Add polynomial interpolation and minimization.
1. polynomial_solver* -> polynomial*.
2. Added support for differentiating polynomials.
2. Added support for interpolating polynomials from function
values and gradients.
3. Added support for minimizing polynomials by solving
for the roots of their derivatives in an interval.
4. Added support for finding the minimum of a polynomial
that interpolates function values and gradients in
an interval.
Change-Id: Id7e6764ad4db09c3edd60f1378c7f50f20dd08dc
diff --git a/internal/ceres/polynomial_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
