symbolic_monomial_test.cc

#include <exception>
#include <sstream>
#include <stdexcept>
#include <unordered_map>
#include <utility>
#include <vector>

#include <fmt/ostream.h>
#include <gtest/gtest.h>

#include "drake/common/eigen_types.h"
#include "drake/common/hash.h"
#include "drake/common/symbolic.h"
#include "drake/common/test_utilities/expect_throws_message.h"
#include "drake/common/test_utilities/symbolic_test_util.h"

using std::map;
using std::ostringstream;
using std::pair;
using std::runtime_error;
using std::unordered_map;
using std::vector;

namespace drake {
namespace symbolic {
namespace {

using test::ExprEqual;
using test::VarLess;

class MonomialTest : public ::testing::Test {
 protected:
  const Variable var_w_{"w"};
  const Variable var_z_{"z"};
  const Variable var_y_{"y"};
  const Variable var_x_{"x"};

  const Expression w_{var_w_};
  const Expression z_{var_z_};
  const Expression y_{var_y_};
  const Expression x_{var_x_};

  vector<Monomial> monomials_;

  void SetUp() override {
    monomials_ = {
        Monomial{},                                         // 1.0
        Monomial{var_x_, 1},                                // x^1
        Monomial{var_y_, 1},                                // y^1
        Monomial{var_z_, 1},                                // z^1
        Monomial{var_x_, 2},                                // x^2
        Monomial{var_y_, 3},                                // y^3
        Monomial{var_z_, 4},                                // z^4
        Monomial{{{var_x_, 1}, {var_y_, 2}}},               // xy^2
        Monomial{{{var_y_, 2}, {var_z_, 5}}},               // y^2z^5
        Monomial{{{var_x_, 1}, {var_y_, 2}, {var_z_, 3}}},  // xy^2z^3
        Monomial{{{var_x_, 2}, {var_y_, 4}, {var_z_, 3}}},  // x^2y^4z^3
    };

    EXPECT_PRED2(VarLess, var_y_, var_x_);
    EXPECT_PRED2(VarLess, var_z_, var_y_);
    EXPECT_PRED2(VarLess, var_w_, var_z_);
  }

  // Helper function to extract Substitution (Variable -> Expression) from a
  // symbolic environment.
  Substitution ExtractSubst(const Environment& env) {
    Substitution subst;
    subst.reserve(env.size());
    for (const pair<const Variable, double>& p : env) {
      subst.emplace(p.first, p.second);
    }
    return subst;
  }

  // Checks if Monomial::Evaluate corresponds to Expression::Evaluate.
  //
  //                            ToExpression
  //                   Monomial ------------> Expression
  //                      ||                      ||
  //   Monomial::Evaluate ||                      || Expression::Evaluate
  //                      ||                      ||
  //                      \/                      \/
  //                double (result2)   ==   double (result1)
  //
  // In one direction (left-to-right), we convert a Monomial to an Expression
  // and evaluate it to a double value (result1). In another direction
  // (top-to-bottom), we directly evaluate a Monomial to double (result2). The
  // two result1 and result2 should be the same.
  //
  // Also confirms that expression always is declared to be pre-expanded.
  void CheckEvaluate(const Monomial& m, const Environment& env) {
    SCOPED_TRACE(fmt::format("m = {}; env = {}", m, env));
    const Expression e{m.ToExpression()};
    EXPECT_TRUE(e.is_expanded());
    const double result1{e.Evaluate(env)};
    const double result2{m.Evaluate(env)};
    EXPECT_EQ(result1, result2);
  }

  // Checks if Monomial::EvaluatePartial corresponds to Expression's
  // EvaluatePartial.
  //
  //                            ToExpression
  //                   Monomial ------------> Expression
  //                      ||                      ||
  //      EvaluatePartial ||                      || EvaluatePartial
  //                      ||                      ||
  //                      \/                      \/
  //          double * Monomial (e2)   ==   Expression (e1)
  //
  // In one direction (left-to-right), first we convert a Monomial to an
  // Expression using Monomial::ToExpression and call
  // Expression::EvaluatePartial to have an Expression (e1). In another
  // direction (top-to-bottom), we call Monomial::EvaluatePartial which returns
  // a pair of double (coefficient part) and Monomial. We obtain e2 by
  // multiplying the two. Then, we check if e1 and e2 are structurally equal.
  void CheckEvaluatePartial(const Monomial& m, const Environment& env) {
    SCOPED_TRACE(fmt::format("m = {}; env = {}", m, env));
    const Expression e1{m.ToExpression().EvaluatePartial(env)};
    const auto [coeff, m2] = m.EvaluatePartial(env);
    const Expression e2{coeff * m2.ToExpression()};
    EXPECT_PRED2(ExprEqual, e1, e2);
  }
};

// Tests that default constructor and EIGEN_INITIALIZE_MATRICES_BY_ZERO
// constructor both create the same value.
TEST_F(MonomialTest, DefaultConstructors) {
  const Monomial m_default;
  const Monomial m_zero(0);
  EXPECT_EQ(m_default.total_degree(), 0);
  EXPECT_EQ(m_zero.total_degree(), 0);
}

TEST_F(MonomialTest, ConstructFromVariable) {
  const Monomial m1{var_x_};
  const std::map<Variable, int> powers{m1.get_powers()};

  // Checks that powers = {x ↦ 1}.
  ASSERT_EQ(powers.size(), 1u);
  EXPECT_EQ(powers.begin()->first, var_x_);
  EXPECT_EQ(powers.begin()->second, 1);
}

TEST_F(MonomialTest, ConstructFromVariablesAndExponents) {
  // [] * [] => 1.
  const VectorX<Variable> vars(0);
  const VectorX<int> exponents(0);
  const Monomial one{Monomial{Expression::One()}};
  EXPECT_EQ(Monomial(vars, exponents), one);

  const Vector3<Variable> vars_xyz{var_x_, var_y_, var_z_};
  // [x, y, z] * [0, 0, 0] => 1.
  const Monomial m1{vars_xyz, Eigen::Vector3i{0, 0, 0}};
  EXPECT_EQ(m1, one);

  // [x, y, z] * [1, 1, 1] => xyz.
  const Monomial m2{vars_xyz, Eigen::Vector3i{1, 1, 1}};
  const Monomial m2_expected{x_ * y_ * z_};
  EXPECT_EQ(m2, m2_expected);

  // [x, y, z] * [2, 0, 3] => x²z³.
  const Monomial m3{vars_xyz, Eigen::Vector3i{2, 0, 3}};
  const Monomial m3_expected{pow(x_, 2) * pow(z_, 3)};
  EXPECT_EQ(m3, m3_expected);

  // [x, y, z] * [2, 0, -1] => Exception!
  DRAKE_EXPECT_THROWS_MESSAGE(Monomial(vars_xyz, Eigen::Vector3i(2, 0, -1)),
                              "The exponent is negative.");
}

TEST_F(MonomialTest, GetVariables) {
  const Monomial m0{};
  EXPECT_EQ(m0.GetVariables(), Variables{});

  const Monomial m1{var_z_, 4};
  EXPECT_EQ(m1.GetVariables(), Variables({var_z_}));

  const Monomial m2{{{var_x_, 1}, {var_y_, 2}}};
  EXPECT_EQ(m2.GetVariables(), Variables({var_x_, var_y_}));

  const Monomial m3{{{var_x_, 1}, {var_y_, 2}, {var_z_, 3}}};
  EXPECT_EQ(m3.GetVariables(), Variables({var_x_, var_y_, var_z_}));
}

// Checks we can have an Eigen matrix of Monomials without compilation
// errors. No assertions in the test.
TEST_F(MonomialTest, EigenMatrixOfMonomials) {
  Eigen::Matrix<Monomial, 2, 2> M;
  // M = | 1   x    |
  //     | y²  x²z³ |
  // clang-format off
  M << Monomial{}, Monomial{var_x_},
       Monomial{{{var_y_, 2}}}, Monomial{{{var_x_, 2}, {var_z_, 3}}};
  // clang-format on

  // The following fails if we do not provide
  // `Eigen::NumTraits<drake::symbolic::Monomial>`.
  ostringstream oss;
  oss << M;
}

TEST_F(MonomialTest, MonomialOne) {
  // Compares monomials all equal to 1, but with different variables.
  Monomial m1{};
  Monomial m2({{var_x_, 0}});
  Monomial m3({{var_x_, 0}, {var_y_, 0}});
  EXPECT_EQ(m1, m2);
  EXPECT_EQ(m1, m3);
  EXPECT_EQ(m2, m3);
}

TEST_F(MonomialTest, MonomialWithZeroExponent) {
  // Compares monomials containing zero exponent, such as x^0 * y^2
  Monomial m1({{var_y_, 2}});
  Monomial m2({{var_x_, 0}, {var_y_, 2}});
  EXPECT_EQ(m1, m2);
  EXPECT_EQ(m2.get_powers().size(), 1);
  std::map<Variable, int> power_expected;
  power_expected.emplace(var_y_, 2);
  EXPECT_EQ(m2.get_powers(), power_expected);
}

TEST_F(MonomialTest, MonomialBasisX0) {
  const drake::VectorX<Monomial> basis1{MonomialBasis({var_x_}, 0)};
  const auto basis2 = MonomialBasis<1, 0>({var_x_});
  EXPECT_EQ(decltype(basis2)::RowsAtCompileTime, 1);
  drake::Vector1<Monomial> expected;

  // MonomialBasis({x}, 0) = {x⁰} = {1}.
  expected << Monomial{};

  EXPECT_EQ(basis1, expected);
  EXPECT_EQ(basis2, expected);
}

TEST_F(MonomialTest, MonomialBasisX2) {
  const drake::VectorX<Monomial> basis1{MonomialBasis({var_x_}, 2)};
  const auto basis2 = MonomialBasis<1, 2>({var_x_});
  EXPECT_EQ(decltype(basis2)::RowsAtCompileTime, 3);
  drake::Vector3<Monomial> expected;

  // MonomialBasis({x}, 2) = {x², x, 1}.
  // clang-format off
  expected << Monomial{var_x_, 2},
              Monomial{var_x_},
              Monomial{};
  // clang-format on
  EXPECT_EQ(basis1, expected);
  EXPECT_EQ(basis2, expected);
}

TEST_F(MonomialTest, MonomialBasisXY0) {
  const drake::VectorX<Monomial> basis1{MonomialBasis({var_x_, var_y_}, 0)};
  const auto basis2 = MonomialBasis<2, 0>({var_x_, var_y_});
  EXPECT_EQ(decltype(basis2)::RowsAtCompileTime, 1);
  drake::Vector1<Monomial> expected;

  // MonomialBasis({x, y}, 0) = {1}.
  expected << Monomial{{{var_x_, 0}, {var_y_, 0}}};  // = {1}.

  EXPECT_EQ(basis1, expected);
  EXPECT_EQ(basis2, expected);
}

TEST_F(MonomialTest, MonomialBasisXY1) {
  const drake::VectorX<Monomial> basis1{MonomialBasis({var_x_, var_y_}, 1)};
  const auto basis2 = MonomialBasis<2, 1>({var_x_, var_y_});
  EXPECT_EQ(decltype(basis2)::RowsAtCompileTime, 3);
  drake::Vector3<Monomial> expected;

  // MonomialBasis({x, y}, 1) = {x, y, 1}.
  // clang-format off
  expected << Monomial{var_x_},
              Monomial{var_y_},
              Monomial{};
  // clang-format on

  EXPECT_EQ(basis1, expected);
  EXPECT_EQ(basis2, expected);
}

TEST_F(MonomialTest, MonomialBasisXY2) {
  const drake::VectorX<Monomial> basis1{MonomialBasis({var_x_, var_y_}, 2)};
  const auto basis2 = MonomialBasis<2, 2>({var_x_, var_y_});
  EXPECT_EQ(decltype(basis2)::RowsAtCompileTime, 6);
  drake::Vector6<Monomial> expected;

  // MonomialBasis({x, y}, 2) = {x², xy, y², x, y, 1}.
  // clang-format off
  expected << Monomial{var_x_, 2},
              Monomial{{{var_x_, 1}, {var_y_, 1}}},
              Monomial{var_y_, 2},
              Monomial{var_x_},
              Monomial{var_y_},
              Monomial{};
  // clang-format on

  EXPECT_EQ(basis1, expected);
  EXPECT_EQ(basis2, expected);
}

TEST_F(MonomialTest, MonomialBasisXY3) {
  const drake::VectorX<Monomial> basis1{MonomialBasis({var_x_, var_y_}, 3)};
  const auto basis2 = MonomialBasis<2, 3>({var_x_, var_y_});
  EXPECT_EQ(decltype(basis2)::RowsAtCompileTime, 10);
  Eigen::Matrix<Monomial, 10, 1> expected;

  // MonomialBasis({x, y}, 3) = {x³, x²y, xy², y³, x², xy, y², x, y, 1}.
  // clang-format off
  expected << Monomial{var_x_, 3},
              Monomial{{{var_x_, 2}, {var_y_, 1}}},
              Monomial{{{var_x_, 1}, {var_y_, 2}}},
              Monomial{var_y_, 3},
              Monomial{var_x_, 2},
              Monomial{{{var_x_, 1}, {var_y_, 1}}},
              Monomial{var_y_, 2},
              Monomial{var_x_},
              Monomial{var_y_},
              Monomial{};
  // clang-format on

  EXPECT_EQ(basis1, expected);
  EXPECT_EQ(basis2, expected);
}

TEST_F(MonomialTest, MonomialBasisXYZ2) {
  const drake::VectorX<Monomial> basis1{
      MonomialBasis({var_x_, var_y_, var_z_}, 2)};
  const auto basis2 = MonomialBasis<3, 2>({var_x_, var_y_, var_z_});
  EXPECT_EQ(decltype(basis2)::RowsAtCompileTime, 10);

  Eigen::Matrix<Monomial, 10, 1> expected;

  // MonomialBasis({x, y, z}, 2)
  // clang-format off
  expected << Monomial{var_x_, 2},
              Monomial{{{var_x_, 1}, {var_y_, 1}}},
              Monomial{var_y_, 2},
              Monomial{{{var_x_, 1}, {var_z_, 1}}},
              Monomial{{{var_y_, 1}, {var_z_, 1}}},
              Monomial{var_z_, 2},
              Monomial{var_x_},
              Monomial{var_y_},
              Monomial{var_z_},
              Monomial{};
  // clang-format on

  EXPECT_EQ(basis1, expected);
  EXPECT_EQ(basis2, expected);
}

TEST_F(MonomialTest, MonomialBasisXYZ3) {
  const drake::VectorX<Monomial> basis1{
      MonomialBasis({var_x_, var_y_, var_z_}, 3)};
  const auto basis2 = MonomialBasis<3, 3>({var_x_, var_y_, var_z_});
  EXPECT_EQ(decltype(basis2)::RowsAtCompileTime, 20);
  Eigen::Matrix<Monomial, 20, 1> expected;

  // MonomialBasis({x, y, z}, 3)
  // = {x³, x²y, xy², y³, x²z, xyz, y²z, xz², yz²,
  //    z³, x², xy, y², xz, yz, z², x, y, z, 1}
  // clang-format off
  expected << Monomial{var_x_, 3},
              Monomial{{{var_x_, 2}, {var_y_, 1}}},
              Monomial{{{var_x_, 1}, {var_y_, 2}}},
              Monomial{var_y_, 3},
              Monomial{{{var_x_, 2}, {var_z_, 1}}},
              Monomial{{{var_x_, 1}, {var_y_, 1}, {var_z_, 1}}},
              Monomial{{{var_y_, 2}, {var_z_, 1}}},
              Monomial{{{var_x_, 1}, {var_z_, 2}}},
              Monomial{{{var_y_, 1}, {var_z_, 2}}},
              Monomial{var_z_, 3},
              Monomial{var_x_, 2},
              Monomial{{{var_x_, 1}, {var_y_, 1}}},
              Monomial{var_y_, 2},
              Monomial{{{var_x_, 1}, {var_z_, 1}}},
              Monomial{{{var_y_, 1}, {var_z_, 1}}},
              Monomial{var_z_, 2},
              Monomial{var_x_},
              Monomial{var_y_},
              Monomial{var_z_},
              Monomial{};
  // clang-format on

  EXPECT_EQ(basis1, expected);
  EXPECT_EQ(basis2, expected);
}

TEST_F(MonomialTest, MonomialBasisXYZW3) {
  const drake::VectorX<Monomial> basis1{
      MonomialBasis({var_x_, var_y_, var_z_, var_w_}, 3)};
  const auto basis2 = MonomialBasis<4, 3>({var_x_, var_y_, var_z_, var_w_});
  EXPECT_EQ(decltype(basis2)::RowsAtCompileTime, 35);
  Eigen::Matrix<Monomial, 35, 1> expected;

  // MonomialBasis({x, y, z, w}, 3)
  // clang-format off
  expected << Monomial{var_x_, 3},
              Monomial{{{var_x_, 2}, {var_y_, 1}}},
              Monomial{{{var_x_, 1}, {var_y_, 2}}},
              Monomial{var_y_, 3},
              Monomial{{{var_x_, 2}, {var_z_, 1}}},
              Monomial{{{var_x_, 1}, {var_y_, 1}, {var_z_, 1}}},
              Monomial{{{var_y_, 2}, {var_z_, 1}}},
              Monomial{{{var_x_, 1}, {var_z_, 2}}},
              Monomial{{{var_y_, 1}, {var_z_, 2}}},
              Monomial{var_z_, 3},
              Monomial{{{var_x_, 2}, {var_w_, 1}}},
              Monomial{{{var_y_, 1}, {var_x_, 1}, {var_w_, 1}}},
              Monomial{{{var_y_, 2}, {var_w_, 1}}},
              Monomial{{{var_z_, 1}, {var_x_, 1}, {var_w_, 1}}},
              Monomial{{{var_z_, 1}, {var_y_, 1}, {var_w_, 1}}},
              Monomial{{{var_z_, 2}, {var_w_, 1}}},
              Monomial{{{var_x_, 1}, {var_w_, 2}}},
              Monomial{{{var_y_, 1}, {var_w_, 2}}},
              Monomial{{{var_z_, 1}, {var_w_, 2}}},
              Monomial{var_w_, 3},
              Monomial{var_x_, 2},
              Monomial{{{var_x_, 1}, {var_y_, 1}}},
              Monomial{var_y_, 2},
              Monomial{{{var_x_, 1}, {var_z_, 1}}},
              Monomial{{{var_y_, 1}, {var_z_, 1}}},
              Monomial{var_z_, 2},
              Monomial{{{var_x_, 1}, {var_w_, 1}}},
              Monomial{{{var_y_, 1}, {var_w_, 1}}},
              Monomial{{{var_z_, 1}, {var_w_, 1}}},
              Monomial{var_w_, 2},
              Monomial{var_x_},
              Monomial{var_y_},
              Monomial{var_z_},
              Monomial{var_w_},
              Monomial{};
  // clang-format on

  EXPECT_EQ(basis1, expected);
  EXPECT_EQ(basis2, expected);
}

TEST_F(MonomialTest, EvenDegreeMonomialBasisX2) {
  const drake::VectorX<Monomial> basis1{EvenDegreeMonomialBasis({var_x_}, 2)};
  drake::Vector2<Monomial> expected;

  // EvenDegreeMonomialBasis({x}, 2) = {x², 1}.
  // clang-format off
  expected << Monomial{var_x_, 2},
              Monomial{};
  // clang-format on
  EXPECT_EQ(basis1, expected);
}

TEST_F(MonomialTest, EvenDegreeMonomialBasisXY01) {
  const drake::VectorX<Monomial> basis1{
      EvenDegreeMonomialBasis({var_x_, var_y_}, 0)};
  const drake::VectorX<Monomial> basis2{
      EvenDegreeMonomialBasis({var_x_, var_y_}, 1)};
  drake::Vector1<Monomial> expected;

  // EvenDegreeMonomialBasis({x, y}, 0) = {1}.
  expected << Monomial{{{var_x_, 0}, {var_y_, 0}}};  // = {1}.

  EXPECT_EQ(basis1, expected);
  EXPECT_EQ(basis2, expected);
}

TEST_F(MonomialTest, EvenDegreeMonomialBasisXY23) {
  const drake::VectorX<Monomial> basis1{
      EvenDegreeMonomialBasis({var_x_, var_y_}, 2)};
  const drake::VectorX<Monomial> basis2{
      EvenDegreeMonomialBasis({var_x_, var_y_}, 3)};
  drake::Vector4<Monomial> expected;

  // EvenDegreeMonomialBasis({x, y}, 2) = {x², xy, y², 1}.
  // clang-format off
  expected << Monomial{{{var_x_, 2}, {var_y_, 0}}},
              Monomial{{{var_x_, 1}, {var_y_, 1}}},
              Monomial{{{var_x_, 0}, {var_y_, 2}}},
              Monomial{{{var_x_, 0}, {var_y_, 0}}};
  // clang-format on

  EXPECT_EQ(basis1, expected);
  EXPECT_EQ(basis2, expected);
}

TEST_F(MonomialTest, EvenDegreeMonomialBasisXYZ45) {
  const drake::VectorX<Monomial> basis1{
      EvenDegreeMonomialBasis({var_x_, var_y_, var_z_}, 4)};
  const drake::VectorX<Monomial> basis2{
      EvenDegreeMonomialBasis({var_x_, var_y_, var_z_}, 5)};
  Eigen::Matrix<Monomial, 22, 1> expected;

  // EvenDegreeMonomialBasis({x, y, z}, 4) = {x⁴, x³y, x²y², xy³, y⁴, x³z, x²yz,
  // xy²z, y³z, x²z², xyz², y²z², xz³, yz³, z⁴, x², xy, y², xz, yz, z², 1}
  // clang-format off
  expected << Monomial{{{var_x_, 4}, {var_y_, 0}, {var_z_, 0}}},
              Monomial{{{var_x_, 3}, {var_y_, 1}, {var_z_, 0}}},
              Monomial{{{var_x_, 2}, {var_y_, 2}, {var_z_, 0}}},
              Monomial{{{var_x_, 1}, {var_y_, 3}, {var_z_, 0}}},
              Monomial{{{var_x_, 0}, {var_y_, 4}, {var_z_, 0}}},
              Monomial{{{var_x_, 3}, {var_y_, 0}, {var_z_, 1}}},
              Monomial{{{var_x_, 2}, {var_y_, 1}, {var_z_, 1}}},
              Monomial{{{var_x_, 1}, {var_y_, 2}, {var_z_, 1}}},
              Monomial{{{var_x_, 0}, {var_y_, 3}, {var_z_, 1}}},
              Monomial{{{var_x_, 2}, {var_y_, 0}, {var_z_, 2}}},
              Monomial{{{var_x_, 1}, {var_y_, 1}, {var_z_, 2}}},
              Monomial{{{var_x_, 0}, {var_y_, 2}, {var_z_, 2}}},
              Monomial{{{var_x_, 1}, {var_y_, 0}, {var_z_, 3}}},
              Monomial{{{var_x_, 0}, {var_y_, 1}, {var_z_, 3}}},
              Monomial{{{var_x_, 0}, {var_y_, 0}, {var_z_, 4}}},
              Monomial{{{var_x_, 2}, {var_y_, 0}, {var_z_, 0}}},
              Monomial{{{var_x_, 1}, {var_y_, 1}, {var_z_, 0}}},
              Monomial{{{var_x_, 0}, {var_y_, 2}, {var_z_, 0}}},
              Monomial{{{var_x_, 1}, {var_y_, 0}, {var_z_, 1}}},
              Monomial{{{var_x_, 0}, {var_y_, 1}, {var_z_, 1}}},
              Monomial{{{var_x_, 0}, {var_y_, 0}, {var_z_, 2}}},
              Monomial{{{var_x_, 0}, {var_y_, 0}, {var_z_, 0}}};
  // clang-format on

  EXPECT_EQ(basis1, expected);
  EXPECT_EQ(basis2, expected);
}

TEST_F(MonomialTest, OddDegreeMonomialBasisX3) {
  const drake::VectorX<Monomial> basis1{OddDegreeMonomialBasis({var_x_}, 3)};
  drake::Vector2<Monomial> expected;

  // OddDegreeMonomialBasis({x}, 3) = {x³, 1}.
  // clang-format off
  expected << Monomial{var_x_, 3},
              Monomial{var_x_, 1};
  // clang-format on
  EXPECT_EQ(basis1, expected);
}

TEST_F(MonomialTest, OddDegreeMonomialBasisXY12) {
  const drake::VectorX<Monomial> basis1{
      OddDegreeMonomialBasis({var_x_, var_y_}, 1)};
  const drake::VectorX<Monomial> basis2{
      OddDegreeMonomialBasis({var_x_, var_y_}, 2)};
  drake::Vector2<Monomial> expected;

  // OddDegreeMonomialBasis({x, y}, 1) = {x, y}.
  expected << Monomial{{{var_x_, 1}, {var_y_, 0}}},
      Monomial{{{var_x_, 0}, {var_y_, 1}}};

  EXPECT_EQ(basis1, expected);
  EXPECT_EQ(basis2, expected);
}

TEST_F(MonomialTest, OddDegreeMonomialBasisXY34) {
  const drake::VectorX<Monomial> basis1{
      OddDegreeMonomialBasis({var_x_, var_y_}, 3)};
  const drake::VectorX<Monomial> basis2{
      OddDegreeMonomialBasis({var_x_, var_y_}, 4)};
  drake::Vector6<Monomial> expected;

  // OddDegreeMonomialBasis({x, y}, 3) = {x³, x²y, xy², y³, x, y}
  // clang-format off
  expected << Monomial{{{var_x_, 3}, {var_y_, 0}}},
              Monomial{{{var_x_, 2}, {var_y_, 1}}},
              Monomial{{{var_x_, 1}, {var_y_, 2}}},
              Monomial{{{var_x_, 0}, {var_y_, 3}}},
              Monomial{{{var_x_, 1}, {var_y_, 0}}},
              Monomial{{{var_x_, 0}, {var_y_, 1}}};
  // clang-format on

  EXPECT_EQ(basis1, expected);
  EXPECT_EQ(basis2, expected);
}

TEST_F(MonomialTest, OddDegreeMonomialBasisXYZ34) {
  const drake::VectorX<Monomial> basis1{
      OddDegreeMonomialBasis({var_x_, var_y_, var_z_}, 3)};
  const drake::VectorX<Monomial> basis2{
      OddDegreeMonomialBasis({var_x_, var_y_, var_z_}, 4)};
  Eigen::Matrix<Monomial, 13, 1> expected;

  // OddDegreeMonomialBasis({x, y, z}, 3) = {x³, x²y, xy², y³, x²z, xyz, y²z,
  // xz², yz², z³, x, y, z}
  // clang-format off
  expected << Monomial{{{var_x_, 3}, {var_y_, 0}, {var_z_, 0}}},
              Monomial{{{var_x_, 2}, {var_y_, 1}, {var_z_, 0}}},
              Monomial{{{var_x_, 1}, {var_y_, 2}, {var_z_, 0}}},
              Monomial{{{var_x_, 0}, {var_y_, 3}, {var_z_, 0}}},
              Monomial{{{var_x_, 2}, {var_y_, 0}, {var_z_, 1}}},
              Monomial{{{var_x_, 1}, {var_y_, 1}, {var_z_, 1}}},
              Monomial{{{var_x_, 0}, {var_y_, 2}, {var_z_, 1}}},
              Monomial{{{var_x_, 1}, {var_y_, 0}, {var_z_, 2}}},
              Monomial{{{var_x_, 0}, {var_y_, 1}, {var_z_, 2}}},
              Monomial{{{var_x_, 0}, {var_y_, 0}, {var_z_, 3}}},
              Monomial{{{var_x_, 1}, {var_y_, 0}, {var_z_, 0}}},
              Monomial{{{var_x_, 0}, {var_y_, 1}, {var_z_, 0}}},
              Monomial{{{var_x_, 0}, {var_y_, 0}, {var_z_, 1}}};
  // clang-format on

  EXPECT_EQ(basis1, expected);
  EXPECT_EQ(basis2, expected);
}

// This test shows that we can have a std::unordered_map whose key is of
// Monomial.
TEST_F(MonomialTest, UnorderedMapOfMonomial) {
  unordered_map<Monomial, double> monomial_to_coeff_map;
  Monomial x_3{var_x_, 3};
  Monomial y_5{var_y_, 5};
  // Add 2 * x^3
  monomial_to_coeff_map.emplace(x_3, 2);
  // Add -7 * y^5
  monomial_to_coeff_map.emplace(y_5, -7);

  const auto it1 = monomial_to_coeff_map.find(x_3);
  ASSERT_TRUE(it1 != monomial_to_coeff_map.end());
  EXPECT_EQ(it1->second, 2);

  const auto it2 = monomial_to_coeff_map.find(y_5);
  ASSERT_TRUE(it2 != monomial_to_coeff_map.end());
  EXPECT_EQ(it2->second, -7);
}

// Converts a constant to monomial.
TEST_F(MonomialTest, ToMonomial0) {
  Monomial expected;
  EXPECT_EQ(Monomial(1), expected);
  EXPECT_EQ(Monomial(pow(x_, 0)), expected);
  EXPECT_THROW(Monomial(2), std::exception);
}

// Converts expression x to monomial.
TEST_F(MonomialTest, ToMonomial1) {
  Monomial expected(var_x_, 1);
  EXPECT_EQ(Monomial(x_), expected);
}

// Converts expression x * y to monomial.
TEST_F(MonomialTest, ToMonomial2) {
  std::map<Variable, int> powers;
  powers.emplace(var_x_, 1);
  powers.emplace(var_y_, 1);
  Monomial expected(powers);
  EXPECT_EQ(Monomial(x_ * y_), expected);
}

// Converts expression x^3 to monomial.
TEST_F(MonomialTest, ToMonomial3) {
  Monomial expected(var_x_, 3);
  EXPECT_EQ(Monomial(pow(x_, 3)), expected);
  EXPECT_EQ(Monomial(pow(x_, 2) * x_), expected);
}

// Converts expression x^3 * y to monomial.
TEST_F(MonomialTest, ToMonomial4) {
  std::map<Variable, int> powers;
  powers.emplace(var_x_, 3);
  powers.emplace(var_y_, 1);
  Monomial expected(powers);
  EXPECT_EQ(Monomial(pow(x_, 3) * y_), expected);
  EXPECT_EQ(Monomial(pow(x_, 2) * y_ * x_), expected);
}

// Converts expression x*(y+z) - x*y to monomial
TEST_F(MonomialTest, ToMonomial5) {
  std::map<Variable, int> powers({{var_x_, 1}, {var_z_, 1}});
  Monomial expected(powers);
  EXPECT_EQ(Monomial(x_ * z_), expected);
  EXPECT_EQ(Monomial(x_ * (y_ + z_) - x_ * y_), expected);
}

// `pow(x * y, 2) * pow(x^2 * y^2, 3)` is a monomial (x^8 * y^8).
TEST_F(MonomialTest, ToMonomial6) {
  const Monomial m1{pow(x_ * y_, 2) * pow(x_ * x_ * y_ * y_, 3)};
  const Monomial m2{{{var_x_, 8}, {var_y_, 8}}};
  EXPECT_EQ(m1, m2);
}

// x^0 is a monomial (1).
TEST_F(MonomialTest, ToMonomial7) {
  const Monomial m1(var_x_, 0);
  const Monomial m2{Expression{1.0}};
  const Monomial m3{pow(x_, 0.0)};
  EXPECT_EQ(m1, m2);
  EXPECT_EQ(m1, m3);
  EXPECT_EQ(m2, m3);
}

// `2 * x` is not a monomial because of its coefficient `2`.
TEST_F(MonomialTest, ToMonomialException1) {
  EXPECT_THROW(Monomial{2 * x_}, runtime_error);
}

// `x + y` is not a monomial.
TEST_F(MonomialTest, ToMonomialException2) {
  EXPECT_THROW(Monomial{x_ + y_}, runtime_error);
}

// `x / 2.0` is not a monomial.
TEST_F(MonomialTest, ToMonomialException3) {
  EXPECT_THROW(Monomial{x_ / 2.0}, runtime_error);
}

// `x ^ -1` is not a monomial.
TEST_F(MonomialTest, ToMonomialException4) {
  // Note: Parentheses are required around macro argument containing braced
  // initializer list.
  DRAKE_EXPECT_THROWS_MESSAGE((Monomial{{{var_x_, -1}}}),
                              "The exponent is negative.");
}

TEST_F(MonomialTest, Multiplication) {
  // m₁ = xy²
  const Monomial m1{{{var_x_, 1}, {var_y_, 2}}};
  // m₂ = y²z⁵
  const Monomial m2{{{var_y_, 2}, {var_z_, 5}}};
  // m₃ = m₁ * m₂ = xy⁴z⁵
  const Monomial m3{{{var_x_, 1}, {var_y_, 4}, {var_z_, 5}}};
  EXPECT_EQ(m1 * m2, m3);

  Monomial m{m1};  // m = m₁
  m *= m2;         // m = m₁ * m₂
  EXPECT_EQ(m, m1 * m2);
}

TEST_F(MonomialTest, Pow) {
  // m₁ = xy²
  const Monomial m1{{{var_x_, 1}, {var_y_, 2}}};
  // m₂ = y²z⁵
  const Monomial m2{{{var_y_, 2}, {var_z_, 5}}};

  // pow(m₁, 0) = 1.0
  EXPECT_EQ(pow(m1, 0), Monomial{});
  // pow(m₂, 0) = 1.0
  EXPECT_EQ(pow(m2, 0), Monomial{});

  // pow(m₁, 1) = m₁
  EXPECT_EQ(pow(m1, 1), m1);
  // pow(m₂, 1) = m₂
  EXPECT_EQ(pow(m2, 1), m2);

  // pow(m₁, 3) = x³y⁶
  const Monomial m1_cube = pow(m1, 3);
  EXPECT_EQ(m1_cube, Monomial{pow(x_, 3) * pow(y_, 6)});
  EXPECT_EQ(m1_cube.total_degree(), 9);
  // pow(m₂, 4) = y⁸z²⁰
  const Monomial m2_4th_power = pow(m2, 4);
  EXPECT_EQ(m2_4th_power, Monomial{pow(y_, 8) * pow(z_, 20)});
  EXPECT_EQ(m2_4th_power.total_degree(), 28);

  // pow(m₁, -1) throws an exception.
  EXPECT_THROW(pow(m1, -1), runtime_error);
  // pow(m₂, -5) throws an exception.
  EXPECT_THROW(pow(m2, -5), runtime_error);
}

TEST_F(MonomialTest, PowInPlace) {
  // m₁ = xy²
  Monomial m1{{{var_x_, 1}, {var_y_, 2}}};
  const Monomial m1_copy{m1};
  EXPECT_EQ(m1, m1_copy);

  // m₁.pow_in_place modifies m₁.
  m1.pow_in_place(2);
  EXPECT_NE(m1, m1_copy);
  EXPECT_EQ(m1, Monomial({{var_x_, 2}, {var_y_, 4}}));
  EXPECT_EQ(m1.total_degree(), 6);

  // m₁ gets m₁⁰, which is 1.
  m1.pow_in_place(0);
  EXPECT_EQ(m1, Monomial());
  EXPECT_EQ(m1.total_degree(), 0);
}

TEST_F(MonomialTest, Evaluate) {
  const vector<Environment> environments{
      {{var_x_, 1.0}, {var_y_, 2.0}, {var_z_, 3.0}},     // + + +
      {{var_x_, -1.0}, {var_y_, 2.0}, {var_z_, 3.0}},    // - + +
      {{var_x_, 1.0}, {var_y_, -2.0}, {var_z_, 3.0}},    // + - +
      {{var_x_, 1.0}, {var_y_, 2.0}, {var_z_, -3.0}},    // + + -
      {{var_x_, -1.0}, {var_y_, -2.0}, {var_z_, 3.0}},   // - - +
      {{var_x_, 1.0}, {var_y_, -2.0}, {var_z_, -3.0}},   // + - -
      {{var_x_, -1.0}, {var_y_, 2.0}, {var_z_, -3.0}},   // - + -
      {{var_x_, -1.0}, {var_y_, -2.0}, {var_z_, -3.0}},  // - - -
  };
  for (const Monomial& m : monomials_) {
    for (const Environment& env : environments) {
      CheckEvaluate(m, env);
    }
  }
}

TEST_F(MonomialTest, EvaluateException) {
  const Monomial m{{{var_x_, 1}, {var_y_, 2}}};  // xy^2
  const Environment env{{{var_x_, 1.0}}};
  EXPECT_THROW(m.Evaluate(env), runtime_error);
}

void CheckEvaluateBatch(
    const Monomial& dut,
    const Eigen::Ref<const VectorX<symbolic::Variable>>& vars,
    const Eigen::Ref<const Eigen::MatrixXd>& vars_val) {
  const Eigen::VectorXd monomial_vals = dut.Evaluate(vars, vars_val);
  EXPECT_EQ(monomial_vals.rows(), vars_val.cols());
  for (int i = 0; i < vars_val.cols(); ++i) {
    symbolic::Environment env;
    env.insert(vars, vars_val.col(i));
    EXPECT_EQ(monomial_vals(i), dut.Evaluate(env));
  }
}

TEST_F(MonomialTest, EvaluateBatch) {
  // Test Evaluate for a batch of data.
  const symbolic::Monomial monomial({{var_x_, 2}, {var_y_, 3}});
  // vars1 contains exactly the same variables as in monomial.
  Vector2<symbolic::Variable> vars1(var_y_, var_x_);
  Eigen::Matrix<double, 2, 3> vars1_val;
  // clang-format off
  vars1_val << 2, 3, 4,
               5, 6, 7;
  // clang-format on
  CheckEvaluateBatch(monomial, vars1, vars1_val);
  CheckEvaluateBatch(monomial, vars1, Eigen::Vector2d(2, 3));

  // vars2 contains more variables than monomial.
  Vector3<symbolic::Variable> vars2(var_y_, var_x_, var_z_);
  Eigen::Matrix<double, 3, 4> vars2_val;
  // clang-format off
  vars2_val << 2, 3, 4, 5,
               -6, -7, 8, 9,
               -1, -3, 2, 4;
  // clang-format on
  CheckEvaluateBatch(monomial, vars2, vars2_val);
}

TEST_F(MonomialTest, EvaluateBatchException) {
  // Test Evaluate for a batch of data with exception.
  const symbolic::Monomial monomial({{var_x_, 2}, {var_y_, 3}});
  DRAKE_EXPECT_THROWS_MESSAGE(
      monomial.Evaluate(Vector3<symbolic::Variable>(var_x_, var_x_, var_y_),
                        Eigen::Vector3d(1, 2, 3)),
      ".* vars contains repeated variables.");

  DRAKE_EXPECT_THROWS_MESSAGE(
      monomial.Evaluate(Vector2<symbolic::Variable>(var_x_, var_z_),
                        Eigen::Vector2d(2, 3)),
      ".* y is not present in vars");
}

TEST_F(MonomialTest, EvaluatePartial) {
  const vector<Environment> environments{
      {{var_x_, 2.0}},
      {{var_y_, 3.0}},
      {{var_z_, 4.0}},
      {{var_x_, 2.0}, {var_y_, -2.0}},
      {{var_y_, -4.0}, {var_z_, 2.5}},
      {{var_x_, -2.3}, {var_z_, 2.6}},
      {{var_x_, -1.0}, {var_y_, 2.0}, {var_z_, 3.0}},
  };
  for (const Monomial& m : monomials_) {
    for (const Environment& env : environments) {
      CheckEvaluatePartial(m, env);
    }
  }
}

TEST_F(MonomialTest, DegreeVariable) {
  EXPECT_EQ(Monomial().degree(var_x_), 0);
  EXPECT_EQ(Monomial().degree(var_y_), 0);

  EXPECT_EQ(Monomial(var_x_).degree(var_x_), 1);
  EXPECT_EQ(Monomial(var_x_).degree(var_y_), 0);

  EXPECT_EQ(Monomial(var_x_, 4).degree(var_x_), 4);
  EXPECT_EQ(Monomial(var_x_, 4).degree(var_y_), 0);

  EXPECT_EQ(Monomial({{var_x_, 1}, {var_y_, 2}}).degree(var_x_), 1);
  EXPECT_EQ(Monomial({{var_x_, 1}, {var_y_, 2}}).degree(var_y_), 2);
}

TEST_F(MonomialTest, TotalDegree) {
  EXPECT_EQ(Monomial().total_degree(), 0);
  EXPECT_EQ(Monomial(var_x_).total_degree(), 1);
  EXPECT_EQ(Monomial(var_x_, 1).total_degree(), 1);
  EXPECT_EQ(Monomial(var_x_, 4).total_degree(), 4);
  EXPECT_EQ(Monomial({{var_x_, 1}, {var_y_, 2}}).total_degree(), 3);
}

}  // namespace
}  // namespace symbolic
}  // namespace drake