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) 2010,2012 Jitse Niesen <jitse@maths.leeds.ac.uk>
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 #ifndef EIGEN_REAL_SCHUR_H
12 #define EIGEN_REAL_SCHUR_H
13
14 #include "./HessenbergDecomposition.h"
15
16 namespace Eigen {
17
18 /** \eigenvalues_module \ingroup Eigenvalues_Module
19 *
20 *
21 * \class RealSchur
22 *
23 * \brief Performs a real Schur decomposition of a square matrix
24 *
25 * \tparam _MatrixType the type of the matrix of which we are computing the
26 * real Schur decomposition; this is expected to be an instantiation of the
27 * Matrix class template.
28 *
29 * Given a real square matrix A, this class computes the real Schur
30 * decomposition: \f$ A = U T U^T \f$ where U is a real orthogonal matrix and
31 * T is a real quasi-triangular matrix. An orthogonal matrix is a matrix whose
32 * inverse is equal to its transpose, \f$ U^{-1} = U^T \f$. A quasi-triangular
33 * matrix is a block-triangular matrix whose diagonal consists of 1-by-1
34 * blocks and 2-by-2 blocks with complex eigenvalues. The eigenvalues of the
35 * blocks on the diagonal of T are the same as the eigenvalues of the matrix
36 * A, and thus the real Schur decomposition is used in EigenSolver to compute
37 * the eigendecomposition of a matrix.
38 *
39 * Call the function compute() to compute the real Schur decomposition of a
40 * given matrix. Alternatively, you can use the RealSchur(const MatrixType&, bool)
41 * constructor which computes the real Schur decomposition at construction
42 * time. Once the decomposition is computed, you can use the matrixU() and
43 * matrixT() functions to retrieve the matrices U and T in the decomposition.
44 *
45 * The documentation of RealSchur(const MatrixType&, bool) contains an example
46 * of the typical use of this class.
47 *
48 * \note The implementation is adapted from
49 * <a href="http://math.nist.gov/javanumerics/jama/">JAMA</a> (public domain).
50 * Their code is based on EISPACK.
51 *
52 * \sa class ComplexSchur, class EigenSolver, class ComplexEigenSolver
53 */
54 template<typename _MatrixType> class RealSchur
55 {
56 public:
57 typedef _MatrixType MatrixType;
58 enum {
59 RowsAtCompileTime = MatrixType::RowsAtCompileTime,
60 ColsAtCompileTime = MatrixType::ColsAtCompileTime,
61 Options = MatrixType::Options,
62 MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
63 MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
64 };
65 typedef typename MatrixType::Scalar Scalar;
66 typedef std::complex<typename NumTraits<Scalar>::Real> ComplexScalar;
67 typedef typename MatrixType::Index Index;
68
69 typedef Matrix<ComplexScalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> EigenvalueType;
70 typedef Matrix<Scalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> ColumnVectorType;
71
72 /** \brief Default constructor.
73 *
74 * \param [in] size Positive integer, size of the matrix whose Schur decomposition will be computed.
75 *
76 * The default constructor is useful in cases in which the user intends to
77 * perform decompositions via compute(). The \p size parameter is only
78 * used as a hint. It is not an error to give a wrong \p size, but it may
79 * impair performance.
80 *
81 * \sa compute() for an example.
82 */
83 RealSchur(Index size = RowsAtCompileTime==Dynamic ? 1 : RowsAtCompileTime)
m_matT(size,size)84 : m_matT(size, size),
85 m_matU(size, size),
86 m_workspaceVector(size),
87 m_hess(size),
88 m_isInitialized(false),
89 m_matUisUptodate(false),
90 m_maxIters(-1)
91 { }
92
93 /** \brief Constructor; computes real Schur decomposition of given matrix.
94 *
95 * \param[in] matrix Square matrix whose Schur decomposition is to be computed.
96 * \param[in] computeU If true, both T and U are computed; if false, only T is computed.
97 *
98 * This constructor calls compute() to compute the Schur decomposition.
99 *
100 * Example: \include RealSchur_RealSchur_MatrixType.cpp
101 * Output: \verbinclude RealSchur_RealSchur_MatrixType.out
102 */
103 RealSchur(const MatrixType& matrix, bool computeU = true)
104 : m_matT(matrix.rows(),matrix.cols()),
105 m_matU(matrix.rows(),matrix.cols()),
106 m_workspaceVector(matrix.rows()),
107 m_hess(matrix.rows()),
108 m_isInitialized(false),
109 m_matUisUptodate(false),
110 m_maxIters(-1)
111 {
112 compute(matrix, computeU);
113 }
114
115 /** \brief Returns the orthogonal matrix in the Schur decomposition.
116 *
117 * \returns A const reference to the matrix U.
118 *
119 * \pre Either the constructor RealSchur(const MatrixType&, bool) or the
120 * member function compute(const MatrixType&, bool) has been called before
121 * to compute the Schur decomposition of a matrix, and \p computeU was set
122 * to true (the default value).
123 *
124 * \sa RealSchur(const MatrixType&, bool) for an example
125 */
matrixU()126 const MatrixType& matrixU() const
127 {
128 eigen_assert(m_isInitialized && "RealSchur is not initialized.");
129 eigen_assert(m_matUisUptodate && "The matrix U has not been computed during the RealSchur decomposition.");
130 return m_matU;
131 }
132
133 /** \brief Returns the quasi-triangular matrix in the Schur decomposition.
134 *
135 * \returns A const reference to the matrix T.
136 *
137 * \pre Either the constructor RealSchur(const MatrixType&, bool) or the
138 * member function compute(const MatrixType&, bool) has been called before
139 * to compute the Schur decomposition of a matrix.
140 *
141 * \sa RealSchur(const MatrixType&, bool) for an example
142 */
matrixT()143 const MatrixType& matrixT() const
144 {
145 eigen_assert(m_isInitialized && "RealSchur is not initialized.");
146 return m_matT;
147 }
148
149 /** \brief Computes Schur decomposition of given matrix.
150 *
151 * \param[in] matrix Square matrix whose Schur decomposition is to be computed.
152 * \param[in] computeU If true, both T and U are computed; if false, only T is computed.
153 * \returns Reference to \c *this
154 *
155 * The Schur decomposition is computed by first reducing the matrix to
156 * Hessenberg form using the class HessenbergDecomposition. The Hessenberg
157 * matrix is then reduced to triangular form by performing Francis QR
158 * iterations with implicit double shift. The cost of computing the Schur
159 * decomposition depends on the number of iterations; as a rough guide, it
160 * may be taken to be \f$25n^3\f$ flops if \a computeU is true and
161 * \f$10n^3\f$ flops if \a computeU is false.
162 *
163 * Example: \include RealSchur_compute.cpp
164 * Output: \verbinclude RealSchur_compute.out
165 *
166 * \sa compute(const MatrixType&, bool, Index)
167 */
168 RealSchur& compute(const MatrixType& matrix, bool computeU = true);
169
170 /** \brief Computes Schur decomposition of a Hessenberg matrix H = Z T Z^T
171 * \param[in] matrixH Matrix in Hessenberg form H
172 * \param[in] matrixQ orthogonal matrix Q that transform a matrix A to H : A = Q H Q^T
173 * \param computeU Computes the matriX U of the Schur vectors
174 * \return Reference to \c *this
175 *
176 * This routine assumes that the matrix is already reduced in Hessenberg form matrixH
177 * using either the class HessenbergDecomposition or another mean.
178 * It computes the upper quasi-triangular matrix T of the Schur decomposition of H
179 * When computeU is true, this routine computes the matrix U such that
180 * A = U T U^T = (QZ) T (QZ)^T = Q H Q^T where A is the initial matrix
181 *
182 * NOTE Q is referenced if computeU is true; so, if the initial orthogonal matrix
183 * is not available, the user should give an identity matrix (Q.setIdentity())
184 *
185 * \sa compute(const MatrixType&, bool)
186 */
187 template<typename HessMatrixType, typename OrthMatrixType>
188 RealSchur& computeFromHessenberg(const HessMatrixType& matrixH, const OrthMatrixType& matrixQ, bool computeU);
189 /** \brief Reports whether previous computation was successful.
190 *
191 * \returns \c Success if computation was succesful, \c NoConvergence otherwise.
192 */
info()193 ComputationInfo info() const
194 {
195 eigen_assert(m_isInitialized && "RealSchur is not initialized.");
196 return m_info;
197 }
198
199 /** \brief Sets the maximum number of iterations allowed.
200 *
201 * If not specified by the user, the maximum number of iterations is m_maxIterationsPerRow times the size
202 * of the matrix.
203 */
setMaxIterations(Index maxIters)204 RealSchur& setMaxIterations(Index maxIters)
205 {
206 m_maxIters = maxIters;
207 return *this;
208 }
209
210 /** \brief Returns the maximum number of iterations. */
getMaxIterations()211 Index getMaxIterations()
212 {
213 return m_maxIters;
214 }
215
216 /** \brief Maximum number of iterations per row.
217 *
218 * If not otherwise specified, the maximum number of iterations is this number times the size of the
219 * matrix. It is currently set to 40.
220 */
221 static const int m_maxIterationsPerRow = 40;
222
223 private:
224
225 MatrixType m_matT;
226 MatrixType m_matU;
227 ColumnVectorType m_workspaceVector;
228 HessenbergDecomposition<MatrixType> m_hess;
229 ComputationInfo m_info;
230 bool m_isInitialized;
231 bool m_matUisUptodate;
232 Index m_maxIters;
233
234 typedef Matrix<Scalar,3,1> Vector3s;
235
236 Scalar computeNormOfT();
237 Index findSmallSubdiagEntry(Index iu, const Scalar& norm);
238 void splitOffTwoRows(Index iu, bool computeU, const Scalar& exshift);
239 void computeShift(Index iu, Index iter, Scalar& exshift, Vector3s& shiftInfo);
240 void initFrancisQRStep(Index il, Index iu, const Vector3s& shiftInfo, Index& im, Vector3s& firstHouseholderVector);
241 void performFrancisQRStep(Index il, Index im, Index iu, bool computeU, const Vector3s& firstHouseholderVector, Scalar* workspace);
242 };
243
244
245 template<typename MatrixType>
compute(const MatrixType & matrix,bool computeU)246 RealSchur<MatrixType>& RealSchur<MatrixType>::compute(const MatrixType& matrix, bool computeU)
247 {
248 eigen_assert(matrix.cols() == matrix.rows());
249 Index maxIters = m_maxIters;
250 if (maxIters == -1)
251 maxIters = m_maxIterationsPerRow * matrix.rows();
252
253 // Step 1. Reduce to Hessenberg form
254 m_hess.compute(matrix);
255
256 // Step 2. Reduce to real Schur form
257 computeFromHessenberg(m_hess.matrixH(), m_hess.matrixQ(), computeU);
258
259 return *this;
260 }
261 template<typename MatrixType>
262 template<typename HessMatrixType, typename OrthMatrixType>
computeFromHessenberg(const HessMatrixType & matrixH,const OrthMatrixType & matrixQ,bool computeU)263 RealSchur<MatrixType>& RealSchur<MatrixType>::computeFromHessenberg(const HessMatrixType& matrixH, const OrthMatrixType& matrixQ, bool computeU)
264 {
265 m_matT = matrixH;
266 if(computeU)
267 m_matU = matrixQ;
268
269 Index maxIters = m_maxIters;
270 if (maxIters == -1)
271 maxIters = m_maxIterationsPerRow * matrixH.rows();
272 m_workspaceVector.resize(m_matT.cols());
273 Scalar* workspace = &m_workspaceVector.coeffRef(0);
274
275 // The matrix m_matT is divided in three parts.
276 // Rows 0,...,il-1 are decoupled from the rest because m_matT(il,il-1) is zero.
277 // Rows il,...,iu is the part we are working on (the active window).
278 // Rows iu+1,...,end are already brought in triangular form.
279 Index iu = m_matT.cols() - 1;
280 Index iter = 0; // iteration count for current eigenvalue
281 Index totalIter = 0; // iteration count for whole matrix
282 Scalar exshift(0); // sum of exceptional shifts
283 Scalar norm = computeNormOfT();
284
285 if(norm!=0)
286 {
287 while (iu >= 0)
288 {
289 Index il = findSmallSubdiagEntry(iu, norm);
290
291 // Check for convergence
292 if (il == iu) // One root found
293 {
294 m_matT.coeffRef(iu,iu) = m_matT.coeff(iu,iu) + exshift;
295 if (iu > 0)
296 m_matT.coeffRef(iu, iu-1) = Scalar(0);
297 iu--;
298 iter = 0;
299 }
300 else if (il == iu-1) // Two roots found
301 {
302 splitOffTwoRows(iu, computeU, exshift);
303 iu -= 2;
304 iter = 0;
305 }
306 else // No convergence yet
307 {
308 // The firstHouseholderVector vector has to be initialized to something to get rid of a silly GCC warning (-O1 -Wall -DNDEBUG )
309 Vector3s firstHouseholderVector(0,0,0), shiftInfo;
310 computeShift(iu, iter, exshift, shiftInfo);
311 iter = iter + 1;
312 totalIter = totalIter + 1;
313 if (totalIter > maxIters) break;
314 Index im;
315 initFrancisQRStep(il, iu, shiftInfo, im, firstHouseholderVector);
316 performFrancisQRStep(il, im, iu, computeU, firstHouseholderVector, workspace);
317 }
318 }
319 }
320 if(totalIter <= maxIters)
321 m_info = Success;
322 else
323 m_info = NoConvergence;
324
325 m_isInitialized = true;
326 m_matUisUptodate = computeU;
327 return *this;
328 }
329
330 /** \internal Computes and returns vector L1 norm of T */
331 template<typename MatrixType>
computeNormOfT()332 inline typename MatrixType::Scalar RealSchur<MatrixType>::computeNormOfT()
333 {
334 const Index size = m_matT.cols();
335 // FIXME to be efficient the following would requires a triangular reduxion code
336 // Scalar norm = m_matT.upper().cwiseAbs().sum()
337 // + m_matT.bottomLeftCorner(size-1,size-1).diagonal().cwiseAbs().sum();
338 Scalar norm(0);
339 for (Index j = 0; j < size; ++j)
340 norm += m_matT.col(j).segment(0, (std::min)(size,j+2)).cwiseAbs().sum();
341 return norm;
342 }
343
344 /** \internal Look for single small sub-diagonal element and returns its index */
345 template<typename MatrixType>
findSmallSubdiagEntry(Index iu,const Scalar & norm)346 inline typename MatrixType::Index RealSchur<MatrixType>::findSmallSubdiagEntry(Index iu, const Scalar& norm)
347 {
348 using std::abs;
349 Index res = iu;
350 while (res > 0)
351 {
352 Scalar s = abs(m_matT.coeff(res-1,res-1)) + abs(m_matT.coeff(res,res));
353 if (s == 0.0)
354 s = norm;
355 if (abs(m_matT.coeff(res,res-1)) < NumTraits<Scalar>::epsilon() * s)
356 break;
357 res--;
358 }
359 return res;
360 }
361
362 /** \internal Update T given that rows iu-1 and iu decouple from the rest. */
363 template<typename MatrixType>
splitOffTwoRows(Index iu,bool computeU,const Scalar & exshift)364 inline void RealSchur<MatrixType>::splitOffTwoRows(Index iu, bool computeU, const Scalar& exshift)
365 {
366 using std::sqrt;
367 using std::abs;
368 const Index size = m_matT.cols();
369
370 // The eigenvalues of the 2x2 matrix [a b; c d] are
371 // trace +/- sqrt(discr/4) where discr = tr^2 - 4*det, tr = a + d, det = ad - bc
372 Scalar p = Scalar(0.5) * (m_matT.coeff(iu-1,iu-1) - m_matT.coeff(iu,iu));
373 Scalar q = p * p + m_matT.coeff(iu,iu-1) * m_matT.coeff(iu-1,iu); // q = tr^2 / 4 - det = discr/4
374 m_matT.coeffRef(iu,iu) += exshift;
375 m_matT.coeffRef(iu-1,iu-1) += exshift;
376
377 if (q >= Scalar(0)) // Two real eigenvalues
378 {
379 Scalar z = sqrt(abs(q));
380 JacobiRotation<Scalar> rot;
381 if (p >= Scalar(0))
382 rot.makeGivens(p + z, m_matT.coeff(iu, iu-1));
383 else
384 rot.makeGivens(p - z, m_matT.coeff(iu, iu-1));
385
386 m_matT.rightCols(size-iu+1).applyOnTheLeft(iu-1, iu, rot.adjoint());
387 m_matT.topRows(iu+1).applyOnTheRight(iu-1, iu, rot);
388 m_matT.coeffRef(iu, iu-1) = Scalar(0);
389 if (computeU)
390 m_matU.applyOnTheRight(iu-1, iu, rot);
391 }
392
393 if (iu > 1)
394 m_matT.coeffRef(iu-1, iu-2) = Scalar(0);
395 }
396
397 /** \internal Form shift in shiftInfo, and update exshift if an exceptional shift is performed. */
398 template<typename MatrixType>
computeShift(Index iu,Index iter,Scalar & exshift,Vector3s & shiftInfo)399 inline void RealSchur<MatrixType>::computeShift(Index iu, Index iter, Scalar& exshift, Vector3s& shiftInfo)
400 {
401 using std::sqrt;
402 using std::abs;
403 shiftInfo.coeffRef(0) = m_matT.coeff(iu,iu);
404 shiftInfo.coeffRef(1) = m_matT.coeff(iu-1,iu-1);
405 shiftInfo.coeffRef(2) = m_matT.coeff(iu,iu-1) * m_matT.coeff(iu-1,iu);
406
407 // Wilkinson's original ad hoc shift
408 if (iter == 10)
409 {
410 exshift += shiftInfo.coeff(0);
411 for (Index i = 0; i <= iu; ++i)
412 m_matT.coeffRef(i,i) -= shiftInfo.coeff(0);
413 Scalar s = abs(m_matT.coeff(iu,iu-1)) + abs(m_matT.coeff(iu-1,iu-2));
414 shiftInfo.coeffRef(0) = Scalar(0.75) * s;
415 shiftInfo.coeffRef(1) = Scalar(0.75) * s;
416 shiftInfo.coeffRef(2) = Scalar(-0.4375) * s * s;
417 }
418
419 // MATLAB's new ad hoc shift
420 if (iter == 30)
421 {
422 Scalar s = (shiftInfo.coeff(1) - shiftInfo.coeff(0)) / Scalar(2.0);
423 s = s * s + shiftInfo.coeff(2);
424 if (s > Scalar(0))
425 {
426 s = sqrt(s);
427 if (shiftInfo.coeff(1) < shiftInfo.coeff(0))
428 s = -s;
429 s = s + (shiftInfo.coeff(1) - shiftInfo.coeff(0)) / Scalar(2.0);
430 s = shiftInfo.coeff(0) - shiftInfo.coeff(2) / s;
431 exshift += s;
432 for (Index i = 0; i <= iu; ++i)
433 m_matT.coeffRef(i,i) -= s;
434 shiftInfo.setConstant(Scalar(0.964));
435 }
436 }
437 }
438
439 /** \internal Compute index im at which Francis QR step starts and the first Householder vector. */
440 template<typename MatrixType>
initFrancisQRStep(Index il,Index iu,const Vector3s & shiftInfo,Index & im,Vector3s & firstHouseholderVector)441 inline void RealSchur<MatrixType>::initFrancisQRStep(Index il, Index iu, const Vector3s& shiftInfo, Index& im, Vector3s& firstHouseholderVector)
442 {
443 using std::abs;
444 Vector3s& v = firstHouseholderVector; // alias to save typing
445
446 for (im = iu-2; im >= il; --im)
447 {
448 const Scalar Tmm = m_matT.coeff(im,im);
449 const Scalar r = shiftInfo.coeff(0) - Tmm;
450 const Scalar s = shiftInfo.coeff(1) - Tmm;
451 v.coeffRef(0) = (r * s - shiftInfo.coeff(2)) / m_matT.coeff(im+1,im) + m_matT.coeff(im,im+1);
452 v.coeffRef(1) = m_matT.coeff(im+1,im+1) - Tmm - r - s;
453 v.coeffRef(2) = m_matT.coeff(im+2,im+1);
454 if (im == il) {
455 break;
456 }
457 const Scalar lhs = m_matT.coeff(im,im-1) * (abs(v.coeff(1)) + abs(v.coeff(2)));
458 const Scalar rhs = v.coeff(0) * (abs(m_matT.coeff(im-1,im-1)) + abs(Tmm) + abs(m_matT.coeff(im+1,im+1)));
459 if (abs(lhs) < NumTraits<Scalar>::epsilon() * rhs)
460 {
461 break;
462 }
463 }
464 }
465
466 /** \internal Perform a Francis QR step involving rows il:iu and columns im:iu. */
467 template<typename MatrixType>
performFrancisQRStep(Index il,Index im,Index iu,bool computeU,const Vector3s & firstHouseholderVector,Scalar * workspace)468 inline void RealSchur<MatrixType>::performFrancisQRStep(Index il, Index im, Index iu, bool computeU, const Vector3s& firstHouseholderVector, Scalar* workspace)
469 {
470 eigen_assert(im >= il);
471 eigen_assert(im <= iu-2);
472
473 const Index size = m_matT.cols();
474
475 for (Index k = im; k <= iu-2; ++k)
476 {
477 bool firstIteration = (k == im);
478
479 Vector3s v;
480 if (firstIteration)
481 v = firstHouseholderVector;
482 else
483 v = m_matT.template block<3,1>(k,k-1);
484
485 Scalar tau, beta;
486 Matrix<Scalar, 2, 1> ess;
487 v.makeHouseholder(ess, tau, beta);
488
489 if (beta != Scalar(0)) // if v is not zero
490 {
491 if (firstIteration && k > il)
492 m_matT.coeffRef(k,k-1) = -m_matT.coeff(k,k-1);
493 else if (!firstIteration)
494 m_matT.coeffRef(k,k-1) = beta;
495
496 // These Householder transformations form the O(n^3) part of the algorithm
497 m_matT.block(k, k, 3, size-k).applyHouseholderOnTheLeft(ess, tau, workspace);
498 m_matT.block(0, k, (std::min)(iu,k+3) + 1, 3).applyHouseholderOnTheRight(ess, tau, workspace);
499 if (computeU)
500 m_matU.block(0, k, size, 3).applyHouseholderOnTheRight(ess, tau, workspace);
501 }
502 }
503
504 Matrix<Scalar, 2, 1> v = m_matT.template block<2,1>(iu-1, iu-2);
505 Scalar tau, beta;
506 Matrix<Scalar, 1, 1> ess;
507 v.makeHouseholder(ess, tau, beta);
508
509 if (beta != Scalar(0)) // if v is not zero
510 {
511 m_matT.coeffRef(iu-1, iu-2) = beta;
512 m_matT.block(iu-1, iu-1, 2, size-iu+1).applyHouseholderOnTheLeft(ess, tau, workspace);
513 m_matT.block(0, iu-1, iu+1, 2).applyHouseholderOnTheRight(ess, tau, workspace);
514 if (computeU)
515 m_matU.block(0, iu-1, size, 2).applyHouseholderOnTheRight(ess, tau, workspace);
516 }
517
518 // clean up pollution due to round-off errors
519 for (Index i = im+2; i <= iu; ++i)
520 {
521 m_matT.coeffRef(i,i-2) = Scalar(0);
522 if (i > im+2)
523 m_matT.coeffRef(i,i-3) = Scalar(0);
524 }
525 }
526
527 } // end namespace Eigen
528
529 #endif // EIGEN_REAL_SCHUR_H
530