1 // This file is part of Eigen, a lightweight C++ template library
2 // for linear algebra.
3 //
4 // Copyright (C) 2008 Gael Guennebaud <[email protected]>
5 // Copyright (C) 2009 Benoit Jacob <[email protected]>
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 #include "solverbase.h"
15
16 template <typename MatrixType>
cod()17 void cod() {
18 STATIC_CHECK(( internal::is_same<typename CompleteOrthogonalDecomposition<MatrixType>::StorageIndex,int>::value ));
19
20 Index rows = internal::random<Index>(2, EIGEN_TEST_MAX_SIZE);
21 Index cols = internal::random<Index>(2, EIGEN_TEST_MAX_SIZE);
22 Index cols2 = internal::random<Index>(2, EIGEN_TEST_MAX_SIZE);
23 Index rank = internal::random<Index>(1, (std::min)(rows, cols) - 1);
24
25 typedef typename MatrixType::Scalar Scalar;
26 typedef Matrix<Scalar, MatrixType::RowsAtCompileTime,
27 MatrixType::RowsAtCompileTime>
28 MatrixQType;
29 MatrixType matrix;
30 createRandomPIMatrixOfRank(rank, rows, cols, matrix);
31 CompleteOrthogonalDecomposition<MatrixType> cod(matrix);
32 VERIFY(rank == cod.rank());
33 VERIFY(cols - cod.rank() == cod.dimensionOfKernel());
34 VERIFY(!cod.isInjective());
35 VERIFY(!cod.isInvertible());
36 VERIFY(!cod.isSurjective());
37
38 MatrixQType q = cod.householderQ();
39 VERIFY_IS_UNITARY(q);
40
41 MatrixType z = cod.matrixZ();
42 VERIFY_IS_UNITARY(z);
43
44 MatrixType t;
45 t.setZero(rows, cols);
46 t.topLeftCorner(rank, rank) =
47 cod.matrixT().topLeftCorner(rank, rank).template triangularView<Upper>();
48
49 MatrixType c = q * t * z * cod.colsPermutation().inverse();
50 VERIFY_IS_APPROX(matrix, c);
51
52 check_solverbase<MatrixType, MatrixType>(matrix, cod, rows, cols, cols2);
53
54 // Verify that we get the same minimum-norm solution as the SVD.
55 MatrixType exact_solution = MatrixType::Random(cols, cols2);
56 MatrixType rhs = matrix * exact_solution;
57 MatrixType cod_solution = cod.solve(rhs);
58 JacobiSVD<MatrixType> svd(matrix, ComputeThinU | ComputeThinV);
59 MatrixType svd_solution = svd.solve(rhs);
60 VERIFY_IS_APPROX(cod_solution, svd_solution);
61
62 MatrixType pinv = cod.pseudoInverse();
63 VERIFY_IS_APPROX(cod_solution, pinv * rhs);
64 }
65
66 template <typename MatrixType, int Cols2>
cod_fixedsize()67 void cod_fixedsize() {
68 enum {
69 Rows = MatrixType::RowsAtCompileTime,
70 Cols = MatrixType::ColsAtCompileTime
71 };
72 typedef typename MatrixType::Scalar Scalar;
73 typedef CompleteOrthogonalDecomposition<Matrix<Scalar, Rows, Cols> > COD;
74 int rank = internal::random<int>(1, (std::min)(int(Rows), int(Cols)) - 1);
75 Matrix<Scalar, Rows, Cols> matrix;
76 createRandomPIMatrixOfRank(rank, Rows, Cols, matrix);
77 COD cod(matrix);
78 VERIFY(rank == cod.rank());
79 VERIFY(Cols - cod.rank() == cod.dimensionOfKernel());
80 VERIFY(cod.isInjective() == (rank == Rows));
81 VERIFY(cod.isSurjective() == (rank == Cols));
82 VERIFY(cod.isInvertible() == (cod.isInjective() && cod.isSurjective()));
83
84 check_solverbase<Matrix<Scalar, Cols, Cols2>, Matrix<Scalar, Rows, Cols2> >(matrix, cod, Rows, Cols, Cols2);
85
86 // Verify that we get the same minimum-norm solution as the SVD.
87 Matrix<Scalar, Cols, Cols2> exact_solution;
88 exact_solution.setRandom(Cols, Cols2);
89 Matrix<Scalar, Rows, Cols2> rhs = matrix * exact_solution;
90 Matrix<Scalar, Cols, Cols2> cod_solution = cod.solve(rhs);
91 JacobiSVD<MatrixType> svd(matrix, ComputeFullU | ComputeFullV);
92 Matrix<Scalar, Cols, Cols2> svd_solution = svd.solve(rhs);
93 VERIFY_IS_APPROX(cod_solution, svd_solution);
94
95 typename Inverse<COD>::PlainObject pinv = cod.pseudoInverse();
96 VERIFY_IS_APPROX(cod_solution, pinv * rhs);
97 }
98
qr()99 template<typename MatrixType> void qr()
100 {
101 using std::sqrt;
102
103 STATIC_CHECK(( internal::is_same<typename ColPivHouseholderQR<MatrixType>::StorageIndex,int>::value ));
104
105 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);
106 Index rank = internal::random<Index>(1, (std::min)(rows, cols)-1);
107
108 typedef typename MatrixType::Scalar Scalar;
109 typedef typename MatrixType::RealScalar RealScalar;
110 typedef Matrix<Scalar, MatrixType::RowsAtCompileTime, MatrixType::RowsAtCompileTime> MatrixQType;
111 MatrixType m1;
112 createRandomPIMatrixOfRank(rank,rows,cols,m1);
113 ColPivHouseholderQR<MatrixType> qr(m1);
114 VERIFY_IS_EQUAL(rank, qr.rank());
115 VERIFY_IS_EQUAL(cols - qr.rank(), qr.dimensionOfKernel());
116 VERIFY(!qr.isInjective());
117 VERIFY(!qr.isInvertible());
118 VERIFY(!qr.isSurjective());
119
120 MatrixQType q = qr.householderQ();
121 VERIFY_IS_UNITARY(q);
122
123 MatrixType r = qr.matrixQR().template triangularView<Upper>();
124 MatrixType c = q * r * qr.colsPermutation().inverse();
125 VERIFY_IS_APPROX(m1, c);
126
127 // Verify that the absolute value of the diagonal elements in R are
128 // non-increasing until they reach the singularity threshold.
129 RealScalar threshold =
130 sqrt(RealScalar(rows)) * numext::abs(r(0, 0)) * NumTraits<Scalar>::epsilon();
131 for (Index i = 0; i < (std::min)(rows, cols) - 1; ++i) {
132 RealScalar x = numext::abs(r(i, i));
133 RealScalar y = numext::abs(r(i + 1, i + 1));
134 if (x < threshold && y < threshold) continue;
135 if (!test_isApproxOrLessThan(y, x)) {
136 for (Index j = 0; j < (std::min)(rows, cols); ++j) {
137 std::cout << "i = " << j << ", |r_ii| = " << numext::abs(r(j, j)) << std::endl;
138 }
139 std::cout << "Failure at i=" << i << ", rank=" << rank
140 << ", threshold=" << threshold << std::endl;
141 }
142 VERIFY_IS_APPROX_OR_LESS_THAN(y, x);
143 }
144
145 check_solverbase<MatrixType, MatrixType>(m1, qr, rows, cols, cols2);
146
147 {
148 MatrixType m2, m3;
149 Index size = rows;
150 do {
151 m1 = MatrixType::Random(size,size);
152 qr.compute(m1);
153 } while(!qr.isInvertible());
154 MatrixType m1_inv = qr.inverse();
155 m3 = m1 * MatrixType::Random(size,cols2);
156 m2 = qr.solve(m3);
157 VERIFY_IS_APPROX(m2, m1_inv*m3);
158 }
159 }
160
qr_fixedsize()161 template<typename MatrixType, int Cols2> void qr_fixedsize()
162 {
163 using std::sqrt;
164 using std::abs;
165 enum { Rows = MatrixType::RowsAtCompileTime, Cols = MatrixType::ColsAtCompileTime };
166 typedef typename MatrixType::Scalar Scalar;
167 typedef typename MatrixType::RealScalar RealScalar;
168 int rank = internal::random<int>(1, (std::min)(int(Rows), int(Cols))-1);
169 Matrix<Scalar,Rows,Cols> m1;
170 createRandomPIMatrixOfRank(rank,Rows,Cols,m1);
171 ColPivHouseholderQR<Matrix<Scalar,Rows,Cols> > qr(m1);
172 VERIFY_IS_EQUAL(rank, qr.rank());
173 VERIFY_IS_EQUAL(Cols - qr.rank(), qr.dimensionOfKernel());
174 VERIFY_IS_EQUAL(qr.isInjective(), (rank == Rows));
175 VERIFY_IS_EQUAL(qr.isSurjective(), (rank == Cols));
176 VERIFY_IS_EQUAL(qr.isInvertible(), (qr.isInjective() && qr.isSurjective()));
177
178 Matrix<Scalar,Rows,Cols> r = qr.matrixQR().template triangularView<Upper>();
179 Matrix<Scalar,Rows,Cols> c = qr.householderQ() * r * qr.colsPermutation().inverse();
180 VERIFY_IS_APPROX(m1, c);
181
182 check_solverbase<Matrix<Scalar,Cols,Cols2>, Matrix<Scalar,Rows,Cols2> >(m1, qr, Rows, Cols, Cols2);
183
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.
qr_kahan_matrix()210 template<typename MatrixType> void qr_kahan_matrix()
211 {
212 using std::sqrt;
213 using std::abs;
214 typedef typename MatrixType::Scalar Scalar;
215 typedef typename MatrixType::RealScalar RealScalar;
216
217 Index rows = 300, cols = rows;
218
219 MatrixType m1;
220 m1.setZero(rows,cols);
221 RealScalar s = std::pow(NumTraits<RealScalar>::epsilon(), 1.0 / rows);
222 RealScalar c = std::sqrt(1 - s*s);
223 RealScalar pow_s_i(1.0); // pow(s,i)
224 for (Index i = 0; i < rows; ++i) {
225 m1(i, i) = pow_s_i;
226 m1.row(i).tail(rows - i - 1) = -pow_s_i * c * MatrixType::Ones(1, rows - i - 1);
227 pow_s_i *= s;
228 }
229 m1 = (m1 + m1.transpose()).eval();
230 ColPivHouseholderQR<MatrixType> qr(m1);
231 MatrixType r = qr.matrixQR().template triangularView<Upper>();
232
233 RealScalar threshold =
234 std::sqrt(RealScalar(rows)) * numext::abs(r(0, 0)) * NumTraits<Scalar>::epsilon();
235 for (Index i = 0; i < (std::min)(rows, cols) - 1; ++i) {
236 RealScalar x = numext::abs(r(i, i));
237 RealScalar y = numext::abs(r(i + 1, i + 1));
238 if (x < threshold && y < threshold) continue;
239 if (!test_isApproxOrLessThan(y, x)) {
240 for (Index j = 0; j < (std::min)(rows, cols); ++j) {
241 std::cout << "i = " << j << ", |r_ii| = " << numext::abs(r(j, j)) << std::endl;
242 }
243 std::cout << "Failure at i=" << i << ", rank=" << qr.rank()
244 << ", threshold=" << threshold << std::endl;
245 }
246 VERIFY_IS_APPROX_OR_LESS_THAN(y, x);
247 }
248 }
249
qr_invertible()250 template<typename MatrixType> void qr_invertible()
251 {
252 using std::log;
253 using std::abs;
254 typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
255 typedef typename MatrixType::Scalar Scalar;
256
257 int size = internal::random<int>(10,50);
258
259 MatrixType m1(size, size), m2(size, size), m3(size, size);
260 m1 = MatrixType::Random(size,size);
261
262 if (internal::is_same<RealScalar,float>::value)
263 {
264 // let's build a matrix more stable to inverse
265 MatrixType a = MatrixType::Random(size,size*2);
266 m1 += a * a.adjoint();
267 }
268
269 ColPivHouseholderQR<MatrixType> qr(m1);
270
271 check_solverbase<MatrixType, MatrixType>(m1, qr, size, size, size);
272
273 // now construct a matrix with prescribed determinant
274 m1.setZero();
275 for(int i = 0; i < size; i++) m1(i,i) = internal::random<Scalar>();
276 RealScalar absdet = abs(m1.diagonal().prod());
277 m3 = qr.householderQ(); // get a unitary
278 m1 = m3 * m1 * m3;
279 qr.compute(m1);
280 VERIFY_IS_APPROX(absdet, qr.absDeterminant());
281 VERIFY_IS_APPROX(log(absdet), qr.logAbsDeterminant());
282 }
283
qr_verify_assert()284 template<typename MatrixType> void qr_verify_assert()
285 {
286 MatrixType tmp;
287
288 ColPivHouseholderQR<MatrixType> qr;
289 VERIFY_RAISES_ASSERT(qr.matrixQR())
290 VERIFY_RAISES_ASSERT(qr.solve(tmp))
291 VERIFY_RAISES_ASSERT(qr.transpose().solve(tmp))
292 VERIFY_RAISES_ASSERT(qr.adjoint().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
cod_verify_assert()303 template<typename MatrixType> void cod_verify_assert()
304 {
305 MatrixType tmp;
306
307 CompleteOrthogonalDecomposition<MatrixType> cod;
308 VERIFY_RAISES_ASSERT(cod.matrixQTZ())
309 VERIFY_RAISES_ASSERT(cod.solve(tmp))
310 VERIFY_RAISES_ASSERT(cod.transpose().solve(tmp))
311 VERIFY_RAISES_ASSERT(cod.adjoint().solve(tmp))
312 VERIFY_RAISES_ASSERT(cod.householderQ())
313 VERIFY_RAISES_ASSERT(cod.dimensionOfKernel())
314 VERIFY_RAISES_ASSERT(cod.isInjective())
315 VERIFY_RAISES_ASSERT(cod.isSurjective())
316 VERIFY_RAISES_ASSERT(cod.isInvertible())
317 VERIFY_RAISES_ASSERT(cod.pseudoInverse())
318 VERIFY_RAISES_ASSERT(cod.absDeterminant())
319 VERIFY_RAISES_ASSERT(cod.logAbsDeterminant())
320 }
321
EIGEN_DECLARE_TEST(qr_colpivoting)322 EIGEN_DECLARE_TEST(qr_colpivoting)
323 {
324 for(int i = 0; i < g_repeat; i++) {
325 CALL_SUBTEST_1( qr<MatrixXf>() );
326 CALL_SUBTEST_2( qr<MatrixXd>() );
327 CALL_SUBTEST_3( qr<MatrixXcd>() );
328 CALL_SUBTEST_4(( qr_fixedsize<Matrix<float,3,5>, 4 >() ));
329 CALL_SUBTEST_5(( qr_fixedsize<Matrix<double,6,2>, 3 >() ));
330 CALL_SUBTEST_5(( qr_fixedsize<Matrix<double,1,1>, 1 >() ));
331 }
332
333 for(int i = 0; i < g_repeat; i++) {
334 CALL_SUBTEST_1( cod<MatrixXf>() );
335 CALL_SUBTEST_2( cod<MatrixXd>() );
336 CALL_SUBTEST_3( cod<MatrixXcd>() );
337 CALL_SUBTEST_4(( cod_fixedsize<Matrix<float,3,5>, 4 >() ));
338 CALL_SUBTEST_5(( cod_fixedsize<Matrix<double,6,2>, 3 >() ));
339 CALL_SUBTEST_5(( cod_fixedsize<Matrix<double,1,1>, 1 >() ));
340 }
341
342 for(int i = 0; i < g_repeat; i++) {
343 CALL_SUBTEST_1( qr_invertible<MatrixXf>() );
344 CALL_SUBTEST_2( qr_invertible<MatrixXd>() );
345 CALL_SUBTEST_6( qr_invertible<MatrixXcf>() );
346 CALL_SUBTEST_3( qr_invertible<MatrixXcd>() );
347 }
348
349 CALL_SUBTEST_7(qr_verify_assert<Matrix3f>());
350 CALL_SUBTEST_8(qr_verify_assert<Matrix3d>());
351 CALL_SUBTEST_1(qr_verify_assert<MatrixXf>());
352 CALL_SUBTEST_2(qr_verify_assert<MatrixXd>());
353 CALL_SUBTEST_6(qr_verify_assert<MatrixXcf>());
354 CALL_SUBTEST_3(qr_verify_assert<MatrixXcd>());
355
356 CALL_SUBTEST_7(cod_verify_assert<Matrix3f>());
357 CALL_SUBTEST_8(cod_verify_assert<Matrix3d>());
358 CALL_SUBTEST_1(cod_verify_assert<MatrixXf>());
359 CALL_SUBTEST_2(cod_verify_assert<MatrixXd>());
360 CALL_SUBTEST_6(cod_verify_assert<MatrixXcf>());
361 CALL_SUBTEST_3(cod_verify_assert<MatrixXcd>());
362
363 // Test problem size constructors
364 CALL_SUBTEST_9(ColPivHouseholderQR<MatrixXf>(10, 20));
365
366 CALL_SUBTEST_1( qr_kahan_matrix<MatrixXf>() );
367 CALL_SUBTEST_2( qr_kahan_matrix<MatrixXd>() );
368 }
369