// Ceres Solver - A fast non-linear least squares minimizer
// Copyright 2015 Google Inc. All rights reserved.
// http://ceres-solver.org/
//
// 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
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// 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
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// 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/function_sample.h"
#include "ceres/test_util.h"
namespace ceres {
namespace internal {
using std::vector;
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, 2 * 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