aboutsummaryrefslogtreecommitdiffhomepage
path: root/test/cholesky.cpp
blob: e1e8b7bf762ed99555701bfe75e1481813384c26 (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2008 Gael Guennebaud <gael.guennebaud@inria.fr>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

#ifndef EIGEN_NO_ASSERTION_CHECKING
#define EIGEN_NO_ASSERTION_CHECKING
#endif

#define TEST_ENABLE_TEMPORARY_TRACKING

#include "main.h"
#include <Eigen/Cholesky>
#include <Eigen/QR>

template<typename MatrixType, int UpLo>
typename MatrixType::RealScalar matrix_l1_norm(const MatrixType& m) {
  if(m.cols()==0) return typename MatrixType::RealScalar(0);
  MatrixType symm = m.template selfadjointView<UpLo>();
  return symm.cwiseAbs().colwise().sum().maxCoeff();
}

template<typename MatrixType,template <typename,int> class CholType> void test_chol_update(const MatrixType& symm)
{
  typedef typename MatrixType::Scalar Scalar;
  typedef typename MatrixType::RealScalar RealScalar;
  typedef Matrix<Scalar, MatrixType::RowsAtCompileTime, 1> VectorType;

  MatrixType symmLo = symm.template triangularView<Lower>();
  MatrixType symmUp = symm.template triangularView<Upper>();
  MatrixType symmCpy = symm;

  CholType<MatrixType,Lower> chollo(symmLo);
  CholType<MatrixType,Upper> cholup(symmUp);

  for (int k=0; k<10; ++k)
  {
    VectorType vec = VectorType::Random(symm.rows());
    RealScalar sigma = internal::random<RealScalar>();
    symmCpy += sigma * vec * vec.adjoint();

    // we are doing some downdates, so it might be the case that the matrix is not SPD anymore
    CholType<MatrixType,Lower> chol(symmCpy);
    if(chol.info()!=Success)
      break;

    chollo.rankUpdate(vec, sigma);
    VERIFY_IS_APPROX(symmCpy, chollo.reconstructedMatrix());

    cholup.rankUpdate(vec, sigma);
    VERIFY_IS_APPROX(symmCpy, cholup.reconstructedMatrix());
  }
}

template<typename MatrixType> void cholesky(const MatrixType& m)
{
  /* this test covers the following files:
     LLT.h LDLT.h
  */
  Index rows = m.rows();
  Index cols = m.cols();

  typedef typename MatrixType::Scalar Scalar;
  typedef typename NumTraits<Scalar>::Real RealScalar;
  typedef Matrix<Scalar, MatrixType::RowsAtCompileTime, MatrixType::RowsAtCompileTime> SquareMatrixType;
  typedef Matrix<Scalar, MatrixType::RowsAtCompileTime, 1> VectorType;

  MatrixType a0 = MatrixType::Random(rows,cols);
  VectorType vecB = VectorType::Random(rows), vecX(rows);
  MatrixType matB = MatrixType::Random(rows,cols), matX(rows,cols);
  SquareMatrixType symm =  a0 * a0.adjoint();
  // let's make sure the matrix is not singular or near singular
  for (int k=0; k<3; ++k)
  {
    MatrixType a1 = MatrixType::Random(rows,cols);
    symm += a1 * a1.adjoint();
  }

  {
    SquareMatrixType symmUp = symm.template triangularView<Upper>();
    SquareMatrixType symmLo = symm.template triangularView<Lower>();

    LLT<SquareMatrixType,Lower> chollo(symmLo);
    VERIFY_IS_APPROX(symm, chollo.reconstructedMatrix());
    vecX = chollo.solve(vecB);
    VERIFY_IS_APPROX(symm * vecX, vecB);
    matX = chollo.solve(matB);
    VERIFY_IS_APPROX(symm * matX, matB);

    const MatrixType symmLo_inverse = chollo.solve(MatrixType::Identity(rows,cols));
    RealScalar rcond = (RealScalar(1) / matrix_l1_norm<MatrixType, Lower>(symmLo)) /
                             matrix_l1_norm<MatrixType, Lower>(symmLo_inverse);
    RealScalar rcond_est = chollo.rcond();
    // Verify that the estimated condition number is within a factor of 10 of the
    // truth.
    VERIFY(rcond_est >= rcond / 10 && rcond_est <= rcond * 10);

    // test the upper mode
    LLT<SquareMatrixType,Upper> cholup(symmUp);
    VERIFY_IS_APPROX(symm, cholup.reconstructedMatrix());
    vecX = cholup.solve(vecB);
    VERIFY_IS_APPROX(symm * vecX, vecB);
    matX = cholup.solve(matB);
    VERIFY_IS_APPROX(symm * matX, matB);

    // Verify that the estimated condition number is within a factor of 10 of the
    // truth.
    const MatrixType symmUp_inverse = cholup.solve(MatrixType::Identity(rows,cols));
    rcond = (RealScalar(1) / matrix_l1_norm<MatrixType, Upper>(symmUp)) /
                             matrix_l1_norm<MatrixType, Upper>(symmUp_inverse);
    rcond_est = cholup.rcond();
    VERIFY(rcond_est >= rcond / 10 && rcond_est <= rcond * 10);


    MatrixType neg = -symmLo;
    chollo.compute(neg);
    VERIFY(neg.size()==0 || chollo.info()==NumericalIssue);

    VERIFY_IS_APPROX(MatrixType(chollo.matrixL().transpose().conjugate()), MatrixType(chollo.matrixU()));
    VERIFY_IS_APPROX(MatrixType(chollo.matrixU().transpose().conjugate()), MatrixType(chollo.matrixL()));
    VERIFY_IS_APPROX(MatrixType(cholup.matrixL().transpose().conjugate()), MatrixType(cholup.matrixU()));
    VERIFY_IS_APPROX(MatrixType(cholup.matrixU().transpose().conjugate()), MatrixType(cholup.matrixL()));

    // test some special use cases of SelfCwiseBinaryOp:
    MatrixType m1 = MatrixType::Random(rows,cols), m2(rows,cols);
    m2 = m1;
    m2 += symmLo.template selfadjointView<Lower>().llt().solve(matB);
    VERIFY_IS_APPROX(m2, m1 + symmLo.template selfadjointView<Lower>().llt().solve(matB));
    m2 = m1;
    m2 -= symmLo.template selfadjointView<Lower>().llt().solve(matB);
    VERIFY_IS_APPROX(m2, m1 - symmLo.template selfadjointView<Lower>().llt().solve(matB));
    m2 = m1;
    m2.noalias() += symmLo.template selfadjointView<Lower>().llt().solve(matB);
    VERIFY_IS_APPROX(m2, m1 + symmLo.template selfadjointView<Lower>().llt().solve(matB));
    m2 = m1;
    m2.noalias() -= symmLo.template selfadjointView<Lower>().llt().solve(matB);
    VERIFY_IS_APPROX(m2, m1 - symmLo.template selfadjointView<Lower>().llt().solve(matB));
  }

  // LDLT
  {
    int sign = internal::random<int>()%2 ? 1 : -1;

    if(sign == -1)
    {
      symm = -symm; // test a negative matrix
    }

    SquareMatrixType symmUp = symm.template triangularView<Upper>();
    SquareMatrixType symmLo = symm.template triangularView<Lower>();

    LDLT<SquareMatrixType,Lower> ldltlo(symmLo);
    VERIFY(ldltlo.info()==Success);
    VERIFY_IS_APPROX(symm, ldltlo.reconstructedMatrix());
    vecX = ldltlo.solve(vecB);
    VERIFY_IS_APPROX(symm * vecX, vecB);
    matX = ldltlo.solve(matB);
    VERIFY_IS_APPROX(symm * matX, matB);

    const MatrixType symmLo_inverse = ldltlo.solve(MatrixType::Identity(rows,cols));
    RealScalar rcond = (RealScalar(1) / matrix_l1_norm<MatrixType, Lower>(symmLo)) /
                             matrix_l1_norm<MatrixType, Lower>(symmLo_inverse);
    RealScalar rcond_est = ldltlo.rcond();
    // Verify that the estimated condition number is within a factor of 10 of the
    // truth.
    VERIFY(rcond_est >= rcond / 10 && rcond_est <= rcond * 10);


    LDLT<SquareMatrixType,Upper> ldltup(symmUp);
    VERIFY(ldltup.info()==Success);
    VERIFY_IS_APPROX(symm, ldltup.reconstructedMatrix());
    vecX = ldltup.solve(vecB);
    VERIFY_IS_APPROX(symm * vecX, vecB);
    matX = ldltup.solve(matB);
    VERIFY_IS_APPROX(symm * matX, matB);

    // Verify that the estimated condition number is within a factor of 10 of the
    // truth.
    const MatrixType symmUp_inverse = ldltup.solve(MatrixType::Identity(rows,cols));
    rcond = (RealScalar(1) / matrix_l1_norm<MatrixType, Upper>(symmUp)) /
                             matrix_l1_norm<MatrixType, Upper>(symmUp_inverse);
    rcond_est = ldltup.rcond();
    VERIFY(rcond_est >= rcond / 10 && rcond_est <= rcond * 10);

    VERIFY_IS_APPROX(MatrixType(ldltlo.matrixL().transpose().conjugate()), MatrixType(ldltlo.matrixU()));
    VERIFY_IS_APPROX(MatrixType(ldltlo.matrixU().transpose().conjugate()), MatrixType(ldltlo.matrixL()));
    VERIFY_IS_APPROX(MatrixType(ldltup.matrixL().transpose().conjugate()), MatrixType(ldltup.matrixU()));
    VERIFY_IS_APPROX(MatrixType(ldltup.matrixU().transpose().conjugate()), MatrixType(ldltup.matrixL()));

    if(MatrixType::RowsAtCompileTime==Dynamic)
    {
      // note : each inplace permutation requires a small temporary vector (mask)

      // check inplace solve
      matX = matB;
      VERIFY_EVALUATION_COUNT(matX = ldltlo.solve(matX), 0);
      VERIFY_IS_APPROX(matX, ldltlo.solve(matB).eval());


      matX = matB;
      VERIFY_EVALUATION_COUNT(matX = ldltup.solve(matX), 0);
      VERIFY_IS_APPROX(matX, ldltup.solve(matB).eval());
    }

    // restore
    if(sign == -1)
      symm = -symm;

    // check matrices coming from linear constraints with Lagrange multipliers
    if(rows>=3)
    {
      SquareMatrixType A = symm;
      Index c = internal::random<Index>(0,rows-2);
      A.bottomRightCorner(c,c).setZero();
      // Make sure a solution exists:
      vecX.setRandom();
      vecB = A * vecX;
      vecX.setZero();
      ldltlo.compute(A);
      VERIFY_IS_APPROX(A, ldltlo.reconstructedMatrix());
      vecX = ldltlo.solve(vecB);
      VERIFY_IS_APPROX(A * vecX, vecB);
    }

    // check non-full rank matrices
    if(rows>=3)
    {
      Index r = internal::random<Index>(1,rows-1);
      Matrix<Scalar,Dynamic,Dynamic> a = Matrix<Scalar,Dynamic,Dynamic>::Random(rows,r);
      SquareMatrixType A = a * a.adjoint();
      // Make sure a solution exists:
      vecX.setRandom();
      vecB = A * vecX;
      vecX.setZero();
      ldltlo.compute(A);
      VERIFY_IS_APPROX(A, ldltlo.reconstructedMatrix());
      vecX = ldltlo.solve(vecB);
      VERIFY_IS_APPROX(A * vecX, vecB);
    }

    // check matrices with a wide spectrum
    if(rows>=3)
    {
      using std::pow;
      using std::sqrt;
      RealScalar s = (std::min)(16,std::numeric_limits<RealScalar>::max_exponent10/8);
      Matrix<Scalar,Dynamic,Dynamic> a = Matrix<Scalar,Dynamic,Dynamic>::Random(rows,rows);
      Matrix<RealScalar,Dynamic,1> d =  Matrix<RealScalar,Dynamic,1>::Random(rows);
      for(Index k=0; k<rows; ++k)
        d(k) = d(k)*pow(RealScalar(10),internal::random<RealScalar>(-s,s));
      SquareMatrixType A = a * d.asDiagonal() * a.adjoint();
      // Make sure a solution exists:
      vecX.setRandom();
      vecB = A * vecX;
      vecX.setZero();
      ldltlo.compute(A);
      VERIFY_IS_APPROX(A, ldltlo.reconstructedMatrix());
      vecX = ldltlo.solve(vecB);

      if(ldltlo.vectorD().real().cwiseAbs().minCoeff()>RealScalar(0))
      {
        VERIFY_IS_APPROX(A * vecX,vecB);
      }
      else
      {
        RealScalar large_tol =  sqrt(test_precision<RealScalar>());
        VERIFY((A * vecX).isApprox(vecB, large_tol));

        ++g_test_level;
        VERIFY_IS_APPROX(A * vecX,vecB);
        --g_test_level;
      }
    }
  }

  // update/downdate
  CALL_SUBTEST(( test_chol_update<SquareMatrixType,LLT>(symm)  ));
  CALL_SUBTEST(( test_chol_update<SquareMatrixType,LDLT>(symm) ));
}

template<typename MatrixType> void cholesky_cplx(const MatrixType& m)
{
  // classic test
  cholesky(m);

  // test mixing real/scalar types

  Index rows = m.rows();
  Index cols = m.cols();

  typedef typename MatrixType::Scalar Scalar;
  typedef typename NumTraits<Scalar>::Real RealScalar;
  typedef Matrix<RealScalar, MatrixType::RowsAtCompileTime, MatrixType::RowsAtCompileTime> RealMatrixType;
  typedef Matrix<Scalar, MatrixType::RowsAtCompileTime, 1> VectorType;

  RealMatrixType a0 = RealMatrixType::Random(rows,cols);
  VectorType vecB = VectorType::Random(rows), vecX(rows);
  MatrixType matB = MatrixType::Random(rows,cols), matX(rows,cols);
  RealMatrixType symm =  a0 * a0.adjoint();
  // let's make sure the matrix is not singular or near singular
  for (int k=0; k<3; ++k)
  {
    RealMatrixType a1 = RealMatrixType::Random(rows,cols);
    symm += a1 * a1.adjoint();
  }

  {
    RealMatrixType symmLo = symm.template triangularView<Lower>();

    LLT<RealMatrixType,Lower> chollo(symmLo);
    VERIFY_IS_APPROX(symm, chollo.reconstructedMatrix());
    vecX = chollo.solve(vecB);
    VERIFY_IS_APPROX(symm * vecX, vecB);
//     matX = chollo.solve(matB);
//     VERIFY_IS_APPROX(symm * matX, matB);
  }

  // LDLT
  {
    int sign = internal::random<int>()%2 ? 1 : -1;

    if(sign == -1)
    {
      symm = -symm; // test a negative matrix
    }

    RealMatrixType symmLo = symm.template triangularView<Lower>();

    LDLT<RealMatrixType,Lower> ldltlo(symmLo);
    VERIFY(ldltlo.info()==Success);
    VERIFY_IS_APPROX(symm, ldltlo.reconstructedMatrix());
    vecX = ldltlo.solve(vecB);
    VERIFY_IS_APPROX(symm * vecX, vecB);
//     matX = ldltlo.solve(matB);
//     VERIFY_IS_APPROX(symm * matX, matB);
  }
}

// regression test for bug 241
template<typename MatrixType> void cholesky_bug241(const MatrixType& m)
{
  eigen_assert(m.rows() == 2 && m.cols() == 2);

  typedef typename MatrixType::Scalar Scalar;
  typedef Matrix<Scalar, MatrixType::RowsAtCompileTime, 1> VectorType;

  MatrixType matA;
  matA << 1, 1, 1, 1;
  VectorType vecB;
  vecB << 1, 1;
  VectorType vecX = matA.ldlt().solve(vecB);
  VERIFY_IS_APPROX(matA * vecX, vecB);
}

// LDLT is not guaranteed to work for indefinite matrices, but happens to work fine if matrix is diagonal.
// This test checks that LDLT reports correctly that matrix is indefinite.
// See http://forum.kde.org/viewtopic.php?f=74&t=106942 and bug 736
template<typename MatrixType> void cholesky_definiteness(const MatrixType& m)
{
  eigen_assert(m.rows() == 2 && m.cols() == 2);
  MatrixType mat;
  LDLT<MatrixType> ldlt(2);

  {
    mat << 1, 0, 0, -1;
    ldlt.compute(mat);
    VERIFY(ldlt.info()==Success);
    VERIFY(!ldlt.isNegative());
    VERIFY(!ldlt.isPositive());
    VERIFY_IS_APPROX(mat,ldlt.reconstructedMatrix());
  }
  {
    mat << 1, 2, 2, 1;
    ldlt.compute(mat);
    VERIFY(ldlt.info()==Success);
    VERIFY(!ldlt.isNegative());
    VERIFY(!ldlt.isPositive());
    VERIFY_IS_APPROX(mat,ldlt.reconstructedMatrix());
  }
  {
    mat << 0, 0, 0, 0;
    ldlt.compute(mat);
    VERIFY(ldlt.info()==Success);
    VERIFY(ldlt.isNegative());
    VERIFY(ldlt.isPositive());
    VERIFY_IS_APPROX(mat,ldlt.reconstructedMatrix());
  }
  {
    mat << 0, 0, 0, 1;
    ldlt.compute(mat);
    VERIFY(ldlt.info()==Success);
    VERIFY(!ldlt.isNegative());
    VERIFY(ldlt.isPositive());
    VERIFY_IS_APPROX(mat,ldlt.reconstructedMatrix());
  }
  {
    mat << -1, 0, 0, 0;
    ldlt.compute(mat);
    VERIFY(ldlt.info()==Success);
    VERIFY(ldlt.isNegative());
    VERIFY(!ldlt.isPositive());
    VERIFY_IS_APPROX(mat,ldlt.reconstructedMatrix());
  }
}

template<typename>
void cholesky_faillure_cases()
{
  MatrixXd mat;
  LDLT<MatrixXd> ldlt;

  {
    mat.resize(2,2);
    mat << 0, 1, 1, 0;
    ldlt.compute(mat);
    VERIFY_IS_NOT_APPROX(mat,ldlt.reconstructedMatrix());
    VERIFY(ldlt.info()==NumericalIssue);
  }
#if (!EIGEN_ARCH_i386) || defined(EIGEN_VECTORIZE_SSE2)
  {
    mat.resize(3,3);
    mat << -1, -3, 3,
           -3, -8.9999999999999999999, 1,
            3, 1, 0;
    ldlt.compute(mat);
    VERIFY(ldlt.info()==NumericalIssue);
    VERIFY_IS_NOT_APPROX(mat,ldlt.reconstructedMatrix());
  }
#endif
  {
    mat.resize(3,3);
    mat <<  1, 2, 3,
            2, 4, 1,
            3, 1, 0;
    ldlt.compute(mat);
    VERIFY(ldlt.info()==NumericalIssue);
    VERIFY_IS_NOT_APPROX(mat,ldlt.reconstructedMatrix());
  }

  {
    mat.resize(8,8);
    mat <<  0.1, 0, -0.1, 0, 0, 0, 1, 0,
            0, 4.24667, 0, 2.00333, 0, 0, 0, 0,
            -0.1, 0, 0.2, 0, -0.1, 0, 0, 0,
            0, 2.00333, 0, 8.49333, 0, 2.00333, 0, 0,
            0, 0, -0.1, 0, 0.1, 0, 0, 1,
            0, 0, 0, 2.00333, 0, 4.24667, 0, 0,
            1, 0, 0, 0, 0, 0, 0, 0,
            0, 0, 0, 0, 1, 0, 0, 0;
    ldlt.compute(mat);
    VERIFY(ldlt.info()==NumericalIssue);
    VERIFY_IS_NOT_APPROX(mat,ldlt.reconstructedMatrix());
  }

  // bug 1479
  {
    mat.resize(4,4);
    mat <<  1, 2, 0, 1,
            2, 4, 0, 2,
            0, 0, 0, 1,
            1, 2, 1, 1;
    ldlt.compute(mat);
    VERIFY(ldlt.info()==NumericalIssue);
    VERIFY_IS_NOT_APPROX(mat,ldlt.reconstructedMatrix());
  }
}

template<typename MatrixType> void cholesky_verify_assert()
{
  MatrixType tmp;

  LLT<MatrixType> llt;
  VERIFY_RAISES_ASSERT(llt.matrixL())
  VERIFY_RAISES_ASSERT(llt.matrixU())
  VERIFY_RAISES_ASSERT(llt.solve(tmp))
  VERIFY_RAISES_ASSERT(llt.solveInPlace(&tmp))

  LDLT<MatrixType> ldlt;
  VERIFY_RAISES_ASSERT(ldlt.matrixL())
  VERIFY_RAISES_ASSERT(ldlt.permutationP())
  VERIFY_RAISES_ASSERT(ldlt.vectorD())
  VERIFY_RAISES_ASSERT(ldlt.isPositive())
  VERIFY_RAISES_ASSERT(ldlt.isNegative())
  VERIFY_RAISES_ASSERT(ldlt.solve(tmp))
  VERIFY_RAISES_ASSERT(ldlt.solveInPlace(&tmp))
}

EIGEN_DECLARE_TEST(cholesky)
{
  int s = 0;
  for(int i = 0; i < g_repeat; i++) {
    CALL_SUBTEST_1( cholesky(Matrix<double,1,1>()) );
    CALL_SUBTEST_3( cholesky(Matrix2d()) );
    CALL_SUBTEST_3( cholesky_bug241(Matrix2d()) );
    CALL_SUBTEST_3( cholesky_definiteness(Matrix2d()) );
    CALL_SUBTEST_4( cholesky(Matrix3f()) );
    CALL_SUBTEST_5( cholesky(Matrix4d()) );

    s = internal::random<int>(1,EIGEN_TEST_MAX_SIZE);
    CALL_SUBTEST_2( cholesky(MatrixXd(s,s)) );
    TEST_SET_BUT_UNUSED_VARIABLE(s)

    s = internal::random<int>(1,EIGEN_TEST_MAX_SIZE/2);
    CALL_SUBTEST_6( cholesky_cplx(MatrixXcd(s,s)) );
    TEST_SET_BUT_UNUSED_VARIABLE(s)
  }
  // empty matrix, regression test for Bug 785:
  CALL_SUBTEST_2( cholesky(MatrixXd(0,0)) );

  // This does not work yet:
  // CALL_SUBTEST_2( cholesky(Matrix<double,0,0>()) );

  CALL_SUBTEST_4( cholesky_verify_assert<Matrix3f>() );
  CALL_SUBTEST_7( cholesky_verify_assert<Matrix3d>() );
  CALL_SUBTEST_8( cholesky_verify_assert<MatrixXf>() );
  CALL_SUBTEST_2( cholesky_verify_assert<MatrixXd>() );

  // Test problem size constructors
  CALL_SUBTEST_9( LLT<MatrixXf>(10) );
  CALL_SUBTEST_9( LDLT<MatrixXf>(10) );

  CALL_SUBTEST_2( cholesky_faillure_cases<void>() );

  TEST_SET_BUT_UNUSED_VARIABLE(nb_temporaries)
}