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/CMakeLists.txt b/internal/ceres/CMakeLists.txt
index adad6dc..0960b95 100644
--- a/internal/ceres/CMakeLists.txt
+++ b/internal/ceres/CMakeLists.txt
@@ -66,7 +66,7 @@
normal_prior.cc
parameter_block_ordering.cc
partitioned_matrix_view.cc
- polynomial_solver.cc
+ polynomial.cc
problem.cc
problem_impl.cc
program.cc
@@ -238,7 +238,7 @@
CERES_TEST(parameter_block)
CERES_TEST(parameter_block_ordering)
CERES_TEST(partitioned_matrix_view)
- CERES_TEST(polynomial_solver)
+ CERES_TEST(polynomial)
CERES_TEST(problem)
CERES_TEST(residual_block)
CERES_TEST(residual_block_utils)
diff --git a/internal/ceres/dogleg_strategy.cc b/internal/ceres/dogleg_strategy.cc
index 668fa54..da861fe 100644
--- a/internal/ceres/dogleg_strategy.cc
+++ b/internal/ceres/dogleg_strategy.cc
@@ -35,7 +35,7 @@
#include "ceres/array_utils.h"
#include "ceres/internal/eigen.h"
#include "ceres/linear_solver.h"
-#include "ceres/polynomial_solver.h"
+#include "ceres/polynomial.h"
#include "ceres/sparse_matrix.h"
#include "ceres/trust_region_strategy.h"
#include "ceres/types.h"
diff --git a/internal/ceres/polynomial_solver.cc b/internal/ceres/polynomial.cc
similarity index 62%
rename from internal/ceres/polynomial_solver.cc
rename to internal/ceres/polynomial.cc
index 0ece7bc..3b88471 100644
--- a/internal/ceres/polynomial_solver.cc
+++ b/internal/ceres/polynomial.cc
@@ -27,11 +27,14 @@
// POSSIBILITY OF SUCH DAMAGE.
//
// Author: moll.markus@arcor.de (Markus Moll)
+// sameeragarwal@google.com (Sameer Agarwal)
-#include "ceres/polynomial_solver.h"
+#include "ceres/polynomial.h"
#include <cmath>
#include <cstddef>
+#include <vector>
+
#include "Eigen/Dense"
#include "ceres/internal/port.h"
#include "glog/logging.h"
@@ -179,5 +182,132 @@
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
diff --git a/internal/ceres/polynomial.h b/internal/ceres/polynomial.h
new file mode 100644
index 0000000..42ffdcb
--- /dev/null
+++ b/internal/ceres/polynomial.h
@@ -0,0 +1,134 @@
+// 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)
+
+#ifndef CERES_INTERNAL_POLYNOMIAL_SOLVER_H_
+#define CERES_INTERNAL_POLYNOMIAL_SOLVER_H_
+
+#include <vector>
+#include "ceres/internal/eigen.h"
+#include "ceres/internal/port.h"
+
+namespace ceres {
+namespace internal {
+
+// All polynomials are assumed to be the form
+//
+// sum_{i=0}^N polynomial(i) x^{N-i}.
+//
+// and are given by a vector of coefficients of size N + 1.
+
+// Evaluate the polynomial at x using the Horner scheme.
+inline double EvaluatePolynomial(const Vector& polynomial, double x) {
+ double v = 0.0;
+ for (int i = 0; i < polynomial.size(); ++i) {
+ v = v * x + polynomial(i);
+ }
+ return v;
+}
+
+// Use the companion matrix eigenvalues to determine the roots of the
+// polynomial.
+//
+// This function returns true on success, false otherwise.
+// Failure indicates that the polynomial is invalid (of size 0) or
+// that the eigenvalues of the companion matrix could not be computed.
+// On failure, a more detailed message will be written to LOG(ERROR).
+// If real is not NULL, the real parts of the roots will be returned in it.
+// Likewise, if imaginary is not NULL, imaginary parts will be returned in it.
+bool FindPolynomialRoots(const Vector& polynomial,
+ Vector* real,
+ Vector* imaginary);
+
+// Return the derivative of the given polynomial. It is assumed that
+// the input polynomial is at least of degree zero.
+Vector DifferentiatePolynomial(const Vector& polynomial);
+
+// Find the minimum value of the polynomial in the interval [x_min,
+// x_max]. The minimum is obtained by computing all the roots of the
+// derivative of the input polynomial. All real roots within the
+// interval [x_min, x_max] are considered as well as the end points
+// x_min and x_max. Since polynomials are differentiable functions,
+// this ensures that the true minimum is found.
+void MinimizePolynomial(const Vector& polynomial,
+ double x_min,
+ double x_max,
+ double* optimal_x,
+ double* optimal_value);
+
+// Structure for storing sample values of a function.
+//
+// Clients can use this struct to communicate the value of the
+// function and or its gradient at a given point x.
+struct FunctionSample {
+ FunctionSample()
+ : x(0.0),
+ value(0.0),
+ value_is_valid(false),
+ gradient(0.0),
+ gradient_is_valid(false) {
+ }
+
+ double x;
+ double value; // value = f(x)
+ bool value_is_valid;
+ double gradient; // gradient = f'(x)
+ bool gradient_is_valid;
+};
+
+// Given a set of function value and/or gradient samples, find a
+// polynomial whose value and gradients are exactly equal to the ones
+// in samples.
+//
+// Generally speaking,
+//
+// degree = # values + # gradients - 1
+//
+// Of course its possible to sample a polynomial any number of times,
+// in which case, generally speaking the spurious higher order
+// coefficients will be zero.
+Vector FindInterpolatingPolynomial(const vector<FunctionSample>& samples);
+
+// Interpolate the function described by samples with a polynomial,
+// and minimize it on the interval [x_min, x_max]. Depending on the
+// input samples, it is possible that the interpolation or the root
+// finding algorithms may fail due to numerical difficulties. But the
+// function is guaranteed to return its best guess of an answer, by
+// considering the samples and the end points as possible solutions.
+void MinimizeInterpolatingPolynomial(const vector<FunctionSample>& samples,
+ double x_min,
+ double x_max,
+ double* optimal_x,
+ double* optimal_value);
+
+} // namespace internal
+} // namespace ceres
+
+#endif // CERES_INTERNAL_POLYNOMIAL_SOLVER_H_
diff --git a/internal/ceres/polynomial_solver.h b/internal/ceres/polynomial_solver.h
deleted file mode 100644
index 1cf07dd..0000000
--- a/internal/ceres/polynomial_solver.h
+++ /dev/null
@@ -1,65 +0,0 @@
-// 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)
-
-#ifndef CERES_INTERNAL_POLYNOMIAL_SOLVER_H_
-#define CERES_INTERNAL_POLYNOMIAL_SOLVER_H_
-
-#include "ceres/internal/eigen.h"
-
-namespace ceres {
-namespace internal {
-
-// Use the companion matrix eigenvalues to determine the roots of the polynomial
-//
-// sum_{i=0}^N polynomial(i) x^{N-i}.
-//
-// This function returns true on success, false otherwise.
-// Failure indicates that the polynomial is invalid (of size 0) or
-// that the eigenvalues of the companion matrix could not be computed.
-// On failure, a more detailed message will be written to LOG(ERROR).
-// If real is not NULL, the real parts of the roots will be returned in it.
-// Likewise, if imaginary is not NULL, imaginary parts will be returned in it.
-bool FindPolynomialRoots(const Vector& polynomial,
- Vector* real,
- Vector* imaginary);
-
-// Evaluate the polynomial at x using the Horner scheme.
-inline double EvaluatePolynomial(const Vector& polynomial, double x) {
- double v = 0.0;
- for (int i = 0; i < polynomial.size(); ++i) {
- v = v * x + polynomial(i);
- }
- return v;
-}
-
-} // namespace internal
-} // namespace ceres
-
-#endif // CERES_INTERNAL_POLYNOMIAL_SOLVER_H_
diff --git a/internal/ceres/polynomial_solver_test.cc b/internal/ceres/polynomial_solver_test.cc
deleted file mode 100644
index ae347fb..0000000
--- a/internal/ceres/polynomial_solver_test.cc
+++ /dev/null
@@ -1,223 +0,0 @@
-// 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 <limits>
-#include <cmath>
-#include <cstddef>
-#include <algorithm>
-#include "gtest/gtest.h"
-#include "ceres/test_util.h"
-
-namespace ceres {
-namespace internal {
-namespace {
-
-// For IEEE-754 doubles, machine precision is about 2e-16.
-const double kEpsilon = 1e-13;
-const double kEpsilonLoose = 1e-9;
-
-// Return the constant polynomial p(x) = 1.23.
-Vector ConstantPolynomial(double value) {
- Vector poly(1);
- poly(0) = value;
- return poly;
-}
-
-// Return the polynomial p(x) = poly(x) * (x - root).
-Vector AddRealRoot(const Vector& poly, double root) {
- Vector poly2(poly.size() + 1);
- poly2.setZero();
- poly2.head(poly.size()) += poly;
- poly2.tail(poly.size()) -= root * poly;
- return poly2;
-}
-
-// Return the polynomial
-// p(x) = poly(x) * (x - real - imag*i) * (x - real + imag*i).
-Vector AddComplexRootPair(const Vector& poly, double real, double imag) {
- Vector poly2(poly.size() + 2);
- poly2.setZero();
- // Multiply poly by x^2 - 2real + abs(real,imag)^2
- poly2.head(poly.size()) += poly;
- poly2.segment(1, poly.size()) -= 2 * real * poly;
- poly2.tail(poly.size()) += (real*real + imag*imag) * poly;
- return poly2;
-}
-
-// Sort the entries in a vector.
-// Needed because the roots are not returned in sorted order.
-Vector SortVector(const Vector& in) {
- Vector out(in);
- std::sort(out.data(), out.data() + out.size());
- return out;
-}
-
-// Run a test with the polynomial defined by the N real roots in roots_real.
-// If use_real is false, NULL is passed as the real argument to
-// FindPolynomialRoots. If use_imaginary is false, NULL is passed as the
-// imaginary argument to FindPolynomialRoots.
-template<int N>
-void RunPolynomialTestRealRoots(const double (&real_roots)[N],
- bool use_real,
- bool use_imaginary,
- double epsilon) {
- Vector real;
- Vector imaginary;
- Vector poly = ConstantPolynomial(1.23);
- for (int i = 0; i < N; ++i) {
- poly = AddRealRoot(poly, real_roots[i]);
- }
- Vector* const real_ptr = use_real ? &real : NULL;
- Vector* const imaginary_ptr = use_imaginary ? &imaginary : NULL;
- bool success = FindPolynomialRoots(poly, real_ptr, imaginary_ptr);
-
- EXPECT_EQ(success, true);
- if (use_real) {
- EXPECT_EQ(real.size(), N);
- real = SortVector(real);
- ExpectArraysClose(N, real.data(), real_roots, epsilon);
- }
- if (use_imaginary) {
- EXPECT_EQ(imaginary.size(), N);
- const Vector zeros = Vector::Zero(N);
- ExpectArraysClose(N, imaginary.data(), zeros.data(), epsilon);
- }
-}
-} // namespace
-
-TEST(PolynomialSolver, InvalidPolynomialOfZeroLengthIsRejected) {
- // Vector poly(0) is an ambiguous constructor call, so
- // use the constructor with explicit column count.
- Vector poly(0, 1);
- Vector real;
- Vector imag;
- bool success = FindPolynomialRoots(poly, &real, &imag);
-
- EXPECT_EQ(success, false);
-}
-
-TEST(PolynomialSolver, ConstantPolynomialReturnsNoRoots) {
- Vector poly = ConstantPolynomial(1.23);
- Vector real;
- Vector imag;
- bool success = FindPolynomialRoots(poly, &real, &imag);
-
- EXPECT_EQ(success, true);
- EXPECT_EQ(real.size(), 0);
- EXPECT_EQ(imag.size(), 0);
-}
-
-TEST(PolynomialSolver, LinearPolynomialWithPositiveRootWorks) {
- const double roots[1] = { 42.42 };
- RunPolynomialTestRealRoots(roots, true, true, kEpsilon);
-}
-
-TEST(PolynomialSolver, LinearPolynomialWithNegativeRootWorks) {
- const double roots[1] = { -42.42 };
- RunPolynomialTestRealRoots(roots, true, true, kEpsilon);
-}
-
-TEST(PolynomialSolver, QuadraticPolynomialWithPositiveRootsWorks) {
- const double roots[2] = { 1.0, 42.42 };
- RunPolynomialTestRealRoots(roots, true, true, kEpsilon);
-}
-
-TEST(PolynomialSolver, QuadraticPolynomialWithOneNegativeRootWorks) {
- const double roots[2] = { -42.42, 1.0 };
- RunPolynomialTestRealRoots(roots, true, true, kEpsilon);
-}
-
-TEST(PolynomialSolver, QuadraticPolynomialWithTwoNegativeRootsWorks) {
- const double roots[2] = { -42.42, -1.0 };
- RunPolynomialTestRealRoots(roots, true, true, kEpsilon);
-}
-
-TEST(PolynomialSolver, QuadraticPolynomialWithCloseRootsWorks) {
- const double roots[2] = { 42.42, 42.43 };
- RunPolynomialTestRealRoots(roots, true, false, kEpsilonLoose);
-}
-
-TEST(PolynomialSolver, QuadraticPolynomialWithComplexRootsWorks) {
- Vector real;
- Vector imag;
-
- Vector poly = ConstantPolynomial(1.23);
- poly = AddComplexRootPair(poly, 42.42, 4.2);
- bool success = FindPolynomialRoots(poly, &real, &imag);
-
- EXPECT_EQ(success, true);
- EXPECT_EQ(real.size(), 2);
- EXPECT_EQ(imag.size(), 2);
- ExpectClose(real(0), 42.42, kEpsilon);
- ExpectClose(real(1), 42.42, kEpsilon);
- ExpectClose(std::abs(imag(0)), 4.2, kEpsilon);
- ExpectClose(std::abs(imag(1)), 4.2, kEpsilon);
- ExpectClose(std::abs(imag(0) + imag(1)), 0.0, kEpsilon);
-}
-
-TEST(PolynomialSolver, QuarticPolynomialWorks) {
- const double roots[4] = { 1.23e-4, 1.23e-1, 1.23e+2, 1.23e+5 };
- RunPolynomialTestRealRoots(roots, true, true, kEpsilon);
-}
-
-TEST(PolynomialSolver, QuarticPolynomialWithTwoClustersOfCloseRootsWorks) {
- const double roots[4] = { 1.23e-1, 2.46e-1, 1.23e+5, 2.46e+5 };
- RunPolynomialTestRealRoots(roots, true, true, kEpsilonLoose);
-}
-
-TEST(PolynomialSolver, QuarticPolynomialWithTwoZeroRootsWorks) {
- const double roots[4] = { -42.42, 0.0, 0.0, 42.42 };
- RunPolynomialTestRealRoots(roots, true, true, kEpsilonLoose);
-}
-
-TEST(PolynomialSolver, QuarticMonomialWorks) {
- const double roots[4] = { 0.0, 0.0, 0.0, 0.0 };
- RunPolynomialTestRealRoots(roots, true, true, kEpsilon);
-}
-
-TEST(PolynomialSolver, NullPointerAsImaginaryPartWorks) {
- const double roots[4] = { 1.23e-4, 1.23e-1, 1.23e+2, 1.23e+5 };
- RunPolynomialTestRealRoots(roots, true, false, kEpsilon);
-}
-
-TEST(PolynomialSolver, NullPointerAsRealPartWorks) {
- const double roots[4] = { 1.23e-4, 1.23e-1, 1.23e+2, 1.23e+5 };
- RunPolynomialTestRealRoots(roots, false, true, kEpsilon);
-}
-
-TEST(PolynomialSolver, BothOutputArgumentsNullWorks) {
- const double roots[4] = { 1.23e-4, 1.23e-1, 1.23e+2, 1.23e+5 };
- RunPolynomialTestRealRoots(roots, false, false, kEpsilon);
-}
-
-} // namespace internal
-} // namespace ceres
diff --git a/internal/ceres/polynomial_test.cc b/internal/ceres/polynomial_test.cc
new file mode 100644
index 0000000..f56352f
--- /dev/null
+++ b/internal/ceres/polynomial_test.cc
@@ -0,0 +1,512 @@
+// 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 <limits>
+#include <cmath>
+#include <cstddef>
+#include <algorithm>
+#include "gtest/gtest.h"
+#include "ceres/test_util.h"
+
+namespace ceres {
+namespace internal {
+namespace {
+
+// For IEEE-754 doubles, machine precision is about 2e-16.
+const double kEpsilon = 1e-13;
+const double kEpsilonLoose = 1e-9;
+
+// Return the constant polynomial p(x) = 1.23.
+Vector ConstantPolynomial(double value) {
+ Vector poly(1);
+ poly(0) = value;
+ return poly;
+}
+
+// Return the polynomial p(x) = poly(x) * (x - root).
+Vector AddRealRoot(const Vector& poly, double root) {
+ Vector poly2(poly.size() + 1);
+ poly2.setZero();
+ poly2.head(poly.size()) += poly;
+ poly2.tail(poly.size()) -= root * poly;
+ return poly2;
+}
+
+// Return the polynomial
+// p(x) = poly(x) * (x - real - imag*i) * (x - real + imag*i).
+Vector AddComplexRootPair(const Vector& poly, double real, double imag) {
+ Vector poly2(poly.size() + 2);
+ poly2.setZero();
+ // Multiply poly by x^2 - 2real + abs(real,imag)^2
+ poly2.head(poly.size()) += poly;
+ poly2.segment(1, poly.size()) -= 2 * real * poly;
+ poly2.tail(poly.size()) += (real*real + imag*imag) * poly;
+ return poly2;
+}
+
+// Sort the entries in a vector.
+// Needed because the roots are not returned in sorted order.
+Vector SortVector(const Vector& in) {
+ Vector out(in);
+ std::sort(out.data(), out.data() + out.size());
+ return out;
+}
+
+// Run a test with the polynomial defined by the N real roots in roots_real.
+// If use_real is false, NULL is passed as the real argument to
+// FindPolynomialRoots. If use_imaginary is false, NULL is passed as the
+// imaginary argument to FindPolynomialRoots.
+template<int N>
+void RunPolynomialTestRealRoots(const double (&real_roots)[N],
+ bool use_real,
+ bool use_imaginary,
+ double epsilon) {
+ Vector real;
+ Vector imaginary;
+ Vector poly = ConstantPolynomial(1.23);
+ for (int i = 0; i < N; ++i) {
+ poly = AddRealRoot(poly, real_roots[i]);
+ }
+ Vector* const real_ptr = use_real ? &real : NULL;
+ Vector* const imaginary_ptr = use_imaginary ? &imaginary : NULL;
+ bool success = FindPolynomialRoots(poly, real_ptr, imaginary_ptr);
+
+ EXPECT_EQ(success, true);
+ if (use_real) {
+ EXPECT_EQ(real.size(), N);
+ real = SortVector(real);
+ ExpectArraysClose(N, real.data(), real_roots, epsilon);
+ }
+ if (use_imaginary) {
+ EXPECT_EQ(imaginary.size(), N);
+ const Vector zeros = Vector::Zero(N);
+ ExpectArraysClose(N, imaginary.data(), zeros.data(), epsilon);
+ }
+}
+} // namespace
+
+TEST(Polynomial, InvalidPolynomialOfZeroLengthIsRejected) {
+ // Vector poly(0) is an ambiguous constructor call, so
+ // use the constructor with explicit column count.
+ Vector poly(0, 1);
+ Vector real;
+ Vector imag;
+ bool success = FindPolynomialRoots(poly, &real, &imag);
+
+ EXPECT_EQ(success, false);
+}
+
+TEST(Polynomial, ConstantPolynomialReturnsNoRoots) {
+ Vector poly = ConstantPolynomial(1.23);
+ Vector real;
+ Vector imag;
+ bool success = FindPolynomialRoots(poly, &real, &imag);
+
+ EXPECT_EQ(success, true);
+ EXPECT_EQ(real.size(), 0);
+ EXPECT_EQ(imag.size(), 0);
+}
+
+TEST(Polynomial, LinearPolynomialWithPositiveRootWorks) {
+ const double roots[1] = { 42.42 };
+ RunPolynomialTestRealRoots(roots, true, true, kEpsilon);
+}
+
+TEST(Polynomial, LinearPolynomialWithNegativeRootWorks) {
+ const double roots[1] = { -42.42 };
+ RunPolynomialTestRealRoots(roots, true, true, kEpsilon);
+}
+
+TEST(Polynomial, QuadraticPolynomialWithPositiveRootsWorks) {
+ const double roots[2] = { 1.0, 42.42 };
+ RunPolynomialTestRealRoots(roots, true, true, kEpsilon);
+}
+
+TEST(Polynomial, QuadraticPolynomialWithOneNegativeRootWorks) {
+ const double roots[2] = { -42.42, 1.0 };
+ RunPolynomialTestRealRoots(roots, true, true, kEpsilon);
+}
+
+TEST(Polynomial, QuadraticPolynomialWithTwoNegativeRootsWorks) {
+ const double roots[2] = { -42.42, -1.0 };
+ RunPolynomialTestRealRoots(roots, true, true, kEpsilon);
+}
+
+TEST(Polynomial, QuadraticPolynomialWithCloseRootsWorks) {
+ const double roots[2] = { 42.42, 42.43 };
+ RunPolynomialTestRealRoots(roots, true, false, kEpsilonLoose);
+}
+
+TEST(Polynomial, QuadraticPolynomialWithComplexRootsWorks) {
+ Vector real;
+ Vector imag;
+
+ Vector poly = ConstantPolynomial(1.23);
+ poly = AddComplexRootPair(poly, 42.42, 4.2);
+ bool success = FindPolynomialRoots(poly, &real, &imag);
+
+ EXPECT_EQ(success, true);
+ EXPECT_EQ(real.size(), 2);
+ EXPECT_EQ(imag.size(), 2);
+ ExpectClose(real(0), 42.42, kEpsilon);
+ ExpectClose(real(1), 42.42, kEpsilon);
+ ExpectClose(std::abs(imag(0)), 4.2, kEpsilon);
+ ExpectClose(std::abs(imag(1)), 4.2, kEpsilon);
+ ExpectClose(std::abs(imag(0) + imag(1)), 0.0, kEpsilon);
+}
+
+TEST(Polynomial, QuarticPolynomialWorks) {
+ const double roots[4] = { 1.23e-4, 1.23e-1, 1.23e+2, 1.23e+5 };
+ RunPolynomialTestRealRoots(roots, true, true, kEpsilon);
+}
+
+TEST(Polynomial, QuarticPolynomialWithTwoClustersOfCloseRootsWorks) {
+ const double roots[4] = { 1.23e-1, 2.46e-1, 1.23e+5, 2.46e+5 };
+ RunPolynomialTestRealRoots(roots, true, true, kEpsilonLoose);
+}
+
+TEST(Polynomial, QuarticPolynomialWithTwoZeroRootsWorks) {
+ const double roots[4] = { -42.42, 0.0, 0.0, 42.42 };
+ RunPolynomialTestRealRoots(roots, true, true, kEpsilonLoose);
+}
+
+TEST(Polynomial, QuarticMonomialWorks) {
+ const double roots[4] = { 0.0, 0.0, 0.0, 0.0 };
+ RunPolynomialTestRealRoots(roots, true, true, kEpsilon);
+}
+
+TEST(Polynomial, NullPointerAsImaginaryPartWorks) {
+ const double roots[4] = { 1.23e-4, 1.23e-1, 1.23e+2, 1.23e+5 };
+ RunPolynomialTestRealRoots(roots, true, false, kEpsilon);
+}
+
+TEST(Polynomial, NullPointerAsRealPartWorks) {
+ const double roots[4] = { 1.23e-4, 1.23e-1, 1.23e+2, 1.23e+5 };
+ RunPolynomialTestRealRoots(roots, false, true, kEpsilon);
+}
+
+TEST(Polynomial, BothOutputArgumentsNullWorks) {
+ const double roots[4] = { 1.23e-4, 1.23e-1, 1.23e+2, 1.23e+5 };
+ RunPolynomialTestRealRoots(roots, false, false, kEpsilon);
+}
+
+TEST(Polynomial, DifferentiateConstantPolynomial) {
+ // p(x) = 1;
+ Vector polynomial(1);
+ polynomial(0) = 1.0;
+ const Vector derivative = DifferentiatePolynomial(polynomial);
+ EXPECT_EQ(derivative.rows(), 0);
+}
+
+TEST(Polynomial, DifferentiateQuadraticPolynomial) {
+ // p(x) = x^2 + 2x + 3;
+ Vector polynomial(3);
+ polynomial(0) = 1.0;
+ polynomial(1) = 2.0;
+ polynomial(2) = 3.0;
+
+ const Vector derivative = DifferentiatePolynomial(polynomial);
+ EXPECT_EQ(derivative.rows(), 2);
+ EXPECT_EQ(derivative(0), 2.0);
+ EXPECT_EQ(derivative(1), 2.0);
+}
+
+TEST(Polynomial, MinimizeConstantPolynomial) {
+ // p(x) = 1;
+ Vector polynomial(1);
+ polynomial(0) = 1.0;
+
+ double optimal_x = 0.0;
+ double optimal_value = 0.0;
+ double min_x = 0.0;
+ double max_x = 1.0;
+ MinimizePolynomial(polynomial, min_x, max_x, &optimal_x, &optimal_value);
+
+ EXPECT_EQ(optimal_value, 1.0);
+ EXPECT_LE(optimal_x, max_x);
+ EXPECT_GE(optimal_x, min_x);
+}
+
+TEST(Polynomial, MinimizeLinearPolynomial) {
+ // p(x) = x - 2
+ Vector polynomial(2);
+
+ polynomial(0) = 1.0;
+ polynomial(1) = 2.0;
+
+ double optimal_x = 0.0;
+ double optimal_value = 0.0;
+ double min_x = 0.0;
+ double max_x = 1.0;
+ MinimizePolynomial(polynomial, min_x, max_x, &optimal_x, &optimal_value);
+
+ EXPECT_EQ(optimal_x, 0.0);
+ EXPECT_EQ(optimal_value, 2.0);
+}
+
+
+TEST(Polynomial, MinimizeQuadraticPolynomial) {
+ // p(x) = x^2 - 3 x + 2
+ // min_x = 3/2
+ // min_value = -1/4;
+ Vector polynomial(3);
+ polynomial(0) = 1.0;
+ polynomial(1) = -3.0;
+ polynomial(2) = 2.0;
+
+ double optimal_x = 0.0;
+ double optimal_value = 0.0;
+ double min_x = -2.0;
+ double max_x = 2.0;
+ MinimizePolynomial(polynomial, min_x, max_x, &optimal_x, &optimal_value);
+ EXPECT_EQ(optimal_x, 3.0/2.0);
+ EXPECT_EQ(optimal_value, -1.0/4.0);
+
+ min_x = -2.0;
+ max_x = 1.0;
+ MinimizePolynomial(polynomial, min_x, max_x, &optimal_x, &optimal_value);
+ EXPECT_EQ(optimal_x, 1.0);
+ EXPECT_EQ(optimal_value, 0.0);
+
+ min_x = 2.0;
+ max_x = 3.0;
+ MinimizePolynomial(polynomial, min_x, max_x, &optimal_x, &optimal_value);
+ EXPECT_EQ(optimal_x, 2.0);
+ EXPECT_EQ(optimal_value, 0.0);
+}
+
+TEST(Polymomial, ConstantInterpolatingPolynomial) {
+ // p(x) = 1.0
+ Vector true_polynomial(1);
+ true_polynomial << 1.0;
+
+ vector<FunctionSample> samples;
+ FunctionSample sample;
+ sample.x = 1.0;
+ sample.value = 1.0;
+ sample.value_is_valid = true;
+ samples.push_back(sample);
+
+ const Vector polynomial = FindInterpolatingPolynomial(samples);
+ EXPECT_NEAR((true_polynomial - polynomial).norm(), 0.0, 1e-15);
+}
+
+TEST(Polynomial, LinearInterpolatingPolynomial) {
+ // p(x) = 2x - 1
+ Vector true_polynomial(2);
+ true_polynomial << 2.0, -1.0;
+
+ vector<FunctionSample> samples;
+ FunctionSample sample;
+ sample.x = 1.0;
+ sample.value = 1.0;
+ sample.value_is_valid = true;
+ sample.gradient = 2.0;
+ sample.gradient_is_valid = true;
+ samples.push_back(sample);
+
+ const Vector polynomial = FindInterpolatingPolynomial(samples);
+ EXPECT_NEAR((true_polynomial - polynomial).norm(), 0.0, 1e-15);
+}
+
+TEST(Polynomial, QuadraticInterpolatingPolynomial) {
+ // p(x) = 2x^2 + 3x + 2
+ Vector true_polynomial(3);
+ true_polynomial << 2.0, 3.0, 2.0;
+
+ vector<FunctionSample> samples;
+ {
+ FunctionSample sample;
+ sample.x = 1.0;
+ sample.value = 7.0;
+ sample.value_is_valid = true;
+ sample.gradient = 7.0;
+ sample.gradient_is_valid = true;
+ samples.push_back(sample);
+ }
+
+ {
+ FunctionSample sample;
+ sample.x = -3.0;
+ sample.value = 11.0;
+ sample.value_is_valid = true;
+ samples.push_back(sample);
+ }
+
+ Vector polynomial = FindInterpolatingPolynomial(samples);
+ EXPECT_NEAR((true_polynomial - polynomial).norm(), 0.0, 1e-15);
+}
+
+TEST(Polynomial, DeficientCubicInterpolatingPolynomial) {
+ // p(x) = 2x^2 + 3x + 2
+ Vector true_polynomial(4);
+ true_polynomial << 0.0, 2.0, 3.0, 2.0;
+
+ vector<FunctionSample> samples;
+ {
+ FunctionSample sample;
+ sample.x = 1.0;
+ sample.value = 7.0;
+ sample.value_is_valid = true;
+ sample.gradient = 7.0;
+ sample.gradient_is_valid = true;
+ samples.push_back(sample);
+ }
+
+ {
+ FunctionSample sample;
+ sample.x = -3.0;
+ sample.value = 11.0;
+ sample.value_is_valid = true;
+ sample.gradient = -9;
+ sample.gradient_is_valid = true;
+ samples.push_back(sample);
+ }
+
+ const Vector polynomial = FindInterpolatingPolynomial(samples);
+ EXPECT_NEAR((true_polynomial - polynomial).norm(), 0.0, 1e-14);
+}
+
+
+TEST(Polynomial, CubicInterpolatingPolynomialFromValues) {
+ // p(x) = x^3 + 2x^2 + 3x + 2
+ Vector true_polynomial(4);
+ true_polynomial << 1.0, 2.0, 3.0, 2.0;
+
+ vector<FunctionSample> samples;
+ {
+ FunctionSample sample;
+ sample.x = 1.0;
+ sample.value = EvaluatePolynomial(true_polynomial, sample.x);
+ sample.value_is_valid = true;
+ samples.push_back(sample);
+ }
+
+ {
+ FunctionSample sample;
+ sample.x = -3.0;
+ sample.value = EvaluatePolynomial(true_polynomial, sample.x);
+ sample.value_is_valid = true;
+ samples.push_back(sample);
+ }
+
+ {
+ FunctionSample sample;
+ sample.x = 2.0;
+ sample.value = EvaluatePolynomial(true_polynomial, sample.x);
+ sample.value_is_valid = true;
+ samples.push_back(sample);
+ }
+
+ {
+ FunctionSample sample;
+ sample.x = 0.0;
+ sample.value = EvaluatePolynomial(true_polynomial, sample.x);
+ sample.value_is_valid = true;
+ samples.push_back(sample);
+ }
+
+ const Vector polynomial = FindInterpolatingPolynomial(samples);
+ EXPECT_NEAR((true_polynomial - polynomial).norm(), 0.0, 1e-14);
+}
+
+TEST(Polynomial, CubicInterpolatingPolynomialFromValuesAndOneGradient) {
+ // p(x) = x^3 + 2x^2 + 3x + 2
+ Vector true_polynomial(4);
+ true_polynomial << 1.0, 2.0, 3.0, 2.0;
+ Vector true_gradient_polynomial = DifferentiatePolynomial(true_polynomial);
+
+ vector<FunctionSample> samples;
+ {
+ FunctionSample sample;
+ sample.x = 1.0;
+ sample.value = EvaluatePolynomial(true_polynomial, sample.x);
+ sample.value_is_valid = true;
+ samples.push_back(sample);
+ }
+
+ {
+ FunctionSample sample;
+ sample.x = -3.0;
+ sample.value = EvaluatePolynomial(true_polynomial, sample.x);
+ sample.value_is_valid = true;
+ samples.push_back(sample);
+ }
+
+ {
+ FunctionSample sample;
+ sample.x = 2.0;
+ sample.value = EvaluatePolynomial(true_polynomial, sample.x);
+ sample.value_is_valid = true;
+ sample.gradient = EvaluatePolynomial(true_gradient_polynomial, sample.x);
+ sample.gradient_is_valid = true;
+ samples.push_back(sample);
+ }
+
+ const Vector polynomial = FindInterpolatingPolynomial(samples);
+ EXPECT_NEAR((true_polynomial - polynomial).norm(), 0.0, 1e-14);
+}
+
+TEST(Polynomial, CubicInterpolatingPolynomialFromValuesAndGradients) {
+ // p(x) = x^3 + 2x^2 + 3x + 2
+ Vector true_polynomial(4);
+ true_polynomial << 1.0, 2.0, 3.0, 2.0;
+ Vector true_gradient_polynomial = DifferentiatePolynomial(true_polynomial);
+
+ vector<FunctionSample> samples;
+ {
+ FunctionSample sample;
+ sample.x = -3.0;
+ sample.value = EvaluatePolynomial(true_polynomial, sample.x);
+ sample.value_is_valid = true;
+ sample.gradient = EvaluatePolynomial(true_gradient_polynomial, sample.x);
+ sample.gradient_is_valid = true;
+ samples.push_back(sample);
+ }
+
+ {
+ FunctionSample sample;
+ sample.x = 2.0;
+ sample.value = EvaluatePolynomial(true_polynomial, sample.x);
+ sample.value_is_valid = true;
+ sample.gradient = EvaluatePolynomial(true_gradient_polynomial, sample.x);
+ sample.gradient_is_valid = true;
+ samples.push_back(sample);
+ }
+
+ const Vector polynomial = FindInterpolatingPolynomial(samples);
+ EXPECT_NEAR((true_polynomial - polynomial).norm(), 0.0, 1e-14);
+}
+
+} // namespace internal
+} // namespace ceres
