1 /* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * */
2 /* */
3 /* This file is part of the class library */
4 /* SoPlex --- the Sequential object-oriented simPlex. */
5 /* */
6 /* Copyright (c) 1996-2023 Zuse Institute Berlin (ZIB) */
7 /* */
8 /* Licensed under the Apache License, Version 2.0 (the "License"); */
9 /* you may not use this file except in compliance with the License. */
10 /* You may obtain a copy of the License at */
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13 /* */
14 /* Unless required by applicable law or agreed to in writing, software */
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18 /* limitations under the License. */
19 /* */
20 /* You should have received a copy of the Apache-2.0 license */
21 /* along with SoPlex; see the file LICENSE. If not email to soplex@zib.de. */
22 /* */
23 /* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * */
24
25 /**@file ssvectorbase.h
26 * @brief Semi sparse vector.
27 */
28 #ifndef _SSVECTORBASE_H_
29 #define _SSVECTORBASE_H_
30
31 #include <assert.h>
32
33 #include "soplex/spxdefines.h"
34 #include "soplex/vectorbase.h"
35 #include "soplex/idxset.h"
36 #include "soplex/spxalloc.h"
37 #include "soplex/timer.h"
38 #include "soplex/stablesum.h"
39
40 namespace soplex
41 {
42 template < class R > class SVectorBase;
43 template < class R > class SVSetBase;
44
45 /**@brief Semi sparse vector.
46 * @ingroup Algebra
47 *
48 * This class implements semi-sparse vectors. Such are #VectorBase%s where the indices of its nonzero elements can be
49 * stored in an extra IdxSet. Only elements with absolute value > #epsilon are considered to be nonzero. Since really
50 * storing the nonzeros is not always convenient, an SSVectorBase provides two different stati: setup and not setup.
51 * An SSVectorBase being setup means that the nonzero indices are available, otherwise an SSVectorBase is just an
52 * ordinary VectorBase with an empty IdxSet. Note that due to arithmetic operation, zeros can slip in, i.e., it is
53 * only guaranteed that at least every non-zero is in the IdxSet.
54 */
55 template < class R >
56 class SSVectorBase : public VectorBase<R>, protected IdxSet
57 {
58 private:
59
60 friend class VectorBase<R>;
61 template < class S > friend class DSVectorBase;
62
63 // ------------------------------------------------------------------------------------------------------------------
64 /**@name Data */
65 ///@{
66
67 /// Is the SSVectorBase set up?
68 bool setupStatus;
69
70 /// Allocates enough space to accommodate \p newmax values.
71 void setMax(int newmax)
72 {
73 assert(idx != 0);
74 assert(newmax != 0);
75 assert(newmax >= IdxSet::size());
76
77 len = newmax;
78 spx_realloc(idx, len);
79 }
80
81 ///@}
82
83 protected:
84 std::shared_ptr<Tolerances> _tolerances;
85
86 public:
87
88 // ------------------------------------------------------------------------------------------------------------------
89 /**@name Status of an SSVectorBase
90 *
91 * An SSVectorBase can be set up or not. In case it is set up, its IdxSet correctly contains all indices of nonzero
92 * elements of the SSVectorBase. Otherwise, it does not contain any useful data. Whether or not an SSVectorBase is
93 * setup can be determined with the method \ref soplex::SSVectorBase::isSetup() "isSetup()".
94 *
95 * There are three methods for directly affecting the setup status of an SSVectorBase:
96 *
97 * - unSetup(): This method sets the status to ``not setup''.
98 *
99 * - setup(): This method initializes the IdxSet to the SSVectorBase's nonzero indices and sets the status to
100 * ``setup''.
101 *
102 * - forceSetup(): This method sets the status to ``setup'' without verifying that the IdxSet correctly contains all
103 * nonzero indices. It may be used when the nonzero indices have been computed externally.
104 */
105 ///@{
106
107 /// Only used in slufactor.hpp.
108 R* get_ptr()
109 {
110 return VectorBase<R>::get_ptr();
111 }
112
113 /// set the _tolerances member variable
114 virtual void setTolerances(std::shared_ptr<Tolerances> newTolerances)
115 {
116 this->_tolerances = newTolerances;
117 }
118
119 /// returns current tolerances
120 const std::shared_ptr<Tolerances>& tolerances() const
121 {
122 assert(this->_tolerances != nullptr);
123 return this->_tolerances;
124 }
125
126 /// Returns setup status.
127 bool isSetup() const
128 {
129 return setupStatus;
130 }
131
132 R getEpsilon() const
133 {
134 assert(this->_tolerances != nullptr);
135 return this->_tolerances == nullptr ? R(0) : R(this->tolerances()->epsilon());
136 }
137
138 /// Makes SSVectorBase not setup.
139 void unSetup()
140 {
141 setupStatus = false;
142 }
143
144 /// Initializes nonzero indices for elements with absolute values above #epsilon and sets all other elements to 0.
145 void setup()
146 {
147 if(!isSetup())
148 {
149 IdxSet::clear();
150
151 int d = dim();
152 num = 0;
153
154 for(int i = 0; i < d; ++i)
155 {
156 if(VectorBase<R>::val[i] != R(0))
157 {
158 if(spxAbs(VectorBase<R>::val[i]) <= this->getEpsilon())
159 VectorBase<R>::val[i] = R(0);
160 else
161 {
162 idx[num] = i;
163 num++;
164 }
165 }
166 }
167
168 setupStatus = true;
169
170 assert(isConsistent());
171 }
172 }
173
174 /// Forces setup status.
175 void forceSetup()
176 {
177 setupStatus = true;
178 }
179
180 ///@}
181
182 // ------------------------------------------------------------------------------------------------------------------
183 /**@name Methods for setup SSVectorBases */
184 ///@{
185
186 /// Returns index of the \p n 'th nonzero element.
187 int index(int n) const
188 {
189 assert(isSetup());
190
191 return IdxSet::index(n);
192 }
193
194 /// Returns value of the \p n 'th nonzero element.
195 R value(int n) const
196 {
197 assert(isSetup());
198 assert(n >= 0 && n < size());
199
200 return VectorBase<R>::val[idx[n]];
201 }
202
203 /// Finds the position of index \p i in the #IdxSet, or -1 if \p i doesn't exist.
204 int pos(int i) const
205 {
206 assert(isSetup());
207
208 return IdxSet::pos(i);
209 }
210
211 /// Returns the number of nonzeros.
212 int size() const
213 {
214 assert(isSetup());
215
216 return IdxSet::size();
217 }
218
219 /// Adds nonzero (\p i, \p x) to SSVectorBase.
220 /** No nonzero with index \p i must exist in the SSVectorBase. */
221 void add(int i, R x)
222 {
223 assert(VectorBase<R>::val[i] == R(0));
224 assert(pos(i) < 0);
225
226 addIdx(i);
227 VectorBase<R>::val[i] = x;
228 }
229
230 /// Sets \p i 'th element to \p x.
231 void setValue(int i, R x)
232 {
233 assert(i >= 0);
234 assert(i < VectorBase<R>::dim());
235
236 if(isSetup())
237 {
238 int n = pos(i);
239
240 if(n < 0)
241 {
242 if(spxAbs(x) > this->getEpsilon())
243 IdxSet::add(1, &i);
244 }
245 else if(x == R(0))
246 clearNum(n);
247 }
248
249 VectorBase<R>::val[i] = x;
250
251 assert(isConsistent());
252 }
253
254 /// Scale \p i 'th element by a
255 void scaleValue(int i, int scaleExp)
256 {
257 assert(i >= 0);
258 assert(i < VectorBase<R>::dim());
259
260 VectorBase<R>::val[i] = spxLdexp(VectorBase<R>::val[i], scaleExp);
261
262 assert(isConsistent());
263 }
264
265 /// Clears element \p i.
266 void clearIdx(int i)
267 {
268 if(isSetup())
269 {
270 int n = pos(i);
271
272 if(n >= 0)
273 remove(n);
274 }
275
276 VectorBase<R>::val[i] = 0;
277
278 assert(isConsistent());
279 }
280
281 /// Sets \p n 'th nonzero element to 0 (index \p n must exist).
282 void clearNum(int n)
283 {
284 assert(isSetup());
285 assert(index(n) >= 0);
286
287 VectorBase<R>::val[index(n)] = 0;
288 remove(n);
289
290 assert(isConsistent());
291 }
292
293 ///@}
294
295 // ------------------------------------------------------------------------------------------------------------------
296 /**@name Methods independent of the Status */
297 ///@{
298
299 /// Returns \p i 'th value.
300 R operator[](int i) const
301 {
302 return VectorBase<R>::val[i];
303 }
304
305 /// Returns array indices.
306 const int* indexMem() const
307 {
308 return idx;
309 }
310
311 /// Returns array values.
312 const R* values() const
313 {
314 return VectorBase<R>::val.data();
315 }
316
317 /// Returns indices.
318 const IdxSet& indices() const
319 {
320 return *this;
321 }
322
323 /// Returns array indices.
324 int* altIndexMem()
325 {
326 unSetup();
327 return idx;
328 }
329
330 /// Returns array values.
331 R* altValues()
332 {
333 unSetup();
334 return VectorBase<R>::val.data();
335 }
336
337 /// Returns indices.
338 IdxSet& altIndices()
339 {
340 unSetup();
341 return *this;
342 }
343
344 ///@}
345
346 // ------------------------------------------------------------------------------------------------------------------
347 /**@name Arithmetic operations */
348 ///@{
349
350 /// Addition.
351 template < class S >
352 SSVectorBase<R>& operator+=(const VectorBase<S>& vec)
353 {
354 VectorBase<S>::operator+=(vec);
355
356 if(isSetup())
357 {
358 setupStatus = false;
359 setup();
360 }
361
362 return *this;
363 }
364
365 /// Addition.
366 template < class S >
367 SSVectorBase<R>& operator+=(const SVectorBase<S>& vec);
368
369 /// Addition.
370 template < class S >
371 SSVectorBase<R>& operator+=(const SSVectorBase<S>& vec)
372 {
373 assert(vec.isSetup());
374
375 for(int i = vec.size() - 1; i >= 0; --i)
376 VectorBase<R>::val[vec.index(i)] += vec.value(i);
377
378 if(isSetup())
379 {
380 setupStatus = false;
381 setup();
382 }
383
384 return *this;
385 }
386
387 /// Subtraction.
388 template < class S >
389 SSVectorBase<R>& operator-=(const VectorBase<S>& vec)
390 {
391 VectorBase<R>::operator-=(vec);
392
393 if(isSetup())
394 {
395 setupStatus = false;
396 setup();
397 }
398
399 return *this;
400 }
401
402 /// Subtraction.
403 template < class S >
404 SSVectorBase<R>& operator-=(const SVectorBase<S>& vec);
405
406 /// Subtraction.
407 template < class S >
408 SSVectorBase<R>& operator-=(const SSVectorBase<S>& vec)
409 {
410 if(vec.isSetup())
411 {
412 for(int i = vec.size() - 1; i >= 0; --i)
413 VectorBase<R>::val[vec.index(i)] -= vec.value(i);
414 }
415 else
416 VectorBase<R>::operator-=(VectorBase<S>(vec));
417
418 if(isSetup())
419 {
420 setupStatus = false;
421 setup();
422 }
423
424 return *this;
425 }
426
427 /// Scaling.
428 template < class S >
429 SSVectorBase<R>& operator*=(S x)
430 {
431 assert(isSetup());
432 assert(x != S(0));
433
434 for(int i = size() - 1; i >= 0; --i)
435 VectorBase<R>::val[index(i)] *= x;
436
437 assert(isConsistent());
438
439 return *this;
440 }
441
442 // Inner product.
443 template < class S >
444 R operator*(const SSVectorBase<S>& w)
445 {
446 setup();
447
448 StableSum<R> x;
449 int i = size() - 1;
450 int j = w.size() - 1;
451
452 // both *this and w non-zero vectors?
453 if(i >= 0 && j >= 0)
454 {
455 int vi = index(i);
456 int wj = w.index(j);
457
458 while(i != 0 && j != 0)
459 {
460 if(vi == wj)
461 {
462 x += VectorBase<R>::val[vi] * R(w.val[wj]);
463 vi = index(--i);
464 wj = w.index(--j);
465 }
466 else if(vi > wj)
467 vi = index(--i);
468 else
469 wj = w.index(--j);
470 }
471
472 /* check remaining indices */
473
474 while(i != 0 && vi != wj)
475 vi = index(--i);
476
477 while(j != 0 && vi != wj)
478 wj = w.index(--j);
479
480 if(vi == wj)
481 x += VectorBase<R>::val[vi] * R(w.val[wj]);
482 }
483
484 return x;
485 }
486
487 /// Addition of a scaled vector.
488 ///@todo SSVectorBase::multAdd() should be rewritten without pointer arithmetic.
489 template < class S, class T >
490 SSVectorBase<R>& multAdd(S xx, const SVectorBase<T>& vec);
491
492 /// Addition of a scaled vector.
493 template < class S, class T >
494 SSVectorBase<R>& multAdd(S x, const VectorBase<T>& vec)
495 {
496 VectorBase<R>::multAdd(x, vec);
497
498 if(isSetup())
499 {
500 setupStatus = false;
501 setup();
502 }
503
504 return *this;
505 }
506
507 /// Assigns pair wise vector product to SSVectorBase.
508 template < class S, class T >
509 SSVectorBase<R>& assignPWproduct4setup(const SSVectorBase<S>& x, const SSVectorBase<T>& y);
510
511 /// Assigns \f$x^T \cdot A\f$ to SSVectorBase.
512 template < class S, class T >
513 SSVectorBase<R>& assign2product(const SSVectorBase<S>& x, const SVSetBase<T>& A);
514
515 /// Assigns SSVectorBase to \f$A \cdot x\f$ for a setup \p x.
516 template < class S, class T >
517 SSVectorBase<R>& assign2product4setup(const SVSetBase<S>& A, const SSVectorBase<T>& x,
518 Timer* timeSparse, Timer* timeFull, int& nCallsSparse, int& nCallsFull);
519
520 public:
521
522 /// Assigns SSVectorBase to \f$A \cdot x\f$ thereby setting up \p x.
523 template < class S, class T >
524 SSVectorBase<R>& assign2productAndSetup(const SVSetBase<S>& A, SSVectorBase<T>& x);
525
526 /// Maximum absolute value, i.e., infinity norm.
527 R maxAbs() const
528 {
529 if(isSetup())
530 {
531 R maxabs = 0;
532
533 for(int i = 0; i < num; ++i)
534 {
535 R x = spxAbs(VectorBase<R>::val[idx[i]]);
536
537 if(x > maxabs)
538 maxabs = x;
539 }
540
541 return maxabs;
542 }
543 else
544 return VectorBase<R>::maxAbs();
545 }
546
547 /// Squared euclidian norm.
548 R length2() const
549 {
550 R x = 0;
551
552 if(isSetup())
553 {
554 for(int i = 0; i < num; ++i)
555 x += VectorBase<R>::val[idx[i]] * VectorBase<R>::val[idx[i]];
556 }
557 else
558 x = VectorBase<R>::length2();
559
560 return x;
561 }
562
563 /// Floating point approximation of euclidian norm (without any approximation guarantee).
564 R length() const
565 {
566 return spxSqrt(R(length2()));
567 }
568
569 ///@}
570
571 // ------------------------------------------------------------------------------------------------------------------
572 /**@name Miscellaneous */
573 ///@{
574
575 /// Dimension of VectorBase.
576 int dim() const
577 {
578 return VectorBase<R>::dim();
579 }
580
581 /// Resets dimension to \p newdim.
582 void reDim(int newdim)
583 {
584 for(int i = IdxSet::size() - 1; i >= 0; --i)
585 {
586 if(index(i) >= newdim)
587 remove(i);
588 }
589
590 VectorBase<R>::reDim(newdim);
591 setMax(VectorBase<R>::memSize() + 1);
592
593 assert(isConsistent());
594 }
595
596 /// Sets number of nonzeros (thereby unSetup SSVectorBase).
597 void setSize(int n)
598 {
599 assert(n >= 0);
600 assert(n <= IdxSet::max());
601
602 unSetup();
603 num = n;
604 }
605
606 /// Resets memory consumption to \p newsize.
607 void reMem(int newsize)
608 {
609 VectorBase<R>::reSize(newsize);
610 assert(isConsistent());
611
612 setMax(VectorBase<R>::memSize() + 1);
613 }
614
615 /// Clears vector.
616 void clear()
617 {
618 if(isSetup())
619 {
620 for(int i = 0; i < num; ++i)
621 VectorBase<R>::val[idx[i]] = 0;
622 }
623 else
624 VectorBase<R>::clear();
625
626 IdxSet::clear();
627 setupStatus = true;
628
629 assert(isConsistent());
630 }
631
632 /// consistency check.
633 bool isConsistent() const
634 {
635 #ifdef ENABLE_CONSISTENCY_CHECKS
636
637 if(VectorBase<R>::dim() > IdxSet::max())
638 return SPX_MSG_INCONSISTENT("SSVectorBase");
639
640 if(VectorBase<R>::dim() < IdxSet::dim())
641 return SPX_MSG_INCONSISTENT("SSVectorBase");
642
643 if(isSetup())
644 {
645 for(int i = 0; i < VectorBase<R>::dim(); ++i)
646 {
647 int j = pos(i);
648
649 if(j < 0 && spxAbs(VectorBase<R>::val[i]) > 0)
650 {
651 SPX_MSG_ERROR(std::cerr << "ESSVEC01 i = " << i
652 << "\tidx = " << j
653 << "\tval = " << std::setprecision(16) << VectorBase<R>::val[i]
654 << std::endl;)
655
656 return SPX_MSG_INCONSISTENT("SSVectorBase");
657 }
658 }
659 }
660
661 return VectorBase<R>::isConsistent() && IdxSet::isConsistent();
662 #else
663 return true;
664 #endif
665 }
666
667 ///@}
668
669 // ------------------------------------------------------------------------------------------------------------------
670 /**@name Constructors / Destructors */
671 ///@{
672
673 /// Default constructor.
674 explicit SSVectorBase<R>(int p_dim, std::shared_ptr<Tolerances> tol = nullptr)
675 : VectorBase<R>(p_dim)
676 , IdxSet()
677 , setupStatus(true)
678 {
679 len = (p_dim < 1) ? 1 : p_dim;
680 spx_alloc(idx, len);
681 VectorBase<R>::clear();
682 _tolerances = tol;
683
684 assert(isConsistent());
685 }
686
687 /// Copy constructor.
688 template < class S >
689 SSVectorBase(const SSVectorBase<S>& vec)
690 : VectorBase<R>(vec)
691 , IdxSet()
692 , setupStatus(vec.setupStatus)
693 {
694 len = (vec.dim() < 1) ? 1 : vec.dim();
695 spx_alloc(idx, len);
696 IdxSet::operator=(vec);
697 _tolerances = vec._tolerances;
698
699 assert(isConsistent());
700 }
701
702 /// Copy constructor.
703 /** The redundancy with the copy constructor below is necessary since otherwise the compiler doesn't realize that it
704 * could use the more general one with S = R and generates a shallow copy constructor.
705 */
706 SSVectorBase(const SSVectorBase<R>& vec)
707 : VectorBase<R>(vec)
708 , IdxSet()
709 , setupStatus(vec.setupStatus)
710 {
711 len = (vec.dim() < 1) ? 1 : vec.dim();
712 spx_alloc(idx, len);
713 IdxSet::operator=(vec);
714 _tolerances = vec._tolerances;
715
716 assert(isConsistent());
717 }
718
719 /// Constructs nonsetup copy of \p vec.
720 template < class S >
721 explicit SSVectorBase<R>(const VectorBase<S>& vec)
722 : VectorBase<R>(vec)
723 , IdxSet()
724 , setupStatus(false)
725 {
726 len = (vec.dim() < 1) ? 1 : vec.dim();
727 spx_alloc(idx, len);
728
729 assert(isConsistent());
730 }
731
732 /// Sets up \p rhs vector, and assigns it.
733 template < class S >
734 void setup_and_assign(SSVectorBase<S>& rhs)
735 {
736 clear();
737 setMax(rhs.max());
738 VectorBase<R>::reDim(rhs.dim());
739 _tolerances = rhs.tolerances();
740
741 if(rhs.isSetup())
742 {
743 IdxSet::operator=(rhs);
744
745 for(int i = size() - 1; i >= 0; --i)
746 {
747 int j = index(i);
748 VectorBase<R>::val[j] = rhs.val[j];
749 }
750 }
751 else
752 {
753 int d = rhs.dim();
754 num = 0;
755
756 for(int i = 0; i < d; ++i)
757 {
758 if(rhs.val[i] != 0)
759 {
760 if(spxAbs(rhs.val[i]) > this->getEpsilon())
761 {
762 rhs.idx[num] = i;
763 idx[num] = i;
764 VectorBase<R>::val[i] = rhs.val[i];
765 num++;
766 }
767 else
768 rhs.val[i] = 0;
769 }
770 }
771
772 rhs.num = num;
773 rhs.setupStatus = true;
774 }
775
776 setupStatus = true;
777
778 assert(rhs.isConsistent());
779 assert(isConsistent());
780 }
781
782 /// Assigns only the elements of \p rhs.
783 template < class S >
784 SSVectorBase<R>& assign(const SVectorBase<S>& rhs);
785
786 /// Assignment operator.
787 template < class S >
788 SSVectorBase<R>& operator=(const SSVectorBase<S>& rhs)
789 {
790 assert(rhs.isConsistent());
791
792 if(this != &rhs)
793 {
794 clear();
795 _tolerances = rhs._tolerances;
796 setMax(rhs.max());
797 VectorBase<R>::reDim(rhs.dim());
798
799 if(rhs.isSetup())
800 {
801 IdxSet::operator=(rhs);
802
803 for(int i = size() - 1; i >= 0; --i)
804 {
805 int j = index(i);
806 VectorBase<R>::val[j] = rhs.val[j];
807 }
808 }
809 else
810 {
811 int d = rhs.dim();
812 num = 0;
813
814 for(int i = 0; i < d; ++i)
815 {
816 if(spxAbs(rhs.val[i]) > this->getEpsilon())
817 {
818 VectorBase<R>::val[i] = rhs.val[i];
819 idx[num] = i;
820 num++;
821 }
822 }
823 }
824
825 setupStatus = true;
826 }
827
828 assert(isConsistent());
829
830 return *this;
831 }
832
833 /// Assignment operator.
834 SSVectorBase<R>& operator=(const SSVectorBase<R>& rhs)
835 {
836 assert(rhs.isConsistent());
837
838 if(this != &rhs)
839 {
840 clear();
841 _tolerances = rhs._tolerances;
842 setMax(rhs.max());
843 VectorBase<R>::reDim(rhs.dim());
844
845 if(rhs.isSetup())
846 {
847 IdxSet::operator=(rhs);
848
849 for(int i = size() - 1; i >= 0; --i)
850 {
851 int j = index(i);
852 VectorBase<R>::val[j] = rhs.val[j];
853 }
854 }
855 else
856 {
857 num = 0;
858
859 for(int i = 0; i < rhs.dim(); ++i)
860 {
861 if(spxAbs(rhs.val[i]) > this->getEpsilon())
862 {
863 VectorBase<R>::val[i] = rhs.val[i];
864 idx[num] = i;
865 num++;
866 }
867 }
868 }
869
870 setupStatus = true;
871 }
872
873 assert(isConsistent());
874
875 return *this;
876 }
877
878 /// Assignment operator.
879 template < class S >
880 SSVectorBase<R>& operator=(const SVectorBase<S>& rhs);
881
882 /// Assignment operator.
883 template < class S >
884 SSVectorBase<R>& operator=(const VectorBase<S>& rhs)
885 {
886 unSetup();
887 VectorBase<R>::operator=(rhs);
888
889 assert(isConsistent());
890
891 return *this;
892 }
893
894 /// destructor
895 ~SSVectorBase()
896 {
897 if(idx)
898 spx_free(idx);
899 }
900
901 ///@}
902
903 private:
904
905 // ------------------------------------------------------------------------------------------------------------------
906 /**@name Private helpers */
907 ///@{
908
909 /// Assignment helper.
910 template < class S, class T >
911 SSVectorBase<R>& assign2product1(const SVSetBase<S>& A, const SSVectorBase<T>& x);
912
913 /// Assignment helper.
914 template < class S, class T >
915 SSVectorBase<R>& assign2productShort(const SVSetBase<S>& A, const SSVectorBase<T>& x);
916
917 /// Assignment helper.
918 template < class S, class T >
919 SSVectorBase<R>& assign2productFull(const SVSetBase<S>& A, const SSVectorBase<T>& x);
920
921 ///@}
922 };
923
924 } // namespace soplex
925 #endif // _SSVECTORBASE_H_
926