text: // This file is part of Eigen, a lightweight C++ template library
  // for linear algebra.
  //
  // Copyright (C) 2008 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"
  #include <Eigen/QR>
  
  template<typename MatrixType> void qr(const MatrixType& m)
  {
    typedef typename MatrixType::Index Index;
  
    Index rows = m.rows();
    Index cols = m.cols();
  
    typedef typename MatrixType::Scalar Scalar;
    typedef Matrix<Scalar, MatrixType::RowsAtCompileTime, MatrixType::RowsAtCompileTime> MatrixQType;
  
    MatrixType a = MatrixType::Random(rows,cols);
    HouseholderQR<MatrixType> qrOfA(a);
  
    MatrixQType q = qrOfA.householderQ();
    VERIFY_IS_UNITARY(q);
  
    MatrixType r = qrOfA.matrixQR().template triangularView<Upper>();
    VERIFY_IS_APPROX(a, qrOfA.householderQ() * r);
  }
  
  template<typename MatrixType, int Cols2> void qr_fixedsize()
  {
    enum { Rows = MatrixType::RowsAtCompileTime, Cols = MatrixType::ColsAtCompileTime };
    typedef typename MatrixType::Scalar Scalar;
    Matrix<Scalar,Rows,Cols> m1 = Matrix<Scalar,Rows,Cols>::Random();
    HouseholderQR<Matrix<Scalar,Rows,Cols> > qr(m1);
  
    Matrix<Scalar,Rows,Cols> r = qr.matrixQR();
    // FIXME need better way to construct trapezoid
    for(int i = 0; i < Rows; i++) for(int j = 0; j < Cols; j++) if(i>j) r(i,j) = Scalar(0);
  
    VERIFY_IS_APPROX(m1, qr.householderQ() * r);
  
    Matrix<Scalar,Cols,Cols2> m2 = Matrix<Scalar,Cols,Cols2>::Random(Cols,Cols2);
    Matrix<Scalar,Rows,Cols2> m3 = m1*m2;
    m2 = Matrix<Scalar,Cols,Cols2>::Random(Cols,Cols2);
    m2 = qr.solve(m3);
    VERIFY_IS_APPROX(m3, m1*m2);
  }
  
  template<typename MatrixType> void qr_invertible()
  {
    using std::log;
    using std::abs;
    typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
    typedef typename MatrixType::Scalar Scalar;
  
    int size = internal::random<int>(10,50);
  
    MatrixType m1(size, size), m2(size, size), m3(size, size);
    m1 = MatrixType::Random(size,size);
  
    if (internal::is_same<RealScalar,float>::value)
    {
      // let's build a matrix more stable to inverse
      MatrixType a = MatrixType::Random(size,size*2);
      m1 += a * a.adjoint();
    }
  
    HouseholderQR<MatrixType> qr(m1);
    m3 = MatrixType::Random(size,size);
    m2 = qr.solve(m3);
    VERIFY_IS_APPROX(m3, m1*m2);
  
    // now construct a matrix with prescribed determinant
    m1.setZero();
    for(int i = 0; i < size; i++) m1(i,i) = internal::random<Scalar>();
    RealScalar absdet = abs(m1.diagonal().prod());
    m3 = qr.householderQ(); // get a unitary
    m1 = m3 * m1 * m3;
    qr.compute(m1);
    VERIFY_IS_APPROX(absdet, qr.absDeterminant());
    VERIFY_IS_APPROX(log(absdet), qr.logAbsDeterminant());
  }
  
  template<typename MatrixType> void qr_verify_assert()
  {
    MatrixType tmp;
  
    HouseholderQR<MatrixType> qr;
    VERIFY_RAISES_ASSERT(qr.matrixQR())
    VERIFY_RAISES_ASSERT(qr.solve(tmp))
    VERIFY_RAISES_ASSERT(qr.householderQ())
    VERIFY_RAISES_ASSERT(qr.absDeterminant())
    VERIFY_RAISES_ASSERT(qr.logAbsDeterminant())
  }
  
  void test_qr()
  {
    for(int i = 0; i < g_repeat; i++) {
     CALL_SUBTEST_1( qr(MatrixXf(internal::random<int>(1,EIGEN_TEST_MAX_SIZE),internal::random<int>(1,EIGEN_TEST_MAX_SIZE))) );
     CALL_SUBTEST_2( qr(MatrixXcd(internal::random<int>(1,EIGEN_TEST_MAX_SIZE/2),internal::random<int>(1,EIGEN_TEST_MAX_SIZE/2))) );
     CALL_SUBTEST_3(( qr_fixedsize<Matrix<float,3,4>, 2 >() ));
     CALL_SUBTEST_4(( qr_fixedsize<Matrix<double,6,2>, 4 >() ));
     CALL_SUBTEST_5(( qr_fixedsize<Matrix<double,2,5>, 7 >() ));
     CALL_SUBTEST_11( qr(Matrix<float,1,1>()) );
    }
  
    for(int i = 0; i < g_repeat; i++) {
      CALL_SUBTEST_1( qr_invertible<MatrixXf>() );
      CALL_SUBTEST_6( qr_invertible<MatrixXd>() );
      CALL_SUBTEST_7( qr_invertible<MatrixXcf>() );
      CALL_SUBTEST_8( qr_invertible<MatrixXcd>() );
    }
  
    CALL_SUBTEST_9(qr_verify_assert<Matrix3f>());
    CALL_SUBTEST_10(qr_verify_assert<Matrix3d>());
    CALL_SUBTEST_1(qr_verify_assert<MatrixXf>());
    CALL_SUBTEST_6(qr_verify_assert<MatrixXd>());
    CALL_SUBTEST_7(qr_verify_assert<MatrixXcf>());
    CALL_SUBTEST_8(qr_verify_assert<MatrixXcd>());
  
    // Test problem size constructors
    CALL_SUBTEST_12(HouseholderQR<MatrixXf>(10, 20));
  }
  