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