// Ceres Solver - A fast non-linear least squares minimizer
// Copyright 2015 Google Inc. All rights reserved.
// http://ceres-solver.org/
//
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// modification, are permitted provided that the following conditions are met:
//
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// this list of conditions and the following disclaimer.
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// this list of conditions and the following disclaimer in the documentation
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//
// Author: moll.markus@arcor.de (Markus Moll)
// sameeragarwal@google.com (Sameer Agarwal)
#include "ceres/polynomial.h"
#include <cmath>
#include <cstddef>
#include <vector>
#include "Eigen/Dense"
#include "ceres/function_sample.h"
#include "ceres/internal/port.h"
#include "glog/logging.h"
namespace ceres {
namespace internal {
using std::vector;
namespace {
// Balancing function as described by B. N. Parlett and C. Reinsch,
// "Balancing a Matrix for Calculation of Eigenvalues and Eigenvectors".
// In: Numerische Mathematik, Volume 13, Number 4 (1969), 293-304,
// Springer Berlin / Heidelberg. DOI: 10.1007/BF02165404
void BalanceCompanionMatrix(Matrix* companion_matrix_ptr) {
CHECK_NOTNULL(companion_matrix_ptr);
Matrix& companion_matrix = *companion_matrix_ptr;
Matrix companion_matrix_offdiagonal = companion_matrix;
companion_matrix_offdiagonal.diagonal().setZero();
const int degree = companion_matrix.rows();
// gamma <= 1 controls how much a change in the scaling has to
// lower the 1-norm of the companion matrix to be accepted.
//
// gamma = 1 seems to lead to cycles (numerical issues?), so
// we set it slightly lower.
const double gamma = 0.9;
// Greedily scale row/column pairs until there is no change.
bool scaling_has_changed;
do {
scaling_has_changed = false;
for (int i = 0; i < degree; ++i) {
const double row_norm = companion_matrix_offdiagonal.row(i).lpNorm<1>();
const double col_norm = companion_matrix_offdiagonal.col(i).lpNorm<1>();
// Decompose row_norm/col_norm into mantissa * 2^exponent,
// where 0.5 <= mantissa < 1. Discard mantissa (return value
// of frexp), as only the exponent is needed.
int exponent = 0;
std::frexp(row_norm / col_norm, &exponent);
exponent /= 2;
if (exponent != 0) {
const double scaled_col_norm = std::ldexp(col_norm, exponent);
const double scaled_row_norm = std::ldexp(row_norm, -exponent);
if (scaled_col_norm + scaled_row_norm < gamma * (col_norm + row_norm)) {
// Accept the new scaling. (Multiplication by powers of 2 should not
// introduce rounding errors (ignoring non-normalized numbers and
// over- or underflow))
scaling_has_changed = true;
companion_matrix_offdiagonal.row(i) *= std::ldexp(1.0, -exponent);
companion_matrix_offdiagonal.col(i) *= std::ldexp(1.0, exponent);
}
}
}
} while (scaling_has_changed);
companion_matrix_offdiagonal.diagonal() = companion_matrix.diagonal();
companion_matrix = companion_matrix_offdiagonal;
VLOG(3) << "Balanced companion matrix is\n" << companion_matrix;
}
void BuildCompanionMatrix(const Vector& polynomial,
Matrix* companion_matrix_ptr) {
CHECK_NOTNULL(companion_matrix_ptr);
Matrix& companion_matrix = *companion_matrix_ptr;
const int degree = polynomial.size() - 1;
companion_matrix.resize(degree, degree);
companion_matrix.setZero();
companion_matrix.diagonal(-1).setOnes();
companion_matrix.col(degree - 1) = -polynomial.reverse().head(degree);
}
// Remove leading terms with zero coefficients.
Vector RemoveLeadingZeros(const Vector& polynomial_in) {
int i = 0;
while (i < (polynomial_in.size() - 1) && polynomial_in(i) == 0.0) {
++i;
}
return polynomial_in.tail(polynomial_in.size() - i);
}
void FindLinearPolynomialRoots(const Vector& polynomial,
Vector* real,
Vector* imaginary) {
CHECK_EQ(polynomial.size(), 2);
if (real != NULL) {
real->resize(1);
(*real)(0) = -polynomial(1) / polynomial(0);
}
if (imaginary != NULL) {
imaginary->setZero(1);
}
}
void FindQuadraticPolynomialRoots(const Vector& polynomial,
Vector* real,
Vector* imaginary) {
CHECK_EQ(polynomial.size(), 3);
const double a = polynomial(0);
const double b = polynomial(1);
const double c = polynomial(2);
const double D = b * b - 4 * a * c;
const double sqrt_D = sqrt(fabs(D));
if (real != NULL) {
real->setZero(2);
}
if (imaginary != NULL) {
imaginary->setZero(2);
}
// Real roots.
if (D >= 0) {
if (real != NULL) {
// Stable quadratic roots according to BKP Horn.
// http://people.csail.mit.edu/bkph/articles/Quadratics.pdf
if (b >= 0) {
(*real)(0) = (-b - sqrt_D) / (2.0 * a);
(*real)(1) = (2.0 * c) / (-b - sqrt_D);
} else {
(*real)(0) = (2.0 * c) / (-b + sqrt_D);
(*real)(1) = (-b + sqrt_D) / (2.0 * a);
}
}
return;
}
// Use the normal quadratic formula for the complex case.
if (real != NULL) {
(*real)(0) = -b / (2.0 * a);
(*real)(1) = -b / (2.0 * a);
}
if (imaginary != NULL) {
(*imaginary)(0) = sqrt_D / (2.0 * a);
(*imaginary)(1) = -sqrt_D / (2.0 * a);
}
}
} // namespace
bool FindPolynomialRoots(const Vector& polynomial_in,
Vector* real,
Vector* imaginary) {
if (polynomial_in.size() == 0) {
LOG(ERROR) << "Invalid polynomial of size 0 passed to FindPolynomialRoots";
return false;
}
Vector polynomial = RemoveLeadingZeros(polynomial_in);
const int degree = polynomial.size() - 1;
VLOG(3) << "Input polynomial: " << polynomial_in.transpose();
if (polynomial.size() != polynomial_in.size()) {
VLOG(3) << "Trimmed polynomial: " << polynomial.transpose();
}
// Is the polynomial constant?
if (degree == 0) {
LOG(WARNING) << "Trying to extract roots from a constant "
<< "polynomial in FindPolynomialRoots";
// We return true with no roots, not false, as if the polynomial is constant
// it is correct that there are no roots. It is not the case that they were
// there, but that we have failed to extract them.
return true;
}
// Linear
if (degree == 1) {
FindLinearPolynomialRoots(polynomial, real, imaginary);
return true;
}
// Quadratic
if (degree == 2) {
FindQuadraticPolynomialRoots(polynomial, real, imaginary);
return true;
}
// The degree is now known to be at least 3. For cubic or higher
// roots we use the method of companion matrices.
// Divide by leading term
const double leading_term = polynomial(0);
polynomial /= leading_term;
// Build and balance the companion matrix to the polynomial.
Matrix companion_matrix(degree, degree);
BuildCompanionMatrix(polynomial, &companion_matrix);
BalanceCompanionMatrix(&companion_matrix);
// Find its (complex) eigenvalues.
Eigen::EigenSolver<Matrix> solver(companion_matrix, false);
if (solver.info() != Eigen::Success) {
LOG(ERROR) << "Failed to extract eigenvalues from companion matrix.";
return false;
}
// Output roots
if (real != NULL) {
*real = solver.eigenvalues().real();
} else {
LOG(WARNING) << "NULL pointer passed as real argument to "
<< "FindPolynomialRoots. Real parts of the roots will not "
<< "be returned.";
}
if (imaginary != NULL) {
*imaginary = solver.eigenvalues().imag();
}
return true;
}
Vector DifferentiatePolynomial(const Vector& polynomial) {
const int degree = polynomial.rows() - 1;
CHECK_GE(degree, 0);
// Degree zero polynomials are constants, and their derivative does
// not result in a smaller degree polynomial, just a degree zero
// polynomial with value zero.
if (degree == 0) {
return Eigen::VectorXd::Zero(1);
}
Vector derivative(degree);
for (int i = 0; i < degree; ++i) {
derivative(i) = (degree - i) * polynomial(i);
}
return derivative;
}
void MinimizePolynomial(const Vector& polynomial,
const double x_min,
const double x_max,
double* optimal_x,
double* optimal_value) {
// Find the minimum of the polynomial at the two ends.
//
// We start by inspecting the middle of the interval. Technically
// this is not needed, but we do this to make this code as close to
// the minFunc package as possible.
*optimal_x = (x_min + x_max) / 2.0;
*optimal_value = EvaluatePolynomial(polynomial, *optimal_x);
const double x_min_value = EvaluatePolynomial(polynomial, x_min);
if (x_min_value < *optimal_value) {
*optimal_value = x_min_value;
*optimal_x = x_min;
}
const double x_max_value = EvaluatePolynomial(polynomial, x_max);
if (x_max_value < *optimal_value) {
*optimal_value = x_max_value;
*optimal_x = x_max;
}
// If the polynomial is linear or constant, we are done.
if (polynomial.rows() <= 2) {
return;
}
const Vector derivative = DifferentiatePolynomial(polynomial);
Vector roots_real;
if (!FindPolynomialRoots(derivative, &roots_real, NULL)) {
LOG(WARNING) << "Unable to find the critical points of "
<< "the interpolating polynomial.";
return;
}
// This is a bit of an overkill, as some of the roots may actually
// have a complex part, but its simpler to just check these values.
for (int i = 0; i < roots_real.rows(); ++i) {
const double root = roots_real(i);
if ((root < x_min) || (root > x_max)) {
continue;
}
const double value = EvaluatePolynomial(polynomial, root);
if (value < *optimal_value) {
*optimal_value = value;
*optimal_x = root;
}
}
}
Vector FindInterpolatingPolynomial(const vector<FunctionSample>& samples) {
const int num_samples = samples.size();
int num_constraints = 0;
for (int i = 0; i < num_samples; ++i) {
if (samples[i].value_is_valid) {
++num_constraints;
}
if (samples[i].gradient_is_valid) {
++num_constraints;
}
}
const int degree = num_constraints - 1;
Matrix lhs = Matrix::Zero(num_constraints, num_constraints);
Vector rhs = Vector::Zero(num_constraints);
int row = 0;
for (int i = 0; i < num_samples; ++i) {
const FunctionSample& sample = samples[i];
if (sample.value_is_valid) {
for (int j = 0; j <= degree; ++j) {
lhs(row, j) = pow(sample.x, degree - j);
}
rhs(row) = sample.value;
++row;
}
if (sample.gradient_is_valid) {
for (int j = 0; j < degree; ++j) {
lhs(row, j) = (degree - j) * pow(sample.x, degree - j - 1);
}
rhs(row) = sample.gradient;
++row;
}
}
// TODO(sameeragarwal): This is a hack.
// https://github.com/ceres-solver/ceres-solver/issues/248
Eigen::FullPivLU<Matrix> lu(lhs);
return lu.setThreshold(0.0).solve(rhs);
}
void MinimizeInterpolatingPolynomial(const vector<FunctionSample>& samples,
double x_min,
double x_max,
double* optimal_x,
double* optimal_value) {
const Vector polynomial = FindInterpolatingPolynomial(samples);
MinimizePolynomial(polynomial, x_min, x_max, optimal_x, optimal_value);
for (int i = 0; i < samples.size(); ++i) {
const FunctionSample& sample = samples[i];
if ((sample.x < x_min) || (sample.x > x_max)) {
continue;
}
const double value = EvaluatePolynomial(polynomial, sample.x);
if (value < *optimal_value) {
*optimal_x = sample.x;
*optimal_value = value;
}
}
}
} // namespace internal
} // namespace ceres