testMatrix.cpp

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/* ----------------------------------------------------------------------------
 
 * GTSAM Copyright 2010, Georgia Tech Research Corporation,
 * Atlanta, Georgia 30332-0415
 * All Rights Reserved
 * Authors: Frank Dellaert, et al. (see THANKS for the full author list)
 
 * See LICENSE for the license information
 
 * -------------------------------------------------------------------------- */
 
#include <gtsam/base/Matrix.h>
#include <gtsam/base/VectorSpace.h>
#include <gtsam/base/testLie.h>
#include <CppUnitLite/TestHarness.h>
#include <iostream>
#include <sstream>
#include <optional>
#include <functional>
 
using namespace std;
using namespace gtsam;
 
static double inf = std::numeric_limits<double>::infinity();
static const double tol = 1e-9;
 
/* ************************************************************************* */
TEST(Matrix, constructor_data )
{
  Matrix A = (Matrix(2, 2) << -5, 3, 0, -5).finished();
 
  Matrix B(2, 2);
  B(0, 0) = -5;
  B(0, 1) = 3;
  B(1, 0) = 0;
  B(1, 1) = -5;
 
  EQUALITY(A,B);
}
 
/* ************************************************************************* */
TEST(Matrix, Matrix_ )
{
  Matrix A = (Matrix(2, 2) << -5.0, 3.0, 0.0, -5.0).finished();
  Matrix B(2, 2);
  B(0, 0) = -5;
  B(0, 1) = 3;
  B(1, 0) = 0;
  B(1, 1) = -5;
 
  EQUALITY(A,B);
 
}
 
namespace {
  /* ************************************************************************* */
  template<typename Derived>
  Matrix testFcn1(const Eigen::DenseBase<Derived>& in)
  {
    return in;
  }
 
  /* ************************************************************************* */
  template<typename Derived>
  Matrix testFcn2(const Eigen::MatrixBase<Derived>& in)
  {
    return in;
  }
}
 
/* ************************************************************************* */
TEST(Matrix, special_comma_initializer)
{
  Matrix expected(2,2);
  expected(0,0) = 1;
  expected(0,1) = 2;
  expected(1,0) = 3;
  expected(1,1) = 4;
 
  Matrix actual1 = (Matrix(2,2) << 1, 2, 3, 4).finished();
  Matrix actual2((Matrix(2,2) << 1, 2, 3, 4).finished());
 
  Matrix submat1 = (Matrix(1,2) << 3, 4).finished();
  Matrix actual3 = (Matrix(2,2) << 1, 2, submat1).finished();
 
  Matrix submat2 = (Matrix(1,2) << 1, 2).finished();
  Matrix actual4 = (Matrix(2,2) << submat2, 3, 4).finished();
 
  Matrix actual5 = testFcn1((Matrix(2,2) << 1, 2, 3, 4).finished());
  Matrix actual6 = testFcn2((Matrix(2,2) << 1, 2, 3, 4).finished());
 
  EXPECT(assert_equal(expected, actual1));
  EXPECT(assert_equal(expected, actual2));
  EXPECT(assert_equal(expected, actual3));
  EXPECT(assert_equal(expected, actual4));
  EXPECT(assert_equal(expected, actual5));
  EXPECT(assert_equal(expected, actual6));
}
 
/* ************************************************************************* */
TEST(Matrix, col_major )
{
  Matrix A = (Matrix(2, 2) << 1.0, 2.0, 3.0, 4.0).finished();
  const double * const a = &A(0, 0);
  EXPECT_DOUBLES_EQUAL(1, a[0], tol);
  EXPECT_DOUBLES_EQUAL(3, a[1], tol);
  EXPECT_DOUBLES_EQUAL(2, a[2], tol);
  EXPECT_DOUBLES_EQUAL(4, a[3], tol);
}
 
/* ************************************************************************* */
TEST(Matrix, collect1 )
{
  Matrix A = (Matrix(2, 2) << -5.0, 3.0, 00.0, -5.0).finished();
  Matrix B = (Matrix(2, 3) << -0.5, 2.1, 1.1, 3.4, 2.6, 7.1).finished();
  Matrix AB = collect(2, &A, &B);
  Matrix C(2, 5);
  for (int i = 0; i < 2; i++)
    for (int j = 0; j < 2; j++)
      C(i, j) = A(i, j);
  for (int i = 0; i < 2; i++)
    for (int j = 0; j < 3; j++)
      C(i, j + 2) = B(i, j);
 
  EQUALITY(C,AB);
}
 
/* ************************************************************************* */
TEST(Matrix, collect2 )
{
  Matrix A = (Matrix(2, 2) << -5.0, 3.0, 00.0, -5.0).finished();
  Matrix B = (Matrix(2, 3) << -0.5, 2.1, 1.1, 3.4, 2.6, 7.1).finished();
  vector<const Matrix*> matrices;
  matrices.push_back(&A);
  matrices.push_back(&B);
  Matrix AB = collect(matrices);
  Matrix C(2, 5);
  for (int i = 0; i < 2; i++)
    for (int j = 0; j < 2; j++)
      C(i, j) = A(i, j);
  for (int i = 0; i < 2; i++)
    for (int j = 0; j < 3; j++)
      C(i, j + 2) = B(i, j);
 
  EQUALITY(C,AB);
}
 
/* ************************************************************************* */
TEST(Matrix, collect3 )
{
  Matrix A, B;
  A = Matrix::Identity(2,3);
  B = Matrix::Identity(2,3);
  vector<const Matrix*> matrices;
  matrices.push_back(&A);
  matrices.push_back(&B);
  Matrix AB = collect(matrices, 2, 3);
  Matrix exp = (Matrix(2, 6) <<
      1.0, 0.0, 0.0, 1.0, 0.0, 0.0,
      0.0, 1.0, 0.0, 0.0, 1.0, 0.0).finished();
 
  EQUALITY(exp,AB);
}
 
/* ************************************************************************* */
TEST(Matrix, stack )
{
  Matrix A = (Matrix(2, 2) << -5.0, 3.0, 00.0, -5.0).finished();
  Matrix B = (Matrix(3, 2) << -0.5, 2.1, 1.1, 3.4, 2.6, 7.1).finished();
  Matrix AB = gtsam::stack(2, &A, &B);
  Matrix C(5, 2);
  for (int i = 0; i < 2; i++)
    for (int j = 0; j < 2; j++)
      C(i, j) = A(i, j);
  for (int i = 0; i < 3; i++)
    for (int j = 0; j < 2; j++)
      C(i + 2, j) = B(i, j);
 
  EQUALITY(C,AB);
 
  std::vector<gtsam::Matrix> matrices;
  matrices.push_back(A);
  matrices.push_back(B);
  Matrix AB2 = gtsam::stack(matrices);
  EQUALITY(C,AB2);
}
 
/* ************************************************************************* */
TEST(Matrix, column )
{
  Matrix A = (Matrix(4, 7) << -1., 0., 1., 0., 0., 0., -0.2, 0., -1., 0., 1.,
      0., 0., 0.3, 1., 0., 0., 0., -1., 0., 0.2, 0., 1., 0., 0., 0., -1.,
      -0.1).finished();
  Vector a1 = column(A, 0);
  Vector exp1 = (Vector(4) << -1., 0., 1., 0.).finished();
  EXPECT(assert_equal(a1, exp1));
 
  Vector a2 = column(A, 3);
  Vector exp2 = (Vector(4) << 0., 1., 0., 0.).finished();
  EXPECT(assert_equal(a2, exp2));
 
  Vector a3 = column(A, 6);
  Vector exp3 = (Vector(4) << -0.2, 0.3, 0.2, -0.1).finished();
  EXPECT(assert_equal(a3, exp3));
}
 
/* ************************************************************************* */
TEST(Matrix, row )
{
  Matrix A = (Matrix(4, 7) << -1., 0., 1., 0., 0., 0., -0.2, 0., -1., 0., 1.,
      0., 0., 0.3, 1., 0., 0., 0., -1., 0., 0.2, 0., 1., 0., 0., 0., -1.,
      -0.1).finished();
  Vector a1 = row(A, 0);
  Vector exp1 = (Vector(7) << -1., 0., 1., 0., 0., 0., -0.2).finished();
  EXPECT(assert_equal(a1, exp1));
 
  Vector a2 = row(A, 2);
  Vector exp2 = (Vector(7) << 1., 0., 0., 0., -1., 0., 0.2).finished();
  EXPECT(assert_equal(a2, exp2));
 
  Vector a3 = row(A, 3);
  Vector exp3 = (Vector(7) << 0., 1., 0., 0., 0., -1., -0.1).finished();
  EXPECT(assert_equal(a3, exp3));
}
 
/* ************************************************************************* */
TEST(Matrix, insert_sub )
{
  Matrix big = Matrix::Zero(5,6), small = (Matrix(2, 3) << 1.0, 1.0, 1.0, 1.0, 1.0,
      1.0).finished();
 
  insertSub(big, small, 1, 2);
 
  Matrix expected = (Matrix(5, 6) << 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0,
      1.0, 1.0, 1.0, 0.0, 0.0, 0.0, 1.0, 1.0, 1.0, 0.0, 0.0, 0.0, 0.0,
      0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0).finished();
 
  EXPECT(assert_equal(expected, big));
}
 
/* ************************************************************************* */
TEST(Matrix, diagMatrices )
{
  std::vector<Matrix> Hs;
  Hs.push_back(Matrix::Ones(3,3));
  Hs.push_back(Matrix::Ones(4,4)*2);
  Hs.push_back(Matrix::Ones(2,2)*3);
 
  Matrix actual = diag(Hs);
 
  Matrix expected = (Matrix(9, 9) <<
      1.0, 1.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0,
      1.0, 1.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0,
      1.0, 1.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0,
      0.0, 0.0, 0.0, 2.0, 2.0, 2.0, 2.0, 0.0, 0.0,
      0.0, 0.0, 0.0, 2.0, 2.0, 2.0, 2.0, 0.0, 0.0,
      0.0, 0.0, 0.0, 2.0, 2.0, 2.0, 2.0, 0.0, 0.0,
      0.0, 0.0, 0.0, 2.0, 2.0, 2.0, 2.0, 0.0, 0.0,
      0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 3.0, 3.0,
      0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 3.0, 3.0).finished();
 
  EXPECT(assert_equal(expected, actual));
}
 
/* ************************************************************************* */
TEST(Matrix, stream_read ) {
  Matrix expected = (Matrix(3,4) <<
    1.1, 2.3, 4.2, 7.6,
    -0.3, -8e-2, 5.1, 9.0,
    1.2, 3.4, 4.5, 6.7).finished();
 
  string matrixAsString =
    "1.1 2.3 4.2 7.6\n"
    "-0.3 -8e-2 5.1    9.0\n\r" // Test extra spaces and windows newlines
    "1.2 \t 3.4 4.5 6.7"; // Test tab as separator
 
  stringstream asStream(matrixAsString, ios::in);
 
  Matrix actual;
  asStream >> actual;
 
  EXPECT(assert_equal(expected, actual));
}
 
/* ************************************************************************* */
TEST(Matrix, scale_columns )
{
  Matrix A(3, 4);
  A(0, 0) = 1.;
  A(0, 1) = 1.;
  A(0, 2) = 1.;
  A(0, 3) = 1.;
  A(1, 0) = 1.;
  A(1, 1) = 1.;
  A(1, 2) = 1.;
  A(1, 3) = 1.;
  A(2, 0) = 1.;
  A(2, 1) = 1.;
  A(2, 2) = 1.;
  A(2, 3) = 1.;
 
  Vector v = (Vector(4) << 2., 3., 4., 5.).finished();
 
  Matrix actual = vector_scale(A, v);
 
  Matrix expected(3, 4);
  expected(0, 0) = 2.;
  expected(0, 1) = 3.;
  expected(0, 2) = 4.;
  expected(0, 3) = 5.;
  expected(1, 0) = 2.;
  expected(1, 1) = 3.;
  expected(1, 2) = 4.;
  expected(1, 3) = 5.;
  expected(2, 0) = 2.;
  expected(2, 1) = 3.;
  expected(2, 2) = 4.;
  expected(2, 3) = 5.;
 
  EXPECT(assert_equal(actual, expected));
}
 
/* ************************************************************************* */
TEST(Matrix, scale_rows )
{
  Matrix A(3, 4);
  A(0, 0) = 1.;
  A(0, 1) = 1.;
  A(0, 2) = 1.;
  A(0, 3) = 1.;
  A(1, 0) = 1.;
  A(1, 1) = 1.;
  A(1, 2) = 1.;
  A(1, 3) = 1.;
  A(2, 0) = 1.;
  A(2, 1) = 1.;
  A(2, 2) = 1.;
  A(2, 3) = 1.;
 
  Vector v = Vector3(2., 3., 4.);
 
  Matrix actual = vector_scale(v, A);
 
  Matrix expected(3, 4);
  expected(0, 0) = 2.;
  expected(0, 1) = 2.;
  expected(0, 2) = 2.;
  expected(0, 3) = 2.;
  expected(1, 0) = 3.;
  expected(1, 1) = 3.;
  expected(1, 2) = 3.;
  expected(1, 3) = 3.;
  expected(2, 0) = 4.;
  expected(2, 1) = 4.;
  expected(2, 2) = 4.;
  expected(2, 3) = 4.;
 
  EXPECT(assert_equal(actual, expected));
}
 
/* ************************************************************************* */
TEST(Matrix, scale_rows_mask )
{
  Matrix A(3, 4);
  A(0, 0) = 1.;
  A(0, 1) = 1.;
  A(0, 2) = 1.;
  A(0, 3) = 1.;
  A(1, 0) = 1.;
  A(1, 1) = 1.;
  A(1, 2) = 1.;
  A(1, 3) = 1.;
  A(2, 0) = 1.;
  A(2, 1) = 1.;
  A(2, 2) = 1.;
  A(2, 3) = 1.;
 
  Vector v = (Vector(3) << 2., std::numeric_limits<double>::infinity(), 4.).finished();
 
  Matrix actual = vector_scale(v, A, true);
 
  Matrix expected(3, 4);
  expected(0, 0) = 2.;
  expected(0, 1) = 2.;
  expected(0, 2) = 2.;
  expected(0, 3) = 2.;
  expected(1, 0) = 1.;
  expected(1, 1) = 1.;
  expected(1, 2) = 1.;
  expected(1, 3) = 1.;
  expected(2, 0) = 4.;
  expected(2, 1) = 4.;
  expected(2, 2) = 4.;
  expected(2, 3) = 4.;
 
  EXPECT(assert_equal(actual, expected));
}
 
/* ************************************************************************* */
TEST(Matrix, skewSymmetric )
{
  double wx = 1, wy = 2, wz = 3;
  Matrix3 actual = skewSymmetric(wx,wy,wz);
 
  Matrix expected(3,3);
  expected << 0, -3, 2,
      3, 0, -1,
      -2, 1, 0;
 
  EXPECT(assert_equal(actual, expected));
 
}
 
 
/* ************************************************************************* */
TEST(Matrix, equal )
{
  Matrix A(4, 4);
  A(0, 0) = -1;
  A(0, 1) = 1;
  A(0, 2) = 2;
  A(0, 3) = 3;
  A(1, 0) = 1;
  A(1, 1) = -3;
  A(1, 2) = 1;
  A(1, 3) = 3;
  A(2, 0) = 1;
  A(2, 1) = 2;
  A(2, 2) = -1;
  A(2, 3) = 4;
  A(3, 0) = 2;
  A(3, 1) = 1;
  A(3, 2) = 2;
  A(3, 3) = -2;
 
  Matrix A2(A);
 
  Matrix A3(A);
  A3(3, 3) = -2.1;
 
  EXPECT(A==A2);
  EXPECT(A!=A3);
}
 
/* ************************************************************************* */
TEST(Matrix, equal_nan )
{
  Matrix A(4, 4);
  A(0, 0) = -1;
  A(0, 1) = 1;
  A(0, 2) = 2;
  A(0, 3) = 3;
  A(1, 0) = 1;
  A(1, 1) = -3;
  A(1, 2) = 1;
  A(1, 3) = 3;
  A(2, 0) = 1;
  A(2, 1) = 2;
  A(2, 2) = -1;
  A(2, 3) = 4;
  A(3, 0) = 2;
  A(3, 1) = 1;
  A(3, 2) = 2;
  A(3, 3) = -2;
 
  Matrix A2(A);
 
  Matrix A3(A);
  A3(3, 3) = inf;
 
  EXPECT(A!=A3);
}
 
/* ************************************************************************* */
TEST(Matrix, addition )
{
  Matrix A = (Matrix(2, 2) << 1.0, 2.0, 3.0, 4.0).finished();
  Matrix B = (Matrix(2, 2) << 4.0, 3.0, 2.0, 1.0).finished();
  Matrix C = (Matrix(2, 2) << 5.0, 5.0, 5.0, 5.0).finished();
  EQUALITY(A+B,C);
}
 
/* ************************************************************************* */
TEST(Matrix, addition_in_place )
{
  Matrix A = (Matrix(2, 2) << 1.0, 2.0, 3.0, 4.0).finished();
  Matrix B = (Matrix(2, 2) << 4.0, 3.0, 2.0, 1.0).finished();
  Matrix C = (Matrix(2, 2) << 5.0, 5.0, 5.0, 5.0).finished();
  A += B;
  EQUALITY(A,C);
}
 
/* ************************************************************************* */
TEST(Matrix, subtraction )
{
  Matrix A = (Matrix(2, 2) << 1.0, 2.0, 3.0, 4.0).finished();
  Matrix B = (Matrix(2, 2) << 4.0, 3.0, 2.0, 1.0).finished();
  Matrix C = (Matrix(2, 2) << -3.0, -1.0, 1.0, 3.0).finished();
  EQUALITY(A-B,C);
}
 
/* ************************************************************************* */
TEST(Matrix, subtraction_in_place )
{
  Matrix A = (Matrix(2, 2) << 1.0, 2.0, 3.0, 4.0).finished();
  Matrix B = (Matrix(2, 2) << 4.0, 3.0, 2.0, 1.0).finished();
  Matrix C = (Matrix(2, 2) << -3.0, -1.0, 1.0, 3.0).finished();
  A -= B;
  EQUALITY(A,C);
}
 
/* ************************************************************************* */
TEST(Matrix, multiplication )
{
  Matrix A(2, 2);
  A(0, 0) = -1;
  A(1, 0) = 1;
  A(0, 1) = 1;
  A(1, 1) = -3;
 
  Matrix B(2, 1);
  B(0, 0) = 1.2;
  B(1, 0) = 3.4;
 
  Matrix AB(2, 1);
  AB(0, 0) = 2.2;
  AB(1, 0) = -9.;
 
  EQUALITY(A*B,AB);
}
 
/* ************************************************************************* */
TEST(Matrix, scalar_matrix_multiplication )
{
  Vector result(2);
 
  Matrix A(2, 2);
  A(0, 0) = -1;
  A(1, 0) = 1;
  A(0, 1) = 1;
  A(1, 1) = -3;
 
  Matrix B(2, 2);
  B(0, 0) = -10;
  B(1, 0) = 10;
  B(0, 1) = 10;
  B(1, 1) = -30;
 
  EQUALITY((10*A),B);
}
 
/* ************************************************************************* */
TEST(Matrix, matrix_vector_multiplication )
{
  Vector result(2);
 
  Matrix A = (Matrix(2, 3) << 1.0, 2.0, 3.0, 4.0, 5.0, 6.0).finished();
  Vector v = Vector3(1., 2., 3.);
  Vector Av = Vector2(14., 32.);
  Vector AtAv = Vector3(142., 188., 234.);
 
  EQUALITY(A*v,Av);
  EQUALITY(A.transpose() * Av,AtAv);
}
 
/* ************************************************************************* */
TEST(Matrix, nrRowsAndnrCols )
{
  Matrix A(3, 6);
  LONGS_EQUAL( A.rows() , 3 );
  LONGS_EQUAL( A.cols() , 6 );
}
 
/* ************************************************************************* */
TEST(Matrix, scalar_divide )
{
  Matrix A(2, 2);
  A(0, 0) = 10;
  A(1, 0) = 30;
  A(0, 1) = 20;
  A(1, 1) = 40;
 
  Matrix B(2, 2);
  B(0, 0) = 1;
  B(1, 0) = 3;
  B(0, 1) = 2;
  B(1, 1) = 4;
 
  EQUALITY(B,A/10);
}
 
/* ************************************************************************* */
TEST(Matrix, inverse )
{
  Matrix A(3, 3);
  A(0, 0) = 1;
  A(0, 1) = 2;
  A(0, 2) = 3;
  A(1, 0) = 0;
  A(1, 1) = 4;
  A(1, 2) = 5;
  A(2, 0) = 1;
  A(2, 1) = 0;
  A(2, 2) = 6;
 
  Matrix Ainv = A.inverse();
  EXPECT(assert_equal((Matrix) I_3x3, A*Ainv));
  EXPECT(assert_equal((Matrix) I_3x3, Ainv*A));
 
  Matrix expected(3, 3);
  expected(0, 0) = 1.0909;
  expected(0, 1) = -0.5454;
  expected(0, 2) = -0.0909;
  expected(1, 0) = 0.2272;
  expected(1, 1) = 0.1363;
  expected(1, 2) = -0.2272;
  expected(2, 0) = -0.1818;
  expected(2, 1) = 0.0909;
  expected(2, 2) = 0.1818;
 
  EXPECT(assert_equal(expected, Ainv, 1e-4));
 
  // These two matrices failed before version 2003 because we called LU incorrectly
  Matrix lMg((Matrix(3, 3) << 0.0, 1.0, -2.0, -1.0, 0.0, 1.0, 0.0, 0.0, 1.0).finished());
  EXPECT(assert_equal((Matrix(3, 3) <<
      0.0, -1.0, 1.0,
      1.0, 0.0, 2.0,
      0.0, 0.0, 1.0).finished(),
      lMg.inverse()));
  Matrix gMl((Matrix(3, 3) << 0.0, -1.0, 1.0, 1.0, 0.0, 2.0, 0.0, 0.0, 1.0).finished());
  EXPECT(assert_equal((Matrix(3, 3) <<
      0.0, 1.0,-2.0,
      -1.0, 0.0, 1.0,
      0.0, 0.0, 1.0).finished(),
      gMl.inverse()));
}
 
/* ************************************************************************* */
TEST(Matrix, inverse2 )
{
  Matrix A(3, 3);
  A(0, 0) = 0;
  A(0, 1) = -1;
  A(0, 2) = 1;
  A(1, 0) = 1;
  A(1, 1) = 0;
  A(1, 2) = 2;
  A(2, 0) = 0;
  A(2, 1) = 0;
  A(2, 2) = 1;
 
  Matrix Ainv = A.inverse();
 
  Matrix expected(3, 3);
  expected(0, 0) = 0;
  expected(0, 1) = 1;
  expected(0, 2) = -2;
  expected(1, 0) = -1;
  expected(1, 1) = 0;
  expected(1, 2) = 1;
  expected(2, 0) = 0;
  expected(2, 1) = 0;
  expected(2, 2) = 1;
 
  EXPECT(assert_equal(expected, Ainv, 1e-4));
}
 
/* ************************************************************************* */
TEST(Matrix, backsubtitution )
{
  // TEST ONE  2x2 matrix U1*x=b1
  Vector expected1 = Vector2(3.6250, -0.75);
  Matrix U22 = (Matrix(2, 2) << 2., 3., 0., 4.).finished();
  Vector b1 = U22 * expected1;
  EXPECT( assert_equal(expected1 , backSubstituteUpper(U22, b1), 0.000001));
 
  // TEST TWO  3x3 matrix U2*x=b2
  Vector expected2 = Vector3(5.5, -8.5, 5.);
  Matrix U33 = (Matrix(3, 3) << 3., 5., 6., 0., 2., 3., 0., 0., 1.).finished();
  Vector b2 = U33 * expected2;
  EXPECT( assert_equal(expected2 , backSubstituteUpper(U33, b2), 0.000001));
 
  // TEST THREE  Lower triangular 3x3 matrix L3*x=b3
  Vector expected3 = Vector3(1., 1., 1.);
  Matrix L3 = trans(U33);
  Vector b3 = L3 * expected3;
  EXPECT( assert_equal(expected3 , backSubstituteLower(L3, b3), 0.000001));
 
  // TEST FOUR Try the above with transpose backSubstituteUpper
  EXPECT( assert_equal(expected3 , backSubstituteUpper(b3,U33), 0.000001));
}
 
/* ************************************************************************* */
TEST(Matrix, householder )
{
  // check in-place householder, with v vectors below diagonal
 
  Matrix expected1 = (Matrix(4, 7) << 11.1803, 0, -2.2361, 0, -8.9443, 0, 2.236,
      0, 11.1803,  0, -2.2361, 0, -8.9443, -1.565,
      -0.618034, 0, 4.4721, 0, -4.4721,  0, 0,
      0, -0.618034, 0, 4.4721, 0, -4.4721, 0.894).finished();
  Matrix A1 = (Matrix(4, 7) << -5, 0, 5, 0, 0, 0, -1,
      00,-5, 0, 5, 0, 0, 1.5,
      10, 0, 0,  0,-10,0,   2,
      00, 10,0, 0, 0, -10, -1 ).finished();
  householder_(A1, 3);
  EXPECT(assert_equal(expected1, A1, 1e-3));
 
  // in-place, with zeros below diagonal
 
  Matrix expected = (Matrix(4, 7) << 11.1803, 0, -2.2361, 0, -8.9443, 0, 2.236, 0, 11.1803,
      0, -2.2361, 0, -8.9443, -1.565, 0, 0, 4.4721, 0, -4.4721, 0, 0, 0,
      0, 0, 4.4721, 0, -4.4721, 0.894).finished();
  Matrix A2 = (Matrix(4, 7) << -5, 0, 5, 0, 0, 0, -1,
      00,-5, 0, 5, 0, 0, 1.5,
      10, 0, 0,  0,-10,0,   2,
      00, 10,0, 0, 0, -10, -1).finished();
  householder(A2, 3);
  EXPECT(assert_equal(expected, A2, 1e-3));
}
 
/* ************************************************************************* */
TEST(Matrix, householder_colMajor )
{
  // check in-place householder, with v vectors below diagonal
 
  Matrix expected1((Matrix(4, 7) << 11.1803, 0, -2.2361, 0, -8.9443, 0, 2.236,
      0, 11.1803,  0, -2.2361, 0, -8.9443, -1.565,
      -0.618034, 0, 4.4721, 0, -4.4721,  0, 0,
      0, -0.618034, 0, 4.4721, 0, -4.4721, 0.894).finished());
  Matrix A1((Matrix(4, 7) << -5, 0, 5, 0, 0, 0, -1,
      00,-5, 0, 5, 0, 0, 1.5,
      10, 0, 0,  0,-10,0,   2,
      00, 10,0, 0, 0, -10, -1).finished());
  householder_(A1, 3);
  EXPECT(assert_equal(expected1, A1, 1e-3));
 
  // in-place, with zeros below diagonal
 
  Matrix expected((Matrix(4, 7) << 11.1803, 0, -2.2361, 0, -8.9443, 0, 2.236, 0, 11.1803,
      0, -2.2361, 0, -8.9443, -1.565, 0, 0, 4.4721, 0, -4.4721, 0, 0, 0,
      0, 0, 4.4721, 0, -4.4721, 0.894).finished());
  Matrix A2((Matrix(4, 7) << -5, 0, 5, 0, 0, 0, -1,
      00,-5, 0, 5, 0, 0, 1.5,
      10, 0, 0,  0,-10,0,   2,
      00, 10,0, 0, 0, -10, -1).finished());
  householder(A2, 3);
  EXPECT(assert_equal(expected, A2, 1e-3));
}
 
/* ************************************************************************* */
TEST(Matrix, eigen_QR )
{
  // use standard Eigen function to yield a non-in-place QR factorization
 
  // in-place, with zeros below diagonal
 
  Matrix expected((Matrix(4, 7) << 11.1803, 0, -2.2361, 0, -8.9443, 0, 2.236, 0, 11.1803,
      0, -2.2361, 0, -8.9443, -1.565, 0, 0, 4.4721, 0, -4.4721, 0, 0, 0,
      0, 0, 4.4721, 0, -4.4721, 0.894).finished());
  Matrix A((Matrix(4, 7) << -5, 0, 5, 0, 0, 0, -1,
      00,-5, 0, 5, 0, 0, 1.5,
      10, 0, 0,  0,-10,0,   2,
      00, 10,0, 0, 0, -10, -1).finished());
  Matrix actual = A.householderQr().matrixQR();
  actual.triangularView<Eigen::StrictlyLower>().setZero();
 
  EXPECT(assert_equal(expected, actual, 1e-3));
 
  // use shiny new in place QR inside gtsam
  A = Matrix((Matrix(4, 7) << -5, 0, 5, 0, 0, 0, -1,
      00,-5, 0, 5, 0, 0, 1.5,
      10, 0, 0,  0,-10,0,   2,
      00, 10,0, 0, 0, -10, -1).finished());
  inplace_QR(A);
  EXPECT(assert_equal(expected, A, 1e-3));
}
 
/* ************************************************************************* */
// unit test for qr factorization (and hence householder)
// This behaves the same as QR in matlab: [Q,R] = qr(A), except for signs
/* ************************************************************************* */
TEST(Matrix, qr )
{
 
  Matrix A = (Matrix(6, 4) << -5, 0, 5, 0, 00, -5, 0, 5, 10, 0, 0, 0, 00, 10, 0, 0, 00,
      0, 0, -10, 10, 0, -10, 0).finished();
 
 
  Matrix expectedQ = (Matrix(6, 6) << -0.3333, 0, 0.2981, 0, 0, -0.8944, 0000000, -0.4472, 0,
      0.3651, -0.8165, 0, 00.6667, 0, 0.7454, 0, 0, 0, 0000000, 0.8944,
      0, 0.1826, -0.4082, 0, 0000000, 0, 0, -0.9129, -0.4082, 0, 00.6667,
      0, -0.5963, 0, 0, -0.4472).finished();
 
  Matrix expectedR = (Matrix(6, 4) << 15, 0, -8.3333, 0, 00, 11.1803, 0, -2.2361, 00, 0,
      7.4536, 0, 00, 0, 0, 10.9545, 00, 0, 0, 0, 00, 0, 0, 0).finished();
 
  const auto [Q, R] = qr(A);
  EXPECT(assert_equal(expectedQ, Q, 1e-4));
  EXPECT(assert_equal(expectedR, R, 1e-4));
  EXPECT(assert_equal(A, Q*R, 1e-14));
}
 
/* ************************************************************************* */
TEST(Matrix, sub )
{
  Matrix A = (Matrix(4, 6) << -5, 0, 5, 0, 0, 0, 00, -5, 0, 5, 0, 0, 10, 0, 0, 0, -10,
      0, 00, 10, 0, 0, 0, -10).finished();
  Matrix actual = sub(A, 1, 3, 1, 5);
 
  Matrix expected = (Matrix(2, 4) << -5, 0, 5, 0, 00, 0, 0, -10).finished();
 
  EQUALITY(actual,expected);
}
 
/* ************************************************************************* */
TEST(Matrix, trans )
{
  Matrix A = (Matrix(2, 2) << 1.0, 3.0, 2.0, 4.0).finished();
  Matrix B = (Matrix(2, 2) << 1.0, 2.0, 3.0, 4.0).finished();
  EQUALITY(trans(A),B);
}
 
/* ************************************************************************* */
TEST(Matrix, col_major_access )
{
  Matrix A = (Matrix(2, 2) << 1.0, 2.0, 3.0, 4.0).finished();
  const double* a = &A(0, 0);
  DOUBLES_EQUAL(2.0,a[2],1e-9);
}
 
/* ************************************************************************* */
TEST(Matrix, weighted_elimination )
{
  // create a matrix to eliminate
  Matrix A = (Matrix(4, 6) << -1., 0., 1., 0., 0., 0., 0., -1., 0., 1., 0., 0.,
      1., 0., 0., 0., -1., 0., 0., 1., 0., 0., 0., -1.).finished();
  Vector b = (Vector(4) << -0.2, 0.3, 0.2, -0.1).finished();
  Vector sigmas = (Vector(4) << 0.2, 0.2, 0.1, 0.1).finished();
 
  //   expected values
  Matrix expectedR = (Matrix(4, 6) << 1., 0., -0.2, 0., -0.8, 0., 0., 1., 0.,
      -0.2, 0., -0.8, 0., 0., 1., 0., -1., 0., 0., 0., 0., 1., 0., -1.).finished();
  Vector d = (Vector(4) << 0.2, -0.14, 0.0, 0.2).finished();
  Vector newSigmas = (Vector(4) << 0.0894427, 0.0894427, 0.223607, 0.223607).finished();
 
  // perform elimination
  Matrix A1 = A;
  Vector b1 = b;
  std::list<std::tuple<Vector, double, double> > solution =
      weighted_eliminate(A1, b1, sigmas);
 
  // unpack and verify
  size_t i = 0;
  for (const auto& [r, di, sigma] : solution) {
    EXPECT(assert_equal(r, expectedR.row(i))); // verify r
    DOUBLES_EQUAL(d(i), di, 1e-8); // verify d
    DOUBLES_EQUAL(newSigmas(i), sigma, 1e-5); // verify sigma
    i += 1;
  }
}
 
/* ************************************************************************* */
TEST(Matrix, inverse_square_root )
{
  Matrix measurement_covariance = (Matrix(3, 3) << 0.25, 0.0, 0.0, 0.0, 0.25,
      0.0, 0.0, 0.0, 0.01).finished();
  Matrix actual = inverse_square_root(measurement_covariance);
 
  Matrix expected = (Matrix(3, 3) << 2.0, 0.0, 0.0, 0.0, 2.0, 0.0, 0.0, 0.0,
      10.0).finished();
 
  EQUALITY(expected,actual);
  EQUALITY(measurement_covariance,(actual*actual).inverse());
 
  // Randomly generated test.  This test really requires inverse to
  // be working well; if it's not, there's the possibility of a
  // bug in inverse masking a bug in this routine since we
  // use the same inverse routing inside inverse_square_root()
  // as we use here to check it.
 
  Matrix M = (Matrix(5, 5) <<
      0.0785892, 0.0137923, -0.0142219, -0.0171880, 0.0028726,
      0.0137923, 0.0908911, 0.0020775, -0.0101952, 0.0175868,
      -0.0142219, 0.0020775, 0.0973051, 0.0054906, 0.0047064,
      -0.0171880,-0.0101952, 0.0054906, 0.0892453, -0.0059468,
      0.0028726, 0.0175868, 0.0047064, -0.0059468, 0.0816517).finished();
 
  expected = (Matrix(5, 5) <<
      3.567126953241796, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000,
      -0.590030436566913, 3.362022286742925, 0.000000000000000, 0.000000000000000, 0.000000000000000,
      0.618207860252376, -0.168166020746503, 3.253086082942785, 0.000000000000000, 0.000000000000000,
      0.683045380655496, 0.283773848115276, -0.099969232183396, 3.433537147891568, 0.000000000000000,
      -0.006740136923185, -0.669325697387650, -0.169716689114923, 0.171493059476284, 3.583921085468937).finished();
  EQUALITY(expected, inverse_square_root(M));
 
}
 
/* *********************************************************************** */
// M was generated as the covariance of a set of random numbers.  L that
// we are checking against was generated via chol(M)' on octave
namespace cholesky {
Matrix M = (Matrix(5, 5) << 0.0874197, -0.0030860, 0.0116969, 0.0081463,
    0.0048741, -0.0030860, 0.0872727, 0.0183073, 0.0125325, -0.0037363,
    0.0116969, 0.0183073, 0.0966217, 0.0103894, -0.0021113, 0.0081463,
    0.0125325, 0.0103894, 0.0747324, 0.0036415, 0.0048741, -0.0037363,
    -0.0021113, 0.0036415, 0.0909464).finished();
 
Matrix expected = (Matrix(5, 5) <<
    0.295668226226627, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000,
    -0.010437374483502, 0.295235094820875, 0.000000000000000, 0.000000000000000, 0.000000000000000,
    0.039560896175007, 0.063407813693827, 0.301721866387571, 0.000000000000000, 0.000000000000000,
    0.027552165831157, 0.043423266737274, 0.021695600982708, 0.267613525371710, 0.000000000000000,
    0.016485031422565, -0.012072546984405, -0.006621889326331, 0.014405837566082, 0.300462176944247).finished();
}
TEST(Matrix, LLt )
{
  EQUALITY(cholesky::expected, LLt(cholesky::M));
}
TEST(Matrix, RtR )
{
  EQUALITY(cholesky::expected.transpose(), RtR(cholesky::M));
}
 
TEST(Matrix, cholesky_inverse )
{
  EQUALITY(cholesky::M.inverse(), cholesky_inverse(cholesky::M));
}
 
/* ************************************************************************* */
TEST(Matrix, linear_dependent )
{
  Matrix A = (Matrix(2, 3) << 1.0, 2.0, 3.0, 4.0, 5.0, 6.0).finished();
  Matrix B = (Matrix(2, 3) << -1.0, -2.0, -3.0, 8.0, 10.0, 12.0).finished();
  EXPECT(linear_dependent(A, B));
}
 
/* ************************************************************************* */
TEST(Matrix, linear_dependent2 )
{
  Matrix A = (Matrix(2, 3) << 0.0, 2.0, 3.0, 4.0, 5.0, 6.0).finished();
  Matrix B = (Matrix(2, 3) << 0.0, -2.0, -3.0, 8.0, 10.0, 12.0).finished();
  EXPECT(linear_dependent(A, B));
}
 
/* ************************************************************************* */
TEST(Matrix, linear_dependent3 )
{
  Matrix A = (Matrix(2, 3) << 0.0, 2.0, 3.0, 4.0, 5.0, 6.0).finished();
  Matrix B = (Matrix(2, 3) << 0.0, -2.0, -3.0, 8.1, 10.0, 12.0).finished();
  EXPECT(linear_independent(A, B));
}
 
/* ************************************************************************* */
TEST(Matrix, svd1 )
{
  Vector v = Vector3(2., 1., 0.);
  Matrix U1 = Matrix::Identity(4, 3), S1 = v.asDiagonal(), V1 = I_3x3, A = (U1 * S1)
      * Matrix(trans(V1));
  Matrix U, V;
  Vector s;
  svd(A, U, s, V);
  Matrix S = s.asDiagonal();
  EXPECT(assert_equal(U*S*Matrix(trans(V)),A));
  EXPECT(assert_equal(S,S1));
}
 
/* ************************************************************************* */
static Matrix sampleA = (Matrix(3, 2) << 0.,-2., 0., 0., 3., 0.).finished();
static Matrix sampleAt = trans(sampleA);
 
/* ************************************************************************* */
TEST(Matrix, svd2 )
{
  Matrix U, V;
  Vector s;
 
  Matrix expectedU = (Matrix(3, 2) << 0.,-1.,0.,0.,1.,0.).finished();
  Vector expected_s = Vector2(3.,2.);
  Matrix expectedV = (Matrix(2, 2) << 1.,0.,0.,1.).finished();
 
  svd(sampleA, U, s, V);
 
  // take care of sign ambiguity
  if (U(0, 1) > 0) {
    U = -U;
    V = -V;
  }
 
  EXPECT(assert_equal(expectedU,U));
  EXPECT(assert_equal(expected_s,s,1e-9));
  EXPECT(assert_equal(expectedV,V));
}
 
/* ************************************************************************* */
TEST(Matrix, svd3 )
{
  Matrix U, V;
  Vector s;
 
  Matrix expectedU = (Matrix(2, 2) << -1.,0.,0.,-1.).finished();
  Vector expected_s = Vector2(3.0, 2.0);
  Matrix expectedV = (Matrix(3, 2) << 0.,1.,0.,0.,-1.,0.).finished();
 
  svd(sampleAt, U, s, V);
 
  // take care of sign ambiguity
  if (U(0, 0) > 0) {
    U = -U;
    V = -V;
  }
 
  Matrix S = s.asDiagonal();
  Matrix t = U * S;
  Matrix Vt = trans(V);
 
  EXPECT(assert_equal(sampleAt, prod(t, Vt)));
  EXPECT(assert_equal(expectedU,U));
  EXPECT(assert_equal(expected_s,s,1e-9));
  EXPECT(assert_equal(expectedV,V));
}
 
/* ************************************************************************* */
TEST(Matrix, svd4 )
{
  Matrix U, V;
  Vector s;
 
  Matrix A = (Matrix(3, 2) <<
      0.8147,    0.9134,
      0.9058,    0.6324,
      0.1270,    0.0975).finished();
 
  Matrix expectedU = (Matrix(3, 2) <<
     0.7397,   0.6724,
     0.6659,   -0.7370,
     0.0970,   -0.0689).finished();
 
  Vector expected_s = Vector2(1.6455, 0.1910);
 
  Matrix expectedV = (Matrix(2, 2) <<
     0.7403,   -0.6723,
     0.6723,   0.7403).finished();
 
  svd(A, U, s, V);
 
  // take care of sign ambiguity
  if (U(0, 0) < 0) {
    U.col(0) = -U.col(0);
    V.col(0) = -V.col(0);
  }
  if (U(0, 1) < 0) {
    U.col(1) = -U.col(1);
    V.col(1) = -V.col(1);
  }
 
  Matrix reconstructed = U * s.asDiagonal() * trans(V);
 
  EXPECT(assert_equal(A, reconstructed, 1e-4));
  EXPECT(assert_equal(expectedU,U, 1e-3));
  EXPECT(assert_equal(expected_s,s, 1e-4));
  EXPECT(assert_equal(expectedV,V, 1e-4));
}
 
/* ************************************************************************* */
TEST(Matrix, DLT )
{
  Matrix A = (Matrix(8, 9) <<
      0.21,        -0.42,       -10.71,         0.18,        -0.36,        -9.18,        -0.61,         1.22,        31.11,
      0.44,        -0.66,       -15.84,         0.34,        -0.51,       -12.24,        -1.64,         2.46,        59.04,
      0.69,        -8.28,       -12.19,        -0.48,         5.76,         8.48,        -1.89,        22.68,        33.39,
      0.96,         -8.4,       -17.76,         -0.6,         5.25,         11.1,        -3.36,         29.4,        62.16,
      1.25,          0.3,         2.75,         -3.5,        -0.84,         -7.7,        16.25,          3.9,        35.75,
      1.56,         0.42,         4.56,        -3.38,        -0.91,        -9.88,        22.36,         6.02,        65.36,
      1.89,         2.24,         3.99,         3.24,         3.84,         6.84,        18.09,        21.44,        38.19,
      2.24,         2.48,         6.24,         3.08,         3.41,         8.58,        24.64,        27.28,        68.64
  ).finished();
  const auto [rank,error,actual] = DLT(A);
  Vector expected = (Vector(9) << -0.0, 0.2357, 0.4714, -0.2357, 0.0, - 0.4714,-0.4714, 0.4714, 0.0).finished();
  EXPECT_LONGS_EQUAL(8,rank);
  EXPECT_DOUBLES_EQUAL(0,error,1e-8);
  EXPECT(assert_equal(expected, actual, 1e-4));
}
 
//******************************************************************************
TEST(Matrix, Matrix24IsVectorSpace) {
  GTSAM_CONCEPT_ASSERT(IsVectorSpace<Matrix24>);
}
 
TEST(Matrix, RowMajorIsVectorSpace) {
#if GTSAM_USE_BOOST_FEATURES
  typedef Eigen::Matrix<double, 2, 3, Eigen::RowMajor> RowMajor;
  GTSAM_CONCEPT_ASSERT(IsVectorSpace<RowMajor>);
#endif
}
 
TEST(Matrix, MatrixIsVectorSpace) {
  GTSAM_CONCEPT_ASSERT(IsVectorSpace<Matrix>);
}
 
TEST(Matrix, VectorIsVectorSpace) {
  GTSAM_CONCEPT_ASSERT(IsVectorSpace<Vector>);
}
 
TEST(Matrix, RowVectorIsVectorSpace) {
#if GTSAM_USE_BOOST_FEATURES
  typedef Eigen::Matrix<double, 1, -1> RowVector;
  GTSAM_CONCEPT_ASSERT(IsVectorSpace<RowVector>);
  GTSAM_CONCEPT_ASSERT(IsVectorSpace<Vector5>);
#endif
}
 
//******************************************************************************
TEST(Matrix, AbsoluteError) {
  double a = 2000, b = 1997, tol = 1e-1;
  bool isEqual;
 
  // Test only absolute error
  isEqual = fpEqual(a, b, tol, false);
  EXPECT(!isEqual);
 
  // Test relative error as well
  isEqual = fpEqual(a, b, tol);
  EXPECT(isEqual);
}
 
// A test to check if a matrix and an optional reference_wrapper to
// a matrix are equal.
TEST(Matrix, MatrixRef) {
  Matrix A = Matrix::Random(3, 3);
  Matrix B = Matrix::Random(3, 3);
 
  EXPECT(assert_equal(A, A));
  EXPECT(assert_equal(A, std::cref(A)));
  EXPECT(!assert_equal(A, std::cref(B)));
}
 
/* ************************************************************************* */
int main() {
  TestResult tr;
  return TestRegistry::runAllTests(tr);
}
/* ************************************************************************* */