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Robustified C++ version and added callback management.

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Jerome 2023-03-18 16:13:17 +01:00
parent 0e4045f0ff
commit f12e3c45ca
3 changed files with 78 additions and 10 deletions
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23
cpp/include/CubicLagrangeMinimize.hpp
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@ -9,6 +9,24 @@ using Eigen::VectorXd;
using Eigen::Matrix4d;
using Eigen::Vector4d;
/// @brief Returns a callback function that does nothing.
#define CubicLagrangeMinimize_GetEmptyCallback() [](Eigen::VectorXd const&){}
/// @brief Returns a callback function that prints the current best estimate of the minimum and the iteration number.
#define CubicLagrangeMinimize_GetSimpleCallback() [](Eigen::VectorXd const& callback_args){ std::cout << "Iteration " << callback_args[2] << " : f(" << callback_args[0] << ") = " << callback_args[1] << "\n"; }
/// @brief Returns a callback function that prints the current best estimate of the minimum, the iteration number, the current interval, and the cubic polynomial coefficients.
#define CubicLagrangeMinimize_GetDetailedCallback() \
[](Eigen::VectorXd const& callback_args){ \
std::cout << "--------------------------------------------------------------------\n";\
std::cout << " Iteration " << callback_args[2] << "\n";\
std::cout << "--------------------------------------------------------------------\n";\
std::cout << "X values : " << callback_args[8] << ", " << callback_args[9] << ", " << callback_args[10] << ", " << callback_args[11] << "\n";\
std::cout << "Cubic poly coeffs : " << callback_args[12] << ", " << callback_args[13] << ", " << callback_args[14] << ", " << callback_args[15] << "\n";\
std::cout << "Quadratic solution : delta = " << callback_args[3] << ", x_sol_1 = " << callback_args[4] << ", x_sol_2 = " << callback_args[5] << ", y_sol_1 = " << callback_args[6] << ", y_sol_2 = " << callback_args[7] << "\n";\
std::cout << "Current solution : f(" << callback_args[0] << ") = " << callback_args[1] << "\n";\
}
/// @brief Function to find minimum of f over interval [a, b] using cubic Lagrange polynomial interpolation.
/// If the function is monotonic, then the minimum is one of the bounds of the interval, and the minimum is found in a single iteration.
/// The best estimate of the minimum within the current interval is returned once the interval is smaller than the tolerance.
@ -18,7 +36,10 @@ using Eigen::Vector4d;
/// @param a The lower bound of the interval.
/// @param b The upper bound of the interval.
/// @param tol The tolerance on the interval width.
/// @param callback A function to call at each iteration with the current best estimate of the minimum and other internal variables. The vector passed to the callback function contains the following variables : (x_sol, y_sol, i, delta, x_sol_1, x_sol_2, y_sol_1, y_sol_2, x0, x1, x2, x3, a0, a1, a2, a3).
/// @return The best estimate of the minimum within the current interval once its width is smaller than the tolerance.
double CubicLagrangeMinimize(std::function<double(double)> f, double a, double b, double tol=1e-9);
double CubicLagrangeMinimize(std::function<double(double)> f, double a, double b, double tol=1e-9, std::function<void(Eigen::VectorXd const&)> callback = [](Eigen::VectorXd const&){});
// (x_sol, y_sol, i, delta, x_sol_1, x_sol_2, y_sol_1, y_sol_2, x0, x1, x2, x3, a0, a1, a2, a3).
#endif

57
cpp/src/CubicLagrangeMinimize.cpp
View file

@ -1,5 +1,9 @@
#include <CubicLagrangeMinimize.hpp>
#ifndef M_PI
#define M_PI 3.1415926535897932384626433832795028841971693993751
#endif
/// @brief Linear least-squares fit of a cubic polynomial to four points.
/// @param x1 Abscissa of first point.
/// @param x2 Abscissa of second point.
@ -11,7 +15,7 @@
/// @param y4 Ordinate of fourth point.
/// @return 4-dimensional vector of polynomial coefficients from low to high order : [a0 a1 a2 a3] -> a0 + a1*x + a2*x^2 + a3*x^3
Vector4d polyfit4(double x1, double x2, double x3, double x4, double y1, double y2, double y3, double y4) {
Matrix4d A(4, 4);
Matrix4d A;
double x1x1 = x1*x1, x2x2 = x2*x2, x3x3 = x3*x3, x4x4 = x4*x4;
double x1x1x1 = x1x1*x1, x2x2x2 = x2x2*x2, x3x3x3 = x3x3*x3, x4x4x4 = x4x4*x4;
A << 1, x1, x1x1, x1x1x1,
@ -23,6 +27,26 @@ Vector4d polyfit4(double x1, double x2, double x3, double x4, double y1, double
return (A.transpose() * A).colPivHouseholderQr().solve(A.transpose() * y);
}
/// @brief Linear least-squares fit of a cubic polynomial to four points (-pi/3, y1), (-pi/9, y2), (pi/9, y3), (pi/3, y4).
/// @param y1 Ordinate of first point.
/// @param y2 Ordinate of second point.
/// @param y3 Ordinate of third point.
/// @param y4 Ordinate of fourth point.
/// @return 4-dimensional vector of polynomial coefficients from low to high order : [a0 a1 a2 a3] -> a0 + a1*x + a2*x^2 + a3*x^3
Vector4d polyfit4_mpi3_pi3(double y1, double y2, double y3, double y4) {
static Matrix4d AT = (Matrix4d() << 1,1,1,1,
-M_PI/3.,-M_PI/9.,M_PI/9.,M_PI/3.,
pow(M_PI,2)/9.,pow(M_PI,2)/81.,pow(M_PI,2)/81.,pow(M_PI,2)/9.,
-pow(M_PI,3)/27.,-pow(M_PI,3)/729.,pow(M_PI,3)/729.,pow(M_PI,3)/27.).finished();// A^T
static Matrix4d ATAinv = (Matrix4d() << 41./64.,0,-405./(64.*pow(M_PI,2)),0,
0,3285/(64.*pow(M_PI,2)),0,-29889/(64.*pow(M_PI,4)),
-405/(64.*pow(M_PI,2)),0,6561/(64.*pow(M_PI,4)),0,
0,-29889/(64.*pow(M_PI,4)),0,295245/(64.*pow(M_PI,6))).finished();// (A^T * A)^-1
VectorXd y(4);
y << y1, y2, y3, y4;
return ATAinv*(AT*y);
}
/// @brief Cubic polynomial function.
/// @param x Abscissa.
/// @param a0 Constant term.
@ -34,15 +58,26 @@ double cubic_poly(double x, double a0, double a1, double a2, double a3) {
return a0 + a1*x + a2*x*x + a3*x*x*x;
}
double CubicLagrangeMinimize(std::function<double(double)> f, double a, double b, double tol) {
/// @brief Maps a value or array "x" from the interval [a1, b1] to the interval [a2, b2].
/// @param x Value to map.
/// @param a1 Lower bound of the interval to map from.
/// @param b1 Upper bound of the interval to map from.
/// @param a2 Lower bound of the interval to map to.
/// @param b2 Upper bound of the interval to map to.
/// @return Mapped value.
double map_interval(double x, double a1, double b1, double a2, double b2) { return (x - a1) * (b2 - a2) / (b1 - a1) + a2; }
double CubicLagrangeMinimize(std::function<double(double)> f, double a, double b, double tol, std::function<void(Eigen::VectorXd const&)> callback) {
// initialize interval endpoints and function values
double x0 = a, x1 = a*2./3. + b*1./3., x2 = a*1./3. + b*2./3., x3 = b; // endpoints and two points in the interval
double f0 = f(x0), f1 = 0., f2 = 0., f3 = f(x3); // function values at the endpoints and two points in the interval
double x_prev = x0; // previous value of x_sol to track convergence of the solution
double x_prev = (x0 + x3)/2.; // previous value of x_sol to track convergence of the solution
double a0, a1, a2, a3; // coefficients of the cubic Lagrange polynomial
double delta; // determinant of the quadratic equation of the Lagrange polynomial
double x_sol, x_sol_1, x_sol_2, y_sol, y_sol_1, y_sol_2; // solutions of the Lagrange polynomial
constexpr double small_coefficient = 1e-9; // threshold for small coefficients to avoid ill-conditioning of the quadratic equation
constexpr double x0m = -M_PI/3.; // mapped value of x0 from the interval [x0 x3] to [-pi/3 pi/3]
constexpr double x3m = M_PI/3.; // mapped value of x3 from the interval [x0 x3] to [-pi/3 pi/3]
unsigned int Niter = static_cast<unsigned int>(std::ceil(std::log(std::fabs(b - a)/tol)/std::log(3.)));// number of iterations to reduce interval width by a factor of 3
@ -52,15 +87,17 @@ double CubicLagrangeMinimize(std::function<double(double)> f, double a, double b
f1 = f(x1), f2 = f(x2);
// compute Lagrange polynomial using least-squares fit to 4 points, which is equivalent to the cubic Lagrange polynomial
Vector4d A = polyfit4(x0, x1, x2, x3, f0, f1, f2, f3);
// Use x values remapped to the interval [-pi/3, pi/3] to minimize the condition number of the matrix (A^T * A) in the least-squares fit
Vector4d A = polyfit4_mpi3_pi3(f0, f1, f2, f3);// Equivalent to Vector4d A = polyfit4(x0, x1, x2, x3, f0, f1, f2, f3);
a0 = A[0]; a1 = A[1]; a2 = A[2]; a3 = A[3];
x_sol_1 = 0.; x_sol_2 = 0.; y_sol_1 = 0.; y_sol_2 = 0., delta = 0.;
// Solve the first derivative of the Lagrange polynomial for a zero
if(std::fabs(a3) > small_coefficient) {
delta = -3*a1*a3 + a2*a2;
if(delta < 0) {
x_sol = (f0 < f3) ? x0 : x3; // just choose the interval tha contains the minimum of the linear polynomial
x_sol = (f0 < f3) ? x0m : x3m; // just choose the interval tha contains the minimum of the linear polynomial
y_sol = cubic_poly(x_sol, a0, a1, a2, a3);
} else { // solve for the two solutions of the quadratic equation of the first derivative of the Lagrange polynomial
x_sol_1 = (-a2 + std::sqrt(delta))/(3.*a3);
@ -77,10 +114,18 @@ double CubicLagrangeMinimize(std::function<double(double)> f, double a, double b
x_sol = -a1/(2.*a2);
y_sol = cubic_poly(x_sol, a0, a1, a2, a3);
} else { // if a3 and a2 are zero, then the Lagrange polynomial is a linear polynomial
x_sol = (f0 < f3) ? x0 : x3; // just choose the interval tha contains the minimum of the linear polynomial
x_sol = (f0 < f3) ? x0m : x3m; // just choose the interval tha contains the minimum of the linear polynomial
y_sol = cubic_poly(x_sol, a0, a1, a2, a3);
}
// transform the solution back to the original interval
x_sol = map_interval(x_sol, x0m, x3m, x0, x3),
x_sol_1 = map_interval(x_sol_1, x0m, x3m, x0, x3);
x_sol_2 = map_interval(x_sol_2, x0m, x3m, x0, x3);
// Call the callback function to print progress
callback((Eigen::VectorXd(16) << x_sol, y_sol, i, delta, x_sol_1, x_sol_2, y_sol_1, y_sol_2, x0, x1, x2, x3, a0, a1, a2, a3).finished());
// Check convergence
if(std::fabs(x_sol - x_prev) < tol) { break; }

8
cpp/src/tests.cpp
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@ -64,11 +64,13 @@ double dfct_07(double) {
void test_CubicLagrangeMinimize() {
std::vector<std::function<double(double)>> fcts = {fct_01, fct_02, fct_03, fct_04, fct_05, fct_06, fct_07};
std::vector<std::function<double(double)>> dfcts = {dfct_01, dfct_02, dfct_03, dfct_04, dfct_05, dfct_06, dfct_07};
std::vector<double> mins = {-1.2, -10., -1.5, -10., -1., -2., -3.};
std::vector<double> maxs = {1.5, 10., 3., 5., 6., 3., 4.};
std::vector<double> mins = {-1.2, -2., -1.5, -10., -1., -2., -3.};
std::vector<double> maxs = {1.5, 3., 3., 5., 6., 3., 4.};
constexpr double tol = 1e-9;
for(unsigned int i = 0 ; i < fcts.size() ; ++i) {
cout << "----------------------------------------------------------------------------------------------------\n";
auto f = fcts[i]; auto df = dfcts[i];
auto x_min = CubicLagrangeMinimize(f, mins[i], maxs[i]);
auto x_min = CubicLagrangeMinimize(f, mins[i], maxs[i], tol, CubicLagrangeMinimize_GetDetailedCallback());
cout << "[" << mins[i] << " " << maxs[i] << "]\tf(" << x_min << ") = " << f(x_min) << " df(x_min)/dt = " << df(x_min) << endl;
}
}