3 * Implementation of GiNaC's symbolic exponentiation (basis^exponent). */
6 * GiNaC Copyright (C) 1999-2000 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"
33 #include "relational.h"
39 #ifndef NO_NAMESPACE_GINAC
41 #endif // ndef NO_NAMESPACE_GINAC
43 GINAC_IMPLEMENT_REGISTERED_CLASS(power, basic)
45 typedef vector<int> intvector;
48 // default constructor, destructor, copy constructor assignment operator and helpers
53 power::power() : basic(TINFO_power)
55 debugmsg("power default constructor",LOGLEVEL_CONSTRUCT);
60 debugmsg("power destructor",LOGLEVEL_DESTRUCT);
64 power::power(const power & other)
66 debugmsg("power copy constructor",LOGLEVEL_CONSTRUCT);
70 const power & power::operator=(const power & other)
72 debugmsg("power operator=",LOGLEVEL_ASSIGNMENT);
82 void power::copy(const power & other)
84 inherited::copy(other);
86 exponent=other.exponent;
89 void power::destroy(bool call_parent)
91 if (call_parent) inherited::destroy(call_parent);
100 power::power(const ex & lh, const ex & rh) : basic(TINFO_power), basis(lh), exponent(rh)
102 debugmsg("power constructor from ex,ex",LOGLEVEL_CONSTRUCT);
103 GINAC_ASSERT(basis.return_type()==return_types::commutative);
106 power::power(const ex & lh, const numeric & rh) : basic(TINFO_power), basis(lh), exponent(rh)
108 debugmsg("power constructor from ex,numeric",LOGLEVEL_CONSTRUCT);
109 GINAC_ASSERT(basis.return_type()==return_types::commutative);
116 /** Construct object from archive_node. */
117 power::power(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
119 debugmsg("power constructor from archive_node", LOGLEVEL_CONSTRUCT);
120 n.find_ex("basis", basis, sym_lst);
121 n.find_ex("exponent", exponent, sym_lst);
124 /** Unarchive the object. */
125 ex power::unarchive(const archive_node &n, const lst &sym_lst)
127 return (new power(n, sym_lst))->setflag(status_flags::dynallocated);
130 /** Archive the object. */
131 void power::archive(archive_node &n) const
133 inherited::archive(n);
134 n.add_ex("basis", basis);
135 n.add_ex("exponent", exponent);
139 // functions overriding virtual functions from bases classes
144 basic * power::duplicate() const
146 debugmsg("power duplicate",LOGLEVEL_DUPLICATE);
147 return new power(*this);
150 void power::print(ostream & os, unsigned upper_precedence) const
152 debugmsg("power print",LOGLEVEL_PRINT);
153 if (exponent.is_equal(_ex1_2())) {
154 os << "sqrt(" << basis << ")";
156 if (precedence<=upper_precedence) os << "(";
157 basis.print(os,precedence);
159 exponent.print(os,precedence);
160 if (precedence<=upper_precedence) os << ")";
164 void power::printraw(ostream & os) const
166 debugmsg("power printraw",LOGLEVEL_PRINT);
171 exponent.printraw(os);
172 os << ",hash=" << hashvalue << ",flags=" << flags << ")";
175 void power::printtree(ostream & os, unsigned indent) const
177 debugmsg("power printtree",LOGLEVEL_PRINT);
179 os << string(indent,' ') << "power: "
180 << "hash=" << hashvalue << " (0x" << hex << hashvalue << dec << ")"
181 << ", flags=" << flags << endl;
182 basis.printtree(os,indent+delta_indent);
183 exponent.printtree(os,indent+delta_indent);
186 static void print_sym_pow(ostream & os, unsigned type, const symbol &x, int exp)
188 // Optimal output of integer powers of symbols to aid compiler CSE
190 x.printcsrc(os, type, 0);
191 } else if (exp == 2) {
192 x.printcsrc(os, type, 0);
194 x.printcsrc(os, type, 0);
195 } else if (exp & 1) {
198 print_sym_pow(os, type, x, exp-1);
201 print_sym_pow(os, type, x, exp >> 1);
203 print_sym_pow(os, type, x, exp >> 1);
208 void power::printcsrc(ostream & os, unsigned type, unsigned upper_precedence) const
210 debugmsg("power print csrc", LOGLEVEL_PRINT);
212 // Integer powers of symbols are printed in a special, optimized way
213 if (exponent.info(info_flags::integer) &&
214 (is_ex_exactly_of_type(basis, symbol) ||
215 is_ex_exactly_of_type(basis, constant))) {
216 int exp = ex_to_numeric(exponent).to_int();
221 if (type == csrc_types::ctype_cl_N)
226 print_sym_pow(os, type, static_cast<const symbol &>(*basis.bp), exp);
229 // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
230 } else if (exponent.compare(_num_1()) == 0) {
231 if (type == csrc_types::ctype_cl_N)
235 basis.bp->printcsrc(os, type, 0);
238 // Otherwise, use the pow() or expt() (CLN) functions
240 if (type == csrc_types::ctype_cl_N)
244 basis.bp->printcsrc(os, type, 0);
246 exponent.bp->printcsrc(os, type, 0);
251 bool power::info(unsigned inf) const
253 if (inf==info_flags::polynomial ||
254 inf==info_flags::integer_polynomial ||
255 inf==info_flags::cinteger_polynomial ||
256 inf==info_flags::rational_polynomial ||
257 inf==info_flags::crational_polynomial) {
258 return exponent.info(info_flags::nonnegint);
259 } else if (inf==info_flags::rational_function) {
260 return exponent.info(info_flags::integer);
262 return inherited::info(inf);
266 unsigned power::nops() const
271 ex & power::let_op(int i)
276 return i==0 ? basis : exponent;
279 int power::degree(const symbol & s) const
281 if (is_exactly_of_type(*exponent.bp,numeric)) {
282 if ((*basis.bp).compare(s)==0)
283 return ex_to_numeric(exponent).to_int();
285 return basis.degree(s) * ex_to_numeric(exponent).to_int();
290 int power::ldegree(const symbol & s) const
292 if (is_exactly_of_type(*exponent.bp,numeric)) {
293 if ((*basis.bp).compare(s)==0)
294 return ex_to_numeric(exponent).to_int();
296 return basis.ldegree(s) * ex_to_numeric(exponent).to_int();
301 ex power::coeff(const symbol & s, int n) const
303 if ((*basis.bp).compare(s)!=0) {
304 // basis not equal to s
310 } else if (is_exactly_of_type(*exponent.bp,numeric)&&
311 (static_cast<const numeric &>(*exponent.bp).compare(numeric(n))==0)) {
318 ex power::eval(int level) const
320 // simplifications: ^(x,0) -> 1 (0^0 handled here)
322 // ^(0,x) -> 0 (except if x is real and negative, in which case an exception is thrown)
324 // ^(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)
325 // ^(^(x,c1),c2) -> ^(x,c1*c2) (c1, c2 numeric(), c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
326 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
327 // ^(*(x,c1),c2) -> ^(x,c2)*c1^c2 (c1, c2 numeric(), c1>0)
328 // ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2 (c1, c2 numeric(), c1<0)
330 debugmsg("power eval",LOGLEVEL_MEMBER_FUNCTION);
332 if ((level==1)&&(flags & status_flags::evaluated)) {
334 } else if (level == -max_recursion_level) {
335 throw(std::runtime_error("max recursion level reached"));
338 const ex & ebasis = level==1 ? basis : basis.eval(level-1);
339 const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
341 bool basis_is_numerical = 0;
342 bool exponent_is_numerical = 0;
344 numeric * num_exponent;
346 if (is_exactly_of_type(*ebasis.bp,numeric)) {
347 basis_is_numerical = 1;
348 num_basis = static_cast<numeric *>(ebasis.bp);
350 if (is_exactly_of_type(*eexponent.bp,numeric)) {
351 exponent_is_numerical = 1;
352 num_exponent = static_cast<numeric *>(eexponent.bp);
355 // ^(x,0) -> 1 (0^0 also handled here)
356 if (eexponent.is_zero())
357 if (ebasis.is_zero())
358 throw (std::domain_error("power::eval(): pow(0,0) is undefined"));
363 if (eexponent.is_equal(_ex1()))
366 // ^(0,x) -> 0 (except if x is real and negative)
367 if (ebasis.is_zero()) {
368 if (exponent_is_numerical && num_exponent->is_negative()) {
369 throw(std::overflow_error("power::eval(): division by zero"));
375 if (ebasis.is_equal(_ex1()))
378 if (basis_is_numerical && exponent_is_numerical) {
379 // ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
380 // except if c1,c2 are rational, but c1^c2 is not)
381 bool basis_is_crational = num_basis->is_crational();
382 bool exponent_is_crational = num_exponent->is_crational();
383 numeric res = (*num_basis).power(*num_exponent);
385 if ((!basis_is_crational || !exponent_is_crational)
386 || res.is_crational()) {
389 GINAC_ASSERT(!num_exponent->is_integer()); // has been handled by now
390 // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-h)<1, q integer
391 if (basis_is_crational && exponent_is_crational
392 && num_exponent->is_real()
393 && !num_exponent->is_integer()) {
394 numeric n = num_exponent->numer();
395 numeric m = num_exponent->denom();
397 numeric q = iquo(n, m, r);
398 if (r.is_negative()) {
402 if (q.is_zero()) // the exponent was in the allowed range 0<(n/m)<1
406 res.push_back(expair(ebasis,r.div(m)));
407 return (new mul(res,ex(num_basis->power(q))))->setflag(status_flags::dynallocated | status_flags::evaluated);
412 // ^(^(x,c1),c2) -> ^(x,c1*c2)
413 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
414 // case c1==1 should not happen, see below!)
415 if (exponent_is_numerical && is_ex_exactly_of_type(ebasis,power)) {
416 const power & sub_power = ex_to_power(ebasis);
417 const ex & sub_basis = sub_power.basis;
418 const ex & sub_exponent = sub_power.exponent;
419 if (is_ex_exactly_of_type(sub_exponent,numeric)) {
420 const numeric & num_sub_exponent = ex_to_numeric(sub_exponent);
421 GINAC_ASSERT(num_sub_exponent!=numeric(1));
422 if (num_exponent->is_integer() || abs(num_sub_exponent)<1) {
423 return power(sub_basis,num_sub_exponent.mul(*num_exponent));
428 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
429 if (exponent_is_numerical && num_exponent->is_integer() &&
430 is_ex_exactly_of_type(ebasis,mul)) {
431 return expand_mul(ex_to_mul(ebasis), *num_exponent);
434 // ^(*(...,x;c1),c2) -> ^(*(...,x;1),c2)*c1^c2 (c1, c2 numeric(), c1>0)
435 // ^(*(...,x,c1),c2) -> ^(*(...,x;-1),c2)*(-c1)^c2 (c1, c2 numeric(), c1<0)
436 if (exponent_is_numerical && is_ex_exactly_of_type(ebasis,mul)) {
437 GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
438 const mul & mulref=ex_to_mul(ebasis);
439 if (!mulref.overall_coeff.is_equal(_ex1())) {
440 const numeric & num_coeff=ex_to_numeric(mulref.overall_coeff);
441 if (num_coeff.is_real()) {
442 if (num_coeff.is_positive()>0) {
443 mul * mulp=new mul(mulref);
444 mulp->overall_coeff=_ex1();
445 mulp->clearflag(status_flags::evaluated);
446 mulp->clearflag(status_flags::hash_calculated);
447 return (new mul(power(*mulp,exponent),
448 power(num_coeff,*num_exponent)))->
449 setflag(status_flags::dynallocated);
451 GINAC_ASSERT(num_coeff.compare(_num0())<0);
452 if (num_coeff.compare(_num_1())!=0) {
453 mul * mulp=new mul(mulref);
454 mulp->overall_coeff=_ex_1();
455 mulp->clearflag(status_flags::evaluated);
456 mulp->clearflag(status_flags::hash_calculated);
457 return (new mul(power(*mulp,exponent),
458 power(abs(num_coeff),*num_exponent)))->
459 setflag(status_flags::dynallocated);
466 if (are_ex_trivially_equal(ebasis,basis) &&
467 are_ex_trivially_equal(eexponent,exponent)) {
470 return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
471 status_flags::evaluated);
474 ex power::evalf(int level) const
476 debugmsg("power evalf",LOGLEVEL_MEMBER_FUNCTION);
484 } else if (level == -max_recursion_level) {
485 throw(std::runtime_error("max recursion level reached"));
487 ebasis=basis.evalf(level-1);
488 eexponent=exponent.evalf(level-1);
491 return power(ebasis,eexponent);
494 ex power::subs(const lst & ls, const lst & lr) const
496 const ex & subsed_basis=basis.subs(ls,lr);
497 const ex & subsed_exponent=exponent.subs(ls,lr);
499 if (are_ex_trivially_equal(basis,subsed_basis)&&
500 are_ex_trivially_equal(exponent,subsed_exponent)) {
504 return power(subsed_basis, subsed_exponent);
507 ex power::simplify_ncmul(const exvector & v) const
509 return inherited::simplify_ncmul(v);
514 /** Implementation of ex::diff() for a power.
516 ex power::derivative(const symbol & s) const
518 if (exponent.info(info_flags::real)) {
519 // D(b^r) = r * b^(r-1) * D(b) (faster than the formula below)
520 return mul(mul(exponent, power(basis, exponent - _ex1())), basis.diff(s));
522 // D(b^e) = b^e * (D(e)*ln(b) + e*D(b)/b)
523 return mul(power(basis, exponent),
524 add(mul(exponent.diff(s), log(basis)),
525 mul(mul(exponent, basis.diff(s)), power(basis, -1))));
529 int power::compare_same_type(const basic & other) const
531 GINAC_ASSERT(is_exactly_of_type(other, power));
532 const power & o=static_cast<const power &>(const_cast<basic &>(other));
535 cmpval=basis.compare(o.basis);
537 return exponent.compare(o.exponent);
542 unsigned power::return_type(void) const
544 return basis.return_type();
547 unsigned power::return_type_tinfo(void) const
549 return basis.return_type_tinfo();
552 ex power::expand(unsigned options) const
554 ex expanded_basis=basis.expand(options);
556 if (!is_ex_exactly_of_type(exponent,numeric)||
557 !ex_to_numeric(exponent).is_integer()) {
558 if (are_ex_trivially_equal(basis,expanded_basis)) {
561 return (new power(expanded_basis,exponent))->
562 setflag(status_flags::dynallocated);
566 // integer numeric exponent
567 const numeric & num_exponent=ex_to_numeric(exponent);
568 int int_exponent = num_exponent.to_int();
570 if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis,add)) {
571 return expand_add(ex_to_add(expanded_basis), int_exponent);
574 if (is_ex_exactly_of_type(expanded_basis,mul)) {
575 return expand_mul(ex_to_mul(expanded_basis), num_exponent);
578 // cannot expand further
579 if (are_ex_trivially_equal(basis,expanded_basis)) {
582 return (new power(expanded_basis,exponent))->
583 setflag(status_flags::dynallocated);
588 // new virtual functions which can be overridden by derived classes
594 // non-virtual functions in this class
597 ex power::expand_add(const add & a, int n) const
599 // expand a^n where a is an add and n is an integer
602 return expand_add_2(a);
607 sum.reserve((n+1)*(m-1));
609 intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
610 intvector upper_limit(m-1);
613 for (int l=0; l<m-1; l++) {
622 for (l=0; l<m-1; l++) {
623 const ex & b=a.op(l);
624 GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
625 GINAC_ASSERT(!is_ex_exactly_of_type(b,power)||
626 !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric)||
627 !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer());
628 if (is_ex_exactly_of_type(b,mul)) {
629 term.push_back(expand_mul(ex_to_mul(b),numeric(k[l])));
631 term.push_back(power(b,k[l]));
635 const ex & b=a.op(l);
636 GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
637 GINAC_ASSERT(!is_ex_exactly_of_type(b,power)||
638 !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric)||
639 !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer());
640 if (is_ex_exactly_of_type(b,mul)) {
641 term.push_back(expand_mul(ex_to_mul(b),numeric(n-k_cum[m-2])));
643 term.push_back(power(b,n-k_cum[m-2]));
646 numeric f=binomial(numeric(n),numeric(k[0]));
647 for (l=1; l<m-1; l++) {
648 f=f*binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
653 cout << "begin term" << endl;
654 for (int i=0; i<m-1; i++) {
655 cout << "k[" << i << "]=" << k[i] << endl;
656 cout << "k_cum[" << i << "]=" << k_cum[i] << endl;
657 cout << "upper_limit[" << i << "]=" << upper_limit[i] << endl;
659 for (exvector::const_iterator cit=term.begin(); cit!=term.end(); ++cit) {
660 cout << *cit << endl;
662 cout << "end term" << endl;
665 // TODO: optimize this
666 sum.push_back((new mul(term))->setflag(status_flags::dynallocated));
670 while ((l>=0)&&((++k[l])>upper_limit[l])) {
676 // recalc k_cum[] and upper_limit[]
680 k_cum[l]=k_cum[l-1]+k[l];
682 for (int i=l+1; i<m-1; i++) {
683 k_cum[i]=k_cum[i-1]+k[i];
686 for (int i=l+1; i<m-1; i++) {
687 upper_limit[i]=n-k_cum[i-1];
690 return (new add(sum))->setflag(status_flags::dynallocated);
693 ex power::expand_add_2(const add & a) const
695 // special case: expand a^2 where a is an add
698 unsigned a_nops=a.nops();
699 sum.reserve((a_nops*(a_nops+1))/2);
700 epvector::const_iterator last=a.seq.end();
702 // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
703 // first part: ignore overall_coeff and expand other terms
704 for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
705 const ex & r=(*cit0).rest;
706 const ex & c=(*cit0).coeff;
708 GINAC_ASSERT(!is_ex_exactly_of_type(r,add));
709 GINAC_ASSERT(!is_ex_exactly_of_type(r,power)||
710 !is_ex_exactly_of_type(ex_to_power(r).exponent,numeric)||
711 !ex_to_numeric(ex_to_power(r).exponent).is_pos_integer()||
712 !is_ex_exactly_of_type(ex_to_power(r).basis,add)||
713 !is_ex_exactly_of_type(ex_to_power(r).basis,mul)||
714 !is_ex_exactly_of_type(ex_to_power(r).basis,power));
716 if (are_ex_trivially_equal(c,_ex1())) {
717 if (is_ex_exactly_of_type(r,mul)) {
718 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),_ex1()));
720 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
724 if (is_ex_exactly_of_type(r,mul)) {
725 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),
726 ex_to_numeric(c).power_dyn(_num2())));
728 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
729 ex_to_numeric(c).power_dyn(_num2())));
733 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
734 const ex & r1=(*cit1).rest;
735 const ex & c1=(*cit1).coeff;
736 sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
737 _num2().mul(ex_to_numeric(c)).mul_dyn(ex_to_numeric(c1))));
741 GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
743 // second part: add terms coming from overall_factor (if != 0)
744 if (!a.overall_coeff.is_equal(_ex0())) {
745 for (epvector::const_iterator cit=a.seq.begin(); cit!=a.seq.end(); ++cit) {
746 sum.push_back(a.combine_pair_with_coeff_to_pair(*cit,ex_to_numeric(a.overall_coeff).mul_dyn(_num2())));
748 sum.push_back(expair(ex_to_numeric(a.overall_coeff).power_dyn(_num2()),_ex1()));
751 GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
753 return (new add(sum))->setflag(status_flags::dynallocated);
756 ex power::expand_mul(const mul & m, const numeric & n) const
758 // expand m^n where m is a mul and n is and integer
760 if (n.is_equal(_num0())) {
765 distrseq.reserve(m.seq.size());
766 epvector::const_iterator last=m.seq.end();
767 epvector::const_iterator cit=m.seq.begin();
769 if (is_ex_exactly_of_type((*cit).rest,numeric)) {
770 distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit,n));
772 // it is safe not to call mul::combine_pair_with_coeff_to_pair()
773 // since n is an integer
774 distrseq.push_back(expair((*cit).rest,
775 ex_to_numeric((*cit).coeff).mul(n)));
779 return (new mul(distrseq,ex_to_numeric(m.overall_coeff).power_dyn(n)))
780 ->setflag(status_flags::dynallocated);
784 ex power::expand_commutative_3(const ex & basis, const numeric & exponent,
785 unsigned options) const
792 const add & addref=static_cast<const add &>(*basis.bp);
796 ex first_operands=add(splitseq);
797 ex last_operand=addref.recombine_pair_to_ex(*(addref.seq.end()-1));
799 int n=exponent.to_int();
800 for (int k=0; k<=n; k++) {
801 distrseq.push_back(binomial(n,k)*power(first_operands,numeric(k))*
802 power(last_operand,numeric(n-k)));
804 return ex((new add(distrseq))->setflag(status_flags::sub_expanded |
805 status_flags::expanded |
806 status_flags::dynallocated )).
812 ex power::expand_noncommutative(const ex & basis, const numeric & exponent,
813 unsigned options) const
815 ex rest_power=ex(power(basis,exponent.add(_num_1()))).
816 expand(options | expand_options::internal_do_not_expand_power_operands);
818 return ex(mul(rest_power,basis),0).
819 expand(options | expand_options::internal_do_not_expand_mul_operands);
824 // static member variables
829 unsigned power::precedence=60;
835 const power some_power;
836 const type_info & typeid_power=typeid(some_power);
840 ex sqrt(const ex & a)
842 return power(a,_ex1_2());
845 #ifndef NO_NAMESPACE_GINAC
847 #endif // ndef NO_NAMESPACE_GINAC