00001
00002
00003
00004
00005
00006
00007
00008
00009
00010 #ifndef EIGEN_LU_H
00011 #define EIGEN_LU_H
00012
00013 namespace Eigen {
00014
00045 template<typename _MatrixType> class FullPivLU
00046 {
00047 public:
00048 typedef _MatrixType MatrixType;
00049 enum {
00050 RowsAtCompileTime = MatrixType::RowsAtCompileTime,
00051 ColsAtCompileTime = MatrixType::ColsAtCompileTime,
00052 Options = MatrixType::Options,
00053 MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
00054 MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
00055 };
00056 typedef typename MatrixType::Scalar Scalar;
00057 typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
00058 typedef typename internal::traits<MatrixType>::StorageKind StorageKind;
00059 typedef typename MatrixType::Index Index;
00060 typedef typename internal::plain_row_type<MatrixType, Index>::type IntRowVectorType;
00061 typedef typename internal::plain_col_type<MatrixType, Index>::type IntColVectorType;
00062 typedef PermutationMatrix<ColsAtCompileTime, MaxColsAtCompileTime> PermutationQType;
00063 typedef PermutationMatrix<RowsAtCompileTime, MaxRowsAtCompileTime> PermutationPType;
00064
00071 FullPivLU();
00072
00079 FullPivLU(Index rows, Index cols);
00080
00086 FullPivLU(const MatrixType& matrix);
00087
00095 FullPivLU& compute(const MatrixType& matrix);
00096
00103 inline const MatrixType& matrixLU() const
00104 {
00105 eigen_assert(m_isInitialized && "LU is not initialized.");
00106 return m_lu;
00107 }
00108
00116 inline Index nonzeroPivots() const
00117 {
00118 eigen_assert(m_isInitialized && "LU is not initialized.");
00119 return m_nonzero_pivots;
00120 }
00121
00125 RealScalar maxPivot() const { return m_maxpivot; }
00126
00131 inline const PermutationPType& permutationP() const
00132 {
00133 eigen_assert(m_isInitialized && "LU is not initialized.");
00134 return m_p;
00135 }
00136
00141 inline const PermutationQType& permutationQ() const
00142 {
00143 eigen_assert(m_isInitialized && "LU is not initialized.");
00144 return m_q;
00145 }
00146
00161 inline const internal::kernel_retval<FullPivLU> kernel() const
00162 {
00163 eigen_assert(m_isInitialized && "LU is not initialized.");
00164 return internal::kernel_retval<FullPivLU>(*this);
00165 }
00166
00186 inline const internal::image_retval<FullPivLU>
00187 image(const MatrixType& originalMatrix) const
00188 {
00189 eigen_assert(m_isInitialized && "LU is not initialized.");
00190 return internal::image_retval<FullPivLU>(*this, originalMatrix);
00191 }
00192
00212 template<typename Rhs>
00213 inline const internal::solve_retval<FullPivLU, Rhs>
00214 solve(const MatrixBase<Rhs>& b) const
00215 {
00216 eigen_assert(m_isInitialized && "LU is not initialized.");
00217 return internal::solve_retval<FullPivLU, Rhs>(*this, b.derived());
00218 }
00219
00235 typename internal::traits<MatrixType>::Scalar determinant() const;
00236
00254 FullPivLU& setThreshold(const RealScalar& threshold)
00255 {
00256 m_usePrescribedThreshold = true;
00257 m_prescribedThreshold = threshold;
00258 return *this;
00259 }
00260
00269 FullPivLU& setThreshold(Default_t)
00270 {
00271 m_usePrescribedThreshold = false;
00272 return *this;
00273 }
00274
00279 RealScalar threshold() const
00280 {
00281 eigen_assert(m_isInitialized || m_usePrescribedThreshold);
00282 return m_usePrescribedThreshold ? m_prescribedThreshold
00283
00284
00285 : NumTraits<Scalar>::epsilon() * m_lu.diagonalSize();
00286 }
00287
00294 inline Index rank() const
00295 {
00296 using std::abs;
00297 eigen_assert(m_isInitialized && "LU is not initialized.");
00298 RealScalar premultiplied_threshold = abs(m_maxpivot) * threshold();
00299 Index result = 0;
00300 for(Index i = 0; i < m_nonzero_pivots; ++i)
00301 result += (abs(m_lu.coeff(i,i)) > premultiplied_threshold);
00302 return result;
00303 }
00304
00311 inline Index dimensionOfKernel() const
00312 {
00313 eigen_assert(m_isInitialized && "LU is not initialized.");
00314 return cols() - rank();
00315 }
00316
00324 inline bool isInjective() const
00325 {
00326 eigen_assert(m_isInitialized && "LU is not initialized.");
00327 return rank() == cols();
00328 }
00329
00337 inline bool isSurjective() const
00338 {
00339 eigen_assert(m_isInitialized && "LU is not initialized.");
00340 return rank() == rows();
00341 }
00342
00349 inline bool isInvertible() const
00350 {
00351 eigen_assert(m_isInitialized && "LU is not initialized.");
00352 return isInjective() && (m_lu.rows() == m_lu.cols());
00353 }
00354
00362 inline const internal::solve_retval<FullPivLU,typename MatrixType::IdentityReturnType> inverse() const
00363 {
00364 eigen_assert(m_isInitialized && "LU is not initialized.");
00365 eigen_assert(m_lu.rows() == m_lu.cols() && "You can't take the inverse of a non-square matrix!");
00366 return internal::solve_retval<FullPivLU,typename MatrixType::IdentityReturnType>
00367 (*this, MatrixType::Identity(m_lu.rows(), m_lu.cols()));
00368 }
00369
00370 MatrixType reconstructedMatrix() const;
00371
00372 inline Index rows() const { return m_lu.rows(); }
00373 inline Index cols() const { return m_lu.cols(); }
00374
00375 protected:
00376 MatrixType m_lu;
00377 PermutationPType m_p;
00378 PermutationQType m_q;
00379 IntColVectorType m_rowsTranspositions;
00380 IntRowVectorType m_colsTranspositions;
00381 Index m_det_pq, m_nonzero_pivots;
00382 RealScalar m_maxpivot, m_prescribedThreshold;
00383 bool m_isInitialized, m_usePrescribedThreshold;
00384 };
00385
00386 template<typename MatrixType>
00387 FullPivLU<MatrixType>::FullPivLU()
00388 : m_isInitialized(false), m_usePrescribedThreshold(false)
00389 {
00390 }
00391
00392 template<typename MatrixType>
00393 FullPivLU<MatrixType>::FullPivLU(Index rows, Index cols)
00394 : m_lu(rows, cols),
00395 m_p(rows),
00396 m_q(cols),
00397 m_rowsTranspositions(rows),
00398 m_colsTranspositions(cols),
00399 m_isInitialized(false),
00400 m_usePrescribedThreshold(false)
00401 {
00402 }
00403
00404 template<typename MatrixType>
00405 FullPivLU<MatrixType>::FullPivLU(const MatrixType& matrix)
00406 : m_lu(matrix.rows(), matrix.cols()),
00407 m_p(matrix.rows()),
00408 m_q(matrix.cols()),
00409 m_rowsTranspositions(matrix.rows()),
00410 m_colsTranspositions(matrix.cols()),
00411 m_isInitialized(false),
00412 m_usePrescribedThreshold(false)
00413 {
00414 compute(matrix);
00415 }
00416
00417 template<typename MatrixType>
00418 FullPivLU<MatrixType>& FullPivLU<MatrixType>::compute(const MatrixType& matrix)
00419 {
00420
00421 eigen_assert(matrix.rows()<=NumTraits<int>::highest() && matrix.cols()<=NumTraits<int>::highest());
00422
00423 m_isInitialized = true;
00424 m_lu = matrix;
00425
00426 const Index size = matrix.diagonalSize();
00427 const Index rows = matrix.rows();
00428 const Index cols = matrix.cols();
00429
00430
00431
00432 m_rowsTranspositions.resize(matrix.rows());
00433 m_colsTranspositions.resize(matrix.cols());
00434 Index number_of_transpositions = 0;
00435
00436 m_nonzero_pivots = size;
00437 m_maxpivot = RealScalar(0);
00438
00439 for(Index k = 0; k < size; ++k)
00440 {
00441
00442
00443
00444 Index row_of_biggest_in_corner, col_of_biggest_in_corner;
00445 RealScalar biggest_in_corner;
00446 biggest_in_corner = m_lu.bottomRightCorner(rows-k, cols-k)
00447 .cwiseAbs()
00448 .maxCoeff(&row_of_biggest_in_corner, &col_of_biggest_in_corner);
00449 row_of_biggest_in_corner += k;
00450 col_of_biggest_in_corner += k;
00451
00452 if(biggest_in_corner==RealScalar(0))
00453 {
00454
00455
00456 m_nonzero_pivots = k;
00457 for(Index i = k; i < size; ++i)
00458 {
00459 m_rowsTranspositions.coeffRef(i) = i;
00460 m_colsTranspositions.coeffRef(i) = i;
00461 }
00462 break;
00463 }
00464
00465 if(biggest_in_corner > m_maxpivot) m_maxpivot = biggest_in_corner;
00466
00467
00468
00469
00470 m_rowsTranspositions.coeffRef(k) = row_of_biggest_in_corner;
00471 m_colsTranspositions.coeffRef(k) = col_of_biggest_in_corner;
00472 if(k != row_of_biggest_in_corner) {
00473 m_lu.row(k).swap(m_lu.row(row_of_biggest_in_corner));
00474 ++number_of_transpositions;
00475 }
00476 if(k != col_of_biggest_in_corner) {
00477 m_lu.col(k).swap(m_lu.col(col_of_biggest_in_corner));
00478 ++number_of_transpositions;
00479 }
00480
00481
00482
00483
00484 if(k<rows-1)
00485 m_lu.col(k).tail(rows-k-1) /= m_lu.coeff(k,k);
00486 if(k<size-1)
00487 m_lu.block(k+1,k+1,rows-k-1,cols-k-1).noalias() -= m_lu.col(k).tail(rows-k-1) * m_lu.row(k).tail(cols-k-1);
00488 }
00489
00490
00491
00492
00493 m_p.setIdentity(rows);
00494 for(Index k = size-1; k >= 0; --k)
00495 m_p.applyTranspositionOnTheRight(k, m_rowsTranspositions.coeff(k));
00496
00497 m_q.setIdentity(cols);
00498 for(Index k = 0; k < size; ++k)
00499 m_q.applyTranspositionOnTheRight(k, m_colsTranspositions.coeff(k));
00500
00501 m_det_pq = (number_of_transpositions%2) ? -1 : 1;
00502 return *this;
00503 }
00504
00505 template<typename MatrixType>
00506 typename internal::traits<MatrixType>::Scalar FullPivLU<MatrixType>::determinant() const
00507 {
00508 eigen_assert(m_isInitialized && "LU is not initialized.");
00509 eigen_assert(m_lu.rows() == m_lu.cols() && "You can't take the determinant of a non-square matrix!");
00510 return Scalar(m_det_pq) * Scalar(m_lu.diagonal().prod());
00511 }
00512
00516 template<typename MatrixType>
00517 MatrixType FullPivLU<MatrixType>::reconstructedMatrix() const
00518 {
00519 eigen_assert(m_isInitialized && "LU is not initialized.");
00520 const Index smalldim = (std::min)(m_lu.rows(), m_lu.cols());
00521
00522 MatrixType res(m_lu.rows(),m_lu.cols());
00523
00524 res = m_lu.leftCols(smalldim)
00525 .template triangularView<UnitLower>().toDenseMatrix()
00526 * m_lu.topRows(smalldim)
00527 .template triangularView<Upper>().toDenseMatrix();
00528
00529
00530 res = m_p.inverse() * res;
00531
00532
00533 res = res * m_q.inverse();
00534
00535 return res;
00536 }
00537
00538
00539
00540 namespace internal {
00541 template<typename _MatrixType>
00542 struct kernel_retval<FullPivLU<_MatrixType> >
00543 : kernel_retval_base<FullPivLU<_MatrixType> >
00544 {
00545 EIGEN_MAKE_KERNEL_HELPERS(FullPivLU<_MatrixType>)
00546
00547 enum { MaxSmallDimAtCompileTime = EIGEN_SIZE_MIN_PREFER_FIXED(
00548 MatrixType::MaxColsAtCompileTime,
00549 MatrixType::MaxRowsAtCompileTime)
00550 };
00551
00552 template<typename Dest> void evalTo(Dest& dst) const
00553 {
00554 using std::abs;
00555 const Index cols = dec().matrixLU().cols(), dimker = cols - rank();
00556 if(dimker == 0)
00557 {
00558
00559
00560
00561 dst.setZero();
00562 return;
00563 }
00564
00565
00566
00567
00568
00569
00570
00571
00572
00573
00574
00575
00576
00577
00578
00579
00580
00581 Matrix<Index, Dynamic, 1, 0, MaxSmallDimAtCompileTime, 1> pivots(rank());
00582 RealScalar premultiplied_threshold = dec().maxPivot() * dec().threshold();
00583 Index p = 0;
00584 for(Index i = 0; i < dec().nonzeroPivots(); ++i)
00585 if(abs(dec().matrixLU().coeff(i,i)) > premultiplied_threshold)
00586 pivots.coeffRef(p++) = i;
00587 eigen_internal_assert(p == rank());
00588
00589
00590
00591
00592
00593 Matrix<typename MatrixType::Scalar, Dynamic, Dynamic, MatrixType::Options,
00594 MaxSmallDimAtCompileTime, MatrixType::MaxColsAtCompileTime>
00595 m(dec().matrixLU().block(0, 0, rank(), cols));
00596 for(Index i = 0; i < rank(); ++i)
00597 {
00598 if(i) m.row(i).head(i).setZero();
00599 m.row(i).tail(cols-i) = dec().matrixLU().row(pivots.coeff(i)).tail(cols-i);
00600 }
00601 m.block(0, 0, rank(), rank());
00602 m.block(0, 0, rank(), rank()).template triangularView<StrictlyLower>().setZero();
00603 for(Index i = 0; i < rank(); ++i)
00604 m.col(i).swap(m.col(pivots.coeff(i)));
00605
00606
00607
00608
00609 m.topLeftCorner(rank(), rank())
00610 .template triangularView<Upper>().solveInPlace(
00611 m.topRightCorner(rank(), dimker)
00612 );
00613
00614
00615 for(Index i = rank()-1; i >= 0; --i)
00616 m.col(i).swap(m.col(pivots.coeff(i)));
00617
00618
00619 for(Index i = 0; i < rank(); ++i) dst.row(dec().permutationQ().indices().coeff(i)) = -m.row(i).tail(dimker);
00620 for(Index i = rank(); i < cols; ++i) dst.row(dec().permutationQ().indices().coeff(i)).setZero();
00621 for(Index k = 0; k < dimker; ++k) dst.coeffRef(dec().permutationQ().indices().coeff(rank()+k), k) = Scalar(1);
00622 }
00623 };
00624
00625
00626
00627 template<typename _MatrixType>
00628 struct image_retval<FullPivLU<_MatrixType> >
00629 : image_retval_base<FullPivLU<_MatrixType> >
00630 {
00631 EIGEN_MAKE_IMAGE_HELPERS(FullPivLU<_MatrixType>)
00632
00633 enum { MaxSmallDimAtCompileTime = EIGEN_SIZE_MIN_PREFER_FIXED(
00634 MatrixType::MaxColsAtCompileTime,
00635 MatrixType::MaxRowsAtCompileTime)
00636 };
00637
00638 template<typename Dest> void evalTo(Dest& dst) const
00639 {
00640 using std::abs;
00641 if(rank() == 0)
00642 {
00643
00644
00645
00646 dst.setZero();
00647 return;
00648 }
00649
00650 Matrix<Index, Dynamic, 1, 0, MaxSmallDimAtCompileTime, 1> pivots(rank());
00651 RealScalar premultiplied_threshold = dec().maxPivot() * dec().threshold();
00652 Index p = 0;
00653 for(Index i = 0; i < dec().nonzeroPivots(); ++i)
00654 if(abs(dec().matrixLU().coeff(i,i)) > premultiplied_threshold)
00655 pivots.coeffRef(p++) = i;
00656 eigen_internal_assert(p == rank());
00657
00658 for(Index i = 0; i < rank(); ++i)
00659 dst.col(i) = originalMatrix().col(dec().permutationQ().indices().coeff(pivots.coeff(i)));
00660 }
00661 };
00662
00663
00664
00665 template<typename _MatrixType, typename Rhs>
00666 struct solve_retval<FullPivLU<_MatrixType>, Rhs>
00667 : solve_retval_base<FullPivLU<_MatrixType>, Rhs>
00668 {
00669 EIGEN_MAKE_SOLVE_HELPERS(FullPivLU<_MatrixType>,Rhs)
00670
00671 template<typename Dest> void evalTo(Dest& dst) const
00672 {
00673
00674
00675
00676
00677
00678
00679
00680
00681 const Index rows = dec().rows(), cols = dec().cols(),
00682 nonzero_pivots = dec().nonzeroPivots();
00683 eigen_assert(rhs().rows() == rows);
00684 const Index smalldim = (std::min)(rows, cols);
00685
00686 if(nonzero_pivots == 0)
00687 {
00688 dst.setZero();
00689 return;
00690 }
00691
00692 typename Rhs::PlainObject c(rhs().rows(), rhs().cols());
00693
00694
00695 c = dec().permutationP() * rhs();
00696
00697
00698 dec().matrixLU()
00699 .topLeftCorner(smalldim,smalldim)
00700 .template triangularView<UnitLower>()
00701 .solveInPlace(c.topRows(smalldim));
00702 if(rows>cols)
00703 {
00704 c.bottomRows(rows-cols)
00705 -= dec().matrixLU().bottomRows(rows-cols)
00706 * c.topRows(cols);
00707 }
00708
00709
00710 dec().matrixLU()
00711 .topLeftCorner(nonzero_pivots, nonzero_pivots)
00712 .template triangularView<Upper>()
00713 .solveInPlace(c.topRows(nonzero_pivots));
00714
00715
00716 for(Index i = 0; i < nonzero_pivots; ++i)
00717 dst.row(dec().permutationQ().indices().coeff(i)) = c.row(i);
00718 for(Index i = nonzero_pivots; i < dec().matrixLU().cols(); ++i)
00719 dst.row(dec().permutationQ().indices().coeff(i)).setZero();
00720 }
00721 };
00722
00723 }
00724
00725
00726
00733 template<typename Derived>
00734 inline const FullPivLU<typename MatrixBase<Derived>::PlainObject>
00735 MatrixBase<Derived>::fullPivLu() const
00736 {
00737 return FullPivLU<PlainObject>(eval());
00738 }
00739
00740 }
00741
00742 #endif // EIGEN_LU_H