text: // This file is part of Eigen, a lightweight C++ template library
  // for linear algebra.
  //
  // Copyright (C) 2010,2012 Jitse Niesen <jitse@maths.leeds.ac.uk>
  //
  // 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 <limits>
  #include <Eigen/Eigenvalues>
  
  template<typename MatrixType> void schur(int size = MatrixType::ColsAtCompileTime)
  {
    typedef typename ComplexSchur<MatrixType>::ComplexScalar ComplexScalar;
    typedef typename ComplexSchur<MatrixType>::ComplexMatrixType ComplexMatrixType;
  
    // Test basic functionality: T is triangular and A = U T U*
    for(int counter = 0; counter < g_repeat; ++counter) {
      MatrixType A = MatrixType::Random(size, size);
      ComplexSchur<MatrixType> schurOfA(A);
      VERIFY_IS_EQUAL(schurOfA.info(), Success);
      ComplexMatrixType U = schurOfA.matrixU();
      ComplexMatrixType T = schurOfA.matrixT();
      for(int row = 1; row < size; ++row) {
        for(int col = 0; col < row; ++col) {
          VERIFY(T(row,col) == (typename MatrixType::Scalar)0);
        }
      }
      VERIFY_IS_APPROX(A.template cast<ComplexScalar>(), U * T * U.adjoint());
    }
  
    // Test asserts when not initialized
    ComplexSchur<MatrixType> csUninitialized;
    VERIFY_RAISES_ASSERT(csUninitialized.matrixT());
    VERIFY_RAISES_ASSERT(csUninitialized.matrixU());
    VERIFY_RAISES_ASSERT(csUninitialized.info());
    
    // Test whether compute() and constructor returns same result
    MatrixType A = MatrixType::Random(size, size);
    ComplexSchur<MatrixType> cs1;
    cs1.compute(A);
    ComplexSchur<MatrixType> cs2(A);
    VERIFY_IS_EQUAL(cs1.info(), Success);
    VERIFY_IS_EQUAL(cs2.info(), Success);
    VERIFY_IS_EQUAL(cs1.matrixT(), cs2.matrixT());
    VERIFY_IS_EQUAL(cs1.matrixU(), cs2.matrixU());
  
    // Test maximum number of iterations
    ComplexSchur<MatrixType> cs3;
    cs3.setMaxIterations(ComplexSchur<MatrixType>::m_maxIterationsPerRow * size).compute(A);
    VERIFY_IS_EQUAL(cs3.info(), Success);
    VERIFY_IS_EQUAL(cs3.matrixT(), cs1.matrixT());
    VERIFY_IS_EQUAL(cs3.matrixU(), cs1.matrixU());
    cs3.setMaxIterations(1).compute(A);
    VERIFY_IS_EQUAL(cs3.info(), size > 1 ? NoConvergence : Success);
    VERIFY_IS_EQUAL(cs3.getMaxIterations(), 1);
  
    MatrixType Atriangular = A;
    Atriangular.template triangularView<StrictlyLower>().setZero(); 
    cs3.setMaxIterations(1).compute(Atriangular); // triangular matrices do not need any iterations
    VERIFY_IS_EQUAL(cs3.info(), Success);
    VERIFY_IS_EQUAL(cs3.matrixT(), Atriangular.template cast<ComplexScalar>());
    VERIFY_IS_EQUAL(cs3.matrixU(), ComplexMatrixType::Identity(size, size));
  
    // Test computation of only T, not U
    ComplexSchur<MatrixType> csOnlyT(A, false);
    VERIFY_IS_EQUAL(csOnlyT.info(), Success);
    VERIFY_IS_EQUAL(cs1.matrixT(), csOnlyT.matrixT());
    VERIFY_RAISES_ASSERT(csOnlyT.matrixU());
  
    if (size > 1 && size < 20)
    {
      // Test matrix with NaN
      A(0,0) = std::numeric_limits<typename MatrixType::RealScalar>::quiet_NaN();
      ComplexSchur<MatrixType> csNaN(A);
      VERIFY_IS_EQUAL(csNaN.info(), NoConvergence);
    }
  }
  
  void test_schur_complex()
  {
    CALL_SUBTEST_1(( schur<Matrix4cd>() ));
    CALL_SUBTEST_2(( schur<MatrixXcf>(internal::random<int>(1,EIGEN_TEST_MAX_SIZE/4)) ));
    CALL_SUBTEST_3(( schur<Matrix<std::complex<float>, 1, 1> >() ));
    CALL_SUBTEST_4(( schur<Matrix<float, 3, 3, Eigen::RowMajor> >() ));
  
    // Test problem size constructors
    CALL_SUBTEST_5(ComplexSchur<MatrixXf>(10));
  }
  