text: // This file is part of Eigen, a lightweight C++ template library
  // for linear algebra.
  //
  // Copyright (C) 2008 Gael Guennebaud <gael.guennebaud@inria.fr>
  // Copyright (C) 2009 Benoit Jacob <jacob.benoit.1@gmail.com>
  //
  // 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>
  #include <Eigen/SVD>
  
  template <typename MatrixType>
  void cod() {
    Index rows = internal::random<Index>(2, EIGEN_TEST_MAX_SIZE);
    Index cols = internal::random<Index>(2, EIGEN_TEST_MAX_SIZE);
    Index cols2 = internal::random<Index>(2, EIGEN_TEST_MAX_SIZE);
    Index rank = internal::random<Index>(1, (std::min)(rows, cols) - 1);
  
    typedef typename MatrixType::Scalar Scalar;
    typedef Matrix<Scalar, MatrixType::RowsAtCompileTime,
                   MatrixType::RowsAtCompileTime>
        MatrixQType;
    MatrixType matrix;
    createRandomPIMatrixOfRank(rank, rows, cols, matrix);
    CompleteOrthogonalDecomposition<MatrixType> cod(matrix);
    VERIFY(rank == cod.rank());
    VERIFY(cols - cod.rank() == cod.dimensionOfKernel());
    VERIFY(!cod.isInjective());
    VERIFY(!cod.isInvertible());
    VERIFY(!cod.isSurjective());
  
    MatrixQType q = cod.householderQ();
    VERIFY_IS_UNITARY(q);
  
    MatrixType z = cod.matrixZ();
    VERIFY_IS_UNITARY(z);
  
    MatrixType t;
    t.setZero(rows, cols);
    t.topLeftCorner(rank, rank) =
        cod.matrixT().topLeftCorner(rank, rank).template triangularView<Upper>();
  
    MatrixType c = q * t * z * cod.colsPermutation().inverse();
    VERIFY_IS_APPROX(matrix, c);
  
    MatrixType exact_solution = MatrixType::Random(cols, cols2);
    MatrixType rhs = matrix * exact_solution;
    MatrixType cod_solution = cod.solve(rhs);
    VERIFY_IS_APPROX(rhs, matrix * cod_solution);
  
    // Verify that we get the same minimum-norm solution as the SVD.
    JacobiSVD<MatrixType> svd(matrix, ComputeThinU | ComputeThinV);
    MatrixType svd_solution = svd.solve(rhs);
    VERIFY_IS_APPROX(cod_solution, svd_solution);
  
    MatrixType pinv = cod.pseudoInverse();
    VERIFY_IS_APPROX(cod_solution, pinv * rhs);
  }
  
  template <typename MatrixType, int Cols2>
  void cod_fixedsize() {
    enum {
      Rows = MatrixType::RowsAtCompileTime,
      Cols = MatrixType::ColsAtCompileTime
    };
    typedef typename MatrixType::Scalar Scalar;
    int rank = internal::random<int>(1, (std::min)(int(Rows), int(Cols)) - 1);
    Matrix<Scalar, Rows, Cols> matrix;
    createRandomPIMatrixOfRank(rank, Rows, Cols, matrix);
    CompleteOrthogonalDecomposition<Matrix<Scalar, Rows, Cols> > cod(matrix);
    VERIFY(rank == cod.rank());
    VERIFY(Cols - cod.rank() == cod.dimensionOfKernel());
    VERIFY(cod.isInjective() == (rank == Rows));
    VERIFY(cod.isSurjective() == (rank == Cols));
    VERIFY(cod.isInvertible() == (cod.isInjective() && cod.isSurjective()));
  
    Matrix<Scalar, Cols, Cols2> exact_solution;
    exact_solution.setRandom(Cols, Cols2);
    Matrix<Scalar, Rows, Cols2> rhs = matrix * exact_solution;
    Matrix<Scalar, Cols, Cols2> cod_solution = cod.solve(rhs);
    VERIFY_IS_APPROX(rhs, matrix * cod_solution);
  
    // Verify that we get the same minimum-norm solution as the SVD.
    JacobiSVD<MatrixType> svd(matrix, ComputeFullU | ComputeFullV);
    Matrix<Scalar, Cols, Cols2> svd_solution = svd.solve(rhs);
    VERIFY_IS_APPROX(cod_solution, svd_solution);
  }
  
  template<typename MatrixType> void qr()
  {
    using std::sqrt;
  
    Index rows = internal::random<Index>(2,EIGEN_TEST_MAX_SIZE), cols = internal::random<Index>(2,EIGEN_TEST_MAX_SIZE), cols2 = internal::random<Index>(2,EIGEN_TEST_MAX_SIZE);
    Index rank = internal::random<Index>(1, (std::min)(rows, cols)-1);
  
    typedef typename MatrixType::Scalar Scalar;
    typedef typename MatrixType::RealScalar RealScalar;
    typedef Matrix<Scalar, MatrixType::RowsAtCompileTime, MatrixType::RowsAtCompileTime> MatrixQType;
    MatrixType m1;
    createRandomPIMatrixOfRank(rank,rows,cols,m1);
    ColPivHouseholderQR<MatrixType> qr(m1);
    VERIFY_IS_EQUAL(rank, qr.rank());
    VERIFY_IS_EQUAL(cols - qr.rank(), qr.dimensionOfKernel());
    VERIFY(!qr.isInjective());
    VERIFY(!qr.isInvertible());
    VERIFY(!qr.isSurjective());
  
    MatrixQType q = qr.householderQ();
    VERIFY_IS_UNITARY(q);
  
    MatrixType r = qr.matrixQR().template triangularView<Upper>();
    MatrixType c = q * r * qr.colsPermutation().inverse();
    VERIFY_IS_APPROX(m1, c);
  
    // Verify that the absolute value of the diagonal elements in R are
    // non-increasing until they reach the singularity threshold.
    RealScalar threshold =
        sqrt(RealScalar(rows)) * numext::abs(r(0, 0)) * NumTraits<Scalar>::epsilon();
    for (Index i = 0; i < (std::min)(rows, cols) - 1; ++i) {
      RealScalar x = numext::abs(r(i, i));
      RealScalar y = numext::abs(r(i + 1, i + 1));
      if (x < threshold && y < threshold) continue;
      if (!test_isApproxOrLessThan(y, x)) {
        for (Index j = 0; j < (std::min)(rows, cols); ++j) {
          std::cout << "i = " << j << ", |r_ii| = " << numext::abs(r(j, j)) << std::endl;
        }
        std::cout << "Failure at i=" << i << ", rank=" << rank
                  << ", threshold=" << threshold << std::endl;
      }
      VERIFY_IS_APPROX_OR_LESS_THAN(y, x);
    }
  
    MatrixType m2 = MatrixType::Random(cols,cols2);
    MatrixType m3 = m1*m2;
    m2 = MatrixType::Random(cols,cols2);
    m2 = qr.solve(m3);
    VERIFY_IS_APPROX(m3, m1*m2);
  
    {
      Index size = rows;
      do {
        m1 = MatrixType::Random(size,size);
        qr.compute(m1);
      } while(!qr.isInvertible());
      MatrixType m1_inv = qr.inverse();
      m3 = m1 * MatrixType::Random(size,cols2);
      m2 = qr.solve(m3);
      VERIFY_IS_APPROX(m2, m1_inv*m3);
    }
  }
  
  template<typename MatrixType, int Cols2> void qr_fixedsize()
  {
    using std::sqrt;
    using std::abs;
    enum { Rows = MatrixType::RowsAtCompileTime, Cols = MatrixType::ColsAtCompileTime };
    typedef typename MatrixType::Scalar Scalar;
    typedef typename MatrixType::RealScalar RealScalar;
    int rank = internal::random<int>(1, (std::min)(int(Rows), int(Cols))-1);
    Matrix<Scalar,Rows,Cols> m1;
    createRandomPIMatrixOfRank(rank,Rows,Cols,m1);
    ColPivHouseholderQR<Matrix<Scalar,Rows,Cols> > qr(m1);
    VERIFY_IS_EQUAL(rank, qr.rank());
    VERIFY_IS_EQUAL(Cols - qr.rank(), qr.dimensionOfKernel());
    VERIFY_IS_EQUAL(qr.isInjective(), (rank == Rows));
    VERIFY_IS_EQUAL(qr.isSurjective(), (rank == Cols));
    VERIFY_IS_EQUAL(qr.isInvertible(), (qr.isInjective() && qr.isSurjective()));
  
    Matrix<Scalar,Rows,Cols> r = qr.matrixQR().template triangularView<Upper>();
    Matrix<Scalar,Rows,Cols> c = qr.householderQ() * r * qr.colsPermutation().inverse();
    VERIFY_IS_APPROX(m1, c);
  
    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);
    // Verify that the absolute value of the diagonal elements in R are
    // non-increasing until they reache the singularity threshold.
    RealScalar threshold =
        sqrt(RealScalar(Rows)) * (std::abs)(r(0, 0)) * NumTraits<Scalar>::epsilon();
    for (Index i = 0; i < (std::min)(int(Rows), int(Cols)) - 1; ++i) {
      RealScalar x = numext::abs(r(i, i));
      RealScalar y = numext::abs(r(i + 1, i + 1));
      if (x < threshold && y < threshold) continue;
      if (!test_isApproxOrLessThan(y, x)) {
        for (Index j = 0; j < (std::min)(int(Rows), int(Cols)); ++j) {
          std::cout << "i = " << j << ", |r_ii| = " << numext::abs(r(j, j)) << std::endl;
        }
        std::cout << "Failure at i=" << i << ", rank=" << rank
                  << ", threshold=" << threshold << std::endl;
      }
      VERIFY_IS_APPROX_OR_LESS_THAN(y, x);
    }
  }
  
  // This test is meant to verify that pivots are chosen such that
  // even for a graded matrix, the diagonal of R falls of roughly
  // monotonically until it reaches the threshold for singularity.
  // We use the so-called Kahan matrix, which is a famous counter-example
  // for rank-revealing QR. See
  // http://www.netlib.org/lapack/lawnspdf/lawn176.pdf
  // page 3 for more detail.
  template<typename MatrixType> void qr_kahan_matrix()
  {
    using std::sqrt;
    using std::abs;
    typedef typename MatrixType::Scalar Scalar;
    typedef typename MatrixType::RealScalar RealScalar;
  
    Index rows = 300, cols = rows;
  
    MatrixType m1;
    m1.setZero(rows,cols);
    RealScalar s = std::pow(NumTraits<RealScalar>::epsilon(), 1.0 / rows);
    RealScalar c = std::sqrt(1 - s*s);
    RealScalar pow_s_i(1.0); // pow(s,i)
    for (Index i = 0; i < rows; ++i) {
      m1(i, i) = pow_s_i;
      m1.row(i).tail(rows - i - 1) = -pow_s_i * c * MatrixType::Ones(1, rows - i - 1);
      pow_s_i *= s;
    }
    m1 = (m1 + m1.transpose()).eval();
    ColPivHouseholderQR<MatrixType> qr(m1);
    MatrixType r = qr.matrixQR().template triangularView<Upper>();
  
    RealScalar threshold =
        std::sqrt(RealScalar(rows)) * numext::abs(r(0, 0)) * NumTraits<Scalar>::epsilon();
    for (Index i = 0; i < (std::min)(rows, cols) - 1; ++i) {
      RealScalar x = numext::abs(r(i, i));
      RealScalar y = numext::abs(r(i + 1, i + 1));
      if (x < threshold && y < threshold) continue;
      if (!test_isApproxOrLessThan(y, x)) {
        for (Index j = 0; j < (std::min)(rows, cols); ++j) {
          std::cout << "i = " << j << ", |r_ii| = " << numext::abs(r(j, j)) << std::endl;
        }
        std::cout << "Failure at i=" << i << ", rank=" << qr.rank()
                  << ", threshold=" << threshold << std::endl;
      }
      VERIFY_IS_APPROX_OR_LESS_THAN(y, x);
    }
  }
  
  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();
    }
  
    ColPivHouseholderQR<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;
  
    ColPivHouseholderQR<MatrixType> qr;
    VERIFY_RAISES_ASSERT(qr.matrixQR())
    VERIFY_RAISES_ASSERT(qr.solve(tmp))
    VERIFY_RAISES_ASSERT(qr.householderQ())
    VERIFY_RAISES_ASSERT(qr.dimensionOfKernel())
    VERIFY_RAISES_ASSERT(qr.isInjective())
    VERIFY_RAISES_ASSERT(qr.isSurjective())
    VERIFY_RAISES_ASSERT(qr.isInvertible())
    VERIFY_RAISES_ASSERT(qr.inverse())
    VERIFY_RAISES_ASSERT(qr.absDeterminant())
    VERIFY_RAISES_ASSERT(qr.logAbsDeterminant())
  }
  
  EIGEN_DECLARE_TEST(qr_colpivoting)
  {
    for(int i = 0; i < g_repeat; i++) {
      CALL_SUBTEST_1( qr<MatrixXf>() );
      CALL_SUBTEST_2( qr<MatrixXd>() );
      CALL_SUBTEST_3( qr<MatrixXcd>() );
      CALL_SUBTEST_4(( qr_fixedsize<Matrix<float,3,5>, 4 >() ));
      CALL_SUBTEST_5(( qr_fixedsize<Matrix<double,6,2>, 3 >() ));
      CALL_SUBTEST_5(( qr_fixedsize<Matrix<double,1,1>, 1 >() ));
    }
  
    for(int i = 0; i < g_repeat; i++) {
      CALL_SUBTEST_1( cod<MatrixXf>() );
      CALL_SUBTEST_2( cod<MatrixXd>() );
      CALL_SUBTEST_3( cod<MatrixXcd>() );
      CALL_SUBTEST_4(( cod_fixedsize<Matrix<float,3,5>, 4 >() ));
      CALL_SUBTEST_5(( cod_fixedsize<Matrix<double,6,2>, 3 >() ));
      CALL_SUBTEST_5(( cod_fixedsize<Matrix<double,1,1>, 1 >() ));
    }
  
    for(int i = 0; i < g_repeat; i++) {
      CALL_SUBTEST_1( qr_invertible<MatrixXf>() );
      CALL_SUBTEST_2( qr_invertible<MatrixXd>() );
      CALL_SUBTEST_6( qr_invertible<MatrixXcf>() );
      CALL_SUBTEST_3( qr_invertible<MatrixXcd>() );
    }
  
    CALL_SUBTEST_7(qr_verify_assert<Matrix3f>());
    CALL_SUBTEST_8(qr_verify_assert<Matrix3d>());
    CALL_SUBTEST_1(qr_verify_assert<MatrixXf>());
    CALL_SUBTEST_2(qr_verify_assert<MatrixXd>());
    CALL_SUBTEST_6(qr_verify_assert<MatrixXcf>());
    CALL_SUBTEST_3(qr_verify_assert<MatrixXcd>());
  
    // Test problem size constructors
    CALL_SUBTEST_9(ColPivHouseholderQR<MatrixXf>(10, 20));
  
    CALL_SUBTEST_1( qr_kahan_matrix<MatrixXf>() );
    CALL_SUBTEST_2( qr_kahan_matrix<MatrixXd>() );
  }
  