Eigen  3.2.7
 All Classes Namespaces Functions Variables Typedefs Enumerations Enumerator Friends Groups Pages
RealQZ.h
1 // This file is part of Eigen, a lightweight C++ template library
2 // for linear algebra.
3 //
4 // Copyright (C) 2012 Alexey Korepanov <[email protected]>
5 //
6 // This Source Code Form is subject to the terms of the Mozilla
7 // Public License v. 2.0. If a copy of the MPL was not distributed
8 // with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
9 
10 #ifndef EIGEN_REAL_QZ_H
11 #define EIGEN_REAL_QZ_H
12 
13 namespace Eigen {
14 
57  template<typename _MatrixType> class RealQZ
58  {
59  public:
60  typedef _MatrixType MatrixType;
61  enum {
62  RowsAtCompileTime = MatrixType::RowsAtCompileTime,
63  ColsAtCompileTime = MatrixType::ColsAtCompileTime,
64  Options = MatrixType::Options,
65  MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
66  MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
67  };
68  typedef typename MatrixType::Scalar Scalar;
69  typedef std::complex<typename NumTraits<Scalar>::Real> ComplexScalar;
70  typedef typename MatrixType::Index Index;
71 
72  typedef Matrix<ComplexScalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> EigenvalueType;
73  typedef Matrix<Scalar, ColsAtCompileTime, 1, Options & ~RowMajor, MaxColsAtCompileTime, 1> ColumnVectorType;
74 
86  RealQZ(Index size = RowsAtCompileTime==Dynamic ? 1 : RowsAtCompileTime) :
87  m_S(size, size),
88  m_T(size, size),
89  m_Q(size, size),
90  m_Z(size, size),
91  m_workspace(size*2),
92  m_maxIters(400),
93  m_isInitialized(false)
94  { }
95 
104  RealQZ(const MatrixType& A, const MatrixType& B, bool computeQZ = true) :
105  m_S(A.rows(),A.cols()),
106  m_T(A.rows(),A.cols()),
107  m_Q(A.rows(),A.cols()),
108  m_Z(A.rows(),A.cols()),
109  m_workspace(A.rows()*2),
110  m_maxIters(400),
111  m_isInitialized(false) {
112  compute(A, B, computeQZ);
113  }
114 
119  const MatrixType& matrixQ() const {
120  eigen_assert(m_isInitialized && "RealQZ is not initialized.");
121  eigen_assert(m_computeQZ && "The matrices Q and Z have not been computed during the QZ decomposition.");
122  return m_Q;
123  }
124 
129  const MatrixType& matrixZ() const {
130  eigen_assert(m_isInitialized && "RealQZ is not initialized.");
131  eigen_assert(m_computeQZ && "The matrices Q and Z have not been computed during the QZ decomposition.");
132  return m_Z;
133  }
134 
139  const MatrixType& matrixS() const {
140  eigen_assert(m_isInitialized && "RealQZ is not initialized.");
141  return m_S;
142  }
143 
148  const MatrixType& matrixT() const {
149  eigen_assert(m_isInitialized && "RealQZ is not initialized.");
150  return m_T;
151  }
152 
160  RealQZ& compute(const MatrixType& A, const MatrixType& B, bool computeQZ = true);
161 
166  ComputationInfo info() const
167  {
168  eigen_assert(m_isInitialized && "RealQZ is not initialized.");
169  return m_info;
170  }
171 
174  Index iterations() const
175  {
176  eigen_assert(m_isInitialized && "RealQZ is not initialized.");
177  return m_global_iter;
178  }
179 
183  RealQZ& setMaxIterations(Index maxIters)
184  {
185  m_maxIters = maxIters;
186  return *this;
187  }
188 
189  private:
190 
191  MatrixType m_S, m_T, m_Q, m_Z;
192  Matrix<Scalar,Dynamic,1> m_workspace;
193  ComputationInfo m_info;
194  Index m_maxIters;
195  bool m_isInitialized;
196  bool m_computeQZ;
197  Scalar m_normOfT, m_normOfS;
198  Index m_global_iter;
199 
200  typedef Matrix<Scalar,3,1> Vector3s;
201  typedef Matrix<Scalar,2,1> Vector2s;
202  typedef Matrix<Scalar,2,2> Matrix2s;
203  typedef JacobiRotation<Scalar> JRs;
204 
205  void hessenbergTriangular();
206  void computeNorms();
207  Index findSmallSubdiagEntry(Index iu);
208  Index findSmallDiagEntry(Index f, Index l);
209  void splitOffTwoRows(Index i);
210  void pushDownZero(Index z, Index f, Index l);
211  void step(Index f, Index l, Index iter);
212 
213  }; // RealQZ
214 
216  template<typename MatrixType>
217  void RealQZ<MatrixType>::hessenbergTriangular()
218  {
219 
220  const Index dim = m_S.cols();
221 
222  // perform QR decomposition of T, overwrite T with R, save Q
223  HouseholderQR<MatrixType> qrT(m_T);
224  m_T = qrT.matrixQR();
225  m_T.template triangularView<StrictlyLower>().setZero();
226  m_Q = qrT.householderQ();
227  // overwrite S with Q* S
228  m_S.applyOnTheLeft(m_Q.adjoint());
229  // init Z as Identity
230  if (m_computeQZ)
231  m_Z = MatrixType::Identity(dim,dim);
232  // reduce S to upper Hessenberg with Givens rotations
233  for (Index j=0; j<=dim-3; j++) {
234  for (Index i=dim-1; i>=j+2; i--) {
235  JRs G;
236  // kill S(i,j)
237  if(m_S.coeff(i,j) != 0)
238  {
239  G.makeGivens(m_S.coeff(i-1,j), m_S.coeff(i,j), &m_S.coeffRef(i-1, j));
240  m_S.coeffRef(i,j) = Scalar(0.0);
241  m_S.rightCols(dim-j-1).applyOnTheLeft(i-1,i,G.adjoint());
242  m_T.rightCols(dim-i+1).applyOnTheLeft(i-1,i,G.adjoint());
243  // update Q
244  if (m_computeQZ)
245  m_Q.applyOnTheRight(i-1,i,G);
246  }
247  // kill T(i,i-1)
248  if(m_T.coeff(i,i-1)!=Scalar(0))
249  {
250  G.makeGivens(m_T.coeff(i,i), m_T.coeff(i,i-1), &m_T.coeffRef(i,i));
251  m_T.coeffRef(i,i-1) = Scalar(0.0);
252  m_S.applyOnTheRight(i,i-1,G);
253  m_T.topRows(i).applyOnTheRight(i,i-1,G);
254  // update Z
255  if (m_computeQZ)
256  m_Z.applyOnTheLeft(i,i-1,G.adjoint());
257  }
258  }
259  }
260  }
261 
263  template<typename MatrixType>
264  inline void RealQZ<MatrixType>::computeNorms()
265  {
266  const Index size = m_S.cols();
267  m_normOfS = Scalar(0.0);
268  m_normOfT = Scalar(0.0);
269  for (Index j = 0; j < size; ++j)
270  {
271  m_normOfS += m_S.col(j).segment(0, (std::min)(size,j+2)).cwiseAbs().sum();
272  m_normOfT += m_T.row(j).segment(j, size - j).cwiseAbs().sum();
273  }
274  }
275 
276 
278  template<typename MatrixType>
279  inline typename MatrixType::Index RealQZ<MatrixType>::findSmallSubdiagEntry(Index iu)
280  {
281  using std::abs;
282  Index res = iu;
283  while (res > 0)
284  {
285  Scalar s = abs(m_S.coeff(res-1,res-1)) + abs(m_S.coeff(res,res));
286  if (s == Scalar(0.0))
287  s = m_normOfS;
288  if (abs(m_S.coeff(res,res-1)) < NumTraits<Scalar>::epsilon() * s)
289  break;
290  res--;
291  }
292  return res;
293  }
294 
296  template<typename MatrixType>
297  inline typename MatrixType::Index RealQZ<MatrixType>::findSmallDiagEntry(Index f, Index l)
298  {
299  using std::abs;
300  Index res = l;
301  while (res >= f) {
302  if (abs(m_T.coeff(res,res)) <= NumTraits<Scalar>::epsilon() * m_normOfT)
303  break;
304  res--;
305  }
306  return res;
307  }
308 
310  template<typename MatrixType>
311  inline void RealQZ<MatrixType>::splitOffTwoRows(Index i)
312  {
313  using std::abs;
314  using std::sqrt;
315  const Index dim=m_S.cols();
316  if (abs(m_S.coeff(i+1,i))==Scalar(0))
317  return;
318  Index z = findSmallDiagEntry(i,i+1);
319  if (z==i-1)
320  {
321  // block of (S T^{-1})
322  Matrix2s STi = m_T.template block<2,2>(i,i).template triangularView<Upper>().
323  template solve<OnTheRight>(m_S.template block<2,2>(i,i));
324  Scalar p = Scalar(0.5)*(STi(0,0)-STi(1,1));
325  Scalar q = p*p + STi(1,0)*STi(0,1);
326  if (q>=0) {
327  Scalar z = sqrt(q);
328  // one QR-like iteration for ABi - lambda I
329  // is enough - when we know exact eigenvalue in advance,
330  // convergence is immediate
331  JRs G;
332  if (p>=0)
333  G.makeGivens(p + z, STi(1,0));
334  else
335  G.makeGivens(p - z, STi(1,0));
336  m_S.rightCols(dim-i).applyOnTheLeft(i,i+1,G.adjoint());
337  m_T.rightCols(dim-i).applyOnTheLeft(i,i+1,G.adjoint());
338  // update Q
339  if (m_computeQZ)
340  m_Q.applyOnTheRight(i,i+1,G);
341 
342  G.makeGivens(m_T.coeff(i+1,i+1), m_T.coeff(i+1,i));
343  m_S.topRows(i+2).applyOnTheRight(i+1,i,G);
344  m_T.topRows(i+2).applyOnTheRight(i+1,i,G);
345  // update Z
346  if (m_computeQZ)
347  m_Z.applyOnTheLeft(i+1,i,G.adjoint());
348 
349  m_S.coeffRef(i+1,i) = Scalar(0.0);
350  m_T.coeffRef(i+1,i) = Scalar(0.0);
351  }
352  }
353  else
354  {
355  pushDownZero(z,i,i+1);
356  }
357  }
358 
360  template<typename MatrixType>
361  inline void RealQZ<MatrixType>::pushDownZero(Index z, Index f, Index l)
362  {
363  JRs G;
364  const Index dim = m_S.cols();
365  for (Index zz=z; zz<l; zz++)
366  {
367  // push 0 down
368  Index firstColS = zz>f ? (zz-1) : zz;
369  G.makeGivens(m_T.coeff(zz, zz+1), m_T.coeff(zz+1, zz+1));
370  m_S.rightCols(dim-firstColS).applyOnTheLeft(zz,zz+1,G.adjoint());
371  m_T.rightCols(dim-zz).applyOnTheLeft(zz,zz+1,G.adjoint());
372  m_T.coeffRef(zz+1,zz+1) = Scalar(0.0);
373  // update Q
374  if (m_computeQZ)
375  m_Q.applyOnTheRight(zz,zz+1,G);
376  // kill S(zz+1, zz-1)
377  if (zz>f)
378  {
379  G.makeGivens(m_S.coeff(zz+1, zz), m_S.coeff(zz+1,zz-1));
380  m_S.topRows(zz+2).applyOnTheRight(zz, zz-1,G);
381  m_T.topRows(zz+1).applyOnTheRight(zz, zz-1,G);
382  m_S.coeffRef(zz+1,zz-1) = Scalar(0.0);
383  // update Z
384  if (m_computeQZ)
385  m_Z.applyOnTheLeft(zz,zz-1,G.adjoint());
386  }
387  }
388  // finally kill S(l,l-1)
389  G.makeGivens(m_S.coeff(l,l), m_S.coeff(l,l-1));
390  m_S.applyOnTheRight(l,l-1,G);
391  m_T.applyOnTheRight(l,l-1,G);
392  m_S.coeffRef(l,l-1)=Scalar(0.0);
393  // update Z
394  if (m_computeQZ)
395  m_Z.applyOnTheLeft(l,l-1,G.adjoint());
396  }
397 
399  template<typename MatrixType>
400  inline void RealQZ<MatrixType>::step(Index f, Index l, Index iter)
401  {
402  using std::abs;
403  const Index dim = m_S.cols();
404 
405  // x, y, z
406  Scalar x, y, z;
407  if (iter==10)
408  {
409  // Wilkinson ad hoc shift
410  const Scalar
411  a11=m_S.coeff(f+0,f+0), a12=m_S.coeff(f+0,f+1),
412  a21=m_S.coeff(f+1,f+0), a22=m_S.coeff(f+1,f+1), a32=m_S.coeff(f+2,f+1),
413  b12=m_T.coeff(f+0,f+1),
414  b11i=Scalar(1.0)/m_T.coeff(f+0,f+0),
415  b22i=Scalar(1.0)/m_T.coeff(f+1,f+1),
416  a87=m_S.coeff(l-1,l-2),
417  a98=m_S.coeff(l-0,l-1),
418  b77i=Scalar(1.0)/m_T.coeff(l-2,l-2),
419  b88i=Scalar(1.0)/m_T.coeff(l-1,l-1);
420  Scalar ss = abs(a87*b77i) + abs(a98*b88i),
421  lpl = Scalar(1.5)*ss,
422  ll = ss*ss;
423  x = ll + a11*a11*b11i*b11i - lpl*a11*b11i + a12*a21*b11i*b22i
424  - a11*a21*b12*b11i*b11i*b22i;
425  y = a11*a21*b11i*b11i - lpl*a21*b11i + a21*a22*b11i*b22i
426  - a21*a21*b12*b11i*b11i*b22i;
427  z = a21*a32*b11i*b22i;
428  }
429  else if (iter==16)
430  {
431  // another exceptional shift
432  x = m_S.coeff(f,f)/m_T.coeff(f,f)-m_S.coeff(l,l)/m_T.coeff(l,l) + m_S.coeff(l,l-1)*m_T.coeff(l-1,l) /
433  (m_T.coeff(l-1,l-1)*m_T.coeff(l,l));
434  y = m_S.coeff(f+1,f)/m_T.coeff(f,f);
435  z = 0;
436  }
437  else if (iter>23 && !(iter%8))
438  {
439  // extremely exceptional shift
440  x = internal::random<Scalar>(-1.0,1.0);
441  y = internal::random<Scalar>(-1.0,1.0);
442  z = internal::random<Scalar>(-1.0,1.0);
443  }
444  else
445  {
446  // Compute the shifts: (x,y,z,0...) = (AB^-1 - l1 I) (AB^-1 - l2 I) e1
447  // where l1 and l2 are the eigenvalues of the 2x2 matrix C = U V^-1 where
448  // U and V are 2x2 bottom right sub matrices of A and B. Thus:
449  // = AB^-1AB^-1 + l1 l2 I - (l1+l2)(AB^-1)
450  // = AB^-1AB^-1 + det(M) - tr(M)(AB^-1)
451  // Since we are only interested in having x, y, z with a correct ratio, we have:
452  const Scalar
453  a11 = m_S.coeff(f,f), a12 = m_S.coeff(f,f+1),
454  a21 = m_S.coeff(f+1,f), a22 = m_S.coeff(f+1,f+1),
455  a32 = m_S.coeff(f+2,f+1),
456 
457  a88 = m_S.coeff(l-1,l-1), a89 = m_S.coeff(l-1,l),
458  a98 = m_S.coeff(l,l-1), a99 = m_S.coeff(l,l),
459 
460  b11 = m_T.coeff(f,f), b12 = m_T.coeff(f,f+1),
461  b22 = m_T.coeff(f+1,f+1),
462 
463  b88 = m_T.coeff(l-1,l-1), b89 = m_T.coeff(l-1,l),
464  b99 = m_T.coeff(l,l);
465 
466  x = ( (a88/b88 - a11/b11)*(a99/b99 - a11/b11) - (a89/b99)*(a98/b88) + (a98/b88)*(b89/b99)*(a11/b11) ) * (b11/a21)
467  + a12/b22 - (a11/b11)*(b12/b22);
468  y = (a22/b22-a11/b11) - (a21/b11)*(b12/b22) - (a88/b88-a11/b11) - (a99/b99-a11/b11) + (a98/b88)*(b89/b99);
469  z = a32/b22;
470  }
471 
472  JRs G;
473 
474  for (Index k=f; k<=l-2; k++)
475  {
476  // variables for Householder reflections
477  Vector2s essential2;
478  Scalar tau, beta;
479 
480  Vector3s hr(x,y,z);
481 
482  // Q_k to annihilate S(k+1,k-1) and S(k+2,k-1)
483  hr.makeHouseholderInPlace(tau, beta);
484  essential2 = hr.template bottomRows<2>();
485  Index fc=(std::max)(k-1,Index(0)); // first col to update
486  m_S.template middleRows<3>(k).rightCols(dim-fc).applyHouseholderOnTheLeft(essential2, tau, m_workspace.data());
487  m_T.template middleRows<3>(k).rightCols(dim-fc).applyHouseholderOnTheLeft(essential2, tau, m_workspace.data());
488  if (m_computeQZ)
489  m_Q.template middleCols<3>(k).applyHouseholderOnTheRight(essential2, tau, m_workspace.data());
490  if (k>f)
491  m_S.coeffRef(k+2,k-1) = m_S.coeffRef(k+1,k-1) = Scalar(0.0);
492 
493  // Z_{k1} to annihilate T(k+2,k+1) and T(k+2,k)
494  hr << m_T.coeff(k+2,k+2),m_T.coeff(k+2,k),m_T.coeff(k+2,k+1);
495  hr.makeHouseholderInPlace(tau, beta);
496  essential2 = hr.template bottomRows<2>();
497  {
498  Index lr = (std::min)(k+4,dim); // last row to update
499  Map<Matrix<Scalar,Dynamic,1> > tmp(m_workspace.data(),lr);
500  // S
501  tmp = m_S.template middleCols<2>(k).topRows(lr) * essential2;
502  tmp += m_S.col(k+2).head(lr);
503  m_S.col(k+2).head(lr) -= tau*tmp;
504  m_S.template middleCols<2>(k).topRows(lr) -= (tau*tmp) * essential2.adjoint();
505  // T
506  tmp = m_T.template middleCols<2>(k).topRows(lr) * essential2;
507  tmp += m_T.col(k+2).head(lr);
508  m_T.col(k+2).head(lr) -= tau*tmp;
509  m_T.template middleCols<2>(k).topRows(lr) -= (tau*tmp) * essential2.adjoint();
510  }
511  if (m_computeQZ)
512  {
513  // Z
514  Map<Matrix<Scalar,1,Dynamic> > tmp(m_workspace.data(),dim);
515  tmp = essential2.adjoint()*(m_Z.template middleRows<2>(k));
516  tmp += m_Z.row(k+2);
517  m_Z.row(k+2) -= tau*tmp;
518  m_Z.template middleRows<2>(k) -= essential2 * (tau*tmp);
519  }
520  m_T.coeffRef(k+2,k) = m_T.coeffRef(k+2,k+1) = Scalar(0.0);
521 
522  // Z_{k2} to annihilate T(k+1,k)
523  G.makeGivens(m_T.coeff(k+1,k+1), m_T.coeff(k+1,k));
524  m_S.applyOnTheRight(k+1,k,G);
525  m_T.applyOnTheRight(k+1,k,G);
526  // update Z
527  if (m_computeQZ)
528  m_Z.applyOnTheLeft(k+1,k,G.adjoint());
529  m_T.coeffRef(k+1,k) = Scalar(0.0);
530 
531  // update x,y,z
532  x = m_S.coeff(k+1,k);
533  y = m_S.coeff(k+2,k);
534  if (k < l-2)
535  z = m_S.coeff(k+3,k);
536  } // loop over k
537 
538  // Q_{n-1} to annihilate y = S(l,l-2)
539  G.makeGivens(x,y);
540  m_S.applyOnTheLeft(l-1,l,G.adjoint());
541  m_T.applyOnTheLeft(l-1,l,G.adjoint());
542  if (m_computeQZ)
543  m_Q.applyOnTheRight(l-1,l,G);
544  m_S.coeffRef(l,l-2) = Scalar(0.0);
545 
546  // Z_{n-1} to annihilate T(l,l-1)
547  G.makeGivens(m_T.coeff(l,l),m_T.coeff(l,l-1));
548  m_S.applyOnTheRight(l,l-1,G);
549  m_T.applyOnTheRight(l,l-1,G);
550  if (m_computeQZ)
551  m_Z.applyOnTheLeft(l,l-1,G.adjoint());
552  m_T.coeffRef(l,l-1) = Scalar(0.0);
553  }
554 
555 
556  template<typename MatrixType>
557  RealQZ<MatrixType>& RealQZ<MatrixType>::compute(const MatrixType& A_in, const MatrixType& B_in, bool computeQZ)
558  {
559 
560  const Index dim = A_in.cols();
561 
562  eigen_assert (A_in.rows()==dim && A_in.cols()==dim
563  && B_in.rows()==dim && B_in.cols()==dim
564  && "Need square matrices of the same dimension");
565 
566  m_isInitialized = true;
567  m_computeQZ = computeQZ;
568  m_S = A_in; m_T = B_in;
569  m_workspace.resize(dim*2);
570  m_global_iter = 0;
571 
572  // entrance point: hessenberg triangular decomposition
573  hessenbergTriangular();
574  // compute L1 vector norms of T, S into m_normOfS, m_normOfT
575  computeNorms();
576 
577  Index l = dim-1,
578  f,
579  local_iter = 0;
580 
581  while (l>0 && local_iter<m_maxIters)
582  {
583  f = findSmallSubdiagEntry(l);
584  // now rows and columns f..l (including) decouple from the rest of the problem
585  if (f>0) m_S.coeffRef(f,f-1) = Scalar(0.0);
586  if (f == l) // One root found
587  {
588  l--;
589  local_iter = 0;
590  }
591  else if (f == l-1) // Two roots found
592  {
593  splitOffTwoRows(f);
594  l -= 2;
595  local_iter = 0;
596  }
597  else // No convergence yet
598  {
599  // if there's zero on diagonal of T, we can isolate an eigenvalue with Givens rotations
600  Index z = findSmallDiagEntry(f,l);
601  if (z>=f)
602  {
603  // zero found
604  pushDownZero(z,f,l);
605  }
606  else
607  {
608  // We are sure now that S.block(f,f, l-f+1,l-f+1) is underuced upper-Hessenberg
609  // and T.block(f,f, l-f+1,l-f+1) is invertible uper-triangular, which allows to
610  // apply a QR-like iteration to rows and columns f..l.
611  step(f,l, local_iter);
612  local_iter++;
613  m_global_iter++;
614  }
615  }
616  }
617  // check if we converged before reaching iterations limit
618  m_info = (local_iter<m_maxIters) ? Success : NoConvergence;
619  return *this;
620  } // end compute
621 
622 } // end namespace Eigen
623 
624 #endif //EIGEN_REAL_QZ
Eigen::RealQZ::matrixZ
const MatrixType & matrixZ() const
Returns matrix Z in the QZ decomposition.
Definition: RealQZ.h:129
Eigen::JacobiRotation
Rotation given by a cosine-sine pair.
Definition: ForwardDeclarations.h:228
Eigen::Dynamic
const int Dynamic
Definition: Constants.h:21
Eigen::RealQZ::compute
RealQZ & compute(const MatrixType &A, const MatrixType &B, bool computeQZ=true)
Computes QZ decomposition of given matrix.
Definition: RealQZ.h:557
Eigen::RealQZ::matrixS
const MatrixType & matrixS() const
Returns matrix S in the QZ decomposition.
Definition: RealQZ.h:139
Eigen::RealQZ::setMaxIterations
RealQZ & setMaxIterations(Index maxIters)
Definition: RealQZ.h:183
Eigen::RealQZ::info
ComputationInfo info() const
Reports whether previous computation was successful.
Definition: RealQZ.h:166
Eigen::RealQZ::matrixT
const MatrixType & matrixT() const
Returns matrix S in the QZ decomposition.
Definition: RealQZ.h:148
Eigen::NoConvergence
Definition: Constants.h:380
Eigen::RealQZ::RealQZ
RealQZ(Index size=RowsAtCompileTime==Dynamic?1:RowsAtCompileTime)
Default constructor.
Definition: RealQZ.h:86
Eigen::RealQZ::iterations
Index iterations() const
Returns number of performed QR-like iterations.
Definition: RealQZ.h:174
Eigen::Success
Definition: Constants.h:376
Eigen::RealQZ::matrixQ
const MatrixType & matrixQ() const
Returns matrix Q in the QZ decomposition.
Definition: RealQZ.h:119
Eigen::RealQZ::RealQZ
RealQZ(const MatrixType &A, const MatrixType &B, bool computeQZ=true)
Constructor; computes real QZ decomposition of given matrices.
Definition: RealQZ.h:104
Eigen::ComputationInfo
ComputationInfo
Definition: Constants.h:374
Eigen::RealQZ
Performs a real QZ decomposition of a pair of square matrices.
Definition: RealQZ.h:57