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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)
+// 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(), 1);
+ EXPECT_EQ(derivative(0), 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
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