.gnu.org/licenses/>.
* ---------------------------------------------------------------------
*
*/
#ifndef TK_SPLINE_H
#define TK_SPLINE_H
#include <cstdio>
#include <cassert>
#include <vector>
#include <algorithm>
// unnamed namespace only because the implementation is in this
// header file and we don't want to export symbols to the obj files
namespace {
namespace tk {
// band matrix solver
class band_matrix {
private:
std::vector<std::vector<double> > m_upper; // upper band
std::vector<std::vector<double> > m_lower; // lower band
public:
band_matrix() {}; // constructor
band_matrix(int dim, int n_u, int n_l); // constructor
~band_matrix() {}; // destructor
void resize(int dim, int n_u, int n_l); // init with dim,n_u,n_l
int dim() const; // matrix dimension
int num_upper() const {
return m_upper.size() - 1;
}
int num_lower() const {
return m_lower.size() - 1;
}
// access operator
double &operator()(int i, int j); // write
double operator()(int i, int j) const; // read
// we can store an additional diogonal (in m_lower)
double &saved_diag(int i);
double saved_diag(int i) const;
void lu_decompose();
std::vector<double> r_solve(const std::vector<double> &b) const;
std::vector<double> l_solve(const std::vector<double> &b) const;
std::vector<double> lu_solve(const std::vector<double> &b,
bool is_lu_decomposed = false);
};
// spline interpolation
class spline {
public:
enum bd_type {
first_deriv = 1,
second_deriv = 2
};
private:
std::vector<double> m_x, m_y; // x,y coordinates of points
// interpolation parameters
// f(x) = a*(x-x_i)^3 + b*(x-x_i)^2 + c*(x-x_i) + y_i
std::vector<double> m_a, m_b, m_c; // spline coefficients
double m_b0, m_c0; // for left extrapol
bd_type m_left, m_right;
double m_left_value, m_right_value;
bool m_force_linear_extrapolation;
public:
// set default boundary condition to be zero curvature at both ends
spline() : m_left(second_deriv), m_right(second_deriv),
m_left_value(0.0), m_right_value(0.0),
m_force_linear_extrapolation(false) {
;
}
// optional, but if called it has to come be before set_points()
void set_boundary(bd_type left, double left_value,
bd_type right, double right_value,
bool force_linear_extrapolation = false);
void set_points(const std::vector<double> &x,
const std::vector<double> &y, bool cubic_spline = true);
double operator()(double x) const;
};
// ---------------------------------------------------------------------
// implementation part, which could be separated into a cpp file
// ---------------------------------------------------------------------
// band_matrix implementation
// -------------------------
band_matrix::band_matrix(int dim, int n_u, int n_l) {
resize(dim, n_u, n_l);
}
void band_matrix::resize(int dim, int n_u, int n_l) {
assert(dim > 0);
assert(n_u >= 0);
assert(n_l >= 0);
m_upper.resize(n_u + 1);
m_lower.resize(n_l + 1);
for (size_t i = 0; i < m_upper.size(); i++) {
m_upper[i].resize(dim);
}
for (size_t i = 0; i < m_lower.size(); i++) {
m_lower[i].resize(dim);
}
}
int band_matrix::dim() const {
if (m_upper.size() > 0) {
return m_upper[0].size();
} else {
return 0;
}
}
// defines the new operator (), so that we can access the elements
// by A(i,j), index going from i=0,...,dim()-1
double &band_matrix::operator()(int i, int j) {
int k = j - i; // what band is the entry
assert((i >= 0) && (i < dim()) && (j >= 0) && (j < dim()));
assert((-num_lower() <= k) && (k <= num_upper()));
// k=0 -> diogonal, k<0 lower left part, k>0 upper right part
if (k >= 0) return m_upper[k][i];
else return m_lower[-k][i];
}
double band_matrix::operator()(int i, int j) const {
int k = j - i; // what band is the entry
assert((i >= 0) && (i < dim()) && (j >= 0) && (j < dim()));
assert((-num_lower() <= k) && (k <= num_upper()));
// k=0 -> diogonal, k<0 lower left part, k>0 upper right part
if (k >= 0) return m_upper[k][i];
else return m_lower[-k][i];
}
// second diag (used in LU decomposition), saved in m_lower
double band_matrix::saved_diag(int i) const {
assert((i >= 0) && (i < dim()));
return m_lower[0][i];
}
double &band_matrix::saved_diag(int i) {
assert((i >= 0) && (i < dim()));
return m_lower[0][i];
}
// LR-Decomposition of a band matrix
void band_matrix::lu_decompose() {
int i_max, j_max;
int j_min;
double x;
// preconditioning
// normalize column i so that a_ii=1
for (int i = 0; i < this->dim(); i++) {
assert(this->operator()(i, i) != 0.0);
this->saved_diag(i) = 1.0 / this->operator()(i, i);
j_min = std::max(0, i - this->num_lower());
j_max = std::min(this->dim() - 1, i + this->num_upper());
for (int j = j_min; j <= j_max; j++) {
this->operator()(i, j) *= this->saved_diag(i);
}
this->operator()(i, i) = 1.0; // prevents rounding errors
}
// Gauss LR-Decomposition
for (int k = 0; k < this->dim(); k++) {
i_max = std::min(this->dim() - 1, k + this->num_lower()); // num_lower not a mistake!
for (int i = k + 1; i <= i_max; i++) {
assert(this->operator()(k, k) != 0.0);
x = -this->operator()(i, k) / this->operator()(k, k);
this->operator()(i, k) = -x; // assembly part of L
j_max = std::min(this->dim() - 1, k + this->num_upper());
for (int j = k + 1; j <= j_max; j++) {
// assembly part of R
this->operator()(i, j) = this->operator()(i, j) + x * this->operator()(k, j);
}
}
}
}
// solves Ly=b
std::vector<double> band_matrix::l_solve(const std::vector<double> &b) const {
assert(this->dim() == (int) b.size());
std::vector<double> x(this->dim());
int j_start;
double sum;
for (int i = 0; i < this->dim(); i++) {
sum = 0;
j_start = std::max(0, i - this->num_lower());
for (int j = j_start; j < i; j++) sum += this->operator()(i, j) * x[j];
x[i] = (b[i] * this->saved_diag(i)) - sum;
}
return x;
}
// solves Rx=y
std::vector<double> band_matrix::r_solve(const std::vector<double> &b) const {
assert(this->dim() == (int) b.size());
std::vector<double> x(this->dim());
int j_stop;
double sum;
for (int i = this->dim() - 1; i >= 0; i--) {
sum = 0;
j_stop = std::min(this->dim() - 1, i + this->num_upper());
for (int j = i + 1; j <= j_stop; j++) sum += this->operator()(i, j) * x[j];
x[i] = (b[i] - sum) / this->operator()(i, i);
}
return x;
}
std::vector<double> band_matrix::lu_solve(const std::vector<double> &b,
bool is_lu_decomposed) {
assert(this->dim() == (int) b.size());
std::vector<double> x, y;
if (is_lu_decomposed == false) {
this->lu_decompose();
}
y = this->l_solve(b);
x = this->r_solve(y);
return x;
}
// spline implementation
// -----------------------
void spline::set_boundary(spline::bd_type left, double left_value,
spline::bd_type right, double right_value,
bool force_linear_extrapolation) {
assert(m_x.size() == 0); // set_points() must not have happened yet
m_left = left;
m_right = right;
m_left_value = left_value;
m_right_value = right_value;
m_force_linear_extrapolation = force_linear_extrapolation;
}
void spline::set_points(const std::vector<double> &x,
const std::vector<double> &y, bool cubic_spline) {
assert(x.size() == y.size());
assert(x.size() > 2);
m_x = x;
m_y = y;
int n = x.size();
// TODO: maybe sort x and y, rather than returning an error
for (int i = 0; i < n - 1; i++) {
assert(m_x[i] < m_x[i + 1]);
}
if (cubic_spline == true) { // cubic spline interpolation
// setting up the matrix and right hand side of the equation system
// for the parameters b[]
band_matrix A(n, 1, 1);
std::vector<double> rhs(n);
for (int i = 1; i < n - 1; i++) {
A(i, i - 1) = 1.0 / 3.0 * (x[i] - x[i - 1]);
A(i, i) = 2.0 / 3.0 * (x[i + 1] - x[i - 1]);
A(i, i + 1) = 1.0 / 3.0 * (x[i + 1] - x[i]);
rhs[i] = (y[i + 1] - y[i]) / (x[i + 1] - x[i]) - (y[i] - y[i - 1]) / (x[i] - x[i - 1]);
}
// boundary conditions
if (m_left == spline::second_deriv) {
// 2*b[0] = f''
A(0, 0) = 2.0;
A(0, 1) = 0.0;
rhs[0] = m_left_value;
} else if (m_left == spline::first_deriv) {
// c[0] = f', needs to be re-expressed in terms of b:
// (2b[0]+b[1])(x[1]-x[0]) = 3 ((y[1]-y[0])/(x[1]-x[0]) - f')
A(0, 0) = 2.0 * (x[1] - x[0]);
A(0, 1) = 1.0 * (x[1] - x[0]);
rhs[0] = 3.0 * ((y[1] - y[0]) / (x[1] - x[0]) - m_left_value);
} else {
assert(false);
}
if (m_right == spline::second_deriv) {
// 2*b[n-1] = f''
A(n - 1, n - 1) = 2.0;
A(n - 1, n - 2) = 0.0;
rhs[n - 1] = m_right_value;
} else if (m_right == spline::first_deriv) {
// c[n-1] = f', needs to be re-expressed in terms of b:
// (b[n-2]+2b[n-1])(x[n-1]-x[n-2])
// = 3 (f' - (y[n-1]-y[n-2])/(x[n-1]-x[n-2]))
A(n - 1, n - 1) = 2.0 * (x[n - 1] - x[n - 2]);
A(n - 1, n - 2) = 1.0 * (x[n - 1] - x[n - 2]);
rhs[n - 1] = 3.0 * (m_right_value - (y[n - 1] - y[n - 2]) / (x[n - 1] - x[n - 2]));
} else {
assert(false);
}
// solve the equation system to obtain the parameters b[]
m_b = A.lu_solve(rhs);
// calculate parameters a[] and c[] based on b[]
m_a.resize(n);
m_c.resize(n);
for (int i = 0; i < n - 1; i++) {
m_a[i] = 1.0 / 3.0 * (m_b[i + 1] - m_b[i]) / (x[i + 1] - x[i]);
m_c[i] = (y[i + 1] - y[i]) / (x[i + 1] - x[i])
- 1.0 / 3.0 * (2.0 * m_b[i] + m_b[i + 1]) * (x[i + 1] - x[i]);
}
} else { // linear interpolation
m_a.resize(n);
m_b.resize(n);
m_c.resize(n);
for (int i = 0; i < n - 1; i++) {
m_a[i] = 0.0;
m_b[i] = 0.0;
m_c[i] = (m_y[i + 1] - m_y[i]) / (m_x[i + 1] - m_x[i]);
}
}
// for left extrapolation coefficients
m_b0 = (m_force_linear_extrapolation == false) ? m_b[0] : 0.0;
m_c0 = m_c[0];
// for the right extrapolation coefficients
// f_{n-1}(x) = b*(x-x_{n-1})^2 + c*(x-x_{n-1}) + y_{n-1}
double h = x[n - 1] - x[n - 2];
// m_b[n-1] is determined by the boundary condition
m_a[n - 1] = 0.0;
m_c[n - 1] = 3.0 * m_a[n - 2] * h * h + 2.0 * m_b[n - 2] * h + m_c[n - 2]; // = f'_{n-2}(x_{n-1})
if (m_force_linear_extrapolation == true)
m_b[n - 1] = 0.0;
}
double spline::operator()(double x) const {
size_t n = m_x.size();
// find the closest point m_x[idx] < x, idx=0 even if x<m_x[0]
std::vector<double>::const_iterator it;
it = std::lower_bound(m_x.begin(), m_x.end(), x);
int idx = std::max(int(it - m_x.begin()) - 1, 0);
double h = x - m_x[idx];
double interpol;
if (x < m_x[0]) {
// extrapolation to the left
interpol = (m_b0 * h + m_c0) * h + m_y[0];
} else if (x > m_x[n - 1]) {
// extrapolation to the right
interpol = (m_b[n - 1] * h + m_c[n - 1]) * h + m_y[n - 1];
} else {
// interpolation
interpol = ((m_a[idx] * h + m_b[idx]) * h + m_c[idx]) * h + m_y[idx];
}
return interpol;
}
} // namespace tk
} // namespace
#endif /* TK_SPLINE_H */