46 template<
typename _MatrixType>
class FullPivLU
49 typedef _MatrixType MatrixType;
51 RowsAtCompileTime = MatrixType::RowsAtCompileTime,
52 ColsAtCompileTime = MatrixType::ColsAtCompileTime,
53 Options = MatrixType::Options,
54 MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
55 MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
57 typedef typename MatrixType::Scalar Scalar;
58 typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
59 typedef typename internal::traits<MatrixType>::StorageKind StorageKind;
60 typedef typename MatrixType::Index Index;
61 typedef typename internal::plain_row_type<MatrixType, Index>::type IntRowVectorType;
62 typedef typename internal::plain_col_type<MatrixType, Index>::type IntColVectorType;
63 typedef PermutationMatrix<ColsAtCompileTime, MaxColsAtCompileTime> PermutationQType;
64 typedef PermutationMatrix<RowsAtCompileTime, MaxRowsAtCompileTime> PermutationPType;
106 eigen_assert(m_isInitialized &&
"LU is not initialized.");
119 eigen_assert(m_isInitialized &&
"LU is not initialized.");
120 return m_nonzero_pivots;
134 eigen_assert(m_isInitialized &&
"LU is not initialized.");
144 eigen_assert(m_isInitialized &&
"LU is not initialized.");
162 inline const internal::kernel_retval<FullPivLU>
kernel()
const
164 eigen_assert(m_isInitialized &&
"LU is not initialized.");
165 return internal::kernel_retval<FullPivLU>(*this);
187 inline const internal::image_retval<FullPivLU>
188 image(
const MatrixType& originalMatrix)
const
190 eigen_assert(m_isInitialized &&
"LU is not initialized.");
191 return internal::image_retval<FullPivLU>(*
this, originalMatrix);
213 template<
typename Rhs>
214 inline const internal::solve_retval<FullPivLU, Rhs>
217 eigen_assert(m_isInitialized &&
"LU is not initialized.");
218 return internal::solve_retval<FullPivLU, Rhs>(*
this, b.derived());
236 typename internal::traits<MatrixType>::Scalar
determinant()
const;
257 m_usePrescribedThreshold =
true;
272 m_usePrescribedThreshold =
false;
282 eigen_assert(m_isInitialized || m_usePrescribedThreshold);
283 return m_usePrescribedThreshold ? m_prescribedThreshold
298 eigen_assert(m_isInitialized &&
"LU is not initialized.");
299 RealScalar premultiplied_threshold = abs(m_maxpivot) *
threshold();
301 for(Index i = 0; i < m_nonzero_pivots; ++i)
302 result += (abs(m_lu.coeff(i,i)) > premultiplied_threshold);
314 eigen_assert(m_isInitialized &&
"LU is not initialized.");
315 return cols() -
rank();
327 eigen_assert(m_isInitialized &&
"LU is not initialized.");
328 return rank() == cols();
340 eigen_assert(m_isInitialized &&
"LU is not initialized.");
341 return rank() == rows();
352 eigen_assert(m_isInitialized &&
"LU is not initialized.");
353 return isInjective() && (m_lu.rows() == m_lu.cols());
363 inline const internal::solve_retval<FullPivLU,typename MatrixType::IdentityReturnType>
inverse()
const
365 eigen_assert(m_isInitialized &&
"LU is not initialized.");
366 eigen_assert(m_lu.rows() == m_lu.cols() &&
"You can't take the inverse of a non-square matrix!");
367 return internal::solve_retval<FullPivLU,typename MatrixType::IdentityReturnType>
368 (*
this, MatrixType::Identity(m_lu.rows(), m_lu.cols()));
373 inline Index rows()
const {
return m_lu.rows(); }
374 inline Index cols()
const {
return m_lu.cols(); }
378 PermutationPType m_p;
379 PermutationQType m_q;
380 IntColVectorType m_rowsTranspositions;
381 IntRowVectorType m_colsTranspositions;
382 Index m_det_pq, m_nonzero_pivots;
383 RealScalar m_maxpivot, m_prescribedThreshold;
384 bool m_isInitialized, m_usePrescribedThreshold;
387 template<
typename MatrixType>
389 : m_isInitialized(false), m_usePrescribedThreshold(false)
393 template<
typename MatrixType>
398 m_rowsTranspositions(rows),
399 m_colsTranspositions(cols),
400 m_isInitialized(false),
401 m_usePrescribedThreshold(false)
405 template<
typename MatrixType>
407 : m_lu(matrix.rows(), matrix.cols()),
410 m_rowsTranspositions(matrix.rows()),
411 m_colsTranspositions(matrix.cols()),
412 m_isInitialized(false),
413 m_usePrescribedThreshold(false)
418 template<
typename MatrixType>
424 m_isInitialized =
true;
427 const Index size = matrix.diagonalSize();
428 const Index rows = matrix.rows();
429 const Index cols = matrix.cols();
433 m_rowsTranspositions.resize(matrix.rows());
434 m_colsTranspositions.resize(matrix.cols());
435 Index number_of_transpositions = 0;
437 m_nonzero_pivots = size;
438 m_maxpivot = RealScalar(0);
440 for(Index k = 0; k < size; ++k)
445 Index row_of_biggest_in_corner, col_of_biggest_in_corner;
446 RealScalar biggest_in_corner;
447 biggest_in_corner = m_lu.bottomRightCorner(rows-k, cols-k)
449 .maxCoeff(&row_of_biggest_in_corner, &col_of_biggest_in_corner);
450 row_of_biggest_in_corner += k;
451 col_of_biggest_in_corner += k;
453 if(biggest_in_corner==RealScalar(0))
457 m_nonzero_pivots = k;
458 for(Index i = k; i < size; ++i)
460 m_rowsTranspositions.coeffRef(i) = i;
461 m_colsTranspositions.coeffRef(i) = i;
466 if(biggest_in_corner > m_maxpivot) m_maxpivot = biggest_in_corner;
471 m_rowsTranspositions.coeffRef(k) = row_of_biggest_in_corner;
472 m_colsTranspositions.coeffRef(k) = col_of_biggest_in_corner;
473 if(k != row_of_biggest_in_corner) {
474 m_lu.row(k).swap(m_lu.row(row_of_biggest_in_corner));
475 ++number_of_transpositions;
477 if(k != col_of_biggest_in_corner) {
478 m_lu.col(k).swap(m_lu.col(col_of_biggest_in_corner));
479 ++number_of_transpositions;
486 m_lu.col(k).tail(rows-k-1) /= m_lu.coeff(k,k);
488 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);
494 m_p.setIdentity(rows);
495 for(Index k = size-1; k >= 0; --k)
496 m_p.applyTranspositionOnTheRight(k, m_rowsTranspositions.coeff(k));
498 m_q.setIdentity(cols);
499 for(Index k = 0; k < size; ++k)
500 m_q.applyTranspositionOnTheRight(k, m_colsTranspositions.coeff(k));
502 m_det_pq = (number_of_transpositions%2) ? -1 : 1;
506 template<
typename MatrixType>
509 eigen_assert(m_isInitialized &&
"LU is not initialized.");
510 eigen_assert(m_lu.rows() == m_lu.cols() &&
"You can't take the determinant of a non-square matrix!");
511 return Scalar(m_det_pq) * Scalar(m_lu.diagonal().prod());
517 template<
typename MatrixType>
520 eigen_assert(m_isInitialized &&
"LU is not initialized.");
521 const Index smalldim = (std::min)(m_lu.rows(), m_lu.cols());
523 MatrixType res(m_lu.rows(),m_lu.cols());
525 res = m_lu.leftCols(smalldim)
526 .template triangularView<UnitLower>().toDenseMatrix()
527 * m_lu.topRows(smalldim)
528 .template triangularView<Upper>().toDenseMatrix();
531 res = m_p.inverse() * res;
534 res = res * m_q.inverse();
542 template<
typename _MatrixType>
543 struct kernel_retval<
FullPivLU<_MatrixType> >
544 : kernel_retval_base<FullPivLU<_MatrixType> >
548 enum { MaxSmallDimAtCompileTime = EIGEN_SIZE_MIN_PREFER_FIXED(
549 MatrixType::MaxColsAtCompileTime,
550 MatrixType::MaxRowsAtCompileTime)
553 template<
typename Dest>
void evalTo(Dest& dst)
const
556 const Index cols = dec().matrixLU().cols(), dimker = cols - rank();
582 Matrix<Index, Dynamic, 1, 0, MaxSmallDimAtCompileTime, 1> pivots(rank());
583 RealScalar premultiplied_threshold = dec().maxPivot() * dec().threshold();
585 for(Index i = 0; i < dec().nonzeroPivots(); ++i)
586 if(abs(dec().matrixLU().coeff(i,i)) > premultiplied_threshold)
587 pivots.coeffRef(p++) = i;
588 eigen_internal_assert(p == rank());
594 Matrix<
typename MatrixType::Scalar, Dynamic, Dynamic, MatrixType::Options,
595 MaxSmallDimAtCompileTime, MatrixType::MaxColsAtCompileTime>
596 m(dec().matrixLU().block(0, 0, rank(), cols));
597 for(Index i = 0; i < rank(); ++i)
599 if(i) m.row(i).head(i).setZero();
600 m.row(i).tail(cols-i) = dec().matrixLU().row(pivots.coeff(i)).tail(cols-i);
602 m.block(0, 0, rank(), rank());
603 m.block(0, 0, rank(), rank()).template triangularView<StrictlyLower>().setZero();
604 for(Index i = 0; i < rank(); ++i)
605 m.col(i).swap(m.col(pivots.coeff(i)));
610 m.topLeftCorner(rank(), rank())
611 .template triangularView<Upper>().solveInPlace(
612 m.topRightCorner(rank(), dimker)
616 for(Index i = rank()-1; i >= 0; --i)
617 m.col(i).swap(m.col(pivots.coeff(i)));
620 for(Index i = 0; i < rank(); ++i) dst.row(dec().permutationQ().indices().coeff(i)) = -m.row(i).tail(dimker);
621 for(Index i = rank(); i < cols; ++i) dst.row(dec().permutationQ().indices().coeff(i)).setZero();
622 for(Index k = 0; k < dimker; ++k) dst.coeffRef(dec().permutationQ().indices().coeff(rank()+k), k) = Scalar(1);
628 template<
typename _MatrixType>
629 struct image_retval<FullPivLU<_MatrixType> >
630 : image_retval_base<FullPivLU<_MatrixType> >
632 EIGEN_MAKE_IMAGE_HELPERS(FullPivLU<_MatrixType>)
634 enum { MaxSmallDimAtCompileTime = EIGEN_SIZE_MIN_PREFER_FIXED(
635 MatrixType::MaxColsAtCompileTime,
636 MatrixType::MaxRowsAtCompileTime)
639 template<
typename Dest>
void evalTo(Dest& dst)
const
651 Matrix<Index, Dynamic, 1, 0, MaxSmallDimAtCompileTime, 1> pivots(rank());
652 RealScalar premultiplied_threshold = dec().maxPivot() * dec().threshold();
654 for(Index i = 0; i < dec().nonzeroPivots(); ++i)
655 if(abs(dec().matrixLU().coeff(i,i)) > premultiplied_threshold)
656 pivots.coeffRef(p++) = i;
657 eigen_internal_assert(p == rank());
659 for(Index i = 0; i < rank(); ++i)
660 dst.col(i) = originalMatrix().col(dec().permutationQ().indices().coeff(pivots.coeff(i)));
666 template<
typename _MatrixType,
typename Rhs>
667 struct solve_retval<FullPivLU<_MatrixType>, Rhs>
668 : solve_retval_base<FullPivLU<_MatrixType>, Rhs>
670 EIGEN_MAKE_SOLVE_HELPERS(FullPivLU<_MatrixType>,Rhs)
672 template<typename Dest>
void evalTo(Dest& dst)
const
682 const Index rows = dec().rows(), cols = dec().cols(),
683 nonzero_pivots = dec().nonzeroPivots();
684 eigen_assert(rhs().rows() == rows);
685 const Index smalldim = (std::min)(rows, cols);
687 if(nonzero_pivots == 0)
693 typename Rhs::PlainObject c(rhs().rows(), rhs().cols());
696 c = dec().permutationP() * rhs();
700 .topLeftCorner(smalldim,smalldim)
701 .template triangularView<UnitLower>()
702 .solveInPlace(c.topRows(smalldim));
705 c.bottomRows(rows-cols)
706 -= dec().matrixLU().bottomRows(rows-cols)
712 .topLeftCorner(nonzero_pivots, nonzero_pivots)
713 .template triangularView<Upper>()
714 .solveInPlace(c.topRows(nonzero_pivots));
717 for(Index i = 0; i < nonzero_pivots; ++i)
718 dst.row(dec().permutationQ().indices().coeff(i)) = c.row(i);
719 for(Index i = nonzero_pivots; i < dec().matrixLU().cols(); ++i)
720 dst.row(dec().permutationQ().indices().coeff(i)).setZero();
734 template<
typename Derived>
735 inline const FullPivLU<typename MatrixBase<Derived>::PlainObject>
bool isInvertible() const
Definition: FullPivLU.h:350
RealScalar threshold() const
Definition: FullPivLU.h:280
const MatrixType & matrixLU() const
Definition: FullPivLU.h:104
const FullPivLU< PlainObject > fullPivLu() const
Definition: FullPivLU.h:736
internal::traits< MatrixType >::Scalar determinant() const
Definition: FullPivLU.h:507
MatrixType reconstructedMatrix() const
Definition: FullPivLU.h:518
Holds information about the various numeric (i.e. scalar) types allowed by Eigen. ...
Definition: NumTraits.h:88
bool isInjective() const
Definition: FullPivLU.h:325
FullPivLU()
Default Constructor.
Definition: FullPivLU.h:388
FullPivLU & compute(const MatrixType &matrix)
Definition: FullPivLU.h:419
const PermutationPType & permutationP() const
Definition: FullPivLU.h:132
Index rank() const
Definition: FullPivLU.h:295
const internal::image_retval< FullPivLU > image(const MatrixType &originalMatrix) const
Definition: FullPivLU.h:188
FullPivLU & setThreshold(Default_t)
Definition: FullPivLU.h:270
Definition: Eigen_Colamd.h:54
Index nonzeroPivots() const
Definition: FullPivLU.h:117
Index dimensionOfKernel() const
Definition: FullPivLU.h:312
LU decomposition of a matrix with complete pivoting, and related features.
Definition: ForwardDeclarations.h:216
FullPivLU & setThreshold(const RealScalar &threshold)
Definition: FullPivLU.h:255
const internal::kernel_retval< FullPivLU > kernel() const
Definition: FullPivLU.h:162
RealScalar maxPivot() const
Definition: FullPivLU.h:126
Base class for all dense matrices, vectors, and expressions.
Definition: MatrixBase.h:48
const internal::solve_retval< FullPivLU, Rhs > solve(const MatrixBase< Rhs > &b) const
Definition: FullPivLU.h:215
bool isSurjective() const
Definition: FullPivLU.h:338
const internal::solve_retval< FullPivLU, typename MatrixType::IdentityReturnType > inverse() const
Definition: FullPivLU.h:363
const PermutationQType & permutationQ() const
Definition: FullPivLU.h:142