abacus-develop/source/source_base/cubic_spline.cpp at develop · pplab/abacus-develop · GitHub

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#include "cubic_spline.h"

#include <cassert>

#include <algorithm>

#include <numeric>

#include <functional>

#include <vector>

using ModuleBase::CubicSpline;

extern "C"

{

// solve a tridiagonal linear system

void dgtsv_(int* N, int* NRHS, double* DL, double* D, double* DU, double* B, int* LDB, int* INFO);

};

CubicSpline::BoundaryCondition::BoundaryCondition(BoundaryType type)

: type(type)

{

assert(type == BoundaryType::periodic || type == BoundaryType::not_a_knot);

}

CubicSpline::BoundaryCondition::BoundaryCondition(BoundaryType type, double val)

: type(type), val(val)

{

assert(type == BoundaryType::first_deriv || type == BoundaryType::second_deriv);

}

CubicSpline::CubicSpline(

int n,

const double* x,

const double* y,

const BoundaryCondition& bc_start,

const BoundaryCondition& bc_end

): n_spline_(1), n_(n), xmin_(x[0]), xmax_(x[n - 1]), x_(x, x + n), y_(2 * n)

{

std::copy(y, y + n, y_.begin());

build(n, x, y, bc_start, bc_end, &y_[n]);

}

CubicSpline::CubicSpline(

int n,

double x0,

double dx,

const double* y,

const BoundaryCondition& bc_start,

const BoundaryCondition& bc_end

): n_spline_(1), n_(n), xmin_(x0), xmax_(x0 + (n - 1) * dx), dx_(dx), y_(2 * n)

{

std::copy(y, y + n, y_.begin());

build(n, dx, y, bc_start, bc_end, &y_[n]);

}

CubicSpline::CubicSpline(int n, const double* x)

: n_spline_(0), n_(n), xmin_(x[0]), xmax_(x[n - 1]), x_(x, x + n)

{

}

CubicSpline::CubicSpline(int n, double x0, double dx)

: n_spline_(0), n_(n), xmin_(x0), xmax_(x0 + (n - 1) * dx), dx_(dx)

{

}

void CubicSpline::add(

const double* y,

const BoundaryCondition& bc_start,

const BoundaryCondition& bc_end

)

{

int offset = n_spline_ * 2 * n_;

y_.resize(offset + 2 * n_);

std::copy(y, y + n_, &y_[offset]);

double* dy = &y_[offset + n_];

if (x_.empty()) // evenly spaced knots

{

build(n_, dx_, y, bc_start, bc_end, dy);

}

else

{

build(n_, x_.data(), y, bc_start, bc_end, dy);

}

++n_spline_;

}

void CubicSpline::eval(

int n_interp,

const double* x_interp,

double* y_interp,

double* dy_interp,

double* d2y_interp,

int i_spline

) const

{

assert(0 <= i_spline && i_spline < n_spline_);

const double* y = &y_[i_spline * 2 * n_];

const double* dy = y + n_;

if (x_.empty()) // evenly spaced knots

{

eval(n_, xmin_, dx_, y, dy, n_interp, x_interp, y_interp, dy_interp, d2y_interp);

}

else

{

eval(n_, x_.data(), y, dy, n_interp, x_interp, y_interp, dy_interp, d2y_interp);

}

void CubicSpline::multi_eval(

int n_spline,

const int* i_spline,

double x_interp,

double* y_interp,

double* dy_interp,

double* d2y_interp

) const

{

assert(std::all_of(i_spline, i_spline + n_spline,

[this](int i) { return 0 <= i && i < n_spline_; }));

_validate_eval(n_, {xmin_, dx_}, x_.empty() ? nullptr : x_.data(),

y_.data(), &y_[n_], 1, &x_interp);

int p = 0;

double dx = 0.0, r = 0.0;

if (x_.empty()) // evenly spaced knots

{

p = _index(n_, xmin_, dx_, x_interp);

dx = dx_;

r = (x_interp - xmin_) / dx - p;

}

else

{

p = _index(n_, x_.data(), x_interp);

dx = x_[p + 1] - x_[p];

r = (x_interp - x_[p]) / dx;

}

const double r2 = r * r;

const double r3 = r2 * r;

double wy0 = 0.0, wy1 = 0.0, ws0 = 0.0, ws1 = 0.0;

int offset = 0;

if (y_interp)

{

wy1 = 3.0 * r2 - 2.0 * r3;

wy0 = 1.0 - wy1;

ws0 = (r - 2.0 * r2 + r3) * dx;

ws1 = (r3 - r2) * dx;

for (int i = 0; i < n_spline; ++i)

{

offset = i_spline[i] * 2 * n_ + p;

y_interp[i] = wy0 * y_[offset] + wy1 * y_[offset + 1]

+ ws0 * y_[offset + n_] + ws1 * y_[offset + n_ + 1];

}

if (dy_interp)

{

wy1 = 6.0 * (r - r2) / dx; // wy0 = -wy1

ws0 = 3.0 * r2 - 4.0 * r + 1.0;

ws1 = 3.0 * r2 - 2.0 * r;

for (int i = 0; i < n_spline; ++i)

{

offset = i_spline[i] * 2 * n_ + p;

dy_interp[i] = wy1 * (y_[offset + 1] - y_[offset])

+ ws0 * y_[offset + n_] + ws1 * y_[offset + n_ + 1];

}

if (d2y_interp)

{

wy1 = (6.0 - 12.0 * r) / (dx * dx); // wy0 = -wy1

ws0 = (6.0 * r - 4.0) / dx;

ws1 = (6.0 * r - 2.0) / dx;

for (int i = 0; i < n_spline; ++i)

{

offset = i_spline[i] * 2 * n_ + p;

d2y_interp[i] = wy1 * (y_[offset + 1] - y_[offset])

+ ws0 * y_[offset + n_] + ws1 * y_[offset + n_ + 1];

}

void CubicSpline::multi_eval(

double x,

double* y,

double* dy,

double* d2y

) const

{

std::vector<int> i_spline(n_spline_);

std::iota(i_spline.begin(), i_spline.end(), 0);

multi_eval(i_spline.size(), i_spline.data(), x, y, dy, d2y);

}

void CubicSpline::build(

int n,

const double* x,

const double* y,

const BoundaryCondition& bc_start,

const BoundaryCondition& bc_end,

double* dy

)

{

std::vector<double> dx(n);

std::adjacent_difference(x, x + n, dx.begin());

_build(n, &dx[1], y, bc_start, bc_end, dy);

}

void CubicSpline::build(

int n,

double dx,

const double* y,

const BoundaryCondition& bc_start,

const BoundaryCondition& bc_end,

double* dy

)

{

std::vector<double> dx_(n - 1, dx);

_build(n, dx_.data(), y, bc_start, bc_end, dy);

}

void CubicSpline::eval(

int n,

const double* x,

const double* y,

const double* dy,

int n_interp,

const double* x_interp,

double* y_interp,

double* dy_interp,

double* d2y_interp

)

{

_validate_eval(n, {}, x, y, dy, n_interp, x_interp);

// indices of the polynomial segments that contain x_interp

std::vector<int> _ind(n_interp);

std::transform(x_interp, x_interp + n_interp, _ind.begin(),

[n, x](double x_i) { return _index(n, x, x_i); });

std::vector<double> buffer(n_interp * 5);

double* _w = buffer.data();

double* _c0 = _w + n_interp;

double* _c1 = _c0 + n_interp;

double* _c2 = _c1 + n_interp;

double* _c3 = _c2 + n_interp;

for (int i = 0; i < n_interp; ++i)

{

int p = _ind[i];

double dx = x[p + 1] - x[p];

double inv_dx = 1.0 / dx;

double dd = (y[p + 1] - y[p]) * inv_dx;

_w[i] = x_interp[i] - x[p];

_c0[i] = y[p];

_c1[i] = dy[p];

_c3[i] = (_c1[i] + dy[p + 1] - 2.0 * dd) * inv_dx * inv_dx;

_c2[i] = (dd - _c1[i]) * inv_dx - _c3[i] * dx;

}

_cubic(n_interp, _w, _c0, _c1, _c2, _c3, y_interp, dy_interp, d2y_interp);

}

void CubicSpline::eval(

int n,

double x0,

double dx,

const double* y,

const double* dy,

int n_interp,

const double* x_interp,

double* y_interp,

double* dy_interp,

double* d2y_interp

)

{

_validate_eval(n, {x0, dx}, nullptr, y, dy, n_interp, x_interp);

// indices of the polynomial segments that contain x_interp

std::vector<int> _ind(n_interp);

std::transform(x_interp, x_interp + n_interp, _ind.begin(),

[n, x0, dx](double x_i) { return _index(n, x0, dx, x_i); });

std::vector<double> buffer(n_interp * 5);

double* _w = buffer.data();

double* _c0 = _w + n_interp;

double* _c1 = _c0 + n_interp;

double* _c2 = _c1 + n_interp;

double* _c3 = _c2 + n_interp;

double inv_dx = 1.0 / dx;

double inv_dx2 = inv_dx * inv_dx;

for (int i = 0; i < n_interp; ++i)

{

int p = _ind[i];

double dd = (y[p + 1] - y[p]) * inv_dx;

_w[i] = x_interp[i] - x0 - p * dx;

_c0[i] = y[p];

_c1[i] = dy[p];

_c3[i] = (_c1[i] + dy[p + 1] - 2.0 * dd) * inv_dx2;

_c2[i] = (dd - _c1[i]) * inv_dx - _c3[i] * dx;

}

_cubic(n_interp, _w, _c0, _c1, _c2, _c3, y_interp, dy_interp, d2y_interp);

}

void CubicSpline::_validate_build(

int n,

const double* dx,

const double* y,

const BoundaryCondition& bc_start,

const BoundaryCondition& bc_end

)

{

assert(n > 1);

// if periodic boundary condition is specified, it must be applied to both ends

assert((bc_start.type == BoundaryType::periodic)

== (bc_end.type == BoundaryType::periodic));

// y[0] must equal y[n-1] for periodic boundary condition

assert(bc_start.type != BoundaryType::periodic || y[0] == y[n - 1]);

// not-a-knot boundary condition requires the existence of "internal" knot

// so n must be at least 3

assert((bc_start.type != BoundaryType::not_a_knot &&

bc_end.type != BoundaryType::not_a_knot) || n > 2);

// knots must be strictly increasing

assert(std::all_of(dx, dx + n - 1, [](double d) { return d > 0.0; }));

}

void CubicSpline::_validate_eval(

int n,

const double (&u)[2],

const double* x,

const double* y,

const double* dy,

int n_interp,

const double* x_interp

)

{

assert(n > 1 && y && dy);

assert((x && std::is_sorted(x, x + n, std::less_equal<double>())) || u[1] > 0.0);

assert((n_interp > 0 && x_interp) || n_interp == 0);

double xmin = x ? x[0] : u[0];

double xmax = x ? x[n - 1] : u[0] + (n - 1) * u[1];

assert(std::all_of(x_interp, x_interp + n_interp,

[xmin, xmax](double x_i) { return xmin <= x_i && x_i <= xmax; }));

}

void CubicSpline::_build(

int n,

const double* dx,

const double* y,

const BoundaryCondition& bc_start,

const BoundaryCondition& bc_end,

double* dy

)

{

_validate_build(n, dx, y, bc_start, bc_end);

if (n == 2 && bc_start.type == BoundaryType::periodic)

{

dy[0] = dy[1] = 0.0; // the only possible solution: constant

}

else if (n == 3 && bc_start.type == BoundaryType::not_a_knot

&& bc_end.type == BoundaryType::not_a_knot)

{

// in this case two conditions coincide

// simply build a parabola that passes through the three data points

double dd01 = (y[1] - y[0]) / dx[0]; // divided difference f[x0,x1]

double dd12 = (y[2] - y[1]) / dx[1]; // f[x1,x2]

double dd012 = (dd12 - dd01) / (dx[0] + dx[1]); // f[x0,x1,x2]

dy[0] = dd01 - dd012 * dx[0];

dy[1] = 2.0 * dd01 - dy[0];

dy[2] = dd01 + dd012 * (dx[0] + 2.0 * dx[1]);

}

else

{

std::vector<double> buffer(4 * n);

double* dd = buffer.data(); // divided differences

std::adjacent_difference(y, y + n, dd);

dd += 1; // the first element computed by adjacent_difference is not a difference

std::transform(dd, dd + n - 1, dx, dd, std::divides<double>());

// tridiagonal linear system (cyclic tridiagonal if periodic boundary condition)

double* d = buffer.data() + n; // main diagonal

double* l = buffer.data() + 2 * n; // subdiagonal

double* u = buffer.data() + 3 * n; // superdiagonal

//***********************************************

// common part of the tridiagonal linear system

//***********************************************

std::copy(dx + 1, dx + n - 1, l);

std::copy(dx, dx + n - 2, u + 1);

for (int i = 1; i != n - 1; ++i)

{

d[i] = 2.0 * (dx[i - 1] + dx[i]);

dy[i] = 3.0 * (dd[i - 1] * dx[i] + dd[i] * dx[i - 1]);

}

//***********************************************

// boundary-specific part

//***********************************************

if (bc_start.type == BoundaryType::periodic)

{

// exclude s[n-1] and solve a a cyclic tridiagonal linear system of size n-1

d[0] = 2.0 * (dx[n - 2] + dx[0]);

u[0] = dx[n - 2];

l[n - 2] = dx[0];

dy[0] = 3.0 * (dd[0] * dx[n - 2] + dd[n - 2] * dx[0]);

_solve_cyctri(n - 1, d, u, l, dy);

dy[n - 1] = dy[0];

}

else

{

switch (bc_start.type)

{

case BoundaryType::first_deriv:

d[0] = 1.0 * dx[0];

u[0] = 0.0;

dy[0] = bc_start.val * dx[0];

break;

case BoundaryType::second_deriv:

d[0] = 2.0 * dx[0];

u[0] = 1.0 * dx[0];

dy[0] = (3.0 * dd[0] - 0.5 * bc_start.val * dx[0]) * dx[0];

break;

default: // BoundaryCondition::not_a_knot

d[0] = dx[1];

u[0] = dx[0] + dx[1];

dy[0] = (dd[0] * dx[1] * (dx[0] + 2 * u[0]) + dd[1] * dx[0] * dx[0]) / u[0];

}

switch (bc_end.type)

{

case BoundaryType::first_deriv:

d[n - 1] = 1.0 * dx[n - 2];

l[n - 2] = 0.0;

dy[n - 1] = bc_end.val * dx[n - 2];

break;

case BoundaryType::second_deriv:

d[n - 1] = 2.0 * dx[n - 2];

l[n - 2] = 1.0 * dx[n - 2];

dy[n - 1] = (3.0 * dd[n - 2] + 0.5 * bc_end.val * dx[n - 2]) * dx[n - 2];

break;

default: // BoundaryCondition::not_a_knot

d[n - 1] = dx[n - 3];

l[n - 2] = dx[n - 3] + dx[n - 2];

dy[n - 1] = (dd[n - 2] * dx[n - 3] * (dx[n - 2] + 2 * l[n - 2])

+ dd[n - 3] * dx[n - 2] * dx[n - 2]) / l[n - 2];

}

int nrhs = 1;

int ldb = n;

int info = 0;

dgtsv_(&n, &nrhs, l, d, u, dy, &ldb, &info);

}

void CubicSpline::_cubic(

int n,

const double* w,

const double* c0,

const double* c1,

const double* c2,

const double* c3,

double* y,

double* dy,

double* d2y

)

{

if (y)

{

for (int i = 0; i < n; ++i)

{

y[i] = ((c3[i] * w[i] + c2[i]) * w[i] + c1[i]) * w[i] + c0[i];

}

if (dy)

{

for (int i = 0; i < n; ++i)

{

dy[i] = (3.0 * c3[i] * w[i] + 2.0 * c2[i]) * w[i] + c1[i];

}

if (d2y)

{

for (int i = 0; i < n; ++i)

{

d2y[i] = 6.0 * c3[i] * w[i] + 2.0 * c2[i];

}

int CubicSpline::_index(int n, const double* knots, double x)

{

int i = (std::upper_bound(knots, knots + n, x) - knots) - 1;

return i - (i == n - 1);

}

int CubicSpline::_index(int n, double x0, double dx, double x)

{

int i = (x - x0) / dx;

return i - (i == n - 1);

}

void CubicSpline::_solve_cyctri(int n, double* d, double* u, double* l, double* b)

{

// flexible non-zero parameters that can affect the condition number of the

// tridiagonal linear system

double alpha = 1.0;

double beta = -d[0] / u[n - 1];

std::vector<double> bp(2 * n, 0.0);

std::copy(b, b + n, bp.begin());

bp[n] = 1. / alpha;

bp[2 * n - 1] = 1. / beta;

d[0] -= u[n - 1] * beta / alpha;

d[n - 1] -= l[n - 1] * alpha / beta;

int nrhs = 2;

int info = 0;

int ldb = n;

dgtsv_(&n, &nrhs, l, d, u, bp.data(), &ldb, &info);

double fac = (beta * u[n - 1] * bp[0] + alpha * l[n - 1] * bp[n - 1])

/ (1. + beta * u[n - 1] * bp[n] + alpha * l[n - 1] * bp[2 * n - 1]);

std::transform(bp.begin(), bp.begin() + n, bp.begin() + n, b,

[fac](double yi, double zi) { return yi - fac * zi; });

}

abacus-develop/source/source_base/cubic_spline.cpp at develop · pplab/abacus-develop · GitHub

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