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Eigen  3.4.0
 
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FullPivLU.h
1// This file is part of Eigen, a lightweight C++ template library
2// for linear algebra.
3//
4// Copyright (C) 2006-2009 Benoit Jacob <jacob.benoit.1@gmail.com>
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_LU_H
11#define EIGEN_LU_H
12
13namespace Eigen {
14
15namespace internal {
16template<typename _MatrixType> struct traits<FullPivLU<_MatrixType> >
17 : traits<_MatrixType>
18{
19 typedef MatrixXpr XprKind;
20 typedef SolverStorage StorageKind;
21 typedef int StorageIndex;
22 enum { Flags = 0 };
23};
24
25} // end namespace internal
26
60template<typename _MatrixType> class FullPivLU
61 : public SolverBase<FullPivLU<_MatrixType> >
62{
63 public:
64 typedef _MatrixType MatrixType;
66 friend class SolverBase<FullPivLU>;
67
68 EIGEN_GENERIC_PUBLIC_INTERFACE(FullPivLU)
69 enum {
70 MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
71 MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
72 };
73 typedef typename internal::plain_row_type<MatrixType, StorageIndex>::type IntRowVectorType;
74 typedef typename internal::plain_col_type<MatrixType, StorageIndex>::type IntColVectorType;
77 typedef typename MatrixType::PlainObject PlainObject;
78
85 FullPivLU();
86
93 FullPivLU(Index rows, Index cols);
94
100 template<typename InputType>
101 explicit FullPivLU(const EigenBase<InputType>& matrix);
102
109 template<typename InputType>
110 explicit FullPivLU(EigenBase<InputType>& matrix);
111
119 template<typename InputType>
121 m_lu = matrix.derived();
122 computeInPlace();
123 return *this;
124 }
125
132 inline const MatrixType& matrixLU() const
133 {
134 eigen_assert(m_isInitialized && "LU is not initialized.");
135 return m_lu;
136 }
137
145 inline Index nonzeroPivots() const
146 {
147 eigen_assert(m_isInitialized && "LU is not initialized.");
148 return m_nonzero_pivots;
149 }
150
154 RealScalar maxPivot() const { return m_maxpivot; }
155
160 EIGEN_DEVICE_FUNC inline const PermutationPType& permutationP() const
161 {
162 eigen_assert(m_isInitialized && "LU is not initialized.");
163 return m_p;
164 }
165
170 inline const PermutationQType& permutationQ() const
171 {
172 eigen_assert(m_isInitialized && "LU is not initialized.");
173 return m_q;
174 }
175
190 inline const internal::kernel_retval<FullPivLU> kernel() const
191 {
192 eigen_assert(m_isInitialized && "LU is not initialized.");
193 return internal::kernel_retval<FullPivLU>(*this);
194 }
195
215 inline const internal::image_retval<FullPivLU>
216 image(const MatrixType& originalMatrix) const
217 {
218 eigen_assert(m_isInitialized && "LU is not initialized.");
219 return internal::image_retval<FullPivLU>(*this, originalMatrix);
220 }
221
222 #ifdef EIGEN_PARSED_BY_DOXYGEN
242 template<typename Rhs>
243 inline const Solve<FullPivLU, Rhs>
244 solve(const MatrixBase<Rhs>& b) const;
245 #endif
246
250 inline RealScalar rcond() const
251 {
252 eigen_assert(m_isInitialized && "PartialPivLU is not initialized.");
253 return internal::rcond_estimate_helper(m_l1_norm, *this);
254 }
255
271 typename internal::traits<MatrixType>::Scalar determinant() const;
272
290 FullPivLU& setThreshold(const RealScalar& threshold)
291 {
292 m_usePrescribedThreshold = true;
293 m_prescribedThreshold = threshold;
294 return *this;
295 }
296
306 {
307 m_usePrescribedThreshold = false;
308 return *this;
309 }
310
315 RealScalar threshold() const
316 {
317 eigen_assert(m_isInitialized || m_usePrescribedThreshold);
318 return m_usePrescribedThreshold ? m_prescribedThreshold
319 // this formula comes from experimenting (see "LU precision tuning" thread on the list)
320 // and turns out to be identical to Higham's formula used already in LDLt.
321 : NumTraits<Scalar>::epsilon() * RealScalar(m_lu.diagonalSize());
322 }
323
330 inline Index rank() const
331 {
332 using std::abs;
333 eigen_assert(m_isInitialized && "LU is not initialized.");
334 RealScalar premultiplied_threshold = abs(m_maxpivot) * threshold();
335 Index result = 0;
336 for(Index i = 0; i < m_nonzero_pivots; ++i)
337 result += (abs(m_lu.coeff(i,i)) > premultiplied_threshold);
338 return result;
339 }
340
348 {
349 eigen_assert(m_isInitialized && "LU is not initialized.");
350 return cols() - rank();
351 }
352
360 inline bool isInjective() const
361 {
362 eigen_assert(m_isInitialized && "LU is not initialized.");
363 return rank() == cols();
364 }
365
373 inline bool isSurjective() const
374 {
375 eigen_assert(m_isInitialized && "LU is not initialized.");
376 return rank() == rows();
377 }
378
385 inline bool isInvertible() const
386 {
387 eigen_assert(m_isInitialized && "LU is not initialized.");
388 return isInjective() && (m_lu.rows() == m_lu.cols());
389 }
390
398 inline const Inverse<FullPivLU> inverse() const
399 {
400 eigen_assert(m_isInitialized && "LU is not initialized.");
401 eigen_assert(m_lu.rows() == m_lu.cols() && "You can't take the inverse of a non-square matrix!");
402 return Inverse<FullPivLU>(*this);
403 }
404
405 MatrixType reconstructedMatrix() const;
406
407 EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR
408 inline Index rows() const EIGEN_NOEXCEPT { return m_lu.rows(); }
409 EIGEN_DEVICE_FUNC EIGEN_CONSTEXPR
410 inline Index cols() const EIGEN_NOEXCEPT { return m_lu.cols(); }
411
412 #ifndef EIGEN_PARSED_BY_DOXYGEN
413 template<typename RhsType, typename DstType>
414 void _solve_impl(const RhsType &rhs, DstType &dst) const;
415
416 template<bool Conjugate, typename RhsType, typename DstType>
417 void _solve_impl_transposed(const RhsType &rhs, DstType &dst) const;
418 #endif
419
420 protected:
421
422 static void check_template_parameters()
423 {
424 EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
425 }
426
427 void computeInPlace();
428
429 MatrixType m_lu;
430 PermutationPType m_p;
431 PermutationQType m_q;
432 IntColVectorType m_rowsTranspositions;
433 IntRowVectorType m_colsTranspositions;
434 Index m_nonzero_pivots;
435 RealScalar m_l1_norm;
436 RealScalar m_maxpivot, m_prescribedThreshold;
437 signed char m_det_pq;
438 bool m_isInitialized, m_usePrescribedThreshold;
439};
440
441template<typename MatrixType>
443 : m_isInitialized(false), m_usePrescribedThreshold(false)
444{
445}
446
447template<typename MatrixType>
449 : m_lu(rows, cols),
450 m_p(rows),
451 m_q(cols),
452 m_rowsTranspositions(rows),
453 m_colsTranspositions(cols),
454 m_isInitialized(false),
455 m_usePrescribedThreshold(false)
456{
457}
458
459template<typename MatrixType>
460template<typename InputType>
462 : m_lu(matrix.rows(), matrix.cols()),
463 m_p(matrix.rows()),
464 m_q(matrix.cols()),
465 m_rowsTranspositions(matrix.rows()),
466 m_colsTranspositions(matrix.cols()),
467 m_isInitialized(false),
468 m_usePrescribedThreshold(false)
469{
470 compute(matrix.derived());
471}
472
473template<typename MatrixType>
474template<typename InputType>
476 : m_lu(matrix.derived()),
477 m_p(matrix.rows()),
478 m_q(matrix.cols()),
479 m_rowsTranspositions(matrix.rows()),
480 m_colsTranspositions(matrix.cols()),
481 m_isInitialized(false),
482 m_usePrescribedThreshold(false)
483{
484 computeInPlace();
485}
486
487template<typename MatrixType>
489{
490 check_template_parameters();
491
492 // the permutations are stored as int indices, so just to be sure:
493 eigen_assert(m_lu.rows()<=NumTraits<int>::highest() && m_lu.cols()<=NumTraits<int>::highest());
494
495 m_l1_norm = m_lu.cwiseAbs().colwise().sum().maxCoeff();
496
497 const Index size = m_lu.diagonalSize();
498 const Index rows = m_lu.rows();
499 const Index cols = m_lu.cols();
500
501 // will store the transpositions, before we accumulate them at the end.
502 // can't accumulate on-the-fly because that will be done in reverse order for the rows.
503 m_rowsTranspositions.resize(m_lu.rows());
504 m_colsTranspositions.resize(m_lu.cols());
505 Index number_of_transpositions = 0; // number of NONTRIVIAL transpositions, i.e. m_rowsTranspositions[i]!=i
506
507 m_nonzero_pivots = size; // the generic case is that in which all pivots are nonzero (invertible case)
508 m_maxpivot = RealScalar(0);
509
510 for(Index k = 0; k < size; ++k)
511 {
512 // First, we need to find the pivot.
513
514 // biggest coefficient in the remaining bottom-right corner (starting at row k, col k)
515 Index row_of_biggest_in_corner, col_of_biggest_in_corner;
516 typedef internal::scalar_score_coeff_op<Scalar> Scoring;
517 typedef typename Scoring::result_type Score;
518 Score biggest_in_corner;
519 biggest_in_corner = m_lu.bottomRightCorner(rows-k, cols-k)
520 .unaryExpr(Scoring())
521 .maxCoeff(&row_of_biggest_in_corner, &col_of_biggest_in_corner);
522 row_of_biggest_in_corner += k; // correct the values! since they were computed in the corner,
523 col_of_biggest_in_corner += k; // need to add k to them.
524
525 if(biggest_in_corner==Score(0))
526 {
527 // before exiting, make sure to initialize the still uninitialized transpositions
528 // in a sane state without destroying what we already have.
529 m_nonzero_pivots = k;
530 for(Index i = k; i < size; ++i)
531 {
532 m_rowsTranspositions.coeffRef(i) = internal::convert_index<StorageIndex>(i);
533 m_colsTranspositions.coeffRef(i) = internal::convert_index<StorageIndex>(i);
534 }
535 break;
536 }
537
538 RealScalar abs_pivot = internal::abs_knowing_score<Scalar>()(m_lu(row_of_biggest_in_corner, col_of_biggest_in_corner), biggest_in_corner);
539 if(abs_pivot > m_maxpivot) m_maxpivot = abs_pivot;
540
541 // Now that we've found the pivot, we need to apply the row/col swaps to
542 // bring it to the location (k,k).
543
544 m_rowsTranspositions.coeffRef(k) = internal::convert_index<StorageIndex>(row_of_biggest_in_corner);
545 m_colsTranspositions.coeffRef(k) = internal::convert_index<StorageIndex>(col_of_biggest_in_corner);
546 if(k != row_of_biggest_in_corner) {
547 m_lu.row(k).swap(m_lu.row(row_of_biggest_in_corner));
548 ++number_of_transpositions;
549 }
550 if(k != col_of_biggest_in_corner) {
551 m_lu.col(k).swap(m_lu.col(col_of_biggest_in_corner));
552 ++number_of_transpositions;
553 }
554
555 // Now that the pivot is at the right location, we update the remaining
556 // bottom-right corner by Gaussian elimination.
557
558 if(k<rows-1)
559 m_lu.col(k).tail(rows-k-1) /= m_lu.coeff(k,k);
560 if(k<size-1)
561 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);
562 }
563
564 // the main loop is over, we still have to accumulate the transpositions to find the
565 // permutations P and Q
566
567 m_p.setIdentity(rows);
568 for(Index k = size-1; k >= 0; --k)
569 m_p.applyTranspositionOnTheRight(k, m_rowsTranspositions.coeff(k));
570
571 m_q.setIdentity(cols);
572 for(Index k = 0; k < size; ++k)
573 m_q.applyTranspositionOnTheRight(k, m_colsTranspositions.coeff(k));
574
575 m_det_pq = (number_of_transpositions%2) ? -1 : 1;
576
577 m_isInitialized = true;
578}
579
580template<typename MatrixType>
581typename internal::traits<MatrixType>::Scalar FullPivLU<MatrixType>::determinant() const
582{
583 eigen_assert(m_isInitialized && "LU is not initialized.");
584 eigen_assert(m_lu.rows() == m_lu.cols() && "You can't take the determinant of a non-square matrix!");
585 return Scalar(m_det_pq) * Scalar(m_lu.diagonal().prod());
586}
587
591template<typename MatrixType>
593{
594 eigen_assert(m_isInitialized && "LU is not initialized.");
595 const Index smalldim = (std::min)(m_lu.rows(), m_lu.cols());
596 // LU
597 MatrixType res(m_lu.rows(),m_lu.cols());
598 // FIXME the .toDenseMatrix() should not be needed...
599 res = m_lu.leftCols(smalldim)
600 .template triangularView<UnitLower>().toDenseMatrix()
601 * m_lu.topRows(smalldim)
602 .template triangularView<Upper>().toDenseMatrix();
603
604 // P^{-1}(LU)
605 res = m_p.inverse() * res;
606
607 // (P^{-1}LU)Q^{-1}
608 res = res * m_q.inverse();
609
610 return res;
611}
612
613/********* Implementation of kernel() **************************************************/
614
615namespace internal {
616template<typename _MatrixType>
617struct kernel_retval<FullPivLU<_MatrixType> >
618 : kernel_retval_base<FullPivLU<_MatrixType> >
619{
620 EIGEN_MAKE_KERNEL_HELPERS(FullPivLU<_MatrixType>)
621
622 enum { MaxSmallDimAtCompileTime = EIGEN_SIZE_MIN_PREFER_FIXED(
623 MatrixType::MaxColsAtCompileTime,
624 MatrixType::MaxRowsAtCompileTime)
625 };
626
627 template<typename Dest> void evalTo(Dest& dst) const
628 {
629 using std::abs;
630 const Index cols = dec().matrixLU().cols(), dimker = cols - rank();
631 if(dimker == 0)
632 {
633 // The Kernel is just {0}, so it doesn't have a basis properly speaking, but let's
634 // avoid crashing/asserting as that depends on floating point calculations. Let's
635 // just return a single column vector filled with zeros.
636 dst.setZero();
637 return;
638 }
639
640 /* Let us use the following lemma:
641 *
642 * Lemma: If the matrix A has the LU decomposition PAQ = LU,
643 * then Ker A = Q(Ker U).
644 *
645 * Proof: trivial: just keep in mind that P, Q, L are invertible.
646 */
647
648 /* Thus, all we need to do is to compute Ker U, and then apply Q.
649 *
650 * U is upper triangular, with eigenvalues sorted so that any zeros appear at the end.
651 * Thus, the diagonal of U ends with exactly
652 * dimKer zero's. Let us use that to construct dimKer linearly
653 * independent vectors in Ker U.
654 */
655
656 Matrix<Index, Dynamic, 1, 0, MaxSmallDimAtCompileTime, 1> pivots(rank());
657 RealScalar premultiplied_threshold = dec().maxPivot() * dec().threshold();
658 Index p = 0;
659 for(Index i = 0; i < dec().nonzeroPivots(); ++i)
660 if(abs(dec().matrixLU().coeff(i,i)) > premultiplied_threshold)
661 pivots.coeffRef(p++) = i;
662 eigen_internal_assert(p == rank());
663
664 // we construct a temporaty trapezoid matrix m, by taking the U matrix and
665 // permuting the rows and cols to bring the nonnegligible pivots to the top of
666 // the main diagonal. We need that to be able to apply our triangular solvers.
667 // FIXME when we get triangularView-for-rectangular-matrices, this can be simplified
668 Matrix<typename MatrixType::Scalar, Dynamic, Dynamic, MatrixType::Options,
669 MaxSmallDimAtCompileTime, MatrixType::MaxColsAtCompileTime>
670 m(dec().matrixLU().block(0, 0, rank(), cols));
671 for(Index i = 0; i < rank(); ++i)
672 {
673 if(i) m.row(i).head(i).setZero();
674 m.row(i).tail(cols-i) = dec().matrixLU().row(pivots.coeff(i)).tail(cols-i);
675 }
676 m.block(0, 0, rank(), rank());
677 m.block(0, 0, rank(), rank()).template triangularView<StrictlyLower>().setZero();
678 for(Index i = 0; i < rank(); ++i)
679 m.col(i).swap(m.col(pivots.coeff(i)));
680
681 // ok, we have our trapezoid matrix, we can apply the triangular solver.
682 // notice that the math behind this suggests that we should apply this to the
683 // negative of the RHS, but for performance we just put the negative sign elsewhere, see below.
684 m.topLeftCorner(rank(), rank())
685 .template triangularView<Upper>().solveInPlace(
686 m.topRightCorner(rank(), dimker)
687 );
688
689 // now we must undo the column permutation that we had applied!
690 for(Index i = rank()-1; i >= 0; --i)
691 m.col(i).swap(m.col(pivots.coeff(i)));
692
693 // see the negative sign in the next line, that's what we were talking about above.
694 for(Index i = 0; i < rank(); ++i) dst.row(dec().permutationQ().indices().coeff(i)) = -m.row(i).tail(dimker);
695 for(Index i = rank(); i < cols; ++i) dst.row(dec().permutationQ().indices().coeff(i)).setZero();
696 for(Index k = 0; k < dimker; ++k) dst.coeffRef(dec().permutationQ().indices().coeff(rank()+k), k) = Scalar(1);
697 }
698};
699
700/***** Implementation of image() *****************************************************/
701
702template<typename _MatrixType>
703struct image_retval<FullPivLU<_MatrixType> >
704 : image_retval_base<FullPivLU<_MatrixType> >
705{
706 EIGEN_MAKE_IMAGE_HELPERS(FullPivLU<_MatrixType>)
707
708 enum { MaxSmallDimAtCompileTime = EIGEN_SIZE_MIN_PREFER_FIXED(
709 MatrixType::MaxColsAtCompileTime,
710 MatrixType::MaxRowsAtCompileTime)
711 };
712
713 template<typename Dest> void evalTo(Dest& dst) const
714 {
715 using std::abs;
716 if(rank() == 0)
717 {
718 // The Image is just {0}, so it doesn't have a basis properly speaking, but let's
719 // avoid crashing/asserting as that depends on floating point calculations. Let's
720 // just return a single column vector filled with zeros.
721 dst.setZero();
722 return;
723 }
724
725 Matrix<Index, Dynamic, 1, 0, MaxSmallDimAtCompileTime, 1> pivots(rank());
726 RealScalar premultiplied_threshold = dec().maxPivot() * dec().threshold();
727 Index p = 0;
728 for(Index i = 0; i < dec().nonzeroPivots(); ++i)
729 if(abs(dec().matrixLU().coeff(i,i)) > premultiplied_threshold)
730 pivots.coeffRef(p++) = i;
731 eigen_internal_assert(p == rank());
732
733 for(Index i = 0; i < rank(); ++i)
734 dst.col(i) = originalMatrix().col(dec().permutationQ().indices().coeff(pivots.coeff(i)));
735 }
736};
737
738/***** Implementation of solve() *****************************************************/
739
740} // end namespace internal
741
742#ifndef EIGEN_PARSED_BY_DOXYGEN
743template<typename _MatrixType>
744template<typename RhsType, typename DstType>
745void FullPivLU<_MatrixType>::_solve_impl(const RhsType &rhs, DstType &dst) const
746{
747 /* The decomposition PAQ = LU can be rewritten as A = P^{-1} L U Q^{-1}.
748 * So we proceed as follows:
749 * Step 1: compute c = P * rhs.
750 * Step 2: replace c by the solution x to Lx = c. Exists because L is invertible.
751 * Step 3: replace c by the solution x to Ux = c. May or may not exist.
752 * Step 4: result = Q * c;
753 */
754
755 const Index rows = this->rows(),
756 cols = this->cols(),
757 nonzero_pivots = this->rank();
758 const Index smalldim = (std::min)(rows, cols);
759
760 if(nonzero_pivots == 0)
761 {
762 dst.setZero();
763 return;
764 }
765
766 typename RhsType::PlainObject c(rhs.rows(), rhs.cols());
767
768 // Step 1
769 c = permutationP() * rhs;
770
771 // Step 2
772 m_lu.topLeftCorner(smalldim,smalldim)
773 .template triangularView<UnitLower>()
774 .solveInPlace(c.topRows(smalldim));
775 if(rows>cols)
776 c.bottomRows(rows-cols) -= m_lu.bottomRows(rows-cols) * c.topRows(cols);
777
778 // Step 3
779 m_lu.topLeftCorner(nonzero_pivots, nonzero_pivots)
780 .template triangularView<Upper>()
781 .solveInPlace(c.topRows(nonzero_pivots));
782
783 // Step 4
784 for(Index i = 0; i < nonzero_pivots; ++i)
785 dst.row(permutationQ().indices().coeff(i)) = c.row(i);
786 for(Index i = nonzero_pivots; i < m_lu.cols(); ++i)
787 dst.row(permutationQ().indices().coeff(i)).setZero();
788}
789
790template<typename _MatrixType>
791template<bool Conjugate, typename RhsType, typename DstType>
792void FullPivLU<_MatrixType>::_solve_impl_transposed(const RhsType &rhs, DstType &dst) const
793{
794 /* The decomposition PAQ = LU can be rewritten as A = P^{-1} L U Q^{-1},
795 * and since permutations are real and unitary, we can write this
796 * as A^T = Q U^T L^T P,
797 * So we proceed as follows:
798 * Step 1: compute c = Q^T rhs.
799 * Step 2: replace c by the solution x to U^T x = c. May or may not exist.
800 * Step 3: replace c by the solution x to L^T x = c.
801 * Step 4: result = P^T c.
802 * If Conjugate is true, replace "^T" by "^*" above.
803 */
804
805 const Index rows = this->rows(), cols = this->cols(),
806 nonzero_pivots = this->rank();
807 const Index smalldim = (std::min)(rows, cols);
808
809 if(nonzero_pivots == 0)
810 {
811 dst.setZero();
812 return;
813 }
814
815 typename RhsType::PlainObject c(rhs.rows(), rhs.cols());
816
817 // Step 1
818 c = permutationQ().inverse() * rhs;
819
820 // Step 2
821 m_lu.topLeftCorner(nonzero_pivots, nonzero_pivots)
822 .template triangularView<Upper>()
823 .transpose()
824 .template conjugateIf<Conjugate>()
825 .solveInPlace(c.topRows(nonzero_pivots));
826
827 // Step 3
828 m_lu.topLeftCorner(smalldim, smalldim)
829 .template triangularView<UnitLower>()
830 .transpose()
831 .template conjugateIf<Conjugate>()
832 .solveInPlace(c.topRows(smalldim));
833
834 // Step 4
835 PermutationPType invp = permutationP().inverse().eval();
836 for(Index i = 0; i < smalldim; ++i)
837 dst.row(invp.indices().coeff(i)) = c.row(i);
838 for(Index i = smalldim; i < rows; ++i)
839 dst.row(invp.indices().coeff(i)).setZero();
840}
841
842#endif
843
844namespace internal {
845
846
847/***** Implementation of inverse() *****************************************************/
848template<typename DstXprType, typename MatrixType>
849struct Assignment<DstXprType, Inverse<FullPivLU<MatrixType> >, internal::assign_op<typename DstXprType::Scalar,typename FullPivLU<MatrixType>::Scalar>, Dense2Dense>
850{
851 typedef FullPivLU<MatrixType> LuType;
852 typedef Inverse<LuType> SrcXprType;
853 static void run(DstXprType &dst, const SrcXprType &src, const internal::assign_op<typename DstXprType::Scalar,typename MatrixType::Scalar> &)
854 {
855 dst = src.nestedExpression().solve(MatrixType::Identity(src.rows(), src.cols()));
856 }
857};
858} // end namespace internal
859
860/******* MatrixBase methods *****************************************************************/
861
868template<typename Derived>
869inline const FullPivLU<typename MatrixBase<Derived>::PlainObject>
871{
872 return FullPivLU<PlainObject>(eval());
873}
874
875} // end namespace Eigen
876
877#endif // EIGEN_LU_H
LU decomposition of a matrix with complete pivoting, and related features.
Definition: FullPivLU.h:62
RealScalar rcond() const
Definition: FullPivLU.h:250
MatrixType reconstructedMatrix() const
Definition: FullPivLU.h:592
bool isSurjective() const
Definition: FullPivLU.h:373
FullPivLU & setThreshold(Default_t)
Definition: FullPivLU.h:305
const internal::kernel_retval< FullPivLU > kernel() const
Definition: FullPivLU.h:190
Index dimensionOfKernel() const
Definition: FullPivLU.h:347
Index rank() const
Definition: FullPivLU.h:330
internal::traits< MatrixType >::Scalar determinant() const
Definition: FullPivLU.h:581
const Solve< FullPivLU, Rhs > solve(const MatrixBase< Rhs > &b) const
const PermutationPType & permutationP() const
Definition: FullPivLU.h:160
Index nonzeroPivots() const
Definition: FullPivLU.h:145
bool isInjective() const
Definition: FullPivLU.h:360
FullPivLU & setThreshold(const RealScalar &threshold)
Definition: FullPivLU.h:290
RealScalar maxPivot() const
Definition: FullPivLU.h:154
const Inverse< FullPivLU > inverse() const
Definition: FullPivLU.h:398
const MatrixType & matrixLU() const
Definition: FullPivLU.h:132
RealScalar threshold() const
Definition: FullPivLU.h:315
const PermutationQType & permutationQ() const
Definition: FullPivLU.h:170
FullPivLU & compute(const EigenBase< InputType > &matrix)
Definition: FullPivLU.h:120
const internal::image_retval< FullPivLU > image(const MatrixType &originalMatrix) const
Definition: FullPivLU.h:216
FullPivLU()
Default Constructor.
Definition: FullPivLU.h:442
bool isInvertible() const
Definition: FullPivLU.h:385
Expression of the inverse of another expression.
Definition: Inverse.h:44
Base class for all dense matrices, vectors, and expressions.
Definition: MatrixBase.h:50
Permutation matrix.
Definition: PermutationMatrix.h:298
Pseudo expression representing a solving operation.
Definition: Solve.h:63
A base class for matrix decomposition and solvers.
Definition: SolverBase.h:69
const Solve< Derived, Rhs > solve(const MatrixBase< Rhs > &b) const
Definition: SolverBase.h:106
Namespace containing all symbols from the Eigen library.
Definition: Core:141
EIGEN_DEFAULT_DENSE_INDEX_TYPE Index
The Index type as used for the API.
Definition: Meta.h:74
const int Dynamic
Definition: Constants.h:22
Definition: EigenBase.h:30
Eigen::Index Index
The interface type of indices.
Definition: EigenBase.h:39
Derived & derived()
Definition: EigenBase.h:46
Holds information about the various numeric (i.e. scalar) types allowed by Eigen.
Definition: NumTraits.h:233