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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 
verifyIsQuasiTriangular(const MatrixType & T)14 template<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 
schur(int size=MatrixType::ColsAtCompileTime)40 template<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 
test_schur_real()103 void 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