1// This file is part of Eigen, a lightweight C++ template library
2// for linear algebra.
3//
4// Copyright (C) 2008 Gael Guennebaud <gael.guennebaud@inria.fr>
5// Copyright (C) 2009 Benoit Jacob <jacob.benoit.1@gmail.com>
6//
7// This Source Code Form is subject to the terms of the Mozilla
8// Public License v. 2.0. If a copy of the MPL was not distributed
9// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
10
11#include "main.h"
12#include <Eigen/QR>
13#include <Eigen/SVD>
14
15template <typename MatrixType>
16void cod() {
17  typedef typename MatrixType::Index Index;
18
19  Index rows = internal::random<Index>(2, EIGEN_TEST_MAX_SIZE);
20  Index cols = internal::random<Index>(2, EIGEN_TEST_MAX_SIZE);
21  Index cols2 = internal::random<Index>(2, EIGEN_TEST_MAX_SIZE);
22  Index rank = internal::random<Index>(1, (std::min)(rows, cols) - 1);
23
24  typedef typename MatrixType::Scalar Scalar;
25  typedef Matrix<Scalar, MatrixType::RowsAtCompileTime,
26                 MatrixType::RowsAtCompileTime>
27      MatrixQType;
28  MatrixType matrix;
29  createRandomPIMatrixOfRank(rank, rows, cols, matrix);
30  CompleteOrthogonalDecomposition<MatrixType> cod(matrix);
31  VERIFY(rank == cod.rank());
32  VERIFY(cols - cod.rank() == cod.dimensionOfKernel());
33  VERIFY(!cod.isInjective());
34  VERIFY(!cod.isInvertible());
35  VERIFY(!cod.isSurjective());
36
37  MatrixQType q = cod.householderQ();
38  VERIFY_IS_UNITARY(q);
39
40  MatrixType z = cod.matrixZ();
41  VERIFY_IS_UNITARY(z);
42
43  MatrixType t;
44  t.setZero(rows, cols);
45  t.topLeftCorner(rank, rank) =
46      cod.matrixT().topLeftCorner(rank, rank).template triangularView<Upper>();
47
48  MatrixType c = q * t * z * cod.colsPermutation().inverse();
49  VERIFY_IS_APPROX(matrix, c);
50
51  MatrixType exact_solution = MatrixType::Random(cols, cols2);
52  MatrixType rhs = matrix * exact_solution;
53  MatrixType cod_solution = cod.solve(rhs);
54  VERIFY_IS_APPROX(rhs, matrix * cod_solution);
55
56  // Verify that we get the same minimum-norm solution as the SVD.
57  JacobiSVD<MatrixType> svd(matrix, ComputeThinU | ComputeThinV);
58  MatrixType svd_solution = svd.solve(rhs);
59  VERIFY_IS_APPROX(cod_solution, svd_solution);
60
61  MatrixType pinv = cod.pseudoInverse();
62  VERIFY_IS_APPROX(cod_solution, pinv * rhs);
63}
64
65template <typename MatrixType, int Cols2>
66void cod_fixedsize() {
67  enum {
68    Rows = MatrixType::RowsAtCompileTime,
69    Cols = MatrixType::ColsAtCompileTime
70  };
71  typedef typename MatrixType::Scalar Scalar;
72  int rank = internal::random<int>(1, (std::min)(int(Rows), int(Cols)) - 1);
73  Matrix<Scalar, Rows, Cols> matrix;
74  createRandomPIMatrixOfRank(rank, Rows, Cols, matrix);
75  CompleteOrthogonalDecomposition<Matrix<Scalar, Rows, Cols> > cod(matrix);
76  VERIFY(rank == cod.rank());
77  VERIFY(Cols - cod.rank() == cod.dimensionOfKernel());
78  VERIFY(cod.isInjective() == (rank == Rows));
79  VERIFY(cod.isSurjective() == (rank == Cols));
80  VERIFY(cod.isInvertible() == (cod.isInjective() && cod.isSurjective()));
81
82  Matrix<Scalar, Cols, Cols2> exact_solution;
83  exact_solution.setRandom(Cols, Cols2);
84  Matrix<Scalar, Rows, Cols2> rhs = matrix * exact_solution;
85  Matrix<Scalar, Cols, Cols2> cod_solution = cod.solve(rhs);
86  VERIFY_IS_APPROX(rhs, matrix * cod_solution);
87
88  // Verify that we get the same minimum-norm solution as the SVD.
89  JacobiSVD<MatrixType> svd(matrix, ComputeFullU | ComputeFullV);
90  Matrix<Scalar, Cols, Cols2> svd_solution = svd.solve(rhs);
91  VERIFY_IS_APPROX(cod_solution, svd_solution);
92}
93
94template<typename MatrixType> void qr()
95{
96  using std::sqrt;
97  typedef typename MatrixType::Index Index;
98
99  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);
100  Index rank = internal::random<Index>(1, (std::min)(rows, cols)-1);
101
102  typedef typename MatrixType::Scalar Scalar;
103  typedef typename MatrixType::RealScalar RealScalar;
104  typedef Matrix<Scalar, MatrixType::RowsAtCompileTime, MatrixType::RowsAtCompileTime> MatrixQType;
105  MatrixType m1;
106  createRandomPIMatrixOfRank(rank,rows,cols,m1);
107  ColPivHouseholderQR<MatrixType> qr(m1);
108  VERIFY_IS_EQUAL(rank, qr.rank());
109  VERIFY_IS_EQUAL(cols - qr.rank(), qr.dimensionOfKernel());
110  VERIFY(!qr.isInjective());
111  VERIFY(!qr.isInvertible());
112  VERIFY(!qr.isSurjective());
113
114  MatrixQType q = qr.householderQ();
115  VERIFY_IS_UNITARY(q);
116
117  MatrixType r = qr.matrixQR().template triangularView<Upper>();
118  MatrixType c = q * r * qr.colsPermutation().inverse();
119  VERIFY_IS_APPROX(m1, c);
120
121  // Verify that the absolute value of the diagonal elements in R are
122  // non-increasing until they reach the singularity threshold.
123  RealScalar threshold =
124      sqrt(RealScalar(rows)) * numext::abs(r(0, 0)) * NumTraits<Scalar>::epsilon();
125  for (Index i = 0; i < (std::min)(rows, cols) - 1; ++i) {
126    RealScalar x = numext::abs(r(i, i));
127    RealScalar y = numext::abs(r(i + 1, i + 1));
128    if (x < threshold && y < threshold) continue;
129    if (!test_isApproxOrLessThan(y, x)) {
130      for (Index j = 0; j < (std::min)(rows, cols); ++j) {
131        std::cout << "i = " << j << ", |r_ii| = " << numext::abs(r(j, j)) << std::endl;
132      }
133      std::cout << "Failure at i=" << i << ", rank=" << rank
134                << ", threshold=" << threshold << std::endl;
135    }
136    VERIFY_IS_APPROX_OR_LESS_THAN(y, x);
137  }
138
139  MatrixType m2 = MatrixType::Random(cols,cols2);
140  MatrixType m3 = m1*m2;
141  m2 = MatrixType::Random(cols,cols2);
142  m2 = qr.solve(m3);
143  VERIFY_IS_APPROX(m3, m1*m2);
144
145  {
146    Index size = rows;
147    do {
148      m1 = MatrixType::Random(size,size);
149      qr.compute(m1);
150    } while(!qr.isInvertible());
151    MatrixType m1_inv = qr.inverse();
152    m3 = m1 * MatrixType::Random(size,cols2);
153    m2 = qr.solve(m3);
154    VERIFY_IS_APPROX(m2, m1_inv*m3);
155  }
156}
157
158template<typename MatrixType, int Cols2> void qr_fixedsize()
159{
160  using std::sqrt;
161  using std::abs;
162  enum { Rows = MatrixType::RowsAtCompileTime, Cols = MatrixType::ColsAtCompileTime };
163  typedef typename MatrixType::Scalar Scalar;
164  typedef typename MatrixType::RealScalar RealScalar;
165  int rank = internal::random<int>(1, (std::min)(int(Rows), int(Cols))-1);
166  Matrix<Scalar,Rows,Cols> m1;
167  createRandomPIMatrixOfRank(rank,Rows,Cols,m1);
168  ColPivHouseholderQR<Matrix<Scalar,Rows,Cols> > qr(m1);
169  VERIFY_IS_EQUAL(rank, qr.rank());
170  VERIFY_IS_EQUAL(Cols - qr.rank(), qr.dimensionOfKernel());
171  VERIFY_IS_EQUAL(qr.isInjective(), (rank == Rows));
172  VERIFY_IS_EQUAL(qr.isSurjective(), (rank == Cols));
173  VERIFY_IS_EQUAL(qr.isInvertible(), (qr.isInjective() && qr.isSurjective()));
174
175  Matrix<Scalar,Rows,Cols> r = qr.matrixQR().template triangularView<Upper>();
176  Matrix<Scalar,Rows,Cols> c = qr.householderQ() * r * qr.colsPermutation().inverse();
177  VERIFY_IS_APPROX(m1, c);
178
179  Matrix<Scalar,Cols,Cols2> m2 = Matrix<Scalar,Cols,Cols2>::Random(Cols,Cols2);
180  Matrix<Scalar,Rows,Cols2> m3 = m1*m2;
181  m2 = Matrix<Scalar,Cols,Cols2>::Random(Cols,Cols2);
182  m2 = qr.solve(m3);
183  VERIFY_IS_APPROX(m3, m1*m2);
184  // Verify that the absolute value of the diagonal elements in R are
185  // non-increasing until they reache the singularity threshold.
186  RealScalar threshold =
187      sqrt(RealScalar(Rows)) * (std::abs)(r(0, 0)) * NumTraits<Scalar>::epsilon();
188  for (Index i = 0; i < (std::min)(int(Rows), int(Cols)) - 1; ++i) {
189    RealScalar x = numext::abs(r(i, i));
190    RealScalar y = numext::abs(r(i + 1, i + 1));
191    if (x < threshold && y < threshold) continue;
192    if (!test_isApproxOrLessThan(y, x)) {
193      for (Index j = 0; j < (std::min)(int(Rows), int(Cols)); ++j) {
194        std::cout << "i = " << j << ", |r_ii| = " << numext::abs(r(j, j)) << std::endl;
195      }
196      std::cout << "Failure at i=" << i << ", rank=" << rank
197                << ", threshold=" << threshold << std::endl;
198    }
199    VERIFY_IS_APPROX_OR_LESS_THAN(y, x);
200  }
201}
202
203// This test is meant to verify that pivots are chosen such that
204// even for a graded matrix, the diagonal of R falls of roughly
205// monotonically until it reaches the threshold for singularity.
206// We use the so-called Kahan matrix, which is a famous counter-example
207// for rank-revealing QR. See
208// http://www.netlib.org/lapack/lawnspdf/lawn176.pdf
209// page 3 for more detail.
210template<typename MatrixType> void qr_kahan_matrix()
211{
212  using std::sqrt;
213  using std::abs;
214  typedef typename MatrixType::Index Index;
215  typedef typename MatrixType::Scalar Scalar;
216  typedef typename MatrixType::RealScalar RealScalar;
217
218  Index rows = 300, cols = rows;
219
220  MatrixType m1;
221  m1.setZero(rows,cols);
222  RealScalar s = std::pow(NumTraits<RealScalar>::epsilon(), 1.0 / rows);
223  RealScalar c = std::sqrt(1 - s*s);
224  RealScalar pow_s_i(1.0); // pow(s,i)
225  for (Index i = 0; i < rows; ++i) {
226    m1(i, i) = pow_s_i;
227    m1.row(i).tail(rows - i - 1) = -pow_s_i * c * MatrixType::Ones(1, rows - i - 1);
228    pow_s_i *= s;
229  }
230  m1 = (m1 + m1.transpose()).eval();
231  ColPivHouseholderQR<MatrixType> qr(m1);
232  MatrixType r = qr.matrixQR().template triangularView<Upper>();
233
234  RealScalar threshold =
235      std::sqrt(RealScalar(rows)) * numext::abs(r(0, 0)) * NumTraits<Scalar>::epsilon();
236  for (Index i = 0; i < (std::min)(rows, cols) - 1; ++i) {
237    RealScalar x = numext::abs(r(i, i));
238    RealScalar y = numext::abs(r(i + 1, i + 1));
239    if (x < threshold && y < threshold) continue;
240    if (!test_isApproxOrLessThan(y, x)) {
241      for (Index j = 0; j < (std::min)(rows, cols); ++j) {
242        std::cout << "i = " << j << ", |r_ii| = " << numext::abs(r(j, j)) << std::endl;
243      }
244      std::cout << "Failure at i=" << i << ", rank=" << qr.rank()
245                << ", threshold=" << threshold << std::endl;
246    }
247    VERIFY_IS_APPROX_OR_LESS_THAN(y, x);
248  }
249}
250
251template<typename MatrixType> void qr_invertible()
252{
253  using std::log;
254  using std::abs;
255  typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
256  typedef typename MatrixType::Scalar Scalar;
257
258  int size = internal::random<int>(10,50);
259
260  MatrixType m1(size, size), m2(size, size), m3(size, size);
261  m1 = MatrixType::Random(size,size);
262
263  if (internal::is_same<RealScalar,float>::value)
264  {
265    // let's build a matrix more stable to inverse
266    MatrixType a = MatrixType::Random(size,size*2);
267    m1 += a * a.adjoint();
268  }
269
270  ColPivHouseholderQR<MatrixType> qr(m1);
271  m3 = MatrixType::Random(size,size);
272  m2 = qr.solve(m3);
273  //VERIFY_IS_APPROX(m3, m1*m2);
274
275  // now construct a matrix with prescribed determinant
276  m1.setZero();
277  for(int i = 0; i < size; i++) m1(i,i) = internal::random<Scalar>();
278  RealScalar absdet = abs(m1.diagonal().prod());
279  m3 = qr.householderQ(); // get a unitary
280  m1 = m3 * m1 * m3;
281  qr.compute(m1);
282  VERIFY_IS_APPROX(absdet, qr.absDeterminant());
283  VERIFY_IS_APPROX(log(absdet), qr.logAbsDeterminant());
284}
285
286template<typename MatrixType> void qr_verify_assert()
287{
288  MatrixType tmp;
289
290  ColPivHouseholderQR<MatrixType> qr;
291  VERIFY_RAISES_ASSERT(qr.matrixQR())
292  VERIFY_RAISES_ASSERT(qr.solve(tmp))
293  VERIFY_RAISES_ASSERT(qr.householderQ())
294  VERIFY_RAISES_ASSERT(qr.dimensionOfKernel())
295  VERIFY_RAISES_ASSERT(qr.isInjective())
296  VERIFY_RAISES_ASSERT(qr.isSurjective())
297  VERIFY_RAISES_ASSERT(qr.isInvertible())
298  VERIFY_RAISES_ASSERT(qr.inverse())
299  VERIFY_RAISES_ASSERT(qr.absDeterminant())
300  VERIFY_RAISES_ASSERT(qr.logAbsDeterminant())
301}
302
303void test_qr_colpivoting()
304{
305  for(int i = 0; i < g_repeat; i++) {
306    CALL_SUBTEST_1( qr<MatrixXf>() );
307    CALL_SUBTEST_2( qr<MatrixXd>() );
308    CALL_SUBTEST_3( qr<MatrixXcd>() );
309    CALL_SUBTEST_4(( qr_fixedsize<Matrix<float,3,5>, 4 >() ));
310    CALL_SUBTEST_5(( qr_fixedsize<Matrix<double,6,2>, 3 >() ));
311    CALL_SUBTEST_5(( qr_fixedsize<Matrix<double,1,1>, 1 >() ));
312  }
313
314  for(int i = 0; i < g_repeat; i++) {
315    CALL_SUBTEST_1( cod<MatrixXf>() );
316    CALL_SUBTEST_2( cod<MatrixXd>() );
317    CALL_SUBTEST_3( cod<MatrixXcd>() );
318    CALL_SUBTEST_4(( cod_fixedsize<Matrix<float,3,5>, 4 >() ));
319    CALL_SUBTEST_5(( cod_fixedsize<Matrix<double,6,2>, 3 >() ));
320    CALL_SUBTEST_5(( cod_fixedsize<Matrix<double,1,1>, 1 >() ));
321  }
322
323  for(int i = 0; i < g_repeat; i++) {
324    CALL_SUBTEST_1( qr_invertible<MatrixXf>() );
325    CALL_SUBTEST_2( qr_invertible<MatrixXd>() );
326    CALL_SUBTEST_6( qr_invertible<MatrixXcf>() );
327    CALL_SUBTEST_3( qr_invertible<MatrixXcd>() );
328  }
329
330  CALL_SUBTEST_7(qr_verify_assert<Matrix3f>());
331  CALL_SUBTEST_8(qr_verify_assert<Matrix3d>());
332  CALL_SUBTEST_1(qr_verify_assert<MatrixXf>());
333  CALL_SUBTEST_2(qr_verify_assert<MatrixXd>());
334  CALL_SUBTEST_6(qr_verify_assert<MatrixXcf>());
335  CALL_SUBTEST_3(qr_verify_assert<MatrixXcd>());
336
337  // Test problem size constructors
338  CALL_SUBTEST_9(ColPivHouseholderQR<MatrixXf>(10, 20));
339
340  CALL_SUBTEST_1( qr_kahan_matrix<MatrixXf>() );
341  CALL_SUBTEST_2( qr_kahan_matrix<MatrixXd>() );
342}
343