text: // This file is triangularView of Eigen, a lightweight C++ template library
  // for linear algebra.
  //
  // Copyright (C) 2008-2009 Gael Guennebaud <gael.guennebaud@inria.fr>
  //
  // This Source Code Form is subject to the terms of the Mozilla
  // Public License v. 2.0. If a copy of the MPL was not distributed
  // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
  
  #include "main.h"
  
  
  
  template<typename MatrixType> void triangular_square(const MatrixType& m)
  {
    typedef typename MatrixType::Scalar Scalar;
    typedef typename NumTraits<Scalar>::Real RealScalar;
    typedef Matrix<Scalar, MatrixType::RowsAtCompileTime, 1> VectorType;
  
    RealScalar largerEps = 10*test_precision<RealScalar>();
  
    typename MatrixType::Index rows = m.rows();
    typename MatrixType::Index cols = m.cols();
  
    MatrixType m1 = MatrixType::Random(rows, cols),
               m2 = MatrixType::Random(rows, cols),
               m3(rows, cols),
               m4(rows, cols),
               r1(rows, cols),
               r2(rows, cols);
    VectorType v2 = VectorType::Random(rows);
  
    MatrixType m1up = m1.template triangularView<Upper>();
    MatrixType m2up = m2.template triangularView<Upper>();
  
    if (rows*cols>1)
    {
      VERIFY(m1up.isUpperTriangular());
      VERIFY(m2up.transpose().isLowerTriangular());
      VERIFY(!m2.isLowerTriangular());
    }
  
  //   VERIFY_IS_APPROX(m1up.transpose() * m2, m1.upper().transpose().lower() * m2);
  
    // test overloaded operator+=
    r1.setZero();
    r2.setZero();
    r1.template triangularView<Upper>() +=  m1;
    r2 += m1up;
    VERIFY_IS_APPROX(r1,r2);
  
    // test overloaded operator=
    m1.setZero();
    m1.template triangularView<Upper>() = m2.transpose() + m2;
    m3 = m2.transpose() + m2;
    VERIFY_IS_APPROX(m3.template triangularView<Lower>().transpose().toDenseMatrix(), m1);
  
    // test overloaded operator=
    m1.setZero();
    m1.template triangularView<Lower>() = m2.transpose() + m2;
    VERIFY_IS_APPROX(m3.template triangularView<Lower>().toDenseMatrix(), m1);
  
    VERIFY_IS_APPROX(m3.template triangularView<Lower>().conjugate().toDenseMatrix(),
                     m3.conjugate().template triangularView<Lower>().toDenseMatrix());
  
    m1 = MatrixType::Random(rows, cols);
    for (int i=0; i<rows; ++i)
      while (numext::abs2(m1(i,i))<1e-1) m1(i,i) = internal::random<Scalar>();
  
    Transpose<MatrixType> trm4(m4);
    // test back and forward subsitution with a vector as the rhs
    m3 = m1.template triangularView<Upper>();
    VERIFY(v2.isApprox(m3.adjoint() * (m1.adjoint().template triangularView<Lower>().solve(v2)), largerEps));
    m3 = m1.template triangularView<Lower>();
    VERIFY(v2.isApprox(m3.transpose() * (m1.transpose().template triangularView<Upper>().solve(v2)), largerEps));
    m3 = m1.template triangularView<Upper>();
    VERIFY(v2.isApprox(m3 * (m1.template triangularView<Upper>().solve(v2)), largerEps));
    m3 = m1.template triangularView<Lower>();
    VERIFY(v2.isApprox(m3.conjugate() * (m1.conjugate().template triangularView<Lower>().solve(v2)), largerEps));
  
    // test back and forward subsitution with a matrix as the rhs
    m3 = m1.template triangularView<Upper>();
    VERIFY(m2.isApprox(m3.adjoint() * (m1.adjoint().template triangularView<Lower>().solve(m2)), largerEps));
    m3 = m1.template triangularView<Lower>();
    VERIFY(m2.isApprox(m3.transpose() * (m1.transpose().template triangularView<Upper>().solve(m2)), largerEps));
    m3 = m1.template triangularView<Upper>();
    VERIFY(m2.isApprox(m3 * (m1.template triangularView<Upper>().solve(m2)), largerEps));
    m3 = m1.template triangularView<Lower>();
    VERIFY(m2.isApprox(m3.conjugate() * (m1.conjugate().template triangularView<Lower>().solve(m2)), largerEps));
  
    // check M * inv(L) using in place API
    m4 = m3;
    m1.transpose().template triangularView<Eigen::Upper>().solveInPlace(trm4);
    VERIFY_IS_APPROX(m4 * m1.template triangularView<Eigen::Lower>(), m3);
  
    // check M * inv(U) using in place API
    m3 = m1.template triangularView<Upper>();
    m4 = m3;
    m3.transpose().template triangularView<Eigen::Lower>().solveInPlace(trm4);
    VERIFY_IS_APPROX(m4 * m1.template triangularView<Eigen::Upper>(), m3);
  
    // check solve with unit diagonal
    m3 = m1.template triangularView<UnitUpper>();
    VERIFY(m2.isApprox(m3 * (m1.template triangularView<UnitUpper>().solve(m2)), largerEps));
  
  //   VERIFY((  m1.template triangularView<Upper>()
  //           * m2.template triangularView<Upper>()).isUpperTriangular());
  
    // test swap
    m1.setOnes();
    m2.setZero();
    m2.template triangularView<Upper>().swap(m1);
    m3.setZero();
    m3.template triangularView<Upper>().setOnes();
    VERIFY_IS_APPROX(m2,m3);
    
    m1.setRandom();
    m3 = m1.template triangularView<Upper>();
    Matrix<Scalar, MatrixType::ColsAtCompileTime, Dynamic> m5(cols, internal::random<int>(1,20));  m5.setRandom();
    Matrix<Scalar, Dynamic, MatrixType::RowsAtCompileTime> m6(internal::random<int>(1,20), rows);  m6.setRandom();
    VERIFY_IS_APPROX(m1.template triangularView<Upper>() * m5, m3*m5);
    VERIFY_IS_APPROX(m6*m1.template triangularView<Upper>(), m6*m3);
  
  }
  
  
  template<typename MatrixType> void triangular_rect(const MatrixType& m)
  {
    typedef const typename MatrixType::Index Index;
    typedef typename MatrixType::Scalar Scalar;
    typedef typename NumTraits<Scalar>::Real RealScalar;
    enum { Rows =  MatrixType::RowsAtCompileTime, Cols =  MatrixType::ColsAtCompileTime };
  
    Index rows = m.rows();
    Index cols = m.cols();
  
    MatrixType m1 = MatrixType::Random(rows, cols),
               m2 = MatrixType::Random(rows, cols),
               m3(rows, cols),
               m4(rows, cols),
               r1(rows, cols),
               r2(rows, cols);
  
    MatrixType m1up = m1.template triangularView<Upper>();
    MatrixType m2up = m2.template triangularView<Upper>();
  
    if (rows>1 && cols>1)
    {
      VERIFY(m1up.isUpperTriangular());
      VERIFY(m2up.transpose().isLowerTriangular());
      VERIFY(!m2.isLowerTriangular());
    }
  
    // test overloaded operator+=
    r1.setZero();
    r2.setZero();
    r1.template triangularView<Upper>() +=  m1;
    r2 += m1up;
    VERIFY_IS_APPROX(r1,r2);
  
    // test overloaded operator=
    m1.setZero();
    m1.template triangularView<Upper>() = 3 * m2;
    m3 = 3 * m2;
    VERIFY_IS_APPROX(m3.template triangularView<Upper>().toDenseMatrix(), m1);
  
  
    m1.setZero();
    m1.template triangularView<Lower>() = 3 * m2;
    VERIFY_IS_APPROX(m3.template triangularView<Lower>().toDenseMatrix(), m1);
  
    m1.setZero();
    m1.template triangularView<StrictlyUpper>() = 3 * m2;
    VERIFY_IS_APPROX(m3.template triangularView<StrictlyUpper>().toDenseMatrix(), m1);
  
  
    m1.setZero();
    m1.template triangularView<StrictlyLower>() = 3 * m2;
    VERIFY_IS_APPROX(m3.template triangularView<StrictlyLower>().toDenseMatrix(), m1);
    m1.setRandom();
    m2 = m1.template triangularView<Upper>();
    VERIFY(m2.isUpperTriangular());
    VERIFY(!m2.isLowerTriangular());
    m2 = m1.template triangularView<StrictlyUpper>();
    VERIFY(m2.isUpperTriangular());
    VERIFY(m2.diagonal().isMuchSmallerThan(RealScalar(1)));
    m2 = m1.template triangularView<UnitUpper>();
    VERIFY(m2.isUpperTriangular());
    m2.diagonal().array() -= Scalar(1);
    VERIFY(m2.diagonal().isMuchSmallerThan(RealScalar(1)));
    m2 = m1.template triangularView<Lower>();
    VERIFY(m2.isLowerTriangular());
    VERIFY(!m2.isUpperTriangular());
    m2 = m1.template triangularView<StrictlyLower>();
    VERIFY(m2.isLowerTriangular());
    VERIFY(m2.diagonal().isMuchSmallerThan(RealScalar(1)));
    m2 = m1.template triangularView<UnitLower>();
    VERIFY(m2.isLowerTriangular());
    m2.diagonal().array() -= Scalar(1);
    VERIFY(m2.diagonal().isMuchSmallerThan(RealScalar(1)));
    // test swap
    m1.setOnes();
    m2.setZero();
    m2.template triangularView<Upper>().swap(m1);
    m3.setZero();
    m3.template triangularView<Upper>().setOnes();
    VERIFY_IS_APPROX(m2,m3);
  }
  
  void bug_159()
  {
    Matrix3d m = Matrix3d::Random().triangularView<Lower>();
    EIGEN_UNUSED_VARIABLE(m)
  }
  
  void test_triangular()
  {
    int maxsize = (std::min)(EIGEN_TEST_MAX_SIZE,20);
    for(int i = 0; i < g_repeat ; i++)
    {
      int r = internal::random<int>(2,maxsize); TEST_SET_BUT_UNUSED_VARIABLE(r)
      int c = internal::random<int>(2,maxsize); TEST_SET_BUT_UNUSED_VARIABLE(c)
  
      CALL_SUBTEST_1( triangular_square(Matrix<float, 1, 1>()) );
      CALL_SUBTEST_2( triangular_square(Matrix<float, 2, 2>()) );
      CALL_SUBTEST_3( triangular_square(Matrix3d()) );
      CALL_SUBTEST_4( triangular_square(Matrix<std::complex<float>,8, 8>()) );
      CALL_SUBTEST_5( triangular_square(MatrixXcd(r,r)) );
      CALL_SUBTEST_6( triangular_square(Matrix<float,Dynamic,Dynamic,RowMajor>(r, r)) );
  
      CALL_SUBTEST_7( triangular_rect(Matrix<float, 4, 5>()) );
      CALL_SUBTEST_8( triangular_rect(Matrix<double, 6, 2>()) );
      CALL_SUBTEST_9( triangular_rect(MatrixXcf(r, c)) );
      CALL_SUBTEST_5( triangular_rect(MatrixXcd(r, c)) );
      CALL_SUBTEST_6( triangular_rect(Matrix<float,Dynamic,Dynamic,RowMajor>(r, c)) );
    }
    
    CALL_SUBTEST_1( bug_159() );
  }
  