3 * Implementation of GiNaC's symbolic exponentiation (basis^exponent). */
6 * GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany
8 * This program is free software; you can redistribute it and/or modify
9 * it under the terms of the GNU General Public License as published by
10 * the Free Software Foundation; either version 2 of the License, or
11 * (at your option) any later version.
13 * This program is distributed in the hope that it will be useful,
14 * but WITHOUT ANY WARRANTY; without even the implied warranty of
15 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
16 * GNU General Public License for more details.
18 * You should have received a copy of the GNU General Public License
19 * along with this program; if not, write to the Free Software
20 * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
28 #include "expairseq.h"
34 #include "inifcns.h" // for log() in power::derivative()
44 GINAC_IMPLEMENT_REGISTERED_CLASS(power, basic)
46 typedef std::vector<int> intvector;
49 // default ctor, dtor, copy ctor assignment operator and helpers
52 power::power() : inherited(TINFO_power)
54 debugmsg("power default ctor",LOGLEVEL_CONSTRUCT);
57 void power::copy(const power & other)
59 inherited::copy(other);
61 exponent = other.exponent;
64 DEFAULT_DESTROY(power)
76 power::power(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
78 debugmsg("power ctor from archive_node", LOGLEVEL_CONSTRUCT);
79 n.find_ex("basis", basis, sym_lst);
80 n.find_ex("exponent", exponent, sym_lst);
83 void power::archive(archive_node &n) const
85 inherited::archive(n);
86 n.add_ex("basis", basis);
87 n.add_ex("exponent", exponent);
90 DEFAULT_UNARCHIVE(power)
93 // functions overriding virtual functions from base classes
98 static void print_sym_pow(const print_context & c, const symbol &x, int exp)
100 // Optimal output of integer powers of symbols to aid compiler CSE.
101 // C.f. ISO/IEC 14882:1998, section 1.9 [intro execution], paragraph 15
102 // to learn why such a hack is really necessary.
105 } else if (exp == 2) {
109 } else if (exp & 1) {
112 print_sym_pow(c, x, exp-1);
115 print_sym_pow(c, x, exp >> 1);
117 print_sym_pow(c, x, exp >> 1);
122 void power::print(const print_context & c, unsigned level) const
124 debugmsg("power print", LOGLEVEL_PRINT);
126 if (is_a<print_tree>(c)) {
128 inherited::print(c, level);
130 } else if (is_a<print_csrc>(c)) {
132 // Integer powers of symbols are printed in a special, optimized way
133 if (exponent.info(info_flags::integer)
134 && (is_exactly_a<symbol>(basis) || is_exactly_a<constant>(basis))) {
135 int exp = ex_to<numeric>(exponent).to_int();
140 if (is_a<print_csrc_cl_N>(c))
145 print_sym_pow(c, ex_to<symbol>(basis), exp);
148 // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
149 } else if (exponent.compare(_num_1()) == 0) {
150 if (is_a<print_csrc_cl_N>(c))
157 // Otherwise, use the pow() or expt() (CLN) functions
159 if (is_a<print_csrc_cl_N>(c))
171 if (exponent.is_equal(_ex1_2())) {
172 if (is_a<print_latex>(c))
177 if (is_a<print_latex>(c))
182 if (precedence() <= level) {
183 if (is_a<print_latex>(c))
188 basis.print(c, precedence());
190 if (is_a<print_latex>(c))
192 exponent.print(c, precedence());
193 if (is_a<print_latex>(c))
195 if (precedence() <= level) {
196 if (is_a<print_latex>(c))
205 bool power::info(unsigned inf) const
208 case info_flags::polynomial:
209 case info_flags::integer_polynomial:
210 case info_flags::cinteger_polynomial:
211 case info_flags::rational_polynomial:
212 case info_flags::crational_polynomial:
213 return exponent.info(info_flags::nonnegint);
214 case info_flags::rational_function:
215 return exponent.info(info_flags::integer);
216 case info_flags::algebraic:
217 return (!exponent.info(info_flags::integer) ||
220 return inherited::info(inf);
223 unsigned power::nops() const
228 ex & power::let_op(int i)
233 return i==0 ? basis : exponent;
236 ex power::map(map_function & f) const
238 return (new power(f(basis), f(exponent)))->setflag(status_flags::dynallocated);
241 int power::degree(const ex & s) const
243 if (is_exactly_of_type(*exponent.bp,numeric)) {
244 if (basis.is_equal(s)) {
245 if (ex_to<numeric>(exponent).is_integer())
246 return ex_to<numeric>(exponent).to_int();
250 return basis.degree(s) * ex_to<numeric>(exponent).to_int();
255 int power::ldegree(const ex & s) const
257 if (is_exactly_of_type(*exponent.bp,numeric)) {
258 if (basis.is_equal(s)) {
259 if (ex_to<numeric>(exponent).is_integer())
260 return ex_to<numeric>(exponent).to_int();
264 return basis.ldegree(s) * ex_to<numeric>(exponent).to_int();
269 ex power::coeff(const ex & s, int n) const
271 if (!basis.is_equal(s)) {
272 // basis not equal to s
279 if (is_exactly_of_type(*exponent.bp, numeric) && ex_to<numeric>(exponent).is_integer()) {
281 int int_exp = ex_to<numeric>(exponent).to_int();
287 // non-integer exponents are treated as zero
296 ex power::eval(int level) const
298 // simplifications: ^(x,0) -> 1 (0^0 handled here)
300 // ^(0,c1) -> 0 or exception (depending on real value of c1)
302 // ^(c1,c2) -> *(c1^n,c1^(c2-n)) (c1, c2 numeric(), 0<(c2-n)<1 except if c1,c2 are rational, but c1^c2 is not)
303 // ^(^(x,c1),c2) -> ^(x,c1*c2) (c1, c2 numeric(), c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
304 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
305 // ^(*(x,c1),c2) -> ^(x,c2)*c1^c2 (c1, c2 numeric(), c1>0)
306 // ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2 (c1, c2 numeric(), c1<0)
308 debugmsg("power eval",LOGLEVEL_MEMBER_FUNCTION);
310 if ((level==1) && (flags & status_flags::evaluated))
312 else if (level == -max_recursion_level)
313 throw(std::runtime_error("max recursion level reached"));
315 const ex & ebasis = level==1 ? basis : basis.eval(level-1);
316 const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
318 bool basis_is_numerical = false;
319 bool exponent_is_numerical = false;
321 numeric * num_exponent;
323 if (is_exactly_of_type(*ebasis.bp,numeric)) {
324 basis_is_numerical = true;
325 num_basis = static_cast<numeric *>(ebasis.bp);
327 if (is_exactly_of_type(*eexponent.bp,numeric)) {
328 exponent_is_numerical = true;
329 num_exponent = static_cast<numeric *>(eexponent.bp);
332 // ^(x,0) -> 1 (0^0 also handled here)
333 if (eexponent.is_zero()) {
334 if (ebasis.is_zero())
335 throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
341 if (eexponent.is_equal(_ex1()))
344 // ^(0,c1) -> 0 or exception (depending on real value of c1)
345 if (ebasis.is_zero() && exponent_is_numerical) {
346 if ((num_exponent->real()).is_zero())
347 throw (std::domain_error("power::eval(): pow(0,I) is undefined"));
348 else if ((num_exponent->real()).is_negative())
349 throw (pole_error("power::eval(): division by zero",1));
355 if (ebasis.is_equal(_ex1()))
358 if (exponent_is_numerical) {
360 // ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
361 // except if c1,c2 are rational, but c1^c2 is not)
362 if (basis_is_numerical) {
363 bool basis_is_crational = num_basis->is_crational();
364 bool exponent_is_crational = num_exponent->is_crational();
365 numeric res = num_basis->power(*num_exponent);
367 if ((!basis_is_crational || !exponent_is_crational)
368 || res.is_crational()) {
371 GINAC_ASSERT(!num_exponent->is_integer()); // has been handled by now
373 // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-h)<1, q integer
374 if (basis_is_crational && exponent_is_crational
375 && num_exponent->is_real()
376 && !num_exponent->is_integer()) {
377 numeric n = num_exponent->numer();
378 numeric m = num_exponent->denom();
380 numeric q = iquo(n, m, r);
381 if (r.is_negative()) {
385 if (q.is_zero()) // the exponent was in the allowed range 0<(n/m)<1
389 res.push_back(expair(ebasis,r.div(m)));
390 return (new mul(res,ex(num_basis->power_dyn(q))))->setflag(status_flags::dynallocated | status_flags::evaluated);
395 // ^(^(x,c1),c2) -> ^(x,c1*c2)
396 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
397 // case c1==1 should not happen, see below!)
398 if (is_ex_exactly_of_type(ebasis,power)) {
399 const power & sub_power = ex_to<power>(ebasis);
400 const ex & sub_basis = sub_power.basis;
401 const ex & sub_exponent = sub_power.exponent;
402 if (is_ex_exactly_of_type(sub_exponent,numeric)) {
403 const numeric & num_sub_exponent = ex_to<numeric>(sub_exponent);
404 GINAC_ASSERT(num_sub_exponent!=numeric(1));
405 if (num_exponent->is_integer() || (abs(num_sub_exponent) - _num1()).is_negative())
406 return power(sub_basis,num_sub_exponent.mul(*num_exponent));
410 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
411 if (num_exponent->is_integer() && is_ex_exactly_of_type(ebasis,mul)) {
412 return expand_mul(ex_to<mul>(ebasis), *num_exponent);
415 // ^(*(...,x;c1),c2) -> ^(*(...,x;1),c2)*c1^c2 (c1, c2 numeric(), c1>0)
416 // ^(*(...,x,c1),c2) -> ^(*(...,x;-1),c2)*(-c1)^c2 (c1, c2 numeric(), c1<0)
417 if (is_ex_exactly_of_type(ebasis,mul)) {
418 GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
419 const mul & mulref = ex_to<mul>(ebasis);
420 if (!mulref.overall_coeff.is_equal(_ex1())) {
421 const numeric & num_coeff = ex_to<numeric>(mulref.overall_coeff);
422 if (num_coeff.is_real()) {
423 if (num_coeff.is_positive()) {
424 mul * mulp = new mul(mulref);
425 mulp->overall_coeff = _ex1();
426 mulp->clearflag(status_flags::evaluated);
427 mulp->clearflag(status_flags::hash_calculated);
428 return (new mul(power(*mulp,exponent),
429 power(num_coeff,*num_exponent)))->setflag(status_flags::dynallocated);
431 GINAC_ASSERT(num_coeff.compare(_num0())<0);
432 if (num_coeff.compare(_num_1())!=0) {
433 mul * mulp = new mul(mulref);
434 mulp->overall_coeff = _ex_1();
435 mulp->clearflag(status_flags::evaluated);
436 mulp->clearflag(status_flags::hash_calculated);
437 return (new mul(power(*mulp,exponent),
438 power(abs(num_coeff),*num_exponent)))->setflag(status_flags::dynallocated);
445 // ^(nc,c1) -> ncmul(nc,nc,...) (c1 positive integer, unless nc is a matrix)
446 if (num_exponent->is_pos_integer() &&
447 ebasis.return_type() != return_types::commutative &&
448 !is_ex_of_type(ebasis,matrix)) {
449 return ncmul(exvector(num_exponent->to_int(), ebasis), true);
453 if (are_ex_trivially_equal(ebasis,basis) &&
454 are_ex_trivially_equal(eexponent,exponent)) {
457 return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
458 status_flags::evaluated);
461 ex power::evalf(int level) const
463 debugmsg("power evalf",LOGLEVEL_MEMBER_FUNCTION);
470 eexponent = exponent;
471 } else if (level == -max_recursion_level) {
472 throw(std::runtime_error("max recursion level reached"));
474 ebasis = basis.evalf(level-1);
475 if (!is_ex_exactly_of_type(eexponent,numeric))
476 eexponent = exponent.evalf(level-1);
478 eexponent = exponent;
481 return power(ebasis,eexponent);
484 ex power::evalm(void) const
486 const ex ebasis = basis.evalm();
487 const ex eexponent = exponent.evalm();
488 if (is_ex_of_type(ebasis,matrix)) {
489 if (is_ex_of_type(eexponent,numeric)) {
490 return (new matrix(ex_to<matrix>(ebasis).pow(eexponent)))->setflag(status_flags::dynallocated);
493 return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated);
496 ex power::subs(const lst & ls, const lst & lr, bool no_pattern) const
498 const ex &subsed_basis = basis.subs(ls, lr, no_pattern);
499 const ex &subsed_exponent = exponent.subs(ls, lr, no_pattern);
501 if (are_ex_trivially_equal(basis, subsed_basis)
502 && are_ex_trivially_equal(exponent, subsed_exponent))
503 return basic::subs(ls, lr, no_pattern);
505 return ex(power(subsed_basis, subsed_exponent)).bp->basic::subs(ls, lr, no_pattern);
508 ex power::simplify_ncmul(const exvector & v) const
510 return inherited::simplify_ncmul(v);
515 /** Implementation of ex::diff() for a power.
517 ex power::derivative(const symbol & s) const
519 if (exponent.info(info_flags::real)) {
520 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
523 newseq.push_back(expair(basis, exponent - _ex1()));
524 newseq.push_back(expair(basis.diff(s), _ex1()));
525 return mul(newseq, exponent);
527 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
529 add(mul(exponent.diff(s), log(basis)),
530 mul(mul(exponent, basis.diff(s)), power(basis, _ex_1()))));
534 int power::compare_same_type(const basic & other) const
536 GINAC_ASSERT(is_exactly_of_type(other, power));
537 const power &o = static_cast<const power &>(other);
539 int cmpval = basis.compare(o.basis);
543 return exponent.compare(o.exponent);
546 unsigned power::return_type(void) const
548 return basis.return_type();
551 unsigned power::return_type_tinfo(void) const
553 return basis.return_type_tinfo();
556 ex power::expand(unsigned options) const
558 if (options == 0 && (flags & status_flags::expanded))
561 ex expanded_basis = basis.expand(options);
562 ex expanded_exponent = exponent.expand(options);
564 // x^(a+b) -> x^a * x^b
565 if (is_ex_exactly_of_type(expanded_exponent, add)) {
566 const add &a = ex_to<add>(expanded_exponent);
568 distrseq.reserve(a.seq.size() + 1);
569 epvector::const_iterator last = a.seq.end();
570 epvector::const_iterator cit = a.seq.begin();
572 distrseq.push_back(power(expanded_basis, a.recombine_pair_to_ex(*cit)));
576 // Make sure that e.g. (x+y)^(2+a) expands the (x+y)^2 factor
577 if (ex_to<numeric>(a.overall_coeff).is_integer()) {
578 const numeric &num_exponent = ex_to<numeric>(a.overall_coeff);
579 int int_exponent = num_exponent.to_int();
580 if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis, add))
581 distrseq.push_back(expand_add(ex_to<add>(expanded_basis), int_exponent));
583 distrseq.push_back(power(expanded_basis, a.overall_coeff));
585 distrseq.push_back(power(expanded_basis, a.overall_coeff));
587 // Make sure that e.g. (x+y)^(1+a) -> x*(x+y)^a + y*(x+y)^a
588 ex r = (new mul(distrseq))->setflag(status_flags::dynallocated);
592 if (!is_ex_exactly_of_type(expanded_exponent, numeric) ||
593 !ex_to<numeric>(expanded_exponent).is_integer()) {
594 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent)) {
597 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
601 // integer numeric exponent
602 const numeric & num_exponent = ex_to<numeric>(expanded_exponent);
603 int int_exponent = num_exponent.to_int();
606 if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis,add))
607 return expand_add(ex_to<add>(expanded_basis), int_exponent);
609 // (x*y)^n -> x^n * y^n
610 if (is_ex_exactly_of_type(expanded_basis,mul))
611 return expand_mul(ex_to<mul>(expanded_basis), num_exponent);
613 // cannot expand further
614 if (are_ex_trivially_equal(basis,expanded_basis) && are_ex_trivially_equal(exponent,expanded_exponent))
617 return (new power(expanded_basis,expanded_exponent))->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
621 // new virtual functions which can be overridden by derived classes
627 // non-virtual functions in this class
630 /** expand a^n where a is an add and n is an integer.
631 * @see power::expand */
632 ex power::expand_add(const add & a, int n) const
635 return expand_add_2(a);
639 sum.reserve((n+1)*(m-1));
641 intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
642 intvector upper_limit(m-1);
645 for (int l=0; l<m-1; l++) {
654 for (l=0; l<m-1; l++) {
655 const ex & b = a.op(l);
656 GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
657 GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
658 !is_ex_exactly_of_type(ex_to<power>(b).exponent,numeric) ||
659 !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
660 !is_ex_exactly_of_type(ex_to<power>(b).basis,add) ||
661 !is_ex_exactly_of_type(ex_to<power>(b).basis,mul) ||
662 !is_ex_exactly_of_type(ex_to<power>(b).basis,power));
663 if (is_ex_exactly_of_type(b,mul))
664 term.push_back(expand_mul(ex_to<mul>(b),numeric(k[l])));
666 term.push_back(power(b,k[l]));
669 const ex & b = a.op(l);
670 GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
671 GINAC_ASSERT(!is_ex_exactly_of_type(b,power) ||
672 !is_ex_exactly_of_type(ex_to<power>(b).exponent,numeric) ||
673 !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
674 !is_ex_exactly_of_type(ex_to<power>(b).basis,add) ||
675 !is_ex_exactly_of_type(ex_to<power>(b).basis,mul) ||
676 !is_ex_exactly_of_type(ex_to<power>(b).basis,power));
677 if (is_ex_exactly_of_type(b,mul))
678 term.push_back(expand_mul(ex_to<mul>(b),numeric(n-k_cum[m-2])));
680 term.push_back(power(b,n-k_cum[m-2]));
682 numeric f = binomial(numeric(n),numeric(k[0]));
683 for (l=1; l<m-1; l++)
684 f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
688 // TODO: Can we optimize this? Alex seemed to think so...
689 sum.push_back((new mul(term))->setflag(status_flags::dynallocated));
693 while ((l>=0) && ((++k[l])>upper_limit[l])) {
699 // recalc k_cum[] and upper_limit[]
703 k_cum[l] = k_cum[l-1]+k[l];
705 for (int i=l+1; i<m-1; i++)
706 k_cum[i] = k_cum[i-1]+k[i];
708 for (int i=l+1; i<m-1; i++)
709 upper_limit[i] = n-k_cum[i-1];
711 return (new add(sum))->setflag(status_flags::dynallocated |
712 status_flags::expanded );
716 /** Special case of power::expand_add. Expands a^2 where a is an add.
717 * @see power::expand_add */
718 ex power::expand_add_2(const add & a) const
721 unsigned a_nops = a.nops();
722 sum.reserve((a_nops*(a_nops+1))/2);
723 epvector::const_iterator last = a.seq.end();
725 // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
726 // first part: ignore overall_coeff and expand other terms
727 for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
728 const ex & r = cit0->rest;
729 const ex & c = cit0->coeff;
731 GINAC_ASSERT(!is_ex_exactly_of_type(r,add));
732 GINAC_ASSERT(!is_ex_exactly_of_type(r,power) ||
733 !is_ex_exactly_of_type(ex_to<power>(r).exponent,numeric) ||
734 !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
735 !is_ex_exactly_of_type(ex_to<power>(r).basis,add) ||
736 !is_ex_exactly_of_type(ex_to<power>(r).basis,mul) ||
737 !is_ex_exactly_of_type(ex_to<power>(r).basis,power));
739 if (are_ex_trivially_equal(c,_ex1())) {
740 if (is_ex_exactly_of_type(r,mul)) {
741 sum.push_back(expair(expand_mul(ex_to<mul>(r),_num2()),
744 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
748 if (is_ex_exactly_of_type(r,mul)) {
749 sum.push_back(expair(expand_mul(ex_to<mul>(r),_num2()),
750 ex_to<numeric>(c).power_dyn(_num2())));
752 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
753 ex_to<numeric>(c).power_dyn(_num2())));
757 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
758 const ex & r1 = cit1->rest;
759 const ex & c1 = cit1->coeff;
760 sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
761 _num2().mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
765 GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
767 // second part: add terms coming from overall_factor (if != 0)
768 if (!a.overall_coeff.is_zero()) {
769 epvector::const_iterator i = a.seq.begin(), end = a.seq.end();
771 sum.push_back(a.combine_pair_with_coeff_to_pair(*i, ex_to<numeric>(a.overall_coeff).mul_dyn(_num2())));
774 sum.push_back(expair(ex_to<numeric>(a.overall_coeff).power_dyn(_num2()),_ex1()));
777 GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
779 return (new add(sum))->setflag(status_flags::dynallocated | status_flags::expanded);
782 /** Expand factors of m in m^n where m is a mul and n is and integer
783 * @see power::expand */
784 ex power::expand_mul(const mul & m, const numeric & n) const
790 distrseq.reserve(m.seq.size());
791 epvector::const_iterator last = m.seq.end();
792 epvector::const_iterator cit = m.seq.begin();
794 if (is_ex_exactly_of_type((*cit).rest,numeric)) {
795 distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit,n));
797 // it is safe not to call mul::combine_pair_with_coeff_to_pair()
798 // since n is an integer
799 distrseq.push_back(expair((*cit).rest, ex_to<numeric>((*cit).coeff).mul(n)));
803 return (new mul(distrseq,ex_to<numeric>(m.overall_coeff).power_dyn(n)))->setflag(status_flags::dynallocated);
808 ex sqrt(const ex & a)
810 return power(a,_ex1_2());