```/* <![CDATA[ */
function get_sym_list(){return [["Typedef","xt",[["Index",16],["Scalar",19]]],["Function","xf",[["schur",40],["test_schur_real",103],["verifyIsQuasiTriangular",14]]]];} /* ]]> */1// This file is part of Eigen, a lightweight C++ template library
2// for linear algebra.
3//
4// Copyright (C) 2010,2012 Jitse Niesen <jitse@maths.leeds.ac.uk>
5//
6// This Source Code Form is subject to the terms of the Mozilla
7// Public License v. 2.0. If a copy of the MPL was not distributed
8// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
9
10#include "main.h"
11#include <limits>
12#include <Eigen/Eigenvalues>
13
14template<typename MatrixType> void verifyIsQuasiTriangular(const MatrixType& T)
15{
16  typedef typename MatrixType::Index Index;
17
18  const Index size = T.cols();
19  typedef typename MatrixType::Scalar Scalar;
20
21  // Check T is lower Hessenberg
22  for(int row = 2; row < size; ++row) {
23    for(int col = 0; col < row - 1; ++col) {
24      VERIFY(T(row,col) == Scalar(0));
25    }
26  }
27
28  // Check that any non-zero on the subdiagonal is followed by a zero and is
29  // part of a 2x2 diagonal block with imaginary eigenvalues.
30  for(int row = 1; row < size; ++row) {
31    if (T(row,row-1) != Scalar(0)) {
32      VERIFY(row == size-1 || T(row+1,row) == 0);
33      Scalar tr = T(row-1,row-1) + T(row,row);
34      Scalar det = T(row-1,row-1) * T(row,row) - T(row-1,row) * T(row,row-1);
35      VERIFY(4 * det > tr * tr);
36    }
37  }
38}
39
40template<typename MatrixType> void schur(int size = MatrixType::ColsAtCompileTime)
41{
42  // Test basic functionality: T is quasi-triangular and A = U T U*
43  for(int counter = 0; counter < g_repeat; ++counter) {
44    MatrixType A = MatrixType::Random(size, size);
45    RealSchur<MatrixType> schurOfA(A);
46    VERIFY_IS_EQUAL(schurOfA.info(), Success);
47    MatrixType U = schurOfA.matrixU();
48    MatrixType T = schurOfA.matrixT();
49    verifyIsQuasiTriangular(T);
50    VERIFY_IS_APPROX(A, U * T * U.transpose());
51  }
52
53  // Test asserts when not initialized
54  RealSchur<MatrixType> rsUninitialized;
55  VERIFY_RAISES_ASSERT(rsUninitialized.matrixT());
56  VERIFY_RAISES_ASSERT(rsUninitialized.matrixU());
57  VERIFY_RAISES_ASSERT(rsUninitialized.info());
58
59  // Test whether compute() and constructor returns same result
60  MatrixType A = MatrixType::Random(size, size);
61  RealSchur<MatrixType> rs1;
62  rs1.compute(A);
63  RealSchur<MatrixType> rs2(A);
64  VERIFY_IS_EQUAL(rs1.info(), Success);
65  VERIFY_IS_EQUAL(rs2.info(), Success);
66  VERIFY_IS_EQUAL(rs1.matrixT(), rs2.matrixT());
67  VERIFY_IS_EQUAL(rs1.matrixU(), rs2.matrixU());
68
69  // Test maximum number of iterations
70  RealSchur<MatrixType> rs3;
71  rs3.setMaxIterations(RealSchur<MatrixType>::m_maxIterationsPerRow * size).compute(A);
72  VERIFY_IS_EQUAL(rs3.info(), Success);
73  VERIFY_IS_EQUAL(rs3.matrixT(), rs1.matrixT());
74  VERIFY_IS_EQUAL(rs3.matrixU(), rs1.matrixU());
75  if (size > 2) {
76    rs3.setMaxIterations(1).compute(A);
77    VERIFY_IS_EQUAL(rs3.info(), NoConvergence);
78    VERIFY_IS_EQUAL(rs3.getMaxIterations(), 1);
79  }
80
81  MatrixType Atriangular = A;
82  Atriangular.template triangularView<StrictlyLower>().setZero();
83  rs3.setMaxIterations(1).compute(Atriangular); // triangular matrices do not need any iterations
84  VERIFY_IS_EQUAL(rs3.info(), Success);
85  VERIFY_IS_EQUAL(rs3.matrixT(), Atriangular);
86  VERIFY_IS_EQUAL(rs3.matrixU(), MatrixType::Identity(size, size));
87
88  // Test computation of only T, not U
89  RealSchur<MatrixType> rsOnlyT(A, false);
90  VERIFY_IS_EQUAL(rsOnlyT.info(), Success);
91  VERIFY_IS_EQUAL(rs1.matrixT(), rsOnlyT.matrixT());
92  VERIFY_RAISES_ASSERT(rsOnlyT.matrixU());
93
94  if (size > 2)
95  {
96    // Test matrix with NaN
97    A(0,0) = std::numeric_limits<typename MatrixType::Scalar>::quiet_NaN();
98    RealSchur<MatrixType> rsNaN(A);
99    VERIFY_IS_EQUAL(rsNaN.info(), NoConvergence);
100  }
101}
102
103void test_schur_real()
104{
105  CALL_SUBTEST_1(( schur<Matrix4f>() ));
106  CALL_SUBTEST_2(( schur<MatrixXd>(internal::random<int>(1,EIGEN_TEST_MAX_SIZE/4)) ));
107  CALL_SUBTEST_3(( schur<Matrix<float, 1, 1> >() ));
108  CALL_SUBTEST_4(( schur<Matrix<double, 3, 3, Eigen::RowMajor> >() ));
109
110  // Test problem size constructors
111  CALL_SUBTEST_5(RealSchur<MatrixXf>(10));
112}
113```