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Eigen  3.4.0
 
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LDLT.h
1// This file is part of Eigen, a lightweight C++ template library
2// for linear algebra.
3//
4// Copyright (C) 2008-2011 Gael Guennebaud <gael.guennebaud@inria.fr>
5// Copyright (C) 2009 Keir Mierle <mierle@gmail.com>
6// Copyright (C) 2009 Benoit Jacob <jacob.benoit.1@gmail.com>
7// Copyright (C) 2011 Timothy E. Holy <tim.holy@gmail.com >
8//
9// This Source Code Form is subject to the terms of the Mozilla
10// Public License v. 2.0. If a copy of the MPL was not distributed
11// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
12
13#ifndef EIGEN_LDLT_H
14#define EIGEN_LDLT_H
15
16namespace Eigen {
17
18namespace internal {
19 template<typename _MatrixType, int _UpLo> struct traits<LDLT<_MatrixType, _UpLo> >
20 : traits<_MatrixType>
21 {
22 typedef MatrixXpr XprKind;
23 typedef SolverStorage StorageKind;
24 typedef int StorageIndex;
25 enum { Flags = 0 };
26 };
27
28 template<typename MatrixType, int UpLo> struct LDLT_Traits;
29
30 // PositiveSemiDef means positive semi-definite and non-zero; same for NegativeSemiDef
31 enum SignMatrix { PositiveSemiDef, NegativeSemiDef, ZeroSign, Indefinite };
32}
33
59template<typename _MatrixType, int _UpLo> class LDLT
60 : public SolverBase<LDLT<_MatrixType, _UpLo> >
61{
62 public:
63 typedef _MatrixType MatrixType;
64 typedef SolverBase<LDLT> Base;
65 friend class SolverBase<LDLT>;
66
67 EIGEN_GENERIC_PUBLIC_INTERFACE(LDLT)
68 enum {
69 MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
70 MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime,
71 UpLo = _UpLo
72 };
74
77
78 typedef internal::LDLT_Traits<MatrixType,UpLo> Traits;
79
86 : m_matrix(),
87 m_transpositions(),
88 m_sign(internal::ZeroSign),
89 m_isInitialized(false)
90 {}
91
98 explicit LDLT(Index size)
99 : m_matrix(size, size),
100 m_transpositions(size),
101 m_temporary(size),
102 m_sign(internal::ZeroSign),
103 m_isInitialized(false)
104 {}
105
112 template<typename InputType>
113 explicit LDLT(const EigenBase<InputType>& matrix)
114 : m_matrix(matrix.rows(), matrix.cols()),
115 m_transpositions(matrix.rows()),
116 m_temporary(matrix.rows()),
117 m_sign(internal::ZeroSign),
118 m_isInitialized(false)
119 {
120 compute(matrix.derived());
121 }
122
129 template<typename InputType>
130 explicit LDLT(EigenBase<InputType>& matrix)
131 : m_matrix(matrix.derived()),
132 m_transpositions(matrix.rows()),
133 m_temporary(matrix.rows()),
134 m_sign(internal::ZeroSign),
135 m_isInitialized(false)
136 {
137 compute(matrix.derived());
138 }
139
143 void setZero()
144 {
145 m_isInitialized = false;
146 }
147
149 inline typename Traits::MatrixU matrixU() const
150 {
151 eigen_assert(m_isInitialized && "LDLT is not initialized.");
152 return Traits::getU(m_matrix);
153 }
154
156 inline typename Traits::MatrixL matrixL() const
157 {
158 eigen_assert(m_isInitialized && "LDLT is not initialized.");
159 return Traits::getL(m_matrix);
160 }
161
164 inline const TranspositionType& transpositionsP() const
165 {
166 eigen_assert(m_isInitialized && "LDLT is not initialized.");
167 return m_transpositions;
168 }
169
172 {
173 eigen_assert(m_isInitialized && "LDLT is not initialized.");
174 return m_matrix.diagonal();
175 }
176
178 inline bool isPositive() const
179 {
180 eigen_assert(m_isInitialized && "LDLT is not initialized.");
181 return m_sign == internal::PositiveSemiDef || m_sign == internal::ZeroSign;
182 }
183
185 inline bool isNegative(void) const
186 {
187 eigen_assert(m_isInitialized && "LDLT is not initialized.");
188 return m_sign == internal::NegativeSemiDef || m_sign == internal::ZeroSign;
189 }
190
191 #ifdef EIGEN_PARSED_BY_DOXYGEN
207 template<typename Rhs>
208 inline const Solve<LDLT, Rhs>
209 solve(const MatrixBase<Rhs>& b) const;
210 #endif
211
212 template<typename Derived>
213 bool solveInPlace(MatrixBase<Derived> &bAndX) const;
214
215 template<typename InputType>
216 LDLT& compute(const EigenBase<InputType>& matrix);
217
221 RealScalar rcond() const
222 {
223 eigen_assert(m_isInitialized && "LDLT is not initialized.");
224 return internal::rcond_estimate_helper(m_l1_norm, *this);
225 }
226
227 template <typename Derived>
228 LDLT& rankUpdate(const MatrixBase<Derived>& w, const RealScalar& alpha=1);
229
234 inline const MatrixType& matrixLDLT() const
235 {
236 eigen_assert(m_isInitialized && "LDLT is not initialized.");
237 return m_matrix;
238 }
239
240 MatrixType reconstructedMatrix() const;
241
247 const LDLT& adjoint() const { return *this; };
248
249 EIGEN_DEVICE_FUNC inline EIGEN_CONSTEXPR Index rows() const EIGEN_NOEXCEPT { return m_matrix.rows(); }
250 EIGEN_DEVICE_FUNC inline EIGEN_CONSTEXPR Index cols() const EIGEN_NOEXCEPT { return m_matrix.cols(); }
251
258 {
259 eigen_assert(m_isInitialized && "LDLT is not initialized.");
260 return m_info;
261 }
262
263 #ifndef EIGEN_PARSED_BY_DOXYGEN
264 template<typename RhsType, typename DstType>
265 void _solve_impl(const RhsType &rhs, DstType &dst) const;
266
267 template<bool Conjugate, typename RhsType, typename DstType>
268 void _solve_impl_transposed(const RhsType &rhs, DstType &dst) const;
269 #endif
270
271 protected:
272
273 static void check_template_parameters()
274 {
275 EIGEN_STATIC_ASSERT_NON_INTEGER(Scalar);
276 }
277
284 MatrixType m_matrix;
285 RealScalar m_l1_norm;
286 TranspositionType m_transpositions;
287 TmpMatrixType m_temporary;
288 internal::SignMatrix m_sign;
289 bool m_isInitialized;
290 ComputationInfo m_info;
291};
292
293namespace internal {
294
295template<int UpLo> struct ldlt_inplace;
296
297template<> struct ldlt_inplace<Lower>
298{
299 template<typename MatrixType, typename TranspositionType, typename Workspace>
300 static bool unblocked(MatrixType& mat, TranspositionType& transpositions, Workspace& temp, SignMatrix& sign)
301 {
302 using std::abs;
303 typedef typename MatrixType::Scalar Scalar;
304 typedef typename MatrixType::RealScalar RealScalar;
305 typedef typename TranspositionType::StorageIndex IndexType;
306 eigen_assert(mat.rows()==mat.cols());
307 const Index size = mat.rows();
308 bool found_zero_pivot = false;
309 bool ret = true;
310
311 if (size <= 1)
312 {
313 transpositions.setIdentity();
314 if(size==0) sign = ZeroSign;
315 else if (numext::real(mat.coeff(0,0)) > static_cast<RealScalar>(0) ) sign = PositiveSemiDef;
316 else if (numext::real(mat.coeff(0,0)) < static_cast<RealScalar>(0)) sign = NegativeSemiDef;
317 else sign = ZeroSign;
318 return true;
319 }
320
321 for (Index k = 0; k < size; ++k)
322 {
323 // Find largest diagonal element
324 Index index_of_biggest_in_corner;
325 mat.diagonal().tail(size-k).cwiseAbs().maxCoeff(&index_of_biggest_in_corner);
326 index_of_biggest_in_corner += k;
327
328 transpositions.coeffRef(k) = IndexType(index_of_biggest_in_corner);
329 if(k != index_of_biggest_in_corner)
330 {
331 // apply the transposition while taking care to consider only
332 // the lower triangular part
333 Index s = size-index_of_biggest_in_corner-1; // trailing size after the biggest element
334 mat.row(k).head(k).swap(mat.row(index_of_biggest_in_corner).head(k));
335 mat.col(k).tail(s).swap(mat.col(index_of_biggest_in_corner).tail(s));
336 std::swap(mat.coeffRef(k,k),mat.coeffRef(index_of_biggest_in_corner,index_of_biggest_in_corner));
337 for(Index i=k+1;i<index_of_biggest_in_corner;++i)
338 {
339 Scalar tmp = mat.coeffRef(i,k);
340 mat.coeffRef(i,k) = numext::conj(mat.coeffRef(index_of_biggest_in_corner,i));
341 mat.coeffRef(index_of_biggest_in_corner,i) = numext::conj(tmp);
342 }
343 if(NumTraits<Scalar>::IsComplex)
344 mat.coeffRef(index_of_biggest_in_corner,k) = numext::conj(mat.coeff(index_of_biggest_in_corner,k));
345 }
346
347 // partition the matrix:
348 // A00 | - | -
349 // lu = A10 | A11 | -
350 // A20 | A21 | A22
351 Index rs = size - k - 1;
352 Block<MatrixType,Dynamic,1> A21(mat,k+1,k,rs,1);
353 Block<MatrixType,1,Dynamic> A10(mat,k,0,1,k);
354 Block<MatrixType,Dynamic,Dynamic> A20(mat,k+1,0,rs,k);
356 if(k>0)
357 {
358 temp.head(k) = mat.diagonal().real().head(k).asDiagonal() * A10.adjoint();
359 mat.coeffRef(k,k) -= (A10 * temp.head(k)).value();
360 if(rs>0)
361 A21.noalias() -= A20 * temp.head(k);
362 }
363
364 // In some previous versions of Eigen (e.g., 3.2.1), the scaling was omitted if the pivot
365 // was smaller than the cutoff value. However, since LDLT is not rank-revealing
366 // we should only make sure that we do not introduce INF or NaN values.
367 // Remark that LAPACK also uses 0 as the cutoff value.
368 RealScalar realAkk = numext::real(mat.coeffRef(k,k));
369 bool pivot_is_valid = (abs(realAkk) > RealScalar(0));
370
371 if(k==0 && !pivot_is_valid)
372 {
373 // The entire diagonal is zero, there is nothing more to do
374 // except filling the transpositions, and checking whether the matrix is zero.
375 sign = ZeroSign;
376 for(Index j = 0; j<size; ++j)
377 {
378 transpositions.coeffRef(j) = IndexType(j);
379 ret = ret && (mat.col(j).tail(size-j-1).array()==Scalar(0)).all();
380 }
381 return ret;
382 }
383
384 if((rs>0) && pivot_is_valid)
385 A21 /= realAkk;
386 else if(rs>0)
387 ret = ret && (A21.array()==Scalar(0)).all();
388
389 if(found_zero_pivot && pivot_is_valid) ret = false; // factorization failed
390 else if(!pivot_is_valid) found_zero_pivot = true;
391
392 if (sign == PositiveSemiDef) {
393 if (realAkk < static_cast<RealScalar>(0)) sign = Indefinite;
394 } else if (sign == NegativeSemiDef) {
395 if (realAkk > static_cast<RealScalar>(0)) sign = Indefinite;
396 } else if (sign == ZeroSign) {
397 if (realAkk > static_cast<RealScalar>(0)) sign = PositiveSemiDef;
398 else if (realAkk < static_cast<RealScalar>(0)) sign = NegativeSemiDef;
399 }
400 }
401
402 return ret;
403 }
404
405 // Reference for the algorithm: Davis and Hager, "Multiple Rank
406 // Modifications of a Sparse Cholesky Factorization" (Algorithm 1)
407 // Trivial rearrangements of their computations (Timothy E. Holy)
408 // allow their algorithm to work for rank-1 updates even if the
409 // original matrix is not of full rank.
410 // Here only rank-1 updates are implemented, to reduce the
411 // requirement for intermediate storage and improve accuracy
412 template<typename MatrixType, typename WDerived>
413 static bool updateInPlace(MatrixType& mat, MatrixBase<WDerived>& w, const typename MatrixType::RealScalar& sigma=1)
414 {
415 using numext::isfinite;
416 typedef typename MatrixType::Scalar Scalar;
417 typedef typename MatrixType::RealScalar RealScalar;
418
419 const Index size = mat.rows();
420 eigen_assert(mat.cols() == size && w.size()==size);
421
422 RealScalar alpha = 1;
423
424 // Apply the update
425 for (Index j = 0; j < size; j++)
426 {
427 // Check for termination due to an original decomposition of low-rank
428 if (!(isfinite)(alpha))
429 break;
430
431 // Update the diagonal terms
432 RealScalar dj = numext::real(mat.coeff(j,j));
433 Scalar wj = w.coeff(j);
434 RealScalar swj2 = sigma*numext::abs2(wj);
435 RealScalar gamma = dj*alpha + swj2;
436
437 mat.coeffRef(j,j) += swj2/alpha;
438 alpha += swj2/dj;
439
440
441 // Update the terms of L
442 Index rs = size-j-1;
443 w.tail(rs) -= wj * mat.col(j).tail(rs);
444 if(gamma != 0)
445 mat.col(j).tail(rs) += (sigma*numext::conj(wj)/gamma)*w.tail(rs);
446 }
447 return true;
448 }
449
450 template<typename MatrixType, typename TranspositionType, typename Workspace, typename WType>
451 static bool update(MatrixType& mat, const TranspositionType& transpositions, Workspace& tmp, const WType& w, const typename MatrixType::RealScalar& sigma=1)
452 {
453 // Apply the permutation to the input w
454 tmp = transpositions * w;
455
456 return ldlt_inplace<Lower>::updateInPlace(mat,tmp,sigma);
457 }
458};
459
460template<> struct ldlt_inplace<Upper>
461{
462 template<typename MatrixType, typename TranspositionType, typename Workspace>
463 static EIGEN_STRONG_INLINE bool unblocked(MatrixType& mat, TranspositionType& transpositions, Workspace& temp, SignMatrix& sign)
464 {
465 Transpose<MatrixType> matt(mat);
466 return ldlt_inplace<Lower>::unblocked(matt, transpositions, temp, sign);
467 }
468
469 template<typename MatrixType, typename TranspositionType, typename Workspace, typename WType>
470 static EIGEN_STRONG_INLINE bool update(MatrixType& mat, TranspositionType& transpositions, Workspace& tmp, WType& w, const typename MatrixType::RealScalar& sigma=1)
471 {
472 Transpose<MatrixType> matt(mat);
473 return ldlt_inplace<Lower>::update(matt, transpositions, tmp, w.conjugate(), sigma);
474 }
475};
476
477template<typename MatrixType> struct LDLT_Traits<MatrixType,Lower>
478{
479 typedef const TriangularView<const MatrixType, UnitLower> MatrixL;
480 typedef const TriangularView<const typename MatrixType::AdjointReturnType, UnitUpper> MatrixU;
481 static inline MatrixL getL(const MatrixType& m) { return MatrixL(m); }
482 static inline MatrixU getU(const MatrixType& m) { return MatrixU(m.adjoint()); }
483};
484
485template<typename MatrixType> struct LDLT_Traits<MatrixType,Upper>
486{
487 typedef const TriangularView<const typename MatrixType::AdjointReturnType, UnitLower> MatrixL;
488 typedef const TriangularView<const MatrixType, UnitUpper> MatrixU;
489 static inline MatrixL getL(const MatrixType& m) { return MatrixL(m.adjoint()); }
490 static inline MatrixU getU(const MatrixType& m) { return MatrixU(m); }
491};
492
493} // end namespace internal
494
497template<typename MatrixType, int _UpLo>
498template<typename InputType>
500{
501 check_template_parameters();
502
503 eigen_assert(a.rows()==a.cols());
504 const Index size = a.rows();
505
506 m_matrix = a.derived();
507
508 // Compute matrix L1 norm = max abs column sum.
509 m_l1_norm = RealScalar(0);
510 // TODO move this code to SelfAdjointView
511 for (Index col = 0; col < size; ++col) {
512 RealScalar abs_col_sum;
513 if (_UpLo == Lower)
514 abs_col_sum = m_matrix.col(col).tail(size - col).template lpNorm<1>() + m_matrix.row(col).head(col).template lpNorm<1>();
515 else
516 abs_col_sum = m_matrix.col(col).head(col).template lpNorm<1>() + m_matrix.row(col).tail(size - col).template lpNorm<1>();
517 if (abs_col_sum > m_l1_norm)
518 m_l1_norm = abs_col_sum;
519 }
520
521 m_transpositions.resize(size);
522 m_isInitialized = false;
523 m_temporary.resize(size);
524 m_sign = internal::ZeroSign;
525
526 m_info = internal::ldlt_inplace<UpLo>::unblocked(m_matrix, m_transpositions, m_temporary, m_sign) ? Success : NumericalIssue;
527
528 m_isInitialized = true;
529 return *this;
530}
531
537template<typename MatrixType, int _UpLo>
538template<typename Derived>
540{
541 typedef typename TranspositionType::StorageIndex IndexType;
542 const Index size = w.rows();
543 if (m_isInitialized)
544 {
545 eigen_assert(m_matrix.rows()==size);
546 }
547 else
548 {
549 m_matrix.resize(size,size);
550 m_matrix.setZero();
551 m_transpositions.resize(size);
552 for (Index i = 0; i < size; i++)
553 m_transpositions.coeffRef(i) = IndexType(i);
554 m_temporary.resize(size);
555 m_sign = sigma>=0 ? internal::PositiveSemiDef : internal::NegativeSemiDef;
556 m_isInitialized = true;
557 }
558
559 internal::ldlt_inplace<UpLo>::update(m_matrix, m_transpositions, m_temporary, w, sigma);
560
561 return *this;
562}
563
564#ifndef EIGEN_PARSED_BY_DOXYGEN
565template<typename _MatrixType, int _UpLo>
566template<typename RhsType, typename DstType>
567void LDLT<_MatrixType,_UpLo>::_solve_impl(const RhsType &rhs, DstType &dst) const
568{
569 _solve_impl_transposed<true>(rhs, dst);
570}
571
572template<typename _MatrixType,int _UpLo>
573template<bool Conjugate, typename RhsType, typename DstType>
574void LDLT<_MatrixType,_UpLo>::_solve_impl_transposed(const RhsType &rhs, DstType &dst) const
575{
576 // dst = P b
577 dst = m_transpositions * rhs;
578
579 // dst = L^-1 (P b)
580 // dst = L^-*T (P b)
581 matrixL().template conjugateIf<!Conjugate>().solveInPlace(dst);
582
583 // dst = D^-* (L^-1 P b)
584 // dst = D^-1 (L^-*T P b)
585 // more precisely, use pseudo-inverse of D (see bug 241)
586 using std::abs;
587 const typename Diagonal<const MatrixType>::RealReturnType vecD(vectorD());
588 // In some previous versions, tolerance was set to the max of 1/highest (or rather numeric_limits::min())
589 // and the maximal diagonal entry * epsilon as motivated by LAPACK's xGELSS:
590 // RealScalar tolerance = numext::maxi(vecD.array().abs().maxCoeff() * NumTraits<RealScalar>::epsilon(),RealScalar(1) / NumTraits<RealScalar>::highest());
591 // However, LDLT is not rank revealing, and so adjusting the tolerance wrt to the highest
592 // diagonal element is not well justified and leads to numerical issues in some cases.
593 // Moreover, Lapack's xSYTRS routines use 0 for the tolerance.
594 // Using numeric_limits::min() gives us more robustness to denormals.
595 RealScalar tolerance = (std::numeric_limits<RealScalar>::min)();
596 for (Index i = 0; i < vecD.size(); ++i)
597 {
598 if(abs(vecD(i)) > tolerance)
599 dst.row(i) /= vecD(i);
600 else
601 dst.row(i).setZero();
602 }
603
604 // dst = L^-* (D^-* L^-1 P b)
605 // dst = L^-T (D^-1 L^-*T P b)
606 matrixL().transpose().template conjugateIf<Conjugate>().solveInPlace(dst);
607
608 // dst = P^T (L^-* D^-* L^-1 P b) = A^-1 b
609 // dst = P^-T (L^-T D^-1 L^-*T P b) = A^-1 b
610 dst = m_transpositions.transpose() * dst;
611}
612#endif
613
627template<typename MatrixType,int _UpLo>
628template<typename Derived>
629bool LDLT<MatrixType,_UpLo>::solveInPlace(MatrixBase<Derived> &bAndX) const
630{
631 eigen_assert(m_isInitialized && "LDLT is not initialized.");
632 eigen_assert(m_matrix.rows() == bAndX.rows());
633
634 bAndX = this->solve(bAndX);
635
636 return true;
637}
638
642template<typename MatrixType, int _UpLo>
644{
645 eigen_assert(m_isInitialized && "LDLT is not initialized.");
646 const Index size = m_matrix.rows();
647 MatrixType res(size,size);
648
649 // P
650 res.setIdentity();
651 res = transpositionsP() * res;
652 // L^* P
653 res = matrixU() * res;
654 // D(L^*P)
655 res = vectorD().real().asDiagonal() * res;
656 // L(DL^*P)
657 res = matrixL() * res;
658 // P^T (LDL^*P)
659 res = transpositionsP().transpose() * res;
660
661 return res;
662}
663
668template<typename MatrixType, unsigned int UpLo>
671{
672 return LDLT<PlainObject,UpLo>(m_matrix);
673}
674
679template<typename Derived>
682{
683 return LDLT<PlainObject>(derived());
684}
685
686} // end namespace Eigen
687
688#endif // EIGEN_LDLT_H
EIGEN_CONSTEXPR Index rows() const EIGEN_NOEXCEPT
Definition: EigenBase.h:60
Expression of a diagonal/subdiagonal/superdiagonal in a matrix.
Definition: Diagonal.h:65
Robust Cholesky decomposition of a matrix with pivoting.
Definition: LDLT.h:61
const Solve< LDLT, Rhs > solve(const MatrixBase< Rhs > &b) const
LDLT(Index size)
Default Constructor with memory preallocation.
Definition: LDLT.h:98
LDLT()
Default Constructor.
Definition: LDLT.h:85
Traits::MatrixU matrixU() const
Definition: LDLT.h:149
bool isPositive() const
Definition: LDLT.h:178
ComputationInfo info() const
Reports whether previous computation was successful.
Definition: LDLT.h:257
void setZero()
Definition: LDLT.h:143
const MatrixType & matrixLDLT() const
Definition: LDLT.h:234
bool isNegative(void) const
Definition: LDLT.h:185
Diagonal< const MatrixType > vectorD() const
Definition: LDLT.h:171
LDLT(const EigenBase< InputType > &matrix)
Constructor with decomposition.
Definition: LDLT.h:113
LDLT(EigenBase< InputType > &matrix)
Constructs a LDLT factorization from a given matrix.
Definition: LDLT.h:130
const LDLT & adjoint() const
Definition: LDLT.h:247
MatrixType reconstructedMatrix() const
Definition: LDLT.h:643
RealScalar rcond() const
Definition: LDLT.h:221
Traits::MatrixL matrixL() const
Definition: LDLT.h:156
const TranspositionType & transpositionsP() const
Definition: LDLT.h:164
Base class for all dense matrices, vectors, and expressions.
Definition: MatrixBase.h:50
The matrix class, also used for vectors and row-vectors.
Definition: Matrix.h:180
Permutation matrix.
Definition: PermutationMatrix.h:298
Expression of a selfadjoint matrix from a triangular part of a dense matrix.
Definition: SelfAdjointView.h:51
Pseudo expression representing a solving operation.
Definition: Solve.h:63
A base class for matrix decomposition and solvers.
Definition: SolverBase.h:69
LDLT< _MatrixType, _UpLo > & derived()
Definition: EigenBase.h:46
Represents a sequence of transpositions (row/column interchange)
Definition: Transpositions.h:156
static const Eigen::internal::all_t all
Definition: IndexedViewHelper.h:171
ComputationInfo
Definition: Constants.h:440
@ Lower
Definition: Constants.h:209
@ Upper
Definition: Constants.h:211
@ NumericalIssue
Definition: Constants.h:444
@ Success
Definition: Constants.h:442
Namespace containing all symbols from the Eigen library.
Definition: Core:141
const Eigen::CwiseUnaryOp< Eigen::internal::scalar_sign_op< typename Derived::Scalar >, const Derived > sign(const Eigen::ArrayBase< Derived > &x)
EIGEN_DEFAULT_DENSE_INDEX_TYPE Index
The Index type as used for the API.
Definition: Meta.h:74
const Eigen::CwiseUnaryOp< Eigen::internal::scalar_isfinite_op< typename Derived::Scalar >, const Derived > isfinite(const Eigen::ArrayBase< Derived > &x)
Definition: EigenBase.h:30
EIGEN_CONSTEXPR Index cols() const EIGEN_NOEXCEPT
Definition: EigenBase.h:63
Eigen::Index Index
The interface type of indices.
Definition: EigenBase.h:39
Derived & derived()
Definition: EigenBase.h:46
EIGEN_CONSTEXPR Index rows() const EIGEN_NOEXCEPT
Definition: EigenBase.h:60
EIGEN_CONSTEXPR Index size() const EIGEN_NOEXCEPT
Definition: EigenBase.h:67