Add polynomial interpolation and minimization.
1. polynomial_solver* -> polynomial*.
2. Added support for differentiating polynomials.
2. Added support for interpolating polynomials from function
values and gradients.
3. Added support for minimizing polynomials by solving
for the roots of their derivatives in an interval.
4. Added support for finding the minimum of a polynomial
that interpolates function values and gradients in
an interval.
Change-Id: Id7e6764ad4db09c3edd60f1378c7f50f20dd08dc
diff --git a/internal/ceres/polynomial.cc b/internal/ceres/polynomial.cc
new file mode 100644
index 0000000..3b88471
--- /dev/null
+++ b/internal/ceres/polynomial.cc
@@ -0,0 +1,313 @@
+// 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)
+// sameeragarwal@google.com (Sameer Agarwal)
+
+#include "ceres/polynomial.h"
+
+#include <cmath>
+#include <cstddef>
+#include <vector>
+
+#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;
+}
+
+Vector DifferentiatePolynomial(const Vector& polynomial) {
+ const int degree = polynomial.rows() - 1;
+ CHECK_GE(degree, 0);
+ 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) {
+ LOG(INFO) << "value constraint";
+ 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) {
+ LOG(INFO) << "gradient constraint";
+ lhs(row, j) = (degree - j) * pow(sample.x, degree - j - 1);
+ }
+ rhs(row) = sample.gradient;
+ ++row;
+ }
+ }
+
+ return lhs.fullPivLu().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
