Eigen  3.3.0
 
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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 <kaikaikai@yandex.ru>
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
13namespace 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;
71
74
86 explicit 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
167 {
168 eigen_assert(m_isInitialized && "RealQZ is not initialized.");
169 return m_info;
170 }
171
175 {
176 eigen_assert(m_isInitialized && "RealQZ is not initialized.");
177 return m_global_iter;
178 }
179
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 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 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 j = findSmallDiagEntry(i,i+1);
319 if (j==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(j,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 template<typename MatrixType>
556 RealQZ<MatrixType>& RealQZ<MatrixType>::compute(const MatrixType& A_in, const MatrixType& B_in, bool computeQZ)
557 {
558
559 const Index dim = A_in.cols();
560
561 eigen_assert (A_in.rows()==dim && A_in.cols()==dim
562 && B_in.rows()==dim && B_in.cols()==dim
563 && "Need square matrices of the same dimension");
564
565 m_isInitialized = true;
566 m_computeQZ = computeQZ;
567 m_S = A_in; m_T = B_in;
568 m_workspace.resize(dim*2);
569 m_global_iter = 0;
570
571 // entrance point: hessenberg triangular decomposition
572 hessenbergTriangular();
573 // compute L1 vector norms of T, S into m_normOfS, m_normOfT
574 computeNorms();
575
576 Index l = dim-1,
577 f,
578 local_iter = 0;
579
580 while (l>0 && local_iter<m_maxIters)
581 {
582 f = findSmallSubdiagEntry(l);
583 // now rows and columns f..l (including) decouple from the rest of the problem
584 if (f>0) m_S.coeffRef(f,f-1) = Scalar(0.0);
585 if (f == l) // One root found
586 {
587 l--;
588 local_iter = 0;
589 }
590 else if (f == l-1) // Two roots found
591 {
592 splitOffTwoRows(f);
593 l -= 2;
594 local_iter = 0;
595 }
596 else // No convergence yet
597 {
598 // if there's zero on diagonal of T, we can isolate an eigenvalue with Givens rotations
599 Index z = findSmallDiagEntry(f,l);
600 if (z>=f)
601 {
602 // zero found
603 pushDownZero(z,f,l);
604 }
605 else
606 {
607 // We are sure now that S.block(f,f, l-f+1,l-f+1) is underuced upper-Hessenberg
608 // and T.block(f,f, l-f+1,l-f+1) is invertible uper-triangular, which allows to
609 // apply a QR-like iteration to rows and columns f..l.
610 step(f,l, local_iter);
611 local_iter++;
612 m_global_iter++;
613 }
614 }
615 }
616 // check if we converged before reaching iterations limit
617 m_info = (local_iter<m_maxIters) ? Success : NoConvergence;
618
619 // For each non triangular 2x2 diagonal block of S,
620 // reduce the respective 2x2 diagonal block of T to positive diagonal form using 2x2 SVD.
621 // This step is not mandatory for QZ, but it does help further extraction of eigenvalues/eigenvectors,
622 // and is in par with Lapack/Matlab QZ.
623 if(m_info==Success)
624 {
625 for(Index i=0; i<dim-1; ++i)
626 {
627 if(m_S.coeff(i+1, i) != Scalar(0))
628 {
629 JacobiRotation<Scalar> j_left, j_right;
630 internal::real_2x2_jacobi_svd(m_T, i, i+1, &j_left, &j_right);
631
632 // Apply resulting Jacobi rotations
633 m_S.applyOnTheLeft(i,i+1,j_left);
634 m_S.applyOnTheRight(i,i+1,j_right);
635 m_T.applyOnTheLeft(i,i+1,j_left);
636 m_T.applyOnTheRight(i,i+1,j_right);
637 m_T(i+1,i) = m_T(i,i+1) = Scalar(0);
638
639 if(m_computeQZ) {
640 m_Q.applyOnTheRight(i,i+1,j_left.transpose());
641 m_Z.applyOnTheLeft(i,i+1,j_right.transpose());
642 }
643
644 i++;
645 }
646 }
647 }
648
649 return *this;
650 } // end compute
651
652} // end namespace Eigen
653
654#endif //EIGEN_REAL_QZ
Rotation given by a cosine-sine pair.
Definition: Jacobi.h:35
JacobiRotation transpose() const
Definition: Jacobi.h:59
The matrix class, also used for vectors and row-vectors.
Definition: Matrix.h:180
Performs a real QZ decomposition of a pair of square matrices.
Definition: RealQZ.h:58
const MatrixType & matrixQ() const
Returns matrix Q in the QZ decomposition.
Definition: RealQZ.h:119
RealQZ & compute(const MatrixType &A, const MatrixType &B, bool computeQZ=true)
Computes QZ decomposition of given matrix.
Definition: RealQZ.h:556
ComputationInfo info() const
Reports whether previous computation was successful.
Definition: RealQZ.h:166
Eigen::Index Index
Definition: RealQZ.h:70
const MatrixType & matrixT() const
Returns matrix S in the QZ decomposition.
Definition: RealQZ.h:148
RealQZ & setMaxIterations(Index maxIters)
Definition: RealQZ.h:183
RealQZ(const MatrixType &A, const MatrixType &B, bool computeQZ=true)
Constructor; computes real QZ decomposition of given matrices.
Definition: RealQZ.h:104
RealQZ(Index size=RowsAtCompileTime==Dynamic ? 1 :RowsAtCompileTime)
Default constructor.
Definition: RealQZ.h:86
const MatrixType & matrixS() const
Returns matrix S in the QZ decomposition.
Definition: RealQZ.h:139
const MatrixType & matrixZ() const
Returns matrix Z in the QZ decomposition.
Definition: RealQZ.h:129
Index iterations() const
Returns number of performed QR-like iterations.
Definition: RealQZ.h:174
ComputationInfo
Definition: Constants.h:430
@ Success
Definition: Constants.h:432
@ NoConvergence
Definition: Constants.h:436
Namespace containing all symbols from the Eigen library.
Definition: Core:287
EIGEN_DEFAULT_DENSE_INDEX_TYPE Index
The Index type as used for the API.
Definition: Meta.h:33
const int Dynamic
Definition: Constants.h:21