517 lines
15 KiB
C++
517 lines
15 KiB
C++
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// Ceres Solver - A fast non-linear least squares minimizer
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// Copyright 2015 Google Inc. All rights reserved.
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// http://ceres-solver.org/
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//
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// Redistribution and use in source and binary forms, with or without
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// modification, are permitted provided that the following conditions are met:
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//
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// * Redistributions of source code must retain the above copyright notice,
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// this list of conditions and the following disclaimer.
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// * Redistributions in binary form must reproduce the above copyright notice,
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// this list of conditions and the following disclaimer in the documentation
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// and/or other materials provided with the distribution.
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// * Neither the name of Google Inc. nor the names of its contributors may be
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// used to endorse or promote products derived from this software without
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// specific prior written permission.
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//
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// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
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// AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
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// IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
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// ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
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// LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
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// CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
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// SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
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// INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
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// CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
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// ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
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// POSSIBILITY OF SUCH DAMAGE.
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//
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// Author: moll.markus@arcor.de (Markus Moll)
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// sameeragarwal@google.com (Sameer Agarwal)
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#include "ceres/polynomial.h"
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#include <limits>
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#include <cmath>
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#include <cstddef>
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#include <algorithm>
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#include "gtest/gtest.h"
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#include "ceres/test_util.h"
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namespace ceres {
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namespace internal {
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using std::vector;
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namespace {
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// For IEEE-754 doubles, machine precision is about 2e-16.
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const double kEpsilon = 1e-13;
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const double kEpsilonLoose = 1e-9;
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// Return the constant polynomial p(x) = 1.23.
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Vector ConstantPolynomial(double value) {
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Vector poly(1);
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poly(0) = value;
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return poly;
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}
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// Return the polynomial p(x) = poly(x) * (x - root).
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Vector AddRealRoot(const Vector& poly, double root) {
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Vector poly2(poly.size() + 1);
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poly2.setZero();
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poly2.head(poly.size()) += poly;
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poly2.tail(poly.size()) -= root * poly;
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return poly2;
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}
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// Return the polynomial
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// p(x) = poly(x) * (x - real - imag*i) * (x - real + imag*i).
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Vector AddComplexRootPair(const Vector& poly, double real, double imag) {
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Vector poly2(poly.size() + 2);
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poly2.setZero();
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// Multiply poly by x^2 - 2real + abs(real,imag)^2
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poly2.head(poly.size()) += poly;
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poly2.segment(1, poly.size()) -= 2 * real * poly;
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poly2.tail(poly.size()) += (real*real + imag*imag) * poly;
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return poly2;
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}
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// Sort the entries in a vector.
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// Needed because the roots are not returned in sorted order.
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Vector SortVector(const Vector& in) {
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Vector out(in);
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std::sort(out.data(), out.data() + out.size());
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return out;
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}
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// Run a test with the polynomial defined by the N real roots in roots_real.
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// If use_real is false, NULL is passed as the real argument to
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// FindPolynomialRoots. If use_imaginary is false, NULL is passed as the
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// imaginary argument to FindPolynomialRoots.
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template<int N>
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void RunPolynomialTestRealRoots(const double (&real_roots)[N],
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bool use_real,
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bool use_imaginary,
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double epsilon) {
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Vector real;
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Vector imaginary;
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Vector poly = ConstantPolynomial(1.23);
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for (int i = 0; i < N; ++i) {
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poly = AddRealRoot(poly, real_roots[i]);
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}
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Vector* const real_ptr = use_real ? &real : NULL;
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Vector* const imaginary_ptr = use_imaginary ? &imaginary : NULL;
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bool success = FindPolynomialRoots(poly, real_ptr, imaginary_ptr);
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EXPECT_EQ(success, true);
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if (use_real) {
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EXPECT_EQ(real.size(), N);
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real = SortVector(real);
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ExpectArraysClose(N, real.data(), real_roots, epsilon);
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}
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if (use_imaginary) {
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EXPECT_EQ(imaginary.size(), N);
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const Vector zeros = Vector::Zero(N);
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ExpectArraysClose(N, imaginary.data(), zeros.data(), epsilon);
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}
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}
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} // namespace
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TEST(Polynomial, InvalidPolynomialOfZeroLengthIsRejected) {
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// Vector poly(0) is an ambiguous constructor call, so
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// use the constructor with explicit column count.
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Vector poly(0, 1);
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Vector real;
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Vector imag;
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bool success = FindPolynomialRoots(poly, &real, &imag);
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EXPECT_EQ(success, false);
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}
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TEST(Polynomial, ConstantPolynomialReturnsNoRoots) {
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Vector poly = ConstantPolynomial(1.23);
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Vector real;
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Vector imag;
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bool success = FindPolynomialRoots(poly, &real, &imag);
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EXPECT_EQ(success, true);
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EXPECT_EQ(real.size(), 0);
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EXPECT_EQ(imag.size(), 0);
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}
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TEST(Polynomial, LinearPolynomialWithPositiveRootWorks) {
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const double roots[1] = { 42.42 };
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RunPolynomialTestRealRoots(roots, true, true, kEpsilon);
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}
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TEST(Polynomial, LinearPolynomialWithNegativeRootWorks) {
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const double roots[1] = { -42.42 };
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RunPolynomialTestRealRoots(roots, true, true, kEpsilon);
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}
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TEST(Polynomial, QuadraticPolynomialWithPositiveRootsWorks) {
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const double roots[2] = { 1.0, 42.42 };
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RunPolynomialTestRealRoots(roots, true, true, kEpsilon);
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}
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TEST(Polynomial, QuadraticPolynomialWithOneNegativeRootWorks) {
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const double roots[2] = { -42.42, 1.0 };
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RunPolynomialTestRealRoots(roots, true, true, kEpsilon);
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}
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TEST(Polynomial, QuadraticPolynomialWithTwoNegativeRootsWorks) {
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const double roots[2] = { -42.42, -1.0 };
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RunPolynomialTestRealRoots(roots, true, true, kEpsilon);
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}
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TEST(Polynomial, QuadraticPolynomialWithCloseRootsWorks) {
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const double roots[2] = { 42.42, 42.43 };
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RunPolynomialTestRealRoots(roots, true, false, kEpsilonLoose);
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}
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TEST(Polynomial, QuadraticPolynomialWithComplexRootsWorks) {
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Vector real;
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Vector imag;
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Vector poly = ConstantPolynomial(1.23);
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poly = AddComplexRootPair(poly, 42.42, 4.2);
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bool success = FindPolynomialRoots(poly, &real, &imag);
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EXPECT_EQ(success, true);
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EXPECT_EQ(real.size(), 2);
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EXPECT_EQ(imag.size(), 2);
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ExpectClose(real(0), 42.42, kEpsilon);
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ExpectClose(real(1), 42.42, kEpsilon);
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ExpectClose(std::abs(imag(0)), 4.2, kEpsilon);
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ExpectClose(std::abs(imag(1)), 4.2, kEpsilon);
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ExpectClose(std::abs(imag(0) + imag(1)), 0.0, kEpsilon);
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}
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TEST(Polynomial, QuarticPolynomialWorks) {
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const double roots[4] = { 1.23e-4, 1.23e-1, 1.23e+2, 1.23e+5 };
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RunPolynomialTestRealRoots(roots, true, true, kEpsilon);
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}
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TEST(Polynomial, QuarticPolynomialWithTwoClustersOfCloseRootsWorks) {
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const double roots[4] = { 1.23e-1, 2.46e-1, 1.23e+5, 2.46e+5 };
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RunPolynomialTestRealRoots(roots, true, true, kEpsilonLoose);
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}
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TEST(Polynomial, QuarticPolynomialWithTwoZeroRootsWorks) {
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const double roots[4] = { -42.42, 0.0, 0.0, 42.42 };
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RunPolynomialTestRealRoots(roots, true, true, 2 * kEpsilonLoose);
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}
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TEST(Polynomial, QuarticMonomialWorks) {
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const double roots[4] = { 0.0, 0.0, 0.0, 0.0 };
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RunPolynomialTestRealRoots(roots, true, true, kEpsilon);
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}
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TEST(Polynomial, NullPointerAsImaginaryPartWorks) {
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const double roots[4] = { 1.23e-4, 1.23e-1, 1.23e+2, 1.23e+5 };
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RunPolynomialTestRealRoots(roots, true, false, kEpsilon);
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}
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TEST(Polynomial, NullPointerAsRealPartWorks) {
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const double roots[4] = { 1.23e-4, 1.23e-1, 1.23e+2, 1.23e+5 };
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RunPolynomialTestRealRoots(roots, false, true, kEpsilon);
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}
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TEST(Polynomial, BothOutputArgumentsNullWorks) {
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const double roots[4] = { 1.23e-4, 1.23e-1, 1.23e+2, 1.23e+5 };
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RunPolynomialTestRealRoots(roots, false, false, kEpsilon);
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}
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TEST(Polynomial, DifferentiateConstantPolynomial) {
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// p(x) = 1;
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Vector polynomial(1);
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polynomial(0) = 1.0;
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const Vector derivative = DifferentiatePolynomial(polynomial);
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EXPECT_EQ(derivative.rows(), 1);
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EXPECT_EQ(derivative(0), 0);
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}
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TEST(Polynomial, DifferentiateQuadraticPolynomial) {
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// p(x) = x^2 + 2x + 3;
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Vector polynomial(3);
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polynomial(0) = 1.0;
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polynomial(1) = 2.0;
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polynomial(2) = 3.0;
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const Vector derivative = DifferentiatePolynomial(polynomial);
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EXPECT_EQ(derivative.rows(), 2);
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EXPECT_EQ(derivative(0), 2.0);
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EXPECT_EQ(derivative(1), 2.0);
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}
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TEST(Polynomial, MinimizeConstantPolynomial) {
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// p(x) = 1;
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Vector polynomial(1);
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polynomial(0) = 1.0;
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double optimal_x = 0.0;
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double optimal_value = 0.0;
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double min_x = 0.0;
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double max_x = 1.0;
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MinimizePolynomial(polynomial, min_x, max_x, &optimal_x, &optimal_value);
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EXPECT_EQ(optimal_value, 1.0);
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EXPECT_LE(optimal_x, max_x);
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EXPECT_GE(optimal_x, min_x);
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}
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TEST(Polynomial, MinimizeLinearPolynomial) {
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// p(x) = x - 2
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Vector polynomial(2);
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polynomial(0) = 1.0;
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polynomial(1) = 2.0;
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double optimal_x = 0.0;
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double optimal_value = 0.0;
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double min_x = 0.0;
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double max_x = 1.0;
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MinimizePolynomial(polynomial, min_x, max_x, &optimal_x, &optimal_value);
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EXPECT_EQ(optimal_x, 0.0);
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EXPECT_EQ(optimal_value, 2.0);
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}
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TEST(Polynomial, MinimizeQuadraticPolynomial) {
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// p(x) = x^2 - 3 x + 2
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// min_x = 3/2
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// min_value = -1/4;
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Vector polynomial(3);
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polynomial(0) = 1.0;
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polynomial(1) = -3.0;
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polynomial(2) = 2.0;
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double optimal_x = 0.0;
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double optimal_value = 0.0;
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double min_x = -2.0;
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double max_x = 2.0;
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MinimizePolynomial(polynomial, min_x, max_x, &optimal_x, &optimal_value);
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EXPECT_EQ(optimal_x, 3.0/2.0);
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EXPECT_EQ(optimal_value, -1.0/4.0);
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min_x = -2.0;
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max_x = 1.0;
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MinimizePolynomial(polynomial, min_x, max_x, &optimal_x, &optimal_value);
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EXPECT_EQ(optimal_x, 1.0);
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EXPECT_EQ(optimal_value, 0.0);
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min_x = 2.0;
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max_x = 3.0;
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MinimizePolynomial(polynomial, min_x, max_x, &optimal_x, &optimal_value);
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EXPECT_EQ(optimal_x, 2.0);
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EXPECT_EQ(optimal_value, 0.0);
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}
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TEST(Polymomial, ConstantInterpolatingPolynomial) {
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// p(x) = 1.0
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Vector true_polynomial(1);
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true_polynomial << 1.0;
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vector<FunctionSample> samples;
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FunctionSample sample;
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sample.x = 1.0;
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sample.value = 1.0;
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sample.value_is_valid = true;
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samples.push_back(sample);
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const Vector polynomial = FindInterpolatingPolynomial(samples);
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EXPECT_NEAR((true_polynomial - polynomial).norm(), 0.0, 1e-15);
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}
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TEST(Polynomial, LinearInterpolatingPolynomial) {
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// p(x) = 2x - 1
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Vector true_polynomial(2);
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true_polynomial << 2.0, -1.0;
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vector<FunctionSample> samples;
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FunctionSample sample;
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sample.x = 1.0;
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sample.value = 1.0;
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sample.value_is_valid = true;
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sample.gradient = 2.0;
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sample.gradient_is_valid = true;
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samples.push_back(sample);
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const Vector polynomial = FindInterpolatingPolynomial(samples);
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EXPECT_NEAR((true_polynomial - polynomial).norm(), 0.0, 1e-15);
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}
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TEST(Polynomial, QuadraticInterpolatingPolynomial) {
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// p(x) = 2x^2 + 3x + 2
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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
|