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00028 #ifndef EIGEN_LDLT_H
00029 #define EIGEN_LDLT_H
00030
00031 namespace Eigen {
00032
00033 namespace internal {
00034 template<typename MatrixType, int UpLo> struct LDLT_Traits;
00035 }
00036
00060 template<typename _MatrixType, int _UpLo> class LDLT
00061 {
00062 public:
00063 typedef _MatrixType MatrixType;
00064 enum {
00065 RowsAtCompileTime = MatrixType::RowsAtCompileTime,
00066 ColsAtCompileTime = MatrixType::ColsAtCompileTime,
00067 Options = MatrixType::Options & ~RowMajorBit,
00068 MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
00069 MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime,
00070 UpLo = _UpLo
00071 };
00072 typedef typename MatrixType::Scalar Scalar;
00073 typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
00074 typedef typename MatrixType::Index Index;
00075 typedef Matrix<Scalar, RowsAtCompileTime, 1, Options, MaxRowsAtCompileTime, 1> TmpMatrixType;
00076
00077 typedef Transpositions<RowsAtCompileTime, MaxRowsAtCompileTime> TranspositionType;
00078 typedef PermutationMatrix<RowsAtCompileTime, MaxRowsAtCompileTime> PermutationType;
00079
00080 typedef internal::LDLT_Traits<MatrixType,UpLo> Traits;
00081
00087 LDLT() : m_matrix(), m_transpositions(), m_isInitialized(false) {}
00088
00095 LDLT(Index size)
00096 : m_matrix(size, size),
00097 m_transpositions(size),
00098 m_temporary(size),
00099 m_isInitialized(false)
00100 {}
00101
00107 LDLT(const MatrixType& matrix)
00108 : m_matrix(matrix.rows(), matrix.cols()),
00109 m_transpositions(matrix.rows()),
00110 m_temporary(matrix.rows()),
00111 m_isInitialized(false)
00112 {
00113 compute(matrix);
00114 }
00115
00119 void setZero()
00120 {
00121 m_isInitialized = false;
00122 }
00123
00125 inline typename Traits::MatrixU matrixU() const
00126 {
00127 eigen_assert(m_isInitialized && "LDLT is not initialized.");
00128 return Traits::getU(m_matrix);
00129 }
00130
00132 inline typename Traits::MatrixL matrixL() const
00133 {
00134 eigen_assert(m_isInitialized && "LDLT is not initialized.");
00135 return Traits::getL(m_matrix);
00136 }
00137
00140 inline const TranspositionType& transpositionsP() const
00141 {
00142 eigen_assert(m_isInitialized && "LDLT is not initialized.");
00143 return m_transpositions;
00144 }
00145
00147 inline Diagonal<const MatrixType> vectorD() const
00148 {
00149 eigen_assert(m_isInitialized && "LDLT is not initialized.");
00150 return m_matrix.diagonal();
00151 }
00152
00154 inline bool isPositive() const
00155 {
00156 eigen_assert(m_isInitialized && "LDLT is not initialized.");
00157 return m_sign == 1;
00158 }
00159
00160 #ifdef EIGEN2_SUPPORT
00161 inline bool isPositiveDefinite() const
00162 {
00163 return isPositive();
00164 }
00165 #endif
00166
00168 inline bool isNegative(void) const
00169 {
00170 eigen_assert(m_isInitialized && "LDLT is not initialized.");
00171 return m_sign == -1;
00172 }
00173
00189 template<typename Rhs>
00190 inline const internal::solve_retval<LDLT, Rhs>
00191 solve(const MatrixBase<Rhs>& b) const
00192 {
00193 eigen_assert(m_isInitialized && "LDLT is not initialized.");
00194 eigen_assert(m_matrix.rows()==b.rows()
00195 && "LDLT::solve(): invalid number of rows of the right hand side matrix b");
00196 return internal::solve_retval<LDLT, Rhs>(*this, b.derived());
00197 }
00198
00199 #ifdef EIGEN2_SUPPORT
00200 template<typename OtherDerived, typename ResultType>
00201 bool solve(const MatrixBase<OtherDerived>& b, ResultType *result) const
00202 {
00203 *result = this->solve(b);
00204 return true;
00205 }
00206 #endif
00207
00208 template<typename Derived>
00209 bool solveInPlace(MatrixBase<Derived> &bAndX) const;
00210
00211 LDLT& compute(const MatrixType& matrix);
00212
00213 template <typename Derived>
00214 LDLT& rankUpdate(const MatrixBase<Derived>& w,RealScalar alpha=1);
00215
00220 inline const MatrixType& matrixLDLT() const
00221 {
00222 eigen_assert(m_isInitialized && "LDLT is not initialized.");
00223 return m_matrix;
00224 }
00225
00226 MatrixType reconstructedMatrix() const;
00227
00228 inline Index rows() const { return m_matrix.rows(); }
00229 inline Index cols() const { return m_matrix.cols(); }
00230
00236 ComputationInfo info() const
00237 {
00238 eigen_assert(m_isInitialized && "LDLT is not initialized.");
00239 return Success;
00240 }
00241
00242 protected:
00243
00250 MatrixType m_matrix;
00251 TranspositionType m_transpositions;
00252 TmpMatrixType m_temporary;
00253 int m_sign;
00254 bool m_isInitialized;
00255 };
00256
00257 namespace internal {
00258
00259 template<int UpLo> struct ldlt_inplace;
00260
00261 template<> struct ldlt_inplace<Lower>
00262 {
00263 template<typename MatrixType, typename TranspositionType, typename Workspace>
00264 static bool unblocked(MatrixType& mat, TranspositionType& transpositions, Workspace& temp, int* sign=0)
00265 {
00266 typedef typename MatrixType::Scalar Scalar;
00267 typedef typename MatrixType::RealScalar RealScalar;
00268 typedef typename MatrixType::Index Index;
00269 eigen_assert(mat.rows()==mat.cols());
00270 const Index size = mat.rows();
00271
00272 if (size <= 1)
00273 {
00274 transpositions.setIdentity();
00275 if(sign)
00276 *sign = real(mat.coeff(0,0))>0 ? 1:-1;
00277 return true;
00278 }
00279
00280 RealScalar cutoff(0), biggest_in_corner;
00281
00282 for (Index k = 0; k < size; ++k)
00283 {
00284
00285 Index index_of_biggest_in_corner;
00286 biggest_in_corner = mat.diagonal().tail(size-k).cwiseAbs().maxCoeff(&index_of_biggest_in_corner);
00287 index_of_biggest_in_corner += k;
00288
00289 if(k == 0)
00290 {
00291
00292
00293
00294 cutoff = abs(NumTraits<Scalar>::epsilon() * biggest_in_corner);
00295
00296 if(sign)
00297 *sign = real(mat.diagonal().coeff(index_of_biggest_in_corner)) > 0 ? 1 : -1;
00298 }
00299
00300
00301 if(biggest_in_corner < cutoff)
00302 {
00303 for(Index i = k; i < size; i++) transpositions.coeffRef(i) = i;
00304 break;
00305 }
00306
00307 transpositions.coeffRef(k) = index_of_biggest_in_corner;
00308 if(k != index_of_biggest_in_corner)
00309 {
00310
00311
00312 Index s = size-index_of_biggest_in_corner-1;
00313 mat.row(k).head(k).swap(mat.row(index_of_biggest_in_corner).head(k));
00314 mat.col(k).tail(s).swap(mat.col(index_of_biggest_in_corner).tail(s));
00315 std::swap(mat.coeffRef(k,k),mat.coeffRef(index_of_biggest_in_corner,index_of_biggest_in_corner));
00316 for(int i=k+1;i<index_of_biggest_in_corner;++i)
00317 {
00318 Scalar tmp = mat.coeffRef(i,k);
00319 mat.coeffRef(i,k) = conj(mat.coeffRef(index_of_biggest_in_corner,i));
00320 mat.coeffRef(index_of_biggest_in_corner,i) = conj(tmp);
00321 }
00322 if(NumTraits<Scalar>::IsComplex)
00323 mat.coeffRef(index_of_biggest_in_corner,k) = conj(mat.coeff(index_of_biggest_in_corner,k));
00324 }
00325
00326
00327
00328
00329
00330 Index rs = size - k - 1;
00331 Block<MatrixType,Dynamic,1> A21(mat,k+1,k,rs,1);
00332 Block<MatrixType,1,Dynamic> A10(mat,k,0,1,k);
00333 Block<MatrixType,Dynamic,Dynamic> A20(mat,k+1,0,rs,k);
00334
00335 if(k>0)
00336 {
00337 temp.head(k) = mat.diagonal().head(k).asDiagonal() * A10.adjoint();
00338 mat.coeffRef(k,k) -= (A10 * temp.head(k)).value();
00339 if(rs>0)
00340 A21.noalias() -= A20 * temp.head(k);
00341 }
00342 if((rs>0) && (abs(mat.coeffRef(k,k)) > cutoff))
00343 A21 /= mat.coeffRef(k,k);
00344 }
00345
00346 return true;
00347 }
00348
00349
00350
00351
00352
00353
00354
00355
00356 template<typename MatrixType, typename WDerived>
00357 static bool updateInPlace(MatrixType& mat, MatrixBase<WDerived>& w, typename MatrixType::RealScalar sigma=1)
00358 {
00359 using internal::isfinite;
00360 typedef typename MatrixType::Scalar Scalar;
00361 typedef typename MatrixType::RealScalar RealScalar;
00362 typedef typename MatrixType::Index Index;
00363
00364 const Index size = mat.rows();
00365 eigen_assert(mat.cols() == size && w.size()==size);
00366
00367 RealScalar alpha = 1;
00368
00369
00370 for (Index j = 0; j < size; j++)
00371 {
00372
00373 if (!isfinite(alpha))
00374 break;
00375
00376
00377 RealScalar dj = real(mat.coeff(j,j));
00378 Scalar wj = w.coeff(j);
00379 RealScalar swj2 = sigma*abs2(wj);
00380 RealScalar gamma = dj*alpha + swj2;
00381
00382 mat.coeffRef(j,j) += swj2/alpha;
00383 alpha += swj2/dj;
00384
00385
00386
00387 Index rs = size-j-1;
00388 w.tail(rs) -= wj * mat.col(j).tail(rs);
00389 if(gamma != 0)
00390 mat.col(j).tail(rs) += (sigma*conj(wj)/gamma)*w.tail(rs);
00391 }
00392 return true;
00393 }
00394
00395 template<typename MatrixType, typename TranspositionType, typename Workspace, typename WType>
00396 static bool update(MatrixType& mat, const TranspositionType& transpositions, Workspace& tmp, const WType& w, typename MatrixType::RealScalar sigma=1)
00397 {
00398
00399 tmp = transpositions * w;
00400
00401 return ldlt_inplace<Lower>::updateInPlace(mat,tmp,sigma);
00402 }
00403 };
00404
00405 template<> struct ldlt_inplace<Upper>
00406 {
00407 template<typename MatrixType, typename TranspositionType, typename Workspace>
00408 static EIGEN_STRONG_INLINE bool unblocked(MatrixType& mat, TranspositionType& transpositions, Workspace& temp, int* sign=0)
00409 {
00410 Transpose<MatrixType> matt(mat);
00411 return ldlt_inplace<Lower>::unblocked(matt, transpositions, temp, sign);
00412 }
00413
00414 template<typename MatrixType, typename TranspositionType, typename Workspace, typename WType>
00415 static EIGEN_STRONG_INLINE bool update(MatrixType& mat, TranspositionType& transpositions, Workspace& tmp, WType& w, typename MatrixType::RealScalar sigma=1)
00416 {
00417 Transpose<MatrixType> matt(mat);
00418 return ldlt_inplace<Lower>::update(matt, transpositions, tmp, w.conjugate(), sigma);
00419 }
00420 };
00421
00422 template<typename MatrixType> struct LDLT_Traits<MatrixType,Lower>
00423 {
00424 typedef const TriangularView<const MatrixType, UnitLower> MatrixL;
00425 typedef const TriangularView<const typename MatrixType::AdjointReturnType, UnitUpper> MatrixU;
00426 static inline MatrixL getL(const MatrixType& m) { return m; }
00427 static inline MatrixU getU(const MatrixType& m) { return m.adjoint(); }
00428 };
00429
00430 template<typename MatrixType> struct LDLT_Traits<MatrixType,Upper>
00431 {
00432 typedef const TriangularView<const typename MatrixType::AdjointReturnType, UnitLower> MatrixL;
00433 typedef const TriangularView<const MatrixType, UnitUpper> MatrixU;
00434 static inline MatrixL getL(const MatrixType& m) { return m.adjoint(); }
00435 static inline MatrixU getU(const MatrixType& m) { return m; }
00436 };
00437
00438 }
00439
00442 template<typename MatrixType, int _UpLo>
00443 LDLT<MatrixType,_UpLo>& LDLT<MatrixType,_UpLo>::compute(const MatrixType& a)
00444 {
00445 eigen_assert(a.rows()==a.cols());
00446 const Index size = a.rows();
00447
00448 m_matrix = a;
00449
00450 m_transpositions.resize(size);
00451 m_isInitialized = false;
00452 m_temporary.resize(size);
00453
00454 internal::ldlt_inplace<UpLo>::unblocked(m_matrix, m_transpositions, m_temporary, &m_sign);
00455
00456 m_isInitialized = true;
00457 return *this;
00458 }
00459
00465 template<typename MatrixType, int _UpLo>
00466 template<typename Derived>
00467 LDLT<MatrixType,_UpLo>& LDLT<MatrixType,_UpLo>::rankUpdate(const MatrixBase<Derived>& w,typename NumTraits<typename MatrixType::Scalar>::Real sigma)
00468 {
00469 const Index size = w.rows();
00470 if (m_isInitialized)
00471 {
00472 eigen_assert(m_matrix.rows()==size);
00473 }
00474 else
00475 {
00476 m_matrix.resize(size,size);
00477 m_matrix.setZero();
00478 m_transpositions.resize(size);
00479 for (Index i = 0; i < size; i++)
00480 m_transpositions.coeffRef(i) = i;
00481 m_temporary.resize(size);
00482 m_sign = sigma;
00483 m_isInitialized = true;
00484 }
00485
00486 internal::ldlt_inplace<UpLo>::update(m_matrix, m_transpositions, m_temporary, w, sigma);
00487
00488 return *this;
00489 }
00490
00491 namespace internal {
00492 template<typename _MatrixType, int _UpLo, typename Rhs>
00493 struct solve_retval<LDLT<_MatrixType,_UpLo>, Rhs>
00494 : solve_retval_base<LDLT<_MatrixType,_UpLo>, Rhs>
00495 {
00496 typedef LDLT<_MatrixType,_UpLo> LDLTType;
00497 EIGEN_MAKE_SOLVE_HELPERS(LDLTType,Rhs)
00498
00499 template<typename Dest> void evalTo(Dest& dst) const
00500 {
00501 eigen_assert(rhs().rows() == dec().matrixLDLT().rows());
00502
00503 dst = dec().transpositionsP() * rhs();
00504
00505
00506 dec().matrixL().solveInPlace(dst);
00507
00508
00509
00510 using std::abs;
00511 using std::max;
00512 typedef typename LDLTType::MatrixType MatrixType;
00513 typedef typename LDLTType::Scalar Scalar;
00514 typedef typename LDLTType::RealScalar RealScalar;
00515 const Diagonal<const MatrixType> vectorD = dec().vectorD();
00516 RealScalar tolerance = (max)(vectorD.array().abs().maxCoeff() * NumTraits<Scalar>::epsilon(),
00517 RealScalar(1) / NumTraits<RealScalar>::highest());
00518 for (Index i = 0; i < vectorD.size(); ++i) {
00519 if(abs(vectorD(i)) > tolerance)
00520 dst.row(i) /= vectorD(i);
00521 else
00522 dst.row(i).setZero();
00523 }
00524
00525
00526 dec().matrixU().solveInPlace(dst);
00527
00528
00529 dst = dec().transpositionsP().transpose() * dst;
00530 }
00531 };
00532 }
00533
00547 template<typename MatrixType,int _UpLo>
00548 template<typename Derived>
00549 bool LDLT<MatrixType,_UpLo>::solveInPlace(MatrixBase<Derived> &bAndX) const
00550 {
00551 eigen_assert(m_isInitialized && "LDLT is not initialized.");
00552 const Index size = m_matrix.rows();
00553 eigen_assert(size == bAndX.rows());
00554
00555 bAndX = this->solve(bAndX);
00556
00557 return true;
00558 }
00559
00563 template<typename MatrixType, int _UpLo>
00564 MatrixType LDLT<MatrixType,_UpLo>::reconstructedMatrix() const
00565 {
00566 eigen_assert(m_isInitialized && "LDLT is not initialized.");
00567 const Index size = m_matrix.rows();
00568 MatrixType res(size,size);
00569
00570
00571 res.setIdentity();
00572 res = transpositionsP() * res;
00573
00574 res = matrixU() * res;
00575
00576 res = vectorD().asDiagonal() * res;
00577
00578 res = matrixL() * res;
00579
00580 res = transpositionsP().transpose() * res;
00581
00582 return res;
00583 }
00584
00588 template<typename MatrixType, unsigned int UpLo>
00589 inline const LDLT<typename SelfAdjointView<MatrixType, UpLo>::PlainObject, UpLo>
00590 SelfAdjointView<MatrixType, UpLo>::ldlt() const
00591 {
00592 return LDLT<PlainObject,UpLo>(m_matrix);
00593 }
00594
00598 template<typename Derived>
00599 inline const LDLT<typename MatrixBase<Derived>::PlainObject>
00600 MatrixBase<Derived>::ldlt() const
00601 {
00602 return LDLT<PlainObject>(derived());
00603 }
00604
00605 }
00606
00607 #endif // EIGEN_LDLT_H