1// This file is part of Eigen, a lightweight C++ template library
2// for linear algebra.
3//
4// Copyright (C) 2009 Claire Maurice
5// Copyright (C) 2009 Gael Guennebaud <gael.guennebaud@inria.fr>
6// Copyright (C) 2010,2012 Jitse Niesen <jitse@maths.leeds.ac.uk>
7//
8// This Source Code Form is subject to the terms of the Mozilla
9// Public License v. 2.0. If a copy of the MPL was not distributed
10// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
11
12#ifndef EIGEN_COMPLEX_SCHUR_H
13#define EIGEN_COMPLEX_SCHUR_H
14
15#include "./HessenbergDecomposition.h"
16
17namespace Eigen {
18
19namespace internal {
20template<typename MatrixType, bool IsComplex> struct complex_schur_reduce_to_hessenberg;
21}
22
23/** \eigenvalues_module \ingroup Eigenvalues_Module
24 *
25 *
26 * \class ComplexSchur
27 *
28 * \brief Performs a complex Schur decomposition of a real or complex square matrix
29 *
30 * \tparam _MatrixType the type of the matrix of which we are
31 * computing the Schur decomposition; this is expected to be an
32 * instantiation of the Matrix class template.
33 *
34 * Given a real or complex square matrix A, this class computes the
35 * Schur decomposition: \f$ A = U T U^*\f$ where U is a unitary
36 * complex matrix, and T is a complex upper triangular matrix. The
37 * diagonal of the matrix T corresponds to the eigenvalues of the
38 * matrix A.
39 *
40 * Call the function compute() to compute the Schur decomposition of
41 * a given matrix. Alternatively, you can use the
42 * ComplexSchur(const MatrixType&, bool) constructor which computes
43 * the Schur decomposition at construction time. Once the
44 * decomposition is computed, you can use the matrixU() and matrixT()
45 * functions to retrieve the matrices U and V in the decomposition.
46 *
47 * \note This code is inspired from Jampack
48 *
49 * \sa class RealSchur, class EigenSolver, class ComplexEigenSolver
50 */
51template<typename _MatrixType> class ComplexSchur
52{
53 public:
54 typedef _MatrixType MatrixType;
55 enum {
56 RowsAtCompileTime = MatrixType::RowsAtCompileTime,
57 ColsAtCompileTime = MatrixType::ColsAtCompileTime,
58 Options = MatrixType::Options,
59 MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
60 MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
61 };
62
63 /** \brief Scalar type for matrices of type \p _MatrixType. */
64 typedef typename MatrixType::Scalar Scalar;
65 typedef typename NumTraits<Scalar>::Real RealScalar;
66 typedef Eigen::Index Index; ///< \deprecated since Eigen 3.3
67
68 /** \brief Complex scalar type for \p _MatrixType.
69 *
70 * This is \c std::complex<Scalar> if #Scalar is real (e.g.,
71 * \c float or \c double) and just \c Scalar if #Scalar is
72 * complex.
73 */
74 typedef std::complex<RealScalar> ComplexScalar;
75
76 /** \brief Type for the matrices in the Schur decomposition.
77 *
78 * This is a square matrix with entries of type #ComplexScalar.
79 * The size is the same as the size of \p _MatrixType.
80 */
81 typedef Matrix<ComplexScalar, RowsAtCompileTime, ColsAtCompileTime, Options, MaxRowsAtCompileTime, MaxColsAtCompileTime> ComplexMatrixType;
82
83 /** \brief Default constructor.
84 *
85 * \param [in] size Positive integer, size of the matrix whose Schur decomposition will be computed.
86 *
87 * The default constructor is useful in cases in which the user
88 * intends to perform decompositions via compute(). The \p size
89 * parameter is only used as a hint. It is not an error to give a
90 * wrong \p size, but it may impair performance.
91 *
92 * \sa compute() for an example.
93 */
94 explicit ComplexSchur(Index size = RowsAtCompileTime==Dynamic ? 1 : RowsAtCompileTime)
95 : m_matT(size,size),
96 m_matU(size,size),
97 m_hess(size),
98 m_isInitialized(false),
99 m_matUisUptodate(false),
100 m_maxIters(-1)
101 {}
102
103 /** \brief Constructor; computes Schur decomposition of given matrix.
104 *
105 * \param[in] matrix Square matrix whose Schur decomposition is to be computed.
106 * \param[in] computeU If true, both T and U are computed; if false, only T is computed.
107 *
108 * This constructor calls compute() to compute the Schur decomposition.
109 *
110 * \sa matrixT() and matrixU() for examples.
111 */
112 template<typename InputType>
113 explicit ComplexSchur(const EigenBase<InputType>& matrix, bool computeU = true)
114 : m_matT(matrix.rows(),matrix.cols()),
115 m_matU(matrix.rows(),matrix.cols()),
116 m_hess(matrix.rows()),
117 m_isInitialized(false),
118 m_matUisUptodate(false),
119 m_maxIters(-1)
120 {
121 compute(matrix.derived(), computeU);
122 }
123
124 /** \brief Returns the unitary matrix in the Schur decomposition.
125 *
126 * \returns A const reference to the matrix U.
127 *
128 * It is assumed that either the constructor
129 * ComplexSchur(const MatrixType& matrix, bool computeU) or the
130 * member function compute(const MatrixType& matrix, bool computeU)
131 * has been called before to compute the Schur decomposition of a
132 * matrix, and that \p computeU was set to true (the default
133 * value).
134 *
135 * Example: \include ComplexSchur_matrixU.cpp
136 * Output: \verbinclude ComplexSchur_matrixU.out
137 */
138 const ComplexMatrixType& matrixU() const
139 {
140 eigen_assert(m_isInitialized && "ComplexSchur is not initialized.");
141 eigen_assert(m_matUisUptodate && "The matrix U has not been computed during the ComplexSchur decomposition.");
142 return m_matU;
143 }
144
145 /** \brief Returns the triangular matrix in the Schur decomposition.
146 *
147 * \returns A const reference to the matrix T.
148 *
149 * It is assumed that either the constructor
150 * ComplexSchur(const MatrixType& matrix, bool computeU) or the
151 * member function compute(const MatrixType& matrix, bool computeU)
152 * has been called before to compute the Schur decomposition of a
153 * matrix.
154 *
155 * Note that this function returns a plain square matrix. If you want to reference
156 * only the upper triangular part, use:
157 * \code schur.matrixT().triangularView<Upper>() \endcode
158 *
159 * Example: \include ComplexSchur_matrixT.cpp
160 * Output: \verbinclude ComplexSchur_matrixT.out
161 */
162 const ComplexMatrixType& matrixT() const
163 {
164 eigen_assert(m_isInitialized && "ComplexSchur is not initialized.");
165 return m_matT;
166 }
167
168 /** \brief Computes Schur decomposition of given matrix.
169 *
170 * \param[in] matrix Square matrix whose Schur decomposition is to be computed.
171 * \param[in] computeU If true, both T and U are computed; if false, only T is computed.
172
173 * \returns Reference to \c *this
174 *
175 * The Schur decomposition is computed by first reducing the
176 * matrix to Hessenberg form using the class
177 * HessenbergDecomposition. The Hessenberg matrix is then reduced
178 * to triangular form by performing QR iterations with a single
179 * shift. The cost of computing the Schur decomposition depends
180 * on the number of iterations; as a rough guide, it may be taken
181 * on the number of iterations; as a rough guide, it may be taken
182 * to be \f$25n^3\f$ complex flops, or \f$10n^3\f$ complex flops
183 * if \a computeU is false.
184 *
185 * Example: \include ComplexSchur_compute.cpp
186 * Output: \verbinclude ComplexSchur_compute.out
187 *
188 * \sa compute(const MatrixType&, bool, Index)
189 */
190 template<typename InputType>
191 ComplexSchur& compute(const EigenBase<InputType>& matrix, bool computeU = true);
192
193 /** \brief Compute Schur decomposition from a given Hessenberg matrix
194 * \param[in] matrixH Matrix in Hessenberg form H
195 * \param[in] matrixQ orthogonal matrix Q that transform a matrix A to H : A = Q H Q^T
196 * \param computeU Computes the matriX U of the Schur vectors
197 * \return Reference to \c *this
198 *
199 * This routine assumes that the matrix is already reduced in Hessenberg form matrixH
200 * using either the class HessenbergDecomposition or another mean.
201 * It computes the upper quasi-triangular matrix T of the Schur decomposition of H
202 * When computeU is true, this routine computes the matrix U such that
203 * A = U T U^T = (QZ) T (QZ)^T = Q H Q^T where A is the initial matrix
204 *
205 * NOTE Q is referenced if computeU is true; so, if the initial orthogonal matrix
206 * is not available, the user should give an identity matrix (Q.setIdentity())
207 *
208 * \sa compute(const MatrixType&, bool)
209 */
210 template<typename HessMatrixType, typename OrthMatrixType>
211 ComplexSchur& computeFromHessenberg(const HessMatrixType& matrixH, const OrthMatrixType& matrixQ, bool computeU=true);
212
213 /** \brief Reports whether previous computation was successful.
214 *
215 * \returns \c Success if computation was successful, \c NoConvergence otherwise.
216 */
217 ComputationInfo info() const
218 {
219 eigen_assert(m_isInitialized && "ComplexSchur is not initialized.");
220 return m_info;
221 }
222
223 /** \brief Sets the maximum number of iterations allowed.
224 *
225 * If not specified by the user, the maximum number of iterations is m_maxIterationsPerRow times the size
226 * of the matrix.
227 */
228 ComplexSchur& setMaxIterations(Index maxIters)
229 {
230 m_maxIters = maxIters;
231 return *this;
232 }
233
234 /** \brief Returns the maximum number of iterations. */
235 Index getMaxIterations()
236 {
237 return m_maxIters;
238 }
239
240 /** \brief Maximum number of iterations per row.
241 *
242 * If not otherwise specified, the maximum number of iterations is this number times the size of the
243 * matrix. It is currently set to 30.
244 */
245 static const int m_maxIterationsPerRow = 30;
246
247 protected:
248 ComplexMatrixType m_matT, m_matU;
249 HessenbergDecomposition<MatrixType> m_hess;
250 ComputationInfo m_info;
251 bool m_isInitialized;
252 bool m_matUisUptodate;
253 Index m_maxIters;
254
255 private:
256 bool subdiagonalEntryIsNeglegible(Index i);
257 ComplexScalar computeShift(Index iu, Index iter);
258 void reduceToTriangularForm(bool computeU);
259 friend struct internal::complex_schur_reduce_to_hessenberg<MatrixType, NumTraits<Scalar>::IsComplex>;
260};
261
262/** If m_matT(i+1,i) is neglegible in floating point arithmetic
263 * compared to m_matT(i,i) and m_matT(j,j), then set it to zero and
264 * return true, else return false. */
265template<typename MatrixType>
266inline bool ComplexSchur<MatrixType>::subdiagonalEntryIsNeglegible(Index i)
267{
268 RealScalar d = numext::norm1(m_matT.coeff(i,i)) + numext::norm1(m_matT.coeff(i+1,i+1));
269 RealScalar sd = numext::norm1(m_matT.coeff(i+1,i));
270 if (internal::isMuchSmallerThan(sd, d, NumTraits<RealScalar>::epsilon()))
271 {
272 m_matT.coeffRef(i+1,i) = ComplexScalar(0);
273 return true;
274 }
275 return false;
276}
277
278
279/** Compute the shift in the current QR iteration. */
280template<typename MatrixType>
281typename ComplexSchur<MatrixType>::ComplexScalar ComplexSchur<MatrixType>::computeShift(Index iu, Index iter)
282{
283 using std::abs;
284 if (iter == 10 || iter == 20)
285 {
286 // exceptional shift, taken from http://www.netlib.org/eispack/comqr.f
287 return abs(numext::real(m_matT.coeff(iu,iu-1))) + abs(numext::real(m_matT.coeff(iu-1,iu-2)));
288 }
289
290 // compute the shift as one of the eigenvalues of t, the 2x2
291 // diagonal block on the bottom of the active submatrix
292 Matrix<ComplexScalar,2,2> t = m_matT.template block<2,2>(iu-1,iu-1);
293 RealScalar normt = t.cwiseAbs().sum();
294 t /= normt; // the normalization by sf is to avoid under/overflow
295
296 ComplexScalar b = t.coeff(0,1) * t.coeff(1,0);
297 ComplexScalar c = t.coeff(0,0) - t.coeff(1,1);
298 ComplexScalar disc = sqrt(c*c + RealScalar(4)*b);
299 ComplexScalar det = t.coeff(0,0) * t.coeff(1,1) - b;
300 ComplexScalar trace = t.coeff(0,0) + t.coeff(1,1);
301 ComplexScalar eival1 = (trace + disc) / RealScalar(2);
302 ComplexScalar eival2 = (trace - disc) / RealScalar(2);
303 RealScalar eival1_norm = numext::norm1(eival1);
304 RealScalar eival2_norm = numext::norm1(eival2);
305 // A division by zero can only occur if eival1==eival2==0.
306 // In this case, det==0, and all we have to do is checking that eival2_norm!=0
307 if(eival1_norm > eival2_norm)
308 eival2 = det / eival1;
309 else if(eival2_norm!=RealScalar(0))
310 eival1 = det / eival2;
311
312 // choose the eigenvalue closest to the bottom entry of the diagonal
313 if(numext::norm1(eival1-t.coeff(1,1)) < numext::norm1(eival2-t.coeff(1,1)))
314 return normt * eival1;
315 else
316 return normt * eival2;
317}
318
319
320template<typename MatrixType>
321template<typename InputType>
322ComplexSchur<MatrixType>& ComplexSchur<MatrixType>::compute(const EigenBase<InputType>& matrix, bool computeU)
323{
324 m_matUisUptodate = false;
325 eigen_assert(matrix.cols() == matrix.rows());
326
327 if(matrix.cols() == 1)
328 {
329 m_matT = matrix.derived().template cast<ComplexScalar>();
330 if(computeU) m_matU = ComplexMatrixType::Identity(1,1);
331 m_info = Success;
332 m_isInitialized = true;
333 m_matUisUptodate = computeU;
334 return *this;
335 }
336
337 internal::complex_schur_reduce_to_hessenberg<MatrixType, NumTraits<Scalar>::IsComplex>::run(*this, matrix.derived(), computeU);
338 computeFromHessenberg(m_matT, m_matU, computeU);
339 return *this;
340}
341
342template<typename MatrixType>
343template<typename HessMatrixType, typename OrthMatrixType>
344ComplexSchur<MatrixType>& ComplexSchur<MatrixType>::computeFromHessenberg(const HessMatrixType& matrixH, const OrthMatrixType& matrixQ, bool computeU)
345{
346 m_matT = matrixH;
347 if(computeU)
348 m_matU = matrixQ;
349 reduceToTriangularForm(computeU);
350 return *this;
351}
352namespace internal {
353
354/* Reduce given matrix to Hessenberg form */
355template<typename MatrixType, bool IsComplex>
356struct complex_schur_reduce_to_hessenberg
357{
358 // this is the implementation for the case IsComplex = true
359 static void run(ComplexSchur<MatrixType>& _this, const MatrixType& matrix, bool computeU)
360 {
361 _this.m_hess.compute(matrix);
362 _this.m_matT = _this.m_hess.matrixH();
363 if(computeU) _this.m_matU = _this.m_hess.matrixQ();
364 }
365};
366
367template<typename MatrixType>
368struct complex_schur_reduce_to_hessenberg<MatrixType, false>
369{
370 static void run(ComplexSchur<MatrixType>& _this, const MatrixType& matrix, bool computeU)
371 {
372 typedef typename ComplexSchur<MatrixType>::ComplexScalar ComplexScalar;
373
374 // Note: m_hess is over RealScalar; m_matT and m_matU is over ComplexScalar
375 _this.m_hess.compute(matrix);
376 _this.m_matT = _this.m_hess.matrixH().template cast<ComplexScalar>();
377 if(computeU)
378 {
379 // This may cause an allocation which seems to be avoidable
380 MatrixType Q = _this.m_hess.matrixQ();
381 _this.m_matU = Q.template cast<ComplexScalar>();
382 }
383 }
384};
385
386} // end namespace internal
387
388// Reduce the Hessenberg matrix m_matT to triangular form by QR iteration.
389template<typename MatrixType>
390void ComplexSchur<MatrixType>::reduceToTriangularForm(bool computeU)
391{
392 Index maxIters = m_maxIters;
393 if (maxIters == -1)
394 maxIters = m_maxIterationsPerRow * m_matT.rows();
395
396 // The matrix m_matT is divided in three parts.
397 // Rows 0,...,il-1 are decoupled from the rest because m_matT(il,il-1) is zero.
398 // Rows il,...,iu is the part we are working on (the active submatrix).
399 // Rows iu+1,...,end are already brought in triangular form.
400 Index iu = m_matT.cols() - 1;
401 Index il;
402 Index iter = 0; // number of iterations we are working on the (iu,iu) element
403 Index totalIter = 0; // number of iterations for whole matrix
404
405 while(true)
406 {
407 // find iu, the bottom row of the active submatrix
408 while(iu > 0)
409 {
410 if(!subdiagonalEntryIsNeglegible(i: iu-1)) break;
411 iter = 0;
412 --iu;
413 }
414
415 // if iu is zero then we are done; the whole matrix is triangularized
416 if(iu==0) break;
417
418 // if we spent too many iterations, we give up
419 iter++;
420 totalIter++;
421 if(totalIter > maxIters) break;
422
423 // find il, the top row of the active submatrix
424 il = iu-1;
425 while(il > 0 && !subdiagonalEntryIsNeglegible(i: il-1))
426 {
427 --il;
428 }
429
430 /* perform the QR step using Givens rotations. The first rotation
431 creates a bulge; the (il+2,il) element becomes nonzero. This
432 bulge is chased down to the bottom of the active submatrix. */
433
434 ComplexScalar shift = computeShift(iu, iter);
435 JacobiRotation<ComplexScalar> rot;
436 rot.makeGivens(m_matT.coeff(il,il) - shift, m_matT.coeff(il+1,il));
437 m_matT.rightCols(m_matT.cols()-il).applyOnTheLeft(il, il+1, rot.adjoint());
438 m_matT.topRows((std::min)(a: il+2,b: iu)+1).applyOnTheRight(il, il+1, rot);
439 if(computeU) m_matU.applyOnTheRight(il, il+1, rot);
440
441 for(Index i=il+1 ; i<iu ; i++)
442 {
443 rot.makeGivens(m_matT.coeffRef(i,i-1), m_matT.coeffRef(i+1,i-1), &m_matT.coeffRef(i,i-1));
444 m_matT.coeffRef(i+1,i-1) = ComplexScalar(0);
445 m_matT.rightCols(m_matT.cols()-i).applyOnTheLeft(i, i+1, rot.adjoint());
446 m_matT.topRows((std::min)(a: i+2,b: iu)+1).applyOnTheRight(i, i+1, rot);
447 if(computeU) m_matU.applyOnTheRight(i, i+1, rot);
448 }
449 }
450
451 if(totalIter <= maxIters)
452 m_info = Success;
453 else
454 m_info = NoConvergence;
455
456 m_isInitialized = true;
457 m_matUisUptodate = computeU;
458}
459
460} // end namespace Eigen
461
462#endif // EIGEN_COMPLEX_SCHUR_H
463

source code of qtmultimedia/src/3rdparty/eigen/Eigen/src/Eigenvalues/ComplexSchur.h