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 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
65 template <typename MatrixType, int Cols2>
cod_fixedsize()66 void 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
qr()94 template<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
qr_fixedsize()158 template<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.
qr_kahan_matrix()210 template<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
qr_invertible()251 template<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
qr_verify_assert()286 template<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
test_qr_colpivoting()303 void 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