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"
32 #include "relational.h"
38 #ifndef NO_NAMESPACE_GINAC
40 #endif // ndef NO_NAMESPACE_GINAC
42 GINAC_IMPLEMENT_REGISTERED_CLASS(power, basic)
44 typedef vector<int> intvector;
47 // default constructor, destructor, copy constructor assignment operator and helpers
52 power::power() : basic(TINFO_power)
54 debugmsg("power default constructor",LOGLEVEL_CONSTRUCT);
59 debugmsg("power destructor",LOGLEVEL_DESTRUCT);
63 power::power(const power & other)
65 debugmsg("power copy constructor",LOGLEVEL_CONSTRUCT);
69 const power & power::operator=(const power & other)
71 debugmsg("power operator=",LOGLEVEL_ASSIGNMENT);
81 void power::copy(const power & other)
83 inherited::copy(other);
85 exponent=other.exponent;
88 void power::destroy(bool call_parent)
90 if (call_parent) inherited::destroy(call_parent);
99 power::power(const ex & lh, const ex & rh) : basic(TINFO_power), basis(lh), exponent(rh)
101 debugmsg("power constructor from ex,ex",LOGLEVEL_CONSTRUCT);
102 GINAC_ASSERT(basis.return_type()==return_types::commutative);
105 power::power(const ex & lh, const numeric & rh) : basic(TINFO_power), basis(lh), exponent(rh)
107 debugmsg("power constructor from ex,numeric",LOGLEVEL_CONSTRUCT);
108 GINAC_ASSERT(basis.return_type()==return_types::commutative);
115 /** Construct object from archive_node. */
116 power::power(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
118 debugmsg("power constructor from archive_node", LOGLEVEL_CONSTRUCT);
119 n.find_ex("basis", basis, sym_lst);
120 n.find_ex("exponent", exponent, sym_lst);
123 /** Unarchive the object. */
124 ex power::unarchive(const archive_node &n, const lst &sym_lst)
126 return (new power(n, sym_lst))->setflag(status_flags::dynallocated);
129 /** Archive the object. */
130 void power::archive(archive_node &n) const
132 inherited::archive(n);
133 n.add_ex("basis", basis);
134 n.add_ex("exponent", exponent);
138 // functions overriding virtual functions from bases classes
143 basic * power::duplicate() const
145 debugmsg("power duplicate",LOGLEVEL_DUPLICATE);
146 return new power(*this);
149 void power::print(ostream & os, unsigned upper_precedence) const
151 debugmsg("power print",LOGLEVEL_PRINT);
152 if (exponent.is_equal(_ex1_2())) {
153 os << "sqrt(" << basis << ")";
155 if (precedence<=upper_precedence) os << "(";
156 basis.print(os,precedence);
158 exponent.print(os,precedence);
159 if (precedence<=upper_precedence) os << ")";
163 void power::printraw(ostream & os) const
165 debugmsg("power printraw",LOGLEVEL_PRINT);
170 exponent.printraw(os);
171 os << ",hash=" << hashvalue << ",flags=" << flags << ")";
174 void power::printtree(ostream & os, unsigned indent) const
176 debugmsg("power printtree",LOGLEVEL_PRINT);
178 os << string(indent,' ') << "power: "
179 << "hash=" << hashvalue << " (0x" << hex << hashvalue << dec << ")"
180 << ", flags=" << flags << endl;
181 basis.printtree(os,indent+delta_indent);
182 exponent.printtree(os,indent+delta_indent);
185 static void print_sym_pow(ostream & os, unsigned type, const symbol &x, int exp)
187 // Optimal output of integer powers of symbols to aid compiler CSE
189 x.printcsrc(os, type, 0);
190 } else if (exp == 2) {
191 x.printcsrc(os, type, 0);
193 x.printcsrc(os, type, 0);
194 } else if (exp & 1) {
197 print_sym_pow(os, type, x, exp-1);
200 print_sym_pow(os, type, x, exp >> 1);
202 print_sym_pow(os, type, x, exp >> 1);
207 void power::printcsrc(ostream & os, unsigned type, unsigned upper_precedence) const
209 debugmsg("power print csrc", LOGLEVEL_PRINT);
211 // Integer powers of symbols are printed in a special, optimized way
212 if (exponent.info(info_flags::integer) &&
213 (is_ex_exactly_of_type(basis, symbol) ||
214 is_ex_exactly_of_type(basis, constant))) {
215 int exp = ex_to_numeric(exponent).to_int();
220 if (type == csrc_types::ctype_cl_N)
225 print_sym_pow(os, type, static_cast<const symbol &>(*basis.bp), exp);
228 // <expr>^-1 is printed as "1.0/<expr>" or with the recip() function of CLN
229 } else if (exponent.compare(_num_1()) == 0) {
230 if (type == csrc_types::ctype_cl_N)
234 basis.bp->printcsrc(os, type, 0);
237 // Otherwise, use the pow() or expt() (CLN) functions
239 if (type == csrc_types::ctype_cl_N)
243 basis.bp->printcsrc(os, type, 0);
245 exponent.bp->printcsrc(os, type, 0);
250 bool power::info(unsigned inf) const
252 if (inf==info_flags::polynomial ||
253 inf==info_flags::integer_polynomial ||
254 inf==info_flags::cinteger_polynomial ||
255 inf==info_flags::rational_polynomial ||
256 inf==info_flags::crational_polynomial) {
257 return exponent.info(info_flags::nonnegint);
258 } else if (inf==info_flags::rational_function) {
259 return exponent.info(info_flags::integer);
261 return inherited::info(inf);
265 unsigned power::nops() const
270 ex & power::let_op(int i)
275 return i==0 ? basis : exponent;
278 int power::degree(const symbol & s) const
280 if (is_exactly_of_type(*exponent.bp,numeric)) {
281 if ((*basis.bp).compare(s)==0)
282 return ex_to_numeric(exponent).to_int();
284 return basis.degree(s) * ex_to_numeric(exponent).to_int();
289 int power::ldegree(const symbol & s) const
291 if (is_exactly_of_type(*exponent.bp,numeric)) {
292 if ((*basis.bp).compare(s)==0)
293 return ex_to_numeric(exponent).to_int();
295 return basis.ldegree(s) * ex_to_numeric(exponent).to_int();
300 ex power::coeff(const symbol & s, int n) const
302 if ((*basis.bp).compare(s)!=0) {
303 // basis not equal to s
309 } else if (is_exactly_of_type(*exponent.bp,numeric)&&
310 (static_cast<const numeric &>(*exponent.bp).compare(numeric(n))==0)) {
317 ex power::eval(int level) const
319 // simplifications: ^(x,0) -> 1 (0^0 handled here)
321 // ^(0,x) -> 0 (except if x is real and negative, in which case an exception is thrown)
323 // ^(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)
324 // ^(^(x,c1),c2) -> ^(x,c1*c2) (c1, c2 numeric(), c2 integer or -1 < c1 <= 1, case c1=1 should not happen, see below!)
325 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
326 // ^(*(x,c1),c2) -> ^(x,c2)*c1^c2 (c1, c2 numeric(), c1>0)
327 // ^(*(x,c1),c2) -> ^(-x,c2)*c1^c2 (c1, c2 numeric(), c1<0)
329 debugmsg("power eval",LOGLEVEL_MEMBER_FUNCTION);
331 if ((level==1)&&(flags & status_flags::evaluated)) {
333 } else if (level == -max_recursion_level) {
334 throw(std::runtime_error("max recursion level reached"));
337 const ex & ebasis = level==1 ? basis : basis.eval(level-1);
338 const ex & eexponent = level==1 ? exponent : exponent.eval(level-1);
340 bool basis_is_numerical=0;
341 bool exponent_is_numerical=0;
343 numeric * num_exponent;
345 if (is_exactly_of_type(*ebasis.bp,numeric)) {
346 basis_is_numerical=1;
347 num_basis=static_cast<numeric *>(ebasis.bp);
349 if (is_exactly_of_type(*eexponent.bp,numeric)) {
350 exponent_is_numerical=1;
351 num_exponent=static_cast<numeric *>(eexponent.bp);
354 // ^(x,0) -> 1 (0^0 also handled here)
355 if (eexponent.is_zero())
359 if (eexponent.is_equal(_ex1()))
362 // ^(0,x) -> 0 (except if x is real and negative)
363 if (ebasis.is_zero()) {
364 if (exponent_is_numerical && num_exponent->is_negative()) {
365 throw(std::overflow_error("power::eval(): division by zero"));
371 if (ebasis.is_equal(_ex1()))
374 if (basis_is_numerical && exponent_is_numerical) {
375 // ^(c1,c2) -> c1^c2 (c1, c2 numeric(),
376 // except if c1,c2 are rational, but c1^c2 is not)
377 bool basis_is_crational = num_basis->is_crational();
378 bool exponent_is_crational = num_exponent->is_crational();
379 numeric res = (*num_basis).power(*num_exponent);
381 if ((!basis_is_crational || !exponent_is_crational)
382 || res.is_crational()) {
385 GINAC_ASSERT(!num_exponent->is_integer()); // has been handled by now
386 // ^(c1,n/m) -> *(c1^q,c1^(n/m-q)), 0<(n/m-h)<1, q integer
387 if (basis_is_crational && exponent_is_crational
388 && num_exponent->is_real()
389 && !num_exponent->is_integer()) {
391 n = num_exponent->numer();
392 m = num_exponent->denom();
394 if (r.is_negative()) {
398 if (q.is_zero()) // the exponent was in the allowed range 0<(n/m)<1
402 res.push_back(expair(ebasis,r.div(m)));
403 res.push_back(expair(ex(num_basis->power(q)),_ex1()));
404 return (new mul(res))->setflag(status_flags::dynallocated | status_flags::evaluated);
405 /*return mul(num_basis->power(q),
406 power(ex(*num_basis),ex(r.div(m)))).hold();
408 /* return (new mul(num_basis->power(q),
409 power(*num_basis,r.div(m)).hold()))->setflag(status_flags::dynallocated | status_flags::evaluated);
415 // ^(^(x,c1),c2) -> ^(x,c1*c2)
416 // (c1, c2 numeric(), c2 integer or -1 < c1 <= 1,
417 // case c1=1 should not happen, see below!)
418 if (exponent_is_numerical && is_ex_exactly_of_type(ebasis,power)) {
419 const power & sub_power=ex_to_power(ebasis);
420 const ex & sub_basis=sub_power.basis;
421 const ex & sub_exponent=sub_power.exponent;
422 if (is_ex_exactly_of_type(sub_exponent,numeric)) {
423 const numeric & num_sub_exponent=ex_to_numeric(sub_exponent);
424 GINAC_ASSERT(num_sub_exponent!=numeric(1));
425 if (num_exponent->is_integer() || abs(num_sub_exponent)<1) {
426 return power(sub_basis,num_sub_exponent.mul(*num_exponent));
431 // ^(*(x,y,z),c1) -> *(x^c1,y^c1,z^c1) (c1 integer)
432 if (exponent_is_numerical && num_exponent->is_integer() &&
433 is_ex_exactly_of_type(ebasis,mul)) {
434 return expand_mul(ex_to_mul(ebasis), *num_exponent);
437 // ^(*(...,x;c1),c2) -> ^(*(...,x;1),c2)*c1^c2 (c1, c2 numeric(), c1>0)
438 // ^(*(...,x,c1),c2) -> ^(*(...,x;-1),c2)*(-c1)^c2 (c1, c2 numeric(), c1<0)
439 if (exponent_is_numerical && is_ex_exactly_of_type(ebasis,mul)) {
440 GINAC_ASSERT(!num_exponent->is_integer()); // should have been handled above
441 const mul & mulref=ex_to_mul(ebasis);
442 if (!mulref.overall_coeff.is_equal(_ex1())) {
443 const numeric & num_coeff=ex_to_numeric(mulref.overall_coeff);
444 if (num_coeff.is_real()) {
445 if (num_coeff.is_positive()>0) {
446 mul * mulp=new mul(mulref);
447 mulp->overall_coeff=_ex1();
448 mulp->clearflag(status_flags::evaluated);
449 mulp->clearflag(status_flags::hash_calculated);
450 return (new mul(power(*mulp,exponent),
451 power(num_coeff,*num_exponent)))->
452 setflag(status_flags::dynallocated);
454 GINAC_ASSERT(num_coeff.compare(_num0())<0);
455 if (num_coeff.compare(_num_1())!=0) {
456 mul * mulp=new mul(mulref);
457 mulp->overall_coeff=_ex_1();
458 mulp->clearflag(status_flags::evaluated);
459 mulp->clearflag(status_flags::hash_calculated);
460 return (new mul(power(*mulp,exponent),
461 power(abs(num_coeff),*num_exponent)))->
462 setflag(status_flags::dynallocated);
469 if (are_ex_trivially_equal(ebasis,basis) &&
470 are_ex_trivially_equal(eexponent,exponent)) {
473 return (new power(ebasis, eexponent))->setflag(status_flags::dynallocated |
474 status_flags::evaluated);
477 ex power::evalf(int level) const
479 debugmsg("power evalf",LOGLEVEL_MEMBER_FUNCTION);
487 } else if (level == -max_recursion_level) {
488 throw(std::runtime_error("max recursion level reached"));
490 ebasis=basis.evalf(level-1);
491 eexponent=exponent.evalf(level-1);
494 return power(ebasis,eexponent);
497 ex power::subs(const lst & ls, const lst & lr) const
499 const ex & subsed_basis=basis.subs(ls,lr);
500 const ex & subsed_exponent=exponent.subs(ls,lr);
502 if (are_ex_trivially_equal(basis,subsed_basis)&&
503 are_ex_trivially_equal(exponent,subsed_exponent)) {
507 return power(subsed_basis, subsed_exponent);
510 ex power::simplify_ncmul(const exvector & v) const
512 return inherited::simplify_ncmul(v);
517 int power::compare_same_type(const basic & other) const
519 GINAC_ASSERT(is_exactly_of_type(other, power));
520 const power & o=static_cast<const power &>(const_cast<basic &>(other));
523 cmpval=basis.compare(o.basis);
525 return exponent.compare(o.exponent);
530 unsigned power::return_type(void) const
532 return basis.return_type();
535 unsigned power::return_type_tinfo(void) const
537 return basis.return_type_tinfo();
540 ex power::expand(unsigned options) const
542 ex expanded_basis=basis.expand(options);
544 if (!is_ex_exactly_of_type(exponent,numeric)||
545 !ex_to_numeric(exponent).is_integer()) {
546 if (are_ex_trivially_equal(basis,expanded_basis)) {
549 return (new power(expanded_basis,exponent))->
550 setflag(status_flags::dynallocated);
554 // integer numeric exponent
555 const numeric & num_exponent=ex_to_numeric(exponent);
556 int int_exponent = num_exponent.to_int();
558 if (int_exponent > 0 && is_ex_exactly_of_type(expanded_basis,add)) {
559 return expand_add(ex_to_add(expanded_basis), int_exponent);
562 if (is_ex_exactly_of_type(expanded_basis,mul)) {
563 return expand_mul(ex_to_mul(expanded_basis), num_exponent);
566 // cannot expand further
567 if (are_ex_trivially_equal(basis,expanded_basis)) {
570 return (new power(expanded_basis,exponent))->
571 setflag(status_flags::dynallocated);
576 // new virtual functions which can be overridden by derived classes
582 // non-virtual functions in this class
585 ex power::expand_add(const add & a, int n) const
587 // expand a^n where a is an add and n is an integer
590 return expand_add_2(a);
595 sum.reserve((n+1)*(m-1));
597 intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
598 intvector upper_limit(m-1);
601 for (int l=0; l<m-1; l++) {
610 for (l=0; l<m-1; l++) {
611 const ex & b=a.op(l);
612 GINAC_ASSERT(!is_ex_exactly_of_type(b,add));
613 GINAC_ASSERT(!is_ex_exactly_of_type(b,power)||
614 !is_ex_exactly_of_type(ex_to_power(b).exponent,numeric)||
615 !ex_to_numeric(ex_to_power(b).exponent).is_pos_integer());
616 if (is_ex_exactly_of_type(b,mul)) {
617 term.push_back(expand_mul(ex_to_mul(b),numeric(k[l])));
619 term.push_back(power(b,k[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(n-k_cum[m-2])));
631 term.push_back(power(b,n-k_cum[m-2]));
634 numeric f=binomial(numeric(n),numeric(k[0]));
635 for (l=1; l<m-1; l++) {
636 f=f*binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
641 cout << "begin term" << endl;
642 for (int i=0; i<m-1; i++) {
643 cout << "k[" << i << "]=" << k[i] << endl;
644 cout << "k_cum[" << i << "]=" << k_cum[i] << endl;
645 cout << "upper_limit[" << i << "]=" << upper_limit[i] << endl;
647 for (exvector::const_iterator cit=term.begin(); cit!=term.end(); ++cit) {
648 cout << *cit << endl;
650 cout << "end term" << endl;
653 // TODO: optimize this
654 sum.push_back((new mul(term))->setflag(status_flags::dynallocated));
658 while ((l>=0)&&((++k[l])>upper_limit[l])) {
664 // recalc k_cum[] and upper_limit[]
668 k_cum[l]=k_cum[l-1]+k[l];
670 for (int i=l+1; i<m-1; i++) {
671 k_cum[i]=k_cum[i-1]+k[i];
674 for (int i=l+1; i<m-1; i++) {
675 upper_limit[i]=n-k_cum[i-1];
678 return (new add(sum))->setflag(status_flags::dynallocated);
681 ex power::expand_add_2(const add & a) const
683 // special case: expand a^2 where a is an add
686 unsigned a_nops=a.nops();
687 sum.reserve((a_nops*(a_nops+1))/2);
688 epvector::const_iterator last=a.seq.end();
690 // power(+(x,...,z;c),2)=power(+(x,...,z;0),2)+2*c*+(x,...,z;0)+c*c
691 // first part: ignore overall_coeff and expand other terms
692 for (epvector::const_iterator cit0=a.seq.begin(); cit0!=last; ++cit0) {
693 const ex & r=(*cit0).rest;
694 const ex & c=(*cit0).coeff;
696 GINAC_ASSERT(!is_ex_exactly_of_type(r,add));
697 GINAC_ASSERT(!is_ex_exactly_of_type(r,power)||
698 !is_ex_exactly_of_type(ex_to_power(r).exponent,numeric)||
699 !ex_to_numeric(ex_to_power(r).exponent).is_pos_integer()||
700 !is_ex_exactly_of_type(ex_to_power(r).basis,add)||
701 !is_ex_exactly_of_type(ex_to_power(r).basis,mul)||
702 !is_ex_exactly_of_type(ex_to_power(r).basis,power));
704 if (are_ex_trivially_equal(c,_ex1())) {
705 if (is_ex_exactly_of_type(r,mul)) {
706 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),_ex1()));
708 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
712 if (is_ex_exactly_of_type(r,mul)) {
713 sum.push_back(expair(expand_mul(ex_to_mul(r),_num2()),
714 ex_to_numeric(c).power_dyn(_num2())));
716 sum.push_back(expair((new power(r,_ex2()))->setflag(status_flags::dynallocated),
717 ex_to_numeric(c).power_dyn(_num2())));
721 for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
722 const ex & r1=(*cit1).rest;
723 const ex & c1=(*cit1).coeff;
724 sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
725 _num2().mul(ex_to_numeric(c)).mul_dyn(ex_to_numeric(c1))));
729 GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
731 // second part: add terms coming from overall_factor (if != 0)
732 if (!a.overall_coeff.is_equal(_ex0())) {
733 for (epvector::const_iterator cit=a.seq.begin(); cit!=a.seq.end(); ++cit) {
734 sum.push_back(a.combine_pair_with_coeff_to_pair(*cit,ex_to_numeric(a.overall_coeff).mul_dyn(_num2())));
736 sum.push_back(expair(ex_to_numeric(a.overall_coeff).power_dyn(_num2()),_ex1()));
739 GINAC_ASSERT(sum.size()==(a_nops*(a_nops+1))/2);
741 return (new add(sum))->setflag(status_flags::dynallocated);
744 ex power::expand_mul(const mul & m, const numeric & n) const
746 // expand m^n where m is a mul and n is and integer
748 if (n.is_equal(_num0())) {
753 distrseq.reserve(m.seq.size());
754 epvector::const_iterator last=m.seq.end();
755 epvector::const_iterator cit=m.seq.begin();
757 if (is_ex_exactly_of_type((*cit).rest,numeric)) {
758 distrseq.push_back(m.combine_pair_with_coeff_to_pair(*cit,n));
760 // it is safe not to call mul::combine_pair_with_coeff_to_pair()
761 // since n is an integer
762 distrseq.push_back(expair((*cit).rest,
763 ex_to_numeric((*cit).coeff).mul(n)));
767 return (new mul(distrseq,ex_to_numeric(m.overall_coeff).power_dyn(n)))
768 ->setflag(status_flags::dynallocated);
772 ex power::expand_commutative_3(const ex & basis, const numeric & exponent,
773 unsigned options) const
780 const add & addref=static_cast<const add &>(*basis.bp);
784 ex first_operands=add(splitseq);
785 ex last_operand=addref.recombine_pair_to_ex(*(addref.seq.end()-1));
787 int n=exponent.to_int();
788 for (int k=0; k<=n; k++) {
789 distrseq.push_back(binomial(n,k)*power(first_operands,numeric(k))*
790 power(last_operand,numeric(n-k)));
792 return ex((new add(distrseq))->setflag(status_flags::sub_expanded |
793 status_flags::expanded |
794 status_flags::dynallocated )).
800 ex power::expand_noncommutative(const ex & basis, const numeric & exponent,
801 unsigned options) const
803 ex rest_power=ex(power(basis,exponent.add(_num_1()))).
804 expand(options | expand_options::internal_do_not_expand_power_operands);
806 return ex(mul(rest_power,basis),0).
807 expand(options | expand_options::internal_do_not_expand_mul_operands);
812 // static member variables
817 unsigned power::precedence=60;
823 const power some_power;
824 const type_info & typeid_power=typeid(some_power);
828 ex sqrt(const ex & a)
830 return power(a,_ex1_2());
833 #ifndef NO_NAMESPACE_GINAC
835 #endif // ndef NO_NAMESPACE_GINAC