RealSchur.h (21078B)
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 Eigen::Index Index; ///< \deprecated since Eigen 3.3 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 explicit RealSchur(Index size = RowsAtCompileTime==Dynamic ? 1 : RowsAtCompileTime) 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 template<typename InputType> 104 explicit RealSchur(const EigenBase<InputType>& matrix, bool computeU = true) 105 : m_matT(matrix.rows(),matrix.cols()), 106 m_matU(matrix.rows(),matrix.cols()), 107 m_workspaceVector(matrix.rows()), 108 m_hess(matrix.rows()), 109 m_isInitialized(false), 110 m_matUisUptodate(false), 111 m_maxIters(-1) 112 { 113 compute(matrix.derived(), computeU); 114 } 115 116 /** \brief Returns the orthogonal matrix in the Schur decomposition. 117 * 118 * \returns A const reference to the matrix U. 119 * 120 * \pre Either the constructor RealSchur(const MatrixType&, bool) or the 121 * member function compute(const MatrixType&, bool) has been called before 122 * to compute the Schur decomposition of a matrix, and \p computeU was set 123 * to true (the default value). 124 * 125 * \sa RealSchur(const MatrixType&, bool) for an example 126 */ 127 const MatrixType& matrixU() const 128 { 129 eigen_assert(m_isInitialized && "RealSchur is not initialized."); 130 eigen_assert(m_matUisUptodate && "The matrix U has not been computed during the RealSchur decomposition."); 131 return m_matU; 132 } 133 134 /** \brief Returns the quasi-triangular matrix in the Schur decomposition. 135 * 136 * \returns A const reference to the matrix T. 137 * 138 * \pre Either the constructor RealSchur(const MatrixType&, bool) or the 139 * member function compute(const MatrixType&, bool) has been called before 140 * to compute the Schur decomposition of a matrix. 141 * 142 * \sa RealSchur(const MatrixType&, bool) for an example 143 */ 144 const MatrixType& matrixT() const 145 { 146 eigen_assert(m_isInitialized && "RealSchur is not initialized."); 147 return m_matT; 148 } 149 150 /** \brief Computes Schur decomposition of given matrix. 151 * 152 * \param[in] matrix Square matrix whose Schur decomposition is to be computed. 153 * \param[in] computeU If true, both T and U are computed; if false, only T is computed. 154 * \returns Reference to \c *this 155 * 156 * The Schur decomposition is computed by first reducing the matrix to 157 * Hessenberg form using the class HessenbergDecomposition. The Hessenberg 158 * matrix is then reduced to triangular form by performing Francis QR 159 * iterations with implicit double shift. The cost of computing the Schur 160 * decomposition depends on the number of iterations; as a rough guide, it 161 * may be taken to be \f$25n^3\f$ flops if \a computeU is true and 162 * \f$10n^3\f$ flops if \a computeU is false. 163 * 164 * Example: \include RealSchur_compute.cpp 165 * Output: \verbinclude RealSchur_compute.out 166 * 167 * \sa compute(const MatrixType&, bool, Index) 168 */ 169 template<typename InputType> 170 RealSchur& compute(const EigenBase<InputType>& matrix, bool computeU = true); 171 172 /** \brief Computes Schur decomposition of a Hessenberg matrix H = Z T Z^T 173 * \param[in] matrixH Matrix in Hessenberg form H 174 * \param[in] matrixQ orthogonal matrix Q that transform a matrix A to H : A = Q H Q^T 175 * \param computeU Computes the matriX U of the Schur vectors 176 * \return Reference to \c *this 177 * 178 * This routine assumes that the matrix is already reduced in Hessenberg form matrixH 179 * using either the class HessenbergDecomposition or another mean. 180 * It computes the upper quasi-triangular matrix T of the Schur decomposition of H 181 * When computeU is true, this routine computes the matrix U such that 182 * A = U T U^T = (QZ) T (QZ)^T = Q H Q^T where A is the initial matrix 183 * 184 * NOTE Q is referenced if computeU is true; so, if the initial orthogonal matrix 185 * is not available, the user should give an identity matrix (Q.setIdentity()) 186 * 187 * \sa compute(const MatrixType&, bool) 188 */ 189 template<typename HessMatrixType, typename OrthMatrixType> 190 RealSchur& computeFromHessenberg(const HessMatrixType& matrixH, const OrthMatrixType& matrixQ, bool computeU); 191 /** \brief Reports whether previous computation was successful. 192 * 193 * \returns \c Success if computation was successful, \c NoConvergence otherwise. 194 */ 195 ComputationInfo info() const 196 { 197 eigen_assert(m_isInitialized && "RealSchur is not initialized."); 198 return m_info; 199 } 200 201 /** \brief Sets the maximum number of iterations allowed. 202 * 203 * If not specified by the user, the maximum number of iterations is m_maxIterationsPerRow times the size 204 * of the matrix. 205 */ 206 RealSchur& setMaxIterations(Index maxIters) 207 { 208 m_maxIters = maxIters; 209 return *this; 210 } 211 212 /** \brief Returns the maximum number of iterations. */ 213 Index getMaxIterations() 214 { 215 return m_maxIters; 216 } 217 218 /** \brief Maximum number of iterations per row. 219 * 220 * If not otherwise specified, the maximum number of iterations is this number times the size of the 221 * matrix. It is currently set to 40. 222 */ 223 static const int m_maxIterationsPerRow = 40; 224 225 private: 226 227 MatrixType m_matT; 228 MatrixType m_matU; 229 ColumnVectorType m_workspaceVector; 230 HessenbergDecomposition<MatrixType> m_hess; 231 ComputationInfo m_info; 232 bool m_isInitialized; 233 bool m_matUisUptodate; 234 Index m_maxIters; 235 236 typedef Matrix<Scalar,3,1> Vector3s; 237 238 Scalar computeNormOfT(); 239 Index findSmallSubdiagEntry(Index iu, const Scalar& considerAsZero); 240 void splitOffTwoRows(Index iu, bool computeU, const Scalar& exshift); 241 void computeShift(Index iu, Index iter, Scalar& exshift, Vector3s& shiftInfo); 242 void initFrancisQRStep(Index il, Index iu, const Vector3s& shiftInfo, Index& im, Vector3s& firstHouseholderVector); 243 void performFrancisQRStep(Index il, Index im, Index iu, bool computeU, const Vector3s& firstHouseholderVector, Scalar* workspace); 244 }; 245 246 247 template<typename MatrixType> 248 template<typename InputType> 249 RealSchur<MatrixType>& RealSchur<MatrixType>::compute(const EigenBase<InputType>& matrix, bool computeU) 250 { 251 const Scalar considerAsZero = (std::numeric_limits<Scalar>::min)(); 252 253 eigen_assert(matrix.cols() == matrix.rows()); 254 Index maxIters = m_maxIters; 255 if (maxIters == -1) 256 maxIters = m_maxIterationsPerRow * matrix.rows(); 257 258 Scalar scale = matrix.derived().cwiseAbs().maxCoeff(); 259 if(scale<considerAsZero) 260 { 261 m_matT.setZero(matrix.rows(),matrix.cols()); 262 if(computeU) 263 m_matU.setIdentity(matrix.rows(),matrix.cols()); 264 m_info = Success; 265 m_isInitialized = true; 266 m_matUisUptodate = computeU; 267 return *this; 268 } 269 270 // Step 1. Reduce to Hessenberg form 271 m_hess.compute(matrix.derived()/scale); 272 273 // Step 2. Reduce to real Schur form 274 // Note: we copy m_hess.matrixQ() into m_matU here and not in computeFromHessenberg 275 // to be able to pass our working-space buffer for the Householder to Dense evaluation. 276 m_workspaceVector.resize(matrix.cols()); 277 if(computeU) 278 m_hess.matrixQ().evalTo(m_matU, m_workspaceVector); 279 computeFromHessenberg(m_hess.matrixH(), m_matU, computeU); 280 281 m_matT *= scale; 282 283 return *this; 284 } 285 template<typename MatrixType> 286 template<typename HessMatrixType, typename OrthMatrixType> 287 RealSchur<MatrixType>& RealSchur<MatrixType>::computeFromHessenberg(const HessMatrixType& matrixH, const OrthMatrixType& matrixQ, bool computeU) 288 { 289 using std::abs; 290 291 m_matT = matrixH; 292 m_workspaceVector.resize(m_matT.cols()); 293 if(computeU && !internal::is_same_dense(m_matU,matrixQ)) 294 m_matU = matrixQ; 295 296 Index maxIters = m_maxIters; 297 if (maxIters == -1) 298 maxIters = m_maxIterationsPerRow * matrixH.rows(); 299 Scalar* workspace = &m_workspaceVector.coeffRef(0); 300 301 // The matrix m_matT is divided in three parts. 302 // Rows 0,...,il-1 are decoupled from the rest because m_matT(il,il-1) is zero. 303 // Rows il,...,iu is the part we are working on (the active window). 304 // Rows iu+1,...,end are already brought in triangular form. 305 Index iu = m_matT.cols() - 1; 306 Index iter = 0; // iteration count for current eigenvalue 307 Index totalIter = 0; // iteration count for whole matrix 308 Scalar exshift(0); // sum of exceptional shifts 309 Scalar norm = computeNormOfT(); 310 // sub-diagonal entries smaller than considerAsZero will be treated as zero. 311 // We use eps^2 to enable more precision in small eigenvalues. 312 Scalar considerAsZero = numext::maxi<Scalar>( norm * numext::abs2(NumTraits<Scalar>::epsilon()), 313 (std::numeric_limits<Scalar>::min)() ); 314 315 if(norm!=Scalar(0)) 316 { 317 while (iu >= 0) 318 { 319 Index il = findSmallSubdiagEntry(iu,considerAsZero); 320 321 // Check for convergence 322 if (il == iu) // One root found 323 { 324 m_matT.coeffRef(iu,iu) = m_matT.coeff(iu,iu) + exshift; 325 if (iu > 0) 326 m_matT.coeffRef(iu, iu-1) = Scalar(0); 327 iu--; 328 iter = 0; 329 } 330 else if (il == iu-1) // Two roots found 331 { 332 splitOffTwoRows(iu, computeU, exshift); 333 iu -= 2; 334 iter = 0; 335 } 336 else // No convergence yet 337 { 338 // The firstHouseholderVector vector has to be initialized to something to get rid of a silly GCC warning (-O1 -Wall -DNDEBUG ) 339 Vector3s firstHouseholderVector = Vector3s::Zero(), shiftInfo; 340 computeShift(iu, iter, exshift, shiftInfo); 341 iter = iter + 1; 342 totalIter = totalIter + 1; 343 if (totalIter > maxIters) break; 344 Index im; 345 initFrancisQRStep(il, iu, shiftInfo, im, firstHouseholderVector); 346 performFrancisQRStep(il, im, iu, computeU, firstHouseholderVector, workspace); 347 } 348 } 349 } 350 if(totalIter <= maxIters) 351 m_info = Success; 352 else 353 m_info = NoConvergence; 354 355 m_isInitialized = true; 356 m_matUisUptodate = computeU; 357 return *this; 358 } 359 360 /** \internal Computes and returns vector L1 norm of T */ 361 template<typename MatrixType> 362 inline typename MatrixType::Scalar RealSchur<MatrixType>::computeNormOfT() 363 { 364 const Index size = m_matT.cols(); 365 // FIXME to be efficient the following would requires a triangular reduxion code 366 // Scalar norm = m_matT.upper().cwiseAbs().sum() 367 // + m_matT.bottomLeftCorner(size-1,size-1).diagonal().cwiseAbs().sum(); 368 Scalar norm(0); 369 for (Index j = 0; j < size; ++j) 370 norm += m_matT.col(j).segment(0, (std::min)(size,j+2)).cwiseAbs().sum(); 371 return norm; 372 } 373 374 /** \internal Look for single small sub-diagonal element and returns its index */ 375 template<typename MatrixType> 376 inline Index RealSchur<MatrixType>::findSmallSubdiagEntry(Index iu, const Scalar& considerAsZero) 377 { 378 using std::abs; 379 Index res = iu; 380 while (res > 0) 381 { 382 Scalar s = abs(m_matT.coeff(res-1,res-1)) + abs(m_matT.coeff(res,res)); 383 384 s = numext::maxi<Scalar>(s * NumTraits<Scalar>::epsilon(), considerAsZero); 385 386 if (abs(m_matT.coeff(res,res-1)) <= s) 387 break; 388 res--; 389 } 390 return res; 391 } 392 393 /** \internal Update T given that rows iu-1 and iu decouple from the rest. */ 394 template<typename MatrixType> 395 inline void RealSchur<MatrixType>::splitOffTwoRows(Index iu, bool computeU, const Scalar& exshift) 396 { 397 using std::sqrt; 398 using std::abs; 399 const Index size = m_matT.cols(); 400 401 // The eigenvalues of the 2x2 matrix [a b; c d] are 402 // trace +/- sqrt(discr/4) where discr = tr^2 - 4*det, tr = a + d, det = ad - bc 403 Scalar p = Scalar(0.5) * (m_matT.coeff(iu-1,iu-1) - m_matT.coeff(iu,iu)); 404 Scalar q = p * p + m_matT.coeff(iu,iu-1) * m_matT.coeff(iu-1,iu); // q = tr^2 / 4 - det = discr/4 405 m_matT.coeffRef(iu,iu) += exshift; 406 m_matT.coeffRef(iu-1,iu-1) += exshift; 407 408 if (q >= Scalar(0)) // Two real eigenvalues 409 { 410 Scalar z = sqrt(abs(q)); 411 JacobiRotation<Scalar> rot; 412 if (p >= Scalar(0)) 413 rot.makeGivens(p + z, m_matT.coeff(iu, iu-1)); 414 else 415 rot.makeGivens(p - z, m_matT.coeff(iu, iu-1)); 416 417 m_matT.rightCols(size-iu+1).applyOnTheLeft(iu-1, iu, rot.adjoint()); 418 m_matT.topRows(iu+1).applyOnTheRight(iu-1, iu, rot); 419 m_matT.coeffRef(iu, iu-1) = Scalar(0); 420 if (computeU) 421 m_matU.applyOnTheRight(iu-1, iu, rot); 422 } 423 424 if (iu > 1) 425 m_matT.coeffRef(iu-1, iu-2) = Scalar(0); 426 } 427 428 /** \internal Form shift in shiftInfo, and update exshift if an exceptional shift is performed. */ 429 template<typename MatrixType> 430 inline void RealSchur<MatrixType>::computeShift(Index iu, Index iter, Scalar& exshift, Vector3s& shiftInfo) 431 { 432 using std::sqrt; 433 using std::abs; 434 shiftInfo.coeffRef(0) = m_matT.coeff(iu,iu); 435 shiftInfo.coeffRef(1) = m_matT.coeff(iu-1,iu-1); 436 shiftInfo.coeffRef(2) = m_matT.coeff(iu,iu-1) * m_matT.coeff(iu-1,iu); 437 438 // Wilkinson's original ad hoc shift 439 if (iter == 10) 440 { 441 exshift += shiftInfo.coeff(0); 442 for (Index i = 0; i <= iu; ++i) 443 m_matT.coeffRef(i,i) -= shiftInfo.coeff(0); 444 Scalar s = abs(m_matT.coeff(iu,iu-1)) + abs(m_matT.coeff(iu-1,iu-2)); 445 shiftInfo.coeffRef(0) = Scalar(0.75) * s; 446 shiftInfo.coeffRef(1) = Scalar(0.75) * s; 447 shiftInfo.coeffRef(2) = Scalar(-0.4375) * s * s; 448 } 449 450 // MATLAB's new ad hoc shift 451 if (iter == 30) 452 { 453 Scalar s = (shiftInfo.coeff(1) - shiftInfo.coeff(0)) / Scalar(2.0); 454 s = s * s + shiftInfo.coeff(2); 455 if (s > Scalar(0)) 456 { 457 s = sqrt(s); 458 if (shiftInfo.coeff(1) < shiftInfo.coeff(0)) 459 s = -s; 460 s = s + (shiftInfo.coeff(1) - shiftInfo.coeff(0)) / Scalar(2.0); 461 s = shiftInfo.coeff(0) - shiftInfo.coeff(2) / s; 462 exshift += s; 463 for (Index i = 0; i <= iu; ++i) 464 m_matT.coeffRef(i,i) -= s; 465 shiftInfo.setConstant(Scalar(0.964)); 466 } 467 } 468 } 469 470 /** \internal Compute index im at which Francis QR step starts and the first Householder vector. */ 471 template<typename MatrixType> 472 inline void RealSchur<MatrixType>::initFrancisQRStep(Index il, Index iu, const Vector3s& shiftInfo, Index& im, Vector3s& firstHouseholderVector) 473 { 474 using std::abs; 475 Vector3s& v = firstHouseholderVector; // alias to save typing 476 477 for (im = iu-2; im >= il; --im) 478 { 479 const Scalar Tmm = m_matT.coeff(im,im); 480 const Scalar r = shiftInfo.coeff(0) - Tmm; 481 const Scalar s = shiftInfo.coeff(1) - Tmm; 482 v.coeffRef(0) = (r * s - shiftInfo.coeff(2)) / m_matT.coeff(im+1,im) + m_matT.coeff(im,im+1); 483 v.coeffRef(1) = m_matT.coeff(im+1,im+1) - Tmm - r - s; 484 v.coeffRef(2) = m_matT.coeff(im+2,im+1); 485 if (im == il) { 486 break; 487 } 488 const Scalar lhs = m_matT.coeff(im,im-1) * (abs(v.coeff(1)) + abs(v.coeff(2))); 489 const Scalar rhs = v.coeff(0) * (abs(m_matT.coeff(im-1,im-1)) + abs(Tmm) + abs(m_matT.coeff(im+1,im+1))); 490 if (abs(lhs) < NumTraits<Scalar>::epsilon() * rhs) 491 break; 492 } 493 } 494 495 /** \internal Perform a Francis QR step involving rows il:iu and columns im:iu. */ 496 template<typename MatrixType> 497 inline void RealSchur<MatrixType>::performFrancisQRStep(Index il, Index im, Index iu, bool computeU, const Vector3s& firstHouseholderVector, Scalar* workspace) 498 { 499 eigen_assert(im >= il); 500 eigen_assert(im <= iu-2); 501 502 const Index size = m_matT.cols(); 503 504 for (Index k = im; k <= iu-2; ++k) 505 { 506 bool firstIteration = (k == im); 507 508 Vector3s v; 509 if (firstIteration) 510 v = firstHouseholderVector; 511 else 512 v = m_matT.template block<3,1>(k,k-1); 513 514 Scalar tau, beta; 515 Matrix<Scalar, 2, 1> ess; 516 v.makeHouseholder(ess, tau, beta); 517 518 if (beta != Scalar(0)) // if v is not zero 519 { 520 if (firstIteration && k > il) 521 m_matT.coeffRef(k,k-1) = -m_matT.coeff(k,k-1); 522 else if (!firstIteration) 523 m_matT.coeffRef(k,k-1) = beta; 524 525 // These Householder transformations form the O(n^3) part of the algorithm 526 m_matT.block(k, k, 3, size-k).applyHouseholderOnTheLeft(ess, tau, workspace); 527 m_matT.block(0, k, (std::min)(iu,k+3) + 1, 3).applyHouseholderOnTheRight(ess, tau, workspace); 528 if (computeU) 529 m_matU.block(0, k, size, 3).applyHouseholderOnTheRight(ess, tau, workspace); 530 } 531 } 532 533 Matrix<Scalar, 2, 1> v = m_matT.template block<2,1>(iu-1, iu-2); 534 Scalar tau, beta; 535 Matrix<Scalar, 1, 1> ess; 536 v.makeHouseholder(ess, tau, beta); 537 538 if (beta != Scalar(0)) // if v is not zero 539 { 540 m_matT.coeffRef(iu-1, iu-2) = beta; 541 m_matT.block(iu-1, iu-1, 2, size-iu+1).applyHouseholderOnTheLeft(ess, tau, workspace); 542 m_matT.block(0, iu-1, iu+1, 2).applyHouseholderOnTheRight(ess, tau, workspace); 543 if (computeU) 544 m_matU.block(0, iu-1, size, 2).applyHouseholderOnTheRight(ess, tau, workspace); 545 } 546 547 // clean up pollution due to round-off errors 548 for (Index i = im+2; i <= iu; ++i) 549 { 550 m_matT.coeffRef(i,i-2) = Scalar(0); 551 if (i > im+2) 552 m_matT.coeffRef(i,i-3) = Scalar(0); 553 } 554 } 555 556 } // end namespace Eigen 557 558 #endif // EIGEN_REAL_SCHUR_H