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// Ceres Solver - A fast non-linear least squares minimizer
// Copyright 2012 Google Inc. All rights reserved.
// http://code.google.com/p/ceres-solver/
//
// Redistribution and use in source and binary forms, with or without
// modification, are permitted provided that the following conditions are met:
//
// * Redistributions of source code must retain the above copyright notice,
// this list of conditions and the following disclaimer.
// * Redistributions in binary form must reproduce the above copyright notice,
// this list of conditions and the following disclaimer in the documentation
// and/or other materials provided with the distribution.
// * Neither the name of Google Inc. nor the names of its contributors may be
// used to endorse or promote products derived from this software without
// specific prior written permission.
//
// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
// AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
// IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
// ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
// LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
// CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
// SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
// INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
// CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
// ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
// POSSIBILITY OF SUCH DAMAGE.
//
// Author: moll.markus@arcor.de (Markus Moll)
#include "ceres/polynomial_solver.h"
#include <cmath>
#include <cstddef>
#include "Eigen/Dense"
#include "ceres/internal/port.h"
#include "glog/logging.h"
namespace ceres {
namespace internal {
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);
}
} // 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;
// Is the polynomial constant?
if (degree == 0) {
LOG(WARNING) << "Trying to extract roots from a constant "
<< "polynomial in FindPolynomialRoots";
return true;
}
// Divide by leading term
const double leading_term = polynomial(0);
polynomial /= leading_term;
// Separately handle linear polynomials.
if (degree == 1) {
if (real != NULL) {
real->resize(1);
(*real)(0) = -polynomial(1);
}
if (imaginary != NULL) {
imaginary->resize(1);
imaginary->setZero();
}
}
// The degree is now known to be at least 2.
// 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;
}
} // namespace internal
} // namespace ceres
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