Blaze  3.6
SMatDMatSchurExpr.h
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1 //=================================================================================================
33 //=================================================================================================
34 
35 #ifndef _BLAZE_MATH_EXPRESSIONS_SMATDMATSCHUREXPR_H_
36 #define _BLAZE_MATH_EXPRESSIONS_SMATDMATSCHUREXPR_H_
37 
38 
39 //*************************************************************************************************
40 // Includes
41 //*************************************************************************************************
42 
43 #include <iterator>
44 #include <utility>
45 #include <blaze/math/Aliases.h>
53 #include <blaze/math/Exception.h>
72 #include <blaze/util/Assert.h>
73 #include <blaze/util/DisableIf.h>
74 #include <blaze/util/EnableIf.h>
77 #include <blaze/util/MaybeUnused.h>
78 #include <blaze/util/mpl/If.h>
79 #include <blaze/util/mpl/Max.h>
80 #include <blaze/util/Types.h>
82 
83 
84 namespace blaze {
85 
86 //=================================================================================================
87 //
88 // CLASS SMATDMATSCHUREXPR
89 //
90 //=================================================================================================
91 
92 //*************************************************************************************************
99 template< typename MT1 // Type of the left-hand side sparse matrix
100  , typename MT2 > // Type of the right-hand side dense matrix
101 class SMatDMatSchurExpr
102  : public SchurExpr< SparseMatrix< SMatDMatSchurExpr<MT1,MT2>, false > >
103  , private Computation
104 {
105  private:
106  //**Type definitions****************************************************************************
113  //**********************************************************************************************
114 
115  //**Return type evaluation**********************************************************************
117 
122  static constexpr bool returnExpr = ( !IsTemporary_v<RN1> && !IsTemporary_v<RN2> );
123 
125  using ExprReturnType = decltype( std::declval<RN1>() * std::declval<RN2>() );
126  //**********************************************************************************************
127 
128  //**Evaluation strategy*************************************************************************
130 
136  static constexpr bool useAssign = ( RequiresEvaluation_v<MT1> || RequiresEvaluation_v<MT2> );
137 
139  template< typename MT >
141  static constexpr bool UseAssign_v = useAssign;
143  //**********************************************************************************************
144 
145  public:
146  //**Type definitions****************************************************************************
153 
156 
159 
161  using LeftOperand = If_t< IsExpression_v<MT1>, const MT1, const MT1& >;
162 
164  using RightOperand = If_t< IsExpression_v<MT2>, const MT2, const MT2& >;
165  //**********************************************************************************************
166 
167  //**ConstIterator class definition**************************************************************
171  {
172  public:
173  //**Type definitions*************************************************************************
176 
179 
180  using IteratorCategory = std::forward_iterator_tag;
181  using ValueType = Element;
185 
186  // STL iterator requirements
192  //*******************************************************************************************
193 
194  //**Constructor******************************************************************************
201  inline ConstIterator( LeftIterator left, RightOperand right, size_t row )
202  : left_ ( left ) // Iterator over the elements of the left-hand side sparse matrix expression
203  , right_( right ) // Right-hand side dense matrix expression
204  , row_ ( row ) // The row index of the iterator
205  {}
206  //*******************************************************************************************
207 
208  //**Prefix increment operator****************************************************************
214  ++left_;
215  return *this;
216  }
217  //*******************************************************************************************
218 
219  //**Element access operator******************************************************************
224  inline const Element operator*() const {
225  return Element( left_->value() * right_(row_,left_->index()), left_->index() );
226  }
227  //*******************************************************************************************
228 
229  //**Element access operator******************************************************************
234  inline const ConstIterator* operator->() const {
235  return this;
236  }
237  //*******************************************************************************************
238 
239  //**Value function***************************************************************************
244  inline ReturnType value() const {
245  return left_->value() * right_(row_,left_->index());
246  }
247  //*******************************************************************************************
248 
249  //**Index function***************************************************************************
254  inline size_t index() const {
255  return left_->index();
256  }
257  //*******************************************************************************************
258 
259  //**Equality operator************************************************************************
265  inline bool operator==( const ConstIterator& rhs ) const {
266  return left_ == rhs.left_;
267  }
268  //*******************************************************************************************
269 
270  //**Inequality operator**********************************************************************
276  inline bool operator!=( const ConstIterator& rhs ) const {
277  return left_ != rhs.left_;
278  }
279  //*******************************************************************************************
280 
281  //**Subtraction operator*********************************************************************
287  inline DifferenceType operator-( const ConstIterator& rhs ) const {
288  return left_ - rhs.left_;
289  }
290  //*******************************************************************************************
291 
292  private:
293  //**Member variables*************************************************************************
296  size_t row_;
297  //*******************************************************************************************
298  };
299  //**********************************************************************************************
300 
301  //**Compilation flags***************************************************************************
303  static constexpr bool smpAssignable = false;
304  //**********************************************************************************************
305 
306  //**Constructor*********************************************************************************
312  explicit inline SMatDMatSchurExpr( const MT1& lhs, const MT2& rhs ) noexcept
313  : lhs_( lhs ) // Left-hand side sparse matrix of the Schur product expression
314  , rhs_( rhs ) // Right-hand side dense matrix of the Schur product expression
315  {
316  BLAZE_INTERNAL_ASSERT( lhs.rows() == rhs.rows() , "Invalid number of rows" );
317  BLAZE_INTERNAL_ASSERT( lhs.columns() == rhs.columns(), "Invalid number of columns" );
318  }
319  //**********************************************************************************************
320 
321  //**Access operator*****************************************************************************
328  inline ReturnType operator()( size_t i, size_t j ) const {
329  BLAZE_INTERNAL_ASSERT( i < lhs_.rows() , "Invalid row access index" );
330  BLAZE_INTERNAL_ASSERT( j < lhs_.columns(), "Invalid column access index" );
331  return lhs_(i,j) * rhs_(i,j);
332  }
333  //**********************************************************************************************
334 
335  //**At function*********************************************************************************
343  inline ReturnType at( size_t i, size_t j ) const {
344  if( i >= lhs_.rows() ) {
345  BLAZE_THROW_OUT_OF_RANGE( "Invalid row access index" );
346  }
347  if( j >= lhs_.columns() ) {
348  BLAZE_THROW_OUT_OF_RANGE( "Invalid column access index" );
349  }
350  return (*this)(i,j);
351  }
352  //**********************************************************************************************
353 
354  //**Begin function******************************************************************************
360  inline ConstIterator begin( size_t i ) const {
361  return ConstIterator( lhs_.begin(i), rhs_, i );
362  }
363  //**********************************************************************************************
364 
365  //**End function********************************************************************************
371  inline ConstIterator end( size_t i ) const {
372  return ConstIterator( lhs_.end(i), rhs_, i );
373  }
374  //**********************************************************************************************
375 
376  //**Rows function*******************************************************************************
381  inline size_t rows() const noexcept {
382  return lhs_.rows();
383  }
384  //**********************************************************************************************
385 
386  //**Columns function****************************************************************************
391  inline size_t columns() const noexcept {
392  return lhs_.columns();
393  }
394  //**********************************************************************************************
395 
396  //**NonZeros function***************************************************************************
401  inline size_t nonZeros() const {
402  return lhs_.nonZeros();
403  }
404  //**********************************************************************************************
405 
406  //**NonZeros function***************************************************************************
412  inline size_t nonZeros( size_t i ) const {
413  return lhs_.nonZeros(i);
414  }
415  //**********************************************************************************************
416 
417  //**Find function*******************************************************************************
424  inline ConstIterator find( size_t i, size_t j ) const {
426  return ConstIterator( lhs_.find( i, j ), rhs_, i );
427  }
428  //**********************************************************************************************
429 
430  //**LowerBound function*************************************************************************
437  inline ConstIterator lowerBound( size_t i, size_t j ) const {
439  return ConstIterator( lhs_.lowerBound( i, j ), rhs_, i );
440  }
441  //**********************************************************************************************
442 
443  //**UpperBound function*************************************************************************
450  inline ConstIterator upperBound( size_t i, size_t j ) const {
452  return ConstIterator( lhs_.upperBound( i, j ), rhs_, i );
453  }
454  //**********************************************************************************************
455 
456  //**Left operand access*************************************************************************
461  inline LeftOperand leftOperand() const noexcept {
462  return lhs_;
463  }
464  //**********************************************************************************************
465 
466  //**Right operand access************************************************************************
471  inline RightOperand rightOperand() const noexcept {
472  return rhs_;
473  }
474  //**********************************************************************************************
475 
476  //**********************************************************************************************
482  template< typename T >
483  inline bool canAlias( const T* alias ) const noexcept {
484  return ( lhs_.canAlias( alias ) || rhs_.isAliased( alias ) );
485  }
486  //**********************************************************************************************
487 
488  //**********************************************************************************************
494  template< typename T >
495  inline bool isAliased( const T* alias ) const noexcept {
496  return ( lhs_.isAliased( alias ) || rhs_.isAliased( alias ) );
497  }
498  //**********************************************************************************************
499 
500  private:
501  //**Member variables****************************************************************************
504  //**********************************************************************************************
505 
506  //**Assignment to dense matrices****************************************************************
518  template< typename MT // Type of the target dense matrix
519  , bool SO2 > // Storage order of the target dense matrix
520  friend inline auto assign( DenseMatrix<MT,SO2>& lhs, const SMatDMatSchurExpr& rhs )
522  {
524 
525  BLAZE_INTERNAL_ASSERT( (~lhs).rows() == rhs.rows() , "Invalid number of rows" );
526  BLAZE_INTERNAL_ASSERT( (~lhs).columns() == rhs.columns(), "Invalid number of columns" );
527 
528  CT1 A( serial( rhs.lhs_ ) ); // Evaluation of the left-hand side sparse matrix operand
529  CT2 B( serial( rhs.rhs_ ) ); // Evaluation of the right-hand side dense matrix operand
530 
531  BLAZE_INTERNAL_ASSERT( A.rows() == rhs.lhs_.rows() , "Invalid number of rows" );
532  BLAZE_INTERNAL_ASSERT( A.columns() == rhs.lhs_.columns(), "Invalid number of columns" );
533  BLAZE_INTERNAL_ASSERT( B.rows() == rhs.rhs_.rows() , "Invalid number of rows" );
534  BLAZE_INTERNAL_ASSERT( B.columns() == rhs.rhs_.columns(), "Invalid number of columns" );
535  BLAZE_INTERNAL_ASSERT( A.rows() == (~lhs).rows() , "Invalid number of rows" );
536  BLAZE_INTERNAL_ASSERT( A.columns() == (~lhs).columns() , "Invalid number of columns" );
537 
538  for( size_t i=0UL; i<(~lhs).rows(); ++i ) {
539  for( auto element=A.begin(i); element!=A.end(i); ++element )
540  (~lhs)(i,element->index()) = element->value() * B(i,element->index());
541  }
542  }
544  //**********************************************************************************************
545 
546  //**Assignment to row-major sparse matrices*****************************************************
558  template< typename MT > // Type of the target sparse matrix
559  friend inline auto assign( SparseMatrix<MT,false>& lhs, const SMatDMatSchurExpr& rhs )
561  {
563 
564  BLAZE_INTERNAL_ASSERT( (~lhs).rows() == rhs.rows() , "Invalid number of rows" );
565  BLAZE_INTERNAL_ASSERT( (~lhs).columns() == rhs.columns(), "Invalid number of columns" );
566 
567  CT1 A( serial( rhs.lhs_ ) ); // Evaluation of the left-hand side sparse matrix operand
568  CT2 B( serial( rhs.rhs_ ) ); // Evaluation of the right-hand side dense matrix operand
569 
570  BLAZE_INTERNAL_ASSERT( A.rows() == rhs.lhs_.rows() , "Invalid number of rows" );
571  BLAZE_INTERNAL_ASSERT( A.columns() == rhs.lhs_.columns(), "Invalid number of columns" );
572  BLAZE_INTERNAL_ASSERT( B.rows() == rhs.rhs_.rows() , "Invalid number of rows" );
573  BLAZE_INTERNAL_ASSERT( B.columns() == rhs.rhs_.columns(), "Invalid number of columns" );
574  BLAZE_INTERNAL_ASSERT( A.rows() == (~lhs).rows() , "Invalid number of rows" );
575  BLAZE_INTERNAL_ASSERT( A.columns() == (~lhs).columns() , "Invalid number of columns" );
576 
577  // Final memory allocation (based on the evaluated operands)
578  (~lhs).reserve( A.nonZeros() );
579 
580  // Performing the Schur product
581  for( size_t i=0UL; i<(~lhs).rows(); ++i ) {
582  for( auto element=A.begin(i); element!=A.end(i); ++element )
583  (~lhs).append( i, element->index(), element->value() * B(i,element->index()) );
584  (~lhs).finalize( i );
585  }
586  }
588  //**********************************************************************************************
589 
590  //**Assignment to column-major sparse matrices**************************************************
603  template< typename MT > // Type of the target sparse matrix
604  friend inline auto assign( SparseMatrix<MT,true>& lhs, const SMatDMatSchurExpr& rhs )
605  -> EnableIf_t< UseAssign_v<MT> >
606  {
608 
609  BLAZE_INTERNAL_ASSERT( (~lhs).rows() == rhs.rows() , "Invalid number of rows" );
610  BLAZE_INTERNAL_ASSERT( (~lhs).columns() == rhs.columns(), "Invalid number of columns" );
611 
612  CT1 A( serial( rhs.lhs_ ) ); // Evaluation of the left-hand side sparse matrix operand
613  CT2 B( serial( rhs.rhs_ ) ); // Evaluation of the right-hand side dense matrix operand
614 
615  BLAZE_INTERNAL_ASSERT( A.rows() == rhs.lhs_.rows() , "Invalid number of rows" );
616  BLAZE_INTERNAL_ASSERT( A.columns() == rhs.lhs_.columns(), "Invalid number of columns" );
617  BLAZE_INTERNAL_ASSERT( B.rows() == rhs.rhs_.rows() , "Invalid number of rows" );
618  BLAZE_INTERNAL_ASSERT( B.columns() == rhs.rhs_.columns(), "Invalid number of columns" );
619  BLAZE_INTERNAL_ASSERT( A.rows() == (~lhs).rows() , "Invalid number of rows" );
620  BLAZE_INTERNAL_ASSERT( A.columns() == (~lhs).columns() , "Invalid number of columns" );
621 
622  const size_t m( rhs.rows() );
623  const size_t n( rhs.columns() );
624 
625  // Counting the number of elements per column
626  std::vector<size_t> nonzeros( n, 0UL );
627  for( size_t i=0UL; i<m; ++i ) {
628  const auto lend( A.end(i) );
629  for( auto l=A.begin(i); l!=lend; ++l ) {
630  ++nonzeros[l->index()];
631  }
632  }
633 
634  // Resizing the left-hand side sparse matrix
635  for( size_t j=0UL; j<n; ++j ) {
636  (~lhs).reserve( j, nonzeros[j] );
637  }
638 
639  // Performing the Schur product
640  for( size_t i=0UL; i<m; ++i ) {
641  for( auto element=A.begin(i); element!=A.end(i); ++element )
642  (~lhs).append( i, element->index(), element->value() * B(i,element->index()) );
643  }
644  }
646  //**********************************************************************************************
647 
648  //**Addition assignment to dense matrices*******************************************************
660  template< typename MT // Type of the target dense matrix
661  , bool SO2 > // Storage order of the target dense matrix
662  friend inline auto addAssign( DenseMatrix<MT,SO2>& lhs, const SMatDMatSchurExpr& rhs )
663  -> EnableIf_t< UseAssign_v<MT> >
664  {
666 
667  BLAZE_INTERNAL_ASSERT( (~lhs).rows() == rhs.rows() , "Invalid number of rows" );
668  BLAZE_INTERNAL_ASSERT( (~lhs).columns() == rhs.columns(), "Invalid number of columns" );
669 
670  CT1 A( serial( rhs.lhs_ ) ); // Evaluation of the left-hand side sparse matrix operand
671  CT2 B( serial( rhs.rhs_ ) ); // Evaluation of the right-hand side dense matrix operand
672 
673  BLAZE_INTERNAL_ASSERT( A.rows() == rhs.lhs_.rows() , "Invalid number of rows" );
674  BLAZE_INTERNAL_ASSERT( A.columns() == rhs.lhs_.columns(), "Invalid number of columns" );
675  BLAZE_INTERNAL_ASSERT( B.rows() == rhs.rhs_.rows() , "Invalid number of rows" );
676  BLAZE_INTERNAL_ASSERT( B.columns() == rhs.rhs_.columns(), "Invalid number of columns" );
677  BLAZE_INTERNAL_ASSERT( A.rows() == (~lhs).rows() , "Invalid number of rows" );
678  BLAZE_INTERNAL_ASSERT( A.columns() == (~lhs).columns() , "Invalid number of columns" );
679 
680  for( size_t i=0UL; i<(~lhs).rows(); ++i ) {
681  for( auto element=A.begin(i); element!=A.end(i); ++element )
682  (~lhs)(i,element->index()) += element->value() * B(i,element->index());
683  }
684  }
686  //**********************************************************************************************
687 
688  //**Addition assignment to sparse matrices******************************************************
689  // No special implementation for the addition assignment to sparse matrices.
690  //**********************************************************************************************
691 
692  //**Subtraction assignment to dense matrices****************************************************
704  template< typename MT // Type of the target dense matrix
705  , bool SO2 > // Storage order of the target dense matrix
706  friend inline auto subAssign( DenseMatrix<MT,SO2>& lhs, const SMatDMatSchurExpr& rhs )
707  -> EnableIf_t< UseAssign_v<MT> >
708  {
710 
711  BLAZE_INTERNAL_ASSERT( (~lhs).rows() == rhs.rows() , "Invalid number of rows" );
712  BLAZE_INTERNAL_ASSERT( (~lhs).columns() == rhs.columns(), "Invalid number of columns" );
713 
714  CT1 A( serial( rhs.lhs_ ) ); // Evaluation of the left-hand side sparse matrix operand
715  CT2 B( serial( rhs.rhs_ ) ); // Evaluation of the right-hand side dense matrix operand
716 
717  BLAZE_INTERNAL_ASSERT( A.rows() == rhs.lhs_.rows() , "Invalid number of rows" );
718  BLAZE_INTERNAL_ASSERT( A.columns() == rhs.lhs_.columns(), "Invalid number of columns" );
719  BLAZE_INTERNAL_ASSERT( B.rows() == rhs.rhs_.rows() , "Invalid number of rows" );
720  BLAZE_INTERNAL_ASSERT( B.columns() == rhs.rhs_.columns(), "Invalid number of columns" );
721  BLAZE_INTERNAL_ASSERT( A.rows() == (~lhs).rows() , "Invalid number of rows" );
722  BLAZE_INTERNAL_ASSERT( A.columns() == (~lhs).columns() , "Invalid number of columns" );
723 
724  for( size_t i=0UL; i<(~lhs).rows(); ++i ) {
725  for( auto element=A.begin(i); element!=A.end(i); ++element )
726  (~lhs)(i,element->index()) -= element->value() * B(i,element->index());
727  }
728  }
730  //**********************************************************************************************
731 
732  //**Subtraction assignment to sparse matrices***************************************************
733  // No special implementation for the subtraction assignment to sparse matrices.
734  //**********************************************************************************************
735 
736  //**Schur product assignment to dense matrices**************************************************
749  template< typename MT // Type of the target dense matrix
750  , bool SO2 > // Storage order of the target dense matrix
751  friend inline void schurAssign( DenseMatrix<MT,SO2>& lhs, const SMatDMatSchurExpr& rhs )
752  {
754 
755  BLAZE_INTERNAL_ASSERT( (~lhs).rows() == rhs.rows() , "Invalid number of rows" );
756  BLAZE_INTERNAL_ASSERT( (~lhs).columns() == rhs.columns(), "Invalid number of columns" );
757 
758  CT1 A( serial( rhs.lhs_ ) ); // Evaluation of the left-hand side sparse matrix operand
759  CT2 B( serial( rhs.rhs_ ) ); // Evaluation of the right-hand side dense matrix operand
760 
761  BLAZE_INTERNAL_ASSERT( A.rows() == rhs.lhs_.rows() , "Invalid number of rows" );
762  BLAZE_INTERNAL_ASSERT( A.columns() == rhs.lhs_.columns(), "Invalid number of columns" );
763  BLAZE_INTERNAL_ASSERT( B.rows() == rhs.rhs_.rows() , "Invalid number of rows" );
764  BLAZE_INTERNAL_ASSERT( B.columns() == rhs.rhs_.columns(), "Invalid number of columns" );
765  BLAZE_INTERNAL_ASSERT( A.rows() == (~lhs).rows() , "Invalid number of rows" );
766  BLAZE_INTERNAL_ASSERT( A.columns() == (~lhs).columns() , "Invalid number of columns" );
767 
768  for( size_t i=0UL; i<(~lhs).rows(); ++i )
769  {
770  const auto end( A.end(i) );
771  auto begin( A.begin(i) );
772  size_t j( 0UL );
773 
774  for( ; begin!=end; ++begin ) {
775  const size_t index( begin->index() );
776  for( ; j<index; ++j )
777  reset( (~lhs)(i,j) );
778  (~lhs)(i,index) *= begin->value() * B(i,index);
779  ++j;
780  }
781 
782  for( ; j<(~lhs).columns(); ++j )
783  reset( (~lhs)(i,j) );
784  }
785  }
787  //**********************************************************************************************
788 
789  //**Schur product assignment to sparse matrices*************************************************
790  // No special implementation for the Schur product assignment to sparse matrices.
791  //**********************************************************************************************
792 
793  //**Multiplication assignment to dense matrices*************************************************
794  // No special implementation for the multiplication assignment to dense matrices.
795  //**********************************************************************************************
796 
797  //**Multiplication assignment to sparse matrices************************************************
798  // No special implementation for the multiplication assignment to sparse matrices.
799  //**********************************************************************************************
800 
801  //**SMP assignment to dense matrices************************************************************
802  // No special implementation for the SMP assignment to dense matrices.
803  //**********************************************************************************************
804 
805  //**SMP assignment to sparse matrices***********************************************************
806  // No special implementation for the SMP assignment to sparse matrices.
807  //**********************************************************************************************
808 
809  //**SMP addition assignment to dense matrices***************************************************
810  // No special implementation for the SMP addition assignment to dense matrices.
811  //**********************************************************************************************
812 
813  //**SMP addition assignment to sparse matrices**************************************************
814  // No special implementation for the SMP addition assignment to sparse matrices.
815  //**********************************************************************************************
816 
817  //**SMP subtraction assignment to dense matrices************************************************
818  // No special implementation for the SMP subtraction assignment to dense matrices.
819  //**********************************************************************************************
820 
821  //**SMP subtraction assignment to sparse matrices***********************************************
822  // No special implementation for the SMP subtraction assignment to sparse matrices.
823  //**********************************************************************************************
824 
825  //**SMP Schur product assignment to dense matrices**********************************************
838  template< typename MT // Type of the target dense matrix
839  , bool SO > // Storage order of the target dense matrix
840  friend inline void smpSchurAssign( DenseMatrix<MT,SO>& lhs, const SMatDMatSchurExpr& rhs )
841  {
843 
844  BLAZE_INTERNAL_ASSERT( (~lhs).rows() == rhs.rows() , "Invalid number of rows" );
845  BLAZE_INTERNAL_ASSERT( (~lhs).columns() == rhs.columns(), "Invalid number of columns" );
846 
847  smpSchurAssign( ~lhs, rhs.lhs_ );
848  smpSchurAssign( ~lhs, rhs.rhs_ );
849  }
851  //**********************************************************************************************
852 
853  //**SMP Schur product assignment to sparse matrices*********************************************
854  // No special implementation for the SMP Schur product assignment to sparse matrices.
855  //**********************************************************************************************
856 
857  //**SMP multiplication assignment to dense matrices*********************************************
858  // No special implementation for the SMP multiplication assignment to dense matrices.
859  //**********************************************************************************************
860 
861  //**SMP multiplication assignment to sparse matrices********************************************
862  // No special implementation for the SMP multiplication assignment to sparse matrices.
863  //**********************************************************************************************
864 
865  //**Compile time checks*************************************************************************
873  //**********************************************************************************************
874 };
875 //*************************************************************************************************
876 
877 
878 
879 
880 //=================================================================================================
881 //
882 // GLOBAL BINARY ARITHMETIC OPERATORS
883 //
884 //=================================================================================================
885 
886 //*************************************************************************************************
899 template< typename MT1 // Type of the left-hand side sparse matrix
900  , typename MT2 // Type of the right-hand side dense matrix
901  , DisableIf_t< ( IsUniLower_v<MT1> && IsUniUpper_v<MT2> ) ||
902  ( IsUniUpper_v<MT1> && IsUniLower_v<MT2> ) ||
903  ( IsStrictlyLower_v<MT1> && IsUpper_v<MT2> ) ||
904  ( IsStrictlyUpper_v<MT1> && IsLower_v<MT2> ) ||
905  ( IsLower_v<MT1> && IsStrictlyUpper_v<MT2> ) ||
906  ( IsUpper_v<MT1> && IsStrictlyLower_v<MT2> ) ||
907  IsZero_v<MT1> >* = nullptr >
908 inline const SMatDMatSchurExpr<MT1,MT2>
909  smatdmatschur( const SparseMatrix<MT1,false>& lhs, const DenseMatrix<MT2,false>& rhs )
910 {
912 
913  BLAZE_INTERNAL_ASSERT( (~lhs).rows() == (~rhs).rows() , "Invalid number of rows" );
914  BLAZE_INTERNAL_ASSERT( (~lhs).columns() == (~rhs).columns(), "Invalid number of columns" );
915 
916  return SMatDMatSchurExpr<MT1,MT2>( ~lhs, ~rhs );
917 }
919 //*************************************************************************************************
920 
921 
922 //*************************************************************************************************
936 template< typename MT1 // Type of the left-hand side sparse matrix
937  , typename MT2 // Type of the right-hand side dense matrix
938  , EnableIf_t< ( IsUniLower_v<MT1> && IsUniUpper_v<MT2> ) ||
939  ( IsUniUpper_v<MT1> && IsUniLower_v<MT2> ) >* = nullptr >
940 inline decltype(auto)
941  smatdmatschur( const SparseMatrix<MT1,false>& lhs, const DenseMatrix<MT2,false>& rhs )
942 {
944 
945  MAYBE_UNUSED( rhs );
946 
947  BLAZE_INTERNAL_ASSERT( (~lhs).rows() == (~rhs).rows() , "Invalid number of rows" );
948  BLAZE_INTERNAL_ASSERT( (~lhs).columns() == (~rhs).columns(), "Invalid number of columns" );
949 
950  using ReturnType = const SchurTrait_t< ResultType_t<MT1>, ResultType_t<MT2> >;
951 
954 
955  return ReturnType( (~lhs).rows() );
956 }
958 //*************************************************************************************************
959 
960 
961 //*************************************************************************************************
975 template< typename MT1 // Type of the left-hand side sparse matrix
976  , typename MT2 // Type of the right-hand side dense matrix
977  , EnableIf_t< ( IsStrictlyLower_v<MT1> && IsUpper_v<MT2> ) ||
978  ( IsStrictlyUpper_v<MT1> && IsLower_v<MT2> ) ||
979  ( IsLower_v<MT1> && IsStrictlyUpper_v<MT2> ) ||
980  ( IsUpper_v<MT1> && IsStrictlyLower_v<MT2> ) ||
981  IsZero_v<MT1> >* = nullptr >
982 inline decltype(auto)
983  smatdmatschur( const SparseMatrix<MT1,false>& lhs, const DenseMatrix<MT2,false>& rhs )
984 {
986 
987  MAYBE_UNUSED( rhs );
988 
989  BLAZE_INTERNAL_ASSERT( (~lhs).rows() == (~rhs).rows() , "Invalid number of rows" );
990  BLAZE_INTERNAL_ASSERT( (~lhs).columns() == (~rhs).columns(), "Invalid number of columns" );
991 
992  using ReturnType = const SchurTrait_t< ResultType_t<MT1>, ResultType_t<MT2> >;
993 
996 
997  return ReturnType( (~lhs).rows(), (~lhs).columns() );
998 }
1000 //*************************************************************************************************
1001 
1002 
1003 //*************************************************************************************************
1032 template< typename MT1 // Type of the left-hand side sparse matrix
1033  , typename MT2 > // Type of the right-hand side dense matrix
1034 inline decltype(auto)
1035  operator%( const SparseMatrix<MT1,false>& lhs, const DenseMatrix<MT2,false>& rhs )
1036 {
1038 
1039  if( (~lhs).rows() != (~rhs).rows() || (~lhs).columns() != (~rhs).columns() ) {
1040  BLAZE_THROW_INVALID_ARGUMENT( "Matrix sizes do not match" );
1041  }
1042 
1043  return smatdmatschur( ~lhs, ~rhs );
1044 }
1045 //*************************************************************************************************
1046 
1047 
1048 //*************************************************************************************************
1061 template< typename MT1 // Type of the left-hand side sparse matrix
1062  , typename MT2 // Type of the right-hand side dense matrix
1063  , DisableIf_t< ( IsUniLower_v<MT1> && IsUniUpper_v<MT2> ) ||
1064  ( IsUniUpper_v<MT1> && IsUniLower_v<MT2> ) ||
1065  ( IsStrictlyLower_v<MT1> && IsUpper_v<MT2> ) ||
1066  ( IsStrictlyUpper_v<MT1> && IsLower_v<MT2> ) ||
1067  ( IsLower_v<MT1> && IsStrictlyUpper_v<MT2> ) ||
1068  ( IsUpper_v<MT1> && IsStrictlyLower_v<MT2> ) ||
1069  IsZero_v<MT1> >* = nullptr >
1070 inline const SMatDMatSchurExpr<MT1,MT2>
1071  smattdmatschur( const SparseMatrix<MT1,false>& lhs, const DenseMatrix<MT2,true>& rhs )
1072 {
1074 
1075  BLAZE_INTERNAL_ASSERT( (~lhs).rows() == (~rhs).rows() , "Invalid number of rows" );
1076  BLAZE_INTERNAL_ASSERT( (~lhs).columns() == (~rhs).columns(), "Invalid number of columns" );
1077 
1078  return SMatDMatSchurExpr<MT1,MT2>( ~lhs, ~rhs );
1079 }
1081 //*************************************************************************************************
1082 
1083 
1084 //*************************************************************************************************
1098 template< typename MT1 // Type of the left-hand side sparse matrix
1099  , typename MT2 // Type of the right-hand side dense matrix
1100  , EnableIf_t< ( IsUniLower_v<MT1> && IsUniUpper_v<MT2> ) ||
1101  ( IsUniUpper_v<MT1> && IsUniLower_v<MT2> ) >* = nullptr >
1102 inline decltype(auto)
1103  smattdmatschur( const SparseMatrix<MT1,false>& lhs, const DenseMatrix<MT2,true>& rhs )
1104 {
1106 
1107  MAYBE_UNUSED( rhs );
1108 
1109  BLAZE_INTERNAL_ASSERT( (~lhs).rows() == (~rhs).rows() , "Invalid number of rows" );
1110  BLAZE_INTERNAL_ASSERT( (~lhs).columns() == (~rhs).columns(), "Invalid number of columns" );
1111 
1112  using ReturnType = const SchurTrait_t< ResultType_t<MT1>, ResultType_t<MT2> >;
1113 
1116 
1117  return ReturnType( (~lhs).rows() );
1118 }
1120 //*************************************************************************************************
1121 
1122 
1123 //*************************************************************************************************
1137 template< typename MT1 // Type of the left-hand side sparse matrix
1138  , typename MT2 // Type of the right-hand side dense matrix
1139  , EnableIf_t< ( IsStrictlyLower_v<MT1> && IsUpper_v<MT2> ) ||
1140  ( IsStrictlyUpper_v<MT1> && IsLower_v<MT2> ) ||
1141  ( IsLower_v<MT1> && IsStrictlyUpper_v<MT2> ) ||
1142  ( IsUpper_v<MT1> && IsStrictlyLower_v<MT2> ) ||
1143  IsZero_v<MT1> >* = nullptr >
1144 inline decltype(auto)
1145  smattdmatschur( const SparseMatrix<MT1,false>& lhs, const DenseMatrix<MT2,true>& rhs )
1146 {
1148 
1149  MAYBE_UNUSED( rhs );
1150 
1151  BLAZE_INTERNAL_ASSERT( (~lhs).rows() == (~rhs).rows() , "Invalid number of rows" );
1152  BLAZE_INTERNAL_ASSERT( (~lhs).columns() == (~rhs).columns(), "Invalid number of columns" );
1153 
1154  using ReturnType = const SchurTrait_t< ResultType_t<MT1>, ResultType_t<MT2> >;
1155 
1157  BLAZE_CONSTRAINT_MUST_BE_ZERO_TYPE( ReturnType );
1158 
1159  return ReturnType( (~lhs).rows(), (~lhs).columns() );
1160 }
1162 //*************************************************************************************************
1163 
1164 
1165 //*************************************************************************************************
1195 template< typename MT1 // Type of the left-hand side sparse matrix
1196  , typename MT2 > // Type of the right-hand side dense matrix
1197 inline decltype(auto)
1198  operator%( const SparseMatrix<MT1,false>& lhs, const DenseMatrix<MT2,true>& rhs )
1199 {
1201 
1202  if( (~lhs).rows() != (~rhs).rows() || (~lhs).columns() != (~rhs).columns() ) {
1203  BLAZE_THROW_INVALID_ARGUMENT( "Matrix sizes do not match" );
1204  }
1205 
1206  return smattdmatschur( ~lhs, ~rhs );
1207 }
1208 //*************************************************************************************************
1209 
1210 
1211 
1212 
1213 //=================================================================================================
1214 //
1215 // SIZE SPECIALIZATIONS
1216 //
1217 //=================================================================================================
1218 
1219 //*************************************************************************************************
1221 template< typename MT1, typename MT2 >
1222 struct Size< SMatDMatSchurExpr<MT1,MT2>, 0UL >
1223  : public Max_t< Size<MT1,0UL>, Size<MT2,0UL> >
1224 {};
1225 
1226 template< typename MT1, typename MT2 >
1227 struct Size< SMatDMatSchurExpr<MT1,MT2>, 1UL >
1228  : public Max_t< Size<MT1,1UL>, Size<MT2,1UL> >
1229 {};
1231 //*************************************************************************************************
1232 
1233 } // namespace blaze
1234 
1235 #endif
LeftOperand leftOperand() const noexcept
Returns the left-hand side sparse matrix operand.
Definition: SMatDMatSchurExpr.h:461
Constraint on the data type.
ReturnType_t< MT2 > RN2
Return type of the right-hand side dense matrix expression.
Definition: SMatDMatSchurExpr.h:110
#define BLAZE_THROW_INVALID_ARGUMENT(MESSAGE)
Macro for the emission of a std::invalid_argument exception.This macro encapsulates the default way o...
Definition: Exception.h:235
Pointer difference type of the Blaze library.
RightOperand rightOperand() const noexcept
Returns the right-hand side dense matrix operand.
Definition: SMatDMatSchurExpr.h:471
Header file for auxiliary alias declarations.
size_t rows() const noexcept
Returns the current number of rows of the matrix.
Definition: SMatDMatSchurExpr.h:381
#define BLAZE_CONSTRAINT_MUST_BE_IDENTITY_MATRIX_TYPE(T)
Constraint on the data type.In case the given data type T is not an identity matrix type,...
Definition: Identity.h:60
Header file for the Schur product trait.
Header file for the IsUniUpper type trait.
ConstIterator find(size_t i, size_t j) const
Searches for a specific matrix element.
Definition: SMatDMatSchurExpr.h:424
Header file for basic type definitions.
Header file for all forward declarations for sparse vectors and matrices.
typename If< Condition, T1, T2 >::Type If_t
Auxiliary alias template for the If class template.The If_t alias template provides a convenient shor...
Definition: If.h:109
const If_t< returnExpr, ExprReturnType, ElementType > ReturnType
Return type for expression template evaluations.
Definition: SMatDMatSchurExpr.h:155
bool operator!=(const ConstIterator &rhs) const
Inequality comparison between two ConstIterator objects.
Definition: SMatDMatSchurExpr.h:276
typename T::ResultType ResultType_t
Alias declaration for nested ResultType type definitions.The ResultType_t alias declaration provides ...
Definition: Aliases.h:390
Header file for the serial shim.
ReturnType at(size_t i, size_t j) const
Checked access to the matrix elements.
Definition: SMatDMatSchurExpr.h:343
size_t columns() const noexcept
Returns the current number of columns of the matrix.
Definition: SMatDMatSchurExpr.h:391
#define BLAZE_CONSTRAINT_MUST_BE_DENSE_MATRIX_TYPE(T)
Constraint on the data type.In case the given data type T is not a dense, N-dimensional matrix type,...
Definition: DenseMatrix.h:61
DifferenceType operator-(const ConstIterator &rhs) const
Calculating the number of elements between two expression iterators.
Definition: SMatDMatSchurExpr.h:287
bool operator==(const ConstIterator &rhs) const
Equality comparison between two ConstIterator objects.
Definition: SMatDMatSchurExpr.h:265
Element ValueType
Type of the underlying pointers.
Definition: SMatDMatSchurExpr.h:181
decltype(std::declval< RN1 >() *std::declval< RN2 >()) ExprReturnType
Expression return type for the subscript operator.
Definition: SMatDMatSchurExpr.h:125
SMatDMatSchurExpr(const MT1 &lhs, const MT2 &rhs) noexcept
Constructor for the SMatDMatSchurExpr class.
Definition: SMatDMatSchurExpr.h:312
void reset(const DiagonalProxy< MT > &proxy)
Resetting the represented element to the default initial values.
Definition: DiagonalProxy.h:595
static constexpr bool useAssign
Compilation switch for the evaluation strategy of the Schur product expression.
Definition: SMatDMatSchurExpr.h:136
Constraint on the data type.
OppositeType_t< ResultType > OppositeType
Result type with opposite storage order for expression template evaluations.
Definition: SMatDMatSchurExpr.h:150
Header file for the MAYBE_UNUSED function template.
Header file for the Computation base class.
Constraints on the storage order of matrix types.
typename T::ReturnType ReturnType_t
Alias declaration for nested ReturnType type definitions.The ReturnType_t alias declaration provides ...
Definition: Aliases.h:410
Header file for the IsUniLower type trait.
size_t index() const
Access to the current index of the sparse element.
Definition: SMatDMatSchurExpr.h:254
#define BLAZE_CONSTRAINT_MUST_FORM_VALID_SCHUREXPR(T1, T2)
Constraint on the data type.In case the given data types T1 and T2 do not form a valid Schur product,...
Definition: SchurExpr.h:103
constexpr size_t columns(const Matrix< MT, SO > &matrix) noexcept
Returns the current number of columns of the matrix.
Definition: Matrix.h:514
Base class for dense matrices.The DenseMatrix class is a base class for all dense matrix classes....
Definition: DenseMatrix.h:81
Base class for sparse matrices.The SparseMatrix class is a base class for all sparse matrix classes....
Definition: Forward.h:145
typename T::ElementType ElementType_t
Alias declaration for nested ElementType type definitions.The ElementType_t alias declaration provide...
Definition: Aliases.h:170
const Element operator *() const
Direct access to the sparse matrix element at the current iterator position.
Definition: SMatDMatSchurExpr.h:224
RightOperand rhs_
Right-hand side dense matrix of the Schur product expression.
Definition: SMatDMatSchurExpr.h:503
static constexpr bool smpAssignable
Compilation switch for the expression template assignment strategy.
Definition: SMatDMatSchurExpr.h:303
Header file for the SparseMatrix base class.
LeftIterator left_
Iterator over the elements of the left-hand side sparse matrix expression.
Definition: SMatDMatSchurExpr.h:294
Constraint on the data type.
Constraint on the data type.
typename EnableIf< Condition, T >::Type EnableIf_t
Auxiliary type for the EnableIf class template.The EnableIf_t alias declaration provides a convenient...
Definition: EnableIf.h:138
Header file for the ValueIndexPair class.
Header file for the DisableIf class template.
Header file for the IsTemporary type trait class.
size_t row_
The row index of the iterator.
Definition: SMatDMatSchurExpr.h:296
Header file for the IsStrictlyUpper type trait.
ReturnType value() const
Access to the current value of the sparse element.
Definition: SMatDMatSchurExpr.h:244
Namespace of the Blaze C++ math library.
Definition: Blaze.h:58
std::forward_iterator_tag IteratorCategory
The iterator category.
Definition: SMatDMatSchurExpr.h:180
Header file for the If class template.
TransposeType_t< ResultType > TransposeType
Transpose type for expression template evaluations.
Definition: SMatDMatSchurExpr.h:151
#define BLAZE_CONSTRAINT_MUST_BE_ZERO_TYPE(T)
Constraint on the data type.In case the given data type T is not a zero vector or matrix type,...
Definition: Zero.h:61
#define BLAZE_THROW_OUT_OF_RANGE(MESSAGE)
Macro for the emission of a std::out_of_range exception.This macro encapsulates the default way of Bl...
Definition: Exception.h:331
ResultType_t< MT2 > RT2
Result type of the right-hand side dense matrix expression.
Definition: SMatDMatSchurExpr.h:108
Header file for the IsLower type trait.
ConstIterator upperBound(size_t i, size_t j) const
Returns an iterator to the first index greater then the given index.
Definition: SMatDMatSchurExpr.h:450
constexpr void MAYBE_UNUSED(const Args &...)
Suppression of unused parameter warnings.
Definition: MaybeUnused.h:81
LeftOperand lhs_
Left-hand side sparse matrix of the Schur product expression.
Definition: SMatDMatSchurExpr.h:502
bool isAliased(const T *alias) const noexcept
Returns whether the expression is aliased with the given address alias.
Definition: SMatDMatSchurExpr.h:495
CompositeType_t< MT2 > CT2
Composite type of the right-hand side dense matrix expression.
Definition: SMatDMatSchurExpr.h:112
ValueIndexPair< ElementType > Element
Element type of the sparse matrix expression.
Definition: SMatDMatSchurExpr.h:175
Header file for the exception macros of the math module.
RightOperand right_
Right-hand side dense matrix expression.
Definition: SMatDMatSchurExpr.h:295
Constraint on the data type.
Header file for the EnableIf class template.
Header file for the IsStrictlyLower type trait.
ResultType_t< MT1 > RT1
Result type of the left-hand side sparse matrix expression.
Definition: SMatDMatSchurExpr.h:107
typename T::OppositeType OppositeType_t
Alias declaration for nested OppositeType type definitions.The OppositeType_t alias declaration provi...
Definition: Aliases.h:270
static constexpr bool returnExpr
Compilation switch for the selection of the subscript operator return type.
Definition: SMatDMatSchurExpr.h:122
const ConstIterator * operator->() const
Direct access to the sparse matrix element at the current iterator position.
Definition: SMatDMatSchurExpr.h:234
#define BLAZE_CONSTRAINT_MUST_BE_ROW_MAJOR_MATRIX_TYPE(T)
Constraint on the data type.In case the given data type T is not a row-major dense or sparse matrix t...
Definition: RowMajorMatrix.h:61
size_t nonZeros() const
Returns the number of non-zero elements in the sparse matrix.
Definition: SMatDMatSchurExpr.h:401
typename T::TransposeType TransposeType_t
Alias declaration for nested TransposeType type definitions.The TransposeType_t alias declaration pro...
Definition: Aliases.h:470
Header file for run time assertion macros.
typename T::CompositeType CompositeType_t
Alias declaration for nested CompositeType type definitions.The CompositeType_t alias declaration pro...
Definition: Aliases.h:90
ConstIterator(LeftIterator left, RightOperand right, size_t row)
Constructor for the ConstIterator class.
Definition: SMatDMatSchurExpr.h:201
Header file for the SchurExpr base class.
decltype(auto) row(Matrix< MT, SO > &, RRAs...)
Creating a view on a specific row of the given matrix.
Definition: Row.h:133
typename SchurTrait< T1, T2 >::Type SchurTrait_t
Auxiliary alias declaration for the SchurTrait class template.The SchurTrait_t alias declaration prov...
Definition: SchurTrait.h:163
ConstIterator end(size_t i) const
Returns an iterator just past the last non-zero element of row i.
Definition: SMatDMatSchurExpr.h:371
Header file for the IsZero type trait.
#define BLAZE_FUNCTION_TRACE
Function trace macro.This macro can be used to reliably trace function calls. In case function tracin...
Definition: FunctionTrace.h:94
Header file for all forward declarations for expression class templates.
If_t< useAssign, const ResultType, const SMatDMatSchurExpr & > CompositeType
Data type for composite expression templates.
Definition: SMatDMatSchurExpr.h:158
ConstIterator & operator++()
Pre-increment operator.
Definition: SMatDMatSchurExpr.h:213
ptrdiff_t DifferenceType
Difference between two iterators.
Definition: SMatDMatSchurExpr.h:184
decltype(auto) serial(const DenseMatrix< MT, SO > &dm)
Forces the serial evaluation of the given dense matrix expression dm.
Definition: DMatSerialExpr.h:808
ReturnType_t< MT1 > RN1
Return type of the left-hand side sparse matrix expression.
Definition: SMatDMatSchurExpr.h:109
ValueType & ReferenceType
Reference return type.
Definition: SMatDMatSchurExpr.h:183
#define BLAZE_CONSTRAINT_MUST_NOT_REQUIRE_EVALUATION(T)
Constraint on the data type.In case the given data type T requires an intermediate evaluation within ...
Definition: RequiresEvaluation.h:81
auto smpSchurAssign(Matrix< MT1, SO1 > &lhs, const Matrix< MT2, SO2 > &rhs) -> EnableIf_t< IsDenseMatrix_v< MT1 > >
Default implementation of the SMP Schur product assignment of a matrix to dense matrix.
Definition: DenseMatrix.h:194
constexpr size_t rows(const Matrix< MT, SO > &matrix) noexcept
Returns the current number of rows of the matrix.
Definition: Matrix.h:498
CompositeType_t< MT1 > CT1
Composite type of the left-hand side sparse matrix expression.
Definition: SMatDMatSchurExpr.h:111
Header file for the RemoveReference type trait.
typename T::ConstIterator ConstIterator_t
Alias declaration for nested ConstIterator type definitions.The ConstIterator_t alias declaration pro...
Definition: Aliases.h:110
Iterator over the elements of the sparse matrix/dense matrix Schur product expression.
Definition: SMatDMatSchurExpr.h:170
Index-value-pair for sparse vectors and matrices.The ValueIndexPair class represents a single index-v...
Definition: ValueIndexPair.h:73
SchurTrait_t< RT1, RT2 > ResultType
Result type for expression template evaluations.
Definition: SMatDMatSchurExpr.h:149
Header file for the IntegralConstant class template.
size_t nonZeros(size_t i) const
Returns the number of non-zero elements in the specified row.
Definition: SMatDMatSchurExpr.h:412
bool canAlias(const T *alias) const noexcept
Returns whether the expression can alias with the given address alias.
Definition: SMatDMatSchurExpr.h:483
If_t< IsExpression_v< MT2 >, const MT2, const MT2 & > RightOperand
Composite type of the right-hand side dense matrix expression.
Definition: SMatDMatSchurExpr.h:164
Header file for the Max_t alias template.
If_t< IsExpression_v< MT1 >, const MT1, const MT1 & > LeftOperand
Composite type of the left-hand side sparse matrix expression.
Definition: SMatDMatSchurExpr.h:161
ValueType * PointerType
Pointer return type.
Definition: SMatDMatSchurExpr.h:182
Header file for the IsUpper type trait.
typename DisableIf< Condition, T >::Type DisableIf_t
Auxiliary type for the DisableIf class template.The DisableIf_t alias declaration provides a convenie...
Definition: DisableIf.h:138
ConstIterator_t< RemoveReference_t< LeftOperand > > LeftIterator
Iterator type of the sparse matrix expression.
Definition: SMatDMatSchurExpr.h:178
ElementType_t< ResultType > ElementType
Resulting element type.
Definition: SMatDMatSchurExpr.h:152
Header file for the Size type trait.
#define BLAZE_CONSTRAINT_MUST_NOT_BE_ZERO_TYPE(T)
Constraint on the data type.In case the given data type T is a zero vector or matrix type,...
Definition: Zero.h:81
IteratorCategory iterator_category
The iterator category.
Definition: SMatDMatSchurExpr.h:187
Expression object for sparse matrix-dense matrix Schur product.The SMatDMatSchurExpr class represents...
Definition: Forward.h:122
ConstIterator begin(size_t i) const
Returns an iterator to the first non-zero element of row i.
Definition: SMatDMatSchurExpr.h:360
#define BLAZE_INTERNAL_ASSERT(expr, msg)
Run time assertion macro for internal checks.In case of an invalid run time expression,...
Definition: Assert.h:101
ReturnType operator()(size_t i, size_t j) const
2D-access to the matrix elements.
Definition: SMatDMatSchurExpr.h:328
#define BLAZE_CONSTRAINT_MUST_BE_SPARSE_MATRIX_TYPE(T)
Constraint on the data type.In case the given data type T is not a sparse, N-dimensional matrix type,...
Definition: SparseMatrix.h:61
Header file for the IsExpression type trait class.
Header file for the function trace functionality.
If_t< Less_t< T1, T2 >::value, T2, T1 > Max_t
Compile time value evaluation.The Max_t alias template selects the larger of the two given template a...
Definition: Max.h:73
Constraint on the data type.
ConstIterator lowerBound(size_t i, size_t j) const
Returns an iterator to the first index not less then the given index.
Definition: SMatDMatSchurExpr.h:437