3 * Implementation of class for extended truncated power series and
4 * methods for series expansion. */
7 * GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany
9 * This program is free software; you can redistribute it and/or modify
10 * it under the terms of the GNU General Public License as published by
11 * the Free Software Foundation; either version 2 of the License, or
12 * (at your option) any later version.
14 * This program is distributed in the hope that it will be useful,
15 * but WITHOUT ANY WARRANTY; without even the implied warranty of
16 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
17 * GNU General Public License for more details.
19 * You should have received a copy of the GNU General Public License
20 * along with this program; if not, write to the Free Software
21 * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
32 #include "relational.h"
41 GINAC_IMPLEMENT_REGISTERED_CLASS(pseries, basic)
45 * Default ctor, dtor, copy ctor, assignment operator and helpers
48 pseries::pseries() : basic(TINFO_pseries)
50 debugmsg("pseries default ctor", LOGLEVEL_CONSTRUCT);
53 void pseries::copy(const pseries &other)
55 inherited::copy(other);
61 DEFAULT_DESTROY(pseries)
68 /** Construct pseries from a vector of coefficients and powers.
69 * expair.rest holds the coefficient, expair.coeff holds the power.
70 * The powers must be integers (positive or negative) and in ascending order;
71 * the last coefficient can be Order(_ex1()) to represent a truncated,
72 * non-terminating series.
74 * @param rel_ expansion variable and point (must hold a relational)
75 * @param ops_ vector of {coefficient, power} pairs (coefficient must not be zero)
76 * @return newly constructed pseries */
77 pseries::pseries(const ex &rel_, const epvector &ops_) : basic(TINFO_pseries), seq(ops_)
79 debugmsg("pseries ctor from ex,epvector", LOGLEVEL_CONSTRUCT);
80 GINAC_ASSERT(is_ex_exactly_of_type(rel_, relational));
81 GINAC_ASSERT(is_ex_exactly_of_type(rel_.lhs(),symbol));
83 var = *static_cast<symbol *>(rel_.lhs().bp);
91 pseries::pseries(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
93 debugmsg("pseries ctor from archive_node", LOGLEVEL_CONSTRUCT);
94 for (unsigned int i=0; true; ++i) {
97 if (n.find_ex("coeff", rest, sym_lst, i) && n.find_ex("power", coeff, sym_lst, i))
98 seq.push_back(expair(rest, coeff));
102 n.find_ex("var", var, sym_lst);
103 n.find_ex("point", point, sym_lst);
106 void pseries::archive(archive_node &n) const
108 inherited::archive(n);
109 epvector::const_iterator i = seq.begin(), iend = seq.end();
111 n.add_ex("coeff", i->rest);
112 n.add_ex("power", i->coeff);
115 n.add_ex("var", var);
116 n.add_ex("point", point);
119 DEFAULT_UNARCHIVE(pseries)
122 // functions overriding virtual functions from bases classes
125 void pseries::print(const print_context & c, unsigned level) const
127 debugmsg("pseries print", LOGLEVEL_PRINT);
129 if (is_of_type(c, print_tree)) {
131 c.s << std::string(level, ' ') << class_name()
132 << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
134 unsigned delta_indent = static_cast<const print_tree &>(c).delta_indent;
135 for (unsigned i=0; i<seq.size(); ++i) {
136 seq[i].rest.print(c, level + delta_indent);
137 seq[i].coeff.print(c, level + delta_indent);
138 c.s << std::string(level + delta_indent, ' ') << "-----" << std::endl;
140 var.print(c, level + delta_indent);
141 point.print(c, level + delta_indent);
145 if (precedence <= level)
148 // objects of type pseries must not have any zero entries, so the
149 // trivial (zero) pseries needs a special treatment here:
152 for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
153 // print a sign, if needed
154 if (i != seq.begin())
156 if (!is_order_function(i->rest)) {
157 // print 'rest', i.e. the expansion coefficient
158 if (i->rest.info(info_flags::numeric) &&
159 i->rest.info(info_flags::positive)) {
166 // print 'coeff', something like (x-1)^42
167 if (!i->coeff.is_zero()) {
169 if (!point.is_zero()) {
171 (var-point).print(c);
175 if (i->coeff.compare(_ex1())) {
177 if (i->coeff.info(info_flags::negative)) {
186 Order(power(var-point,i->coeff)).print(c);
189 if (precedence <= level)
194 int pseries::compare_same_type(const basic & other) const
196 GINAC_ASSERT(is_of_type(other, pseries));
197 const pseries &o = static_cast<const pseries &>(other);
199 // first compare the lengths of the series...
200 if (seq.size()>o.seq.size())
202 if (seq.size()<o.seq.size())
205 // ...then the expansion point...
206 int cmpval = var.compare(o.var);
209 cmpval = point.compare(o.point);
213 // ...and if that failed the individual elements
214 epvector::const_iterator it = seq.begin(), o_it = o.seq.begin();
215 while (it!=seq.end() && o_it!=o.seq.end()) {
216 cmpval = it->compare(*o_it);
223 // so they are equal.
227 /** Return the number of operands including a possible order term. */
228 unsigned pseries::nops(void) const
234 /** Return the ith term in the series when represented as a sum. */
235 ex pseries::op(int i) const
237 if (i < 0 || unsigned(i) >= seq.size())
238 throw (std::out_of_range("op() out of range"));
239 return seq[i].rest * power(var - point, seq[i].coeff);
243 ex &pseries::let_op(int i)
245 throw (std::logic_error("let_op not defined for pseries"));
249 /** Return degree of highest power of the series. This is usually the exponent
250 * of the Order term. If s is not the expansion variable of the series, the
251 * series is examined termwise. */
252 int pseries::degree(const ex &s) const
254 if (var.is_equal(s)) {
255 // Return last exponent
257 return ex_to_numeric((*(seq.end() - 1)).coeff).to_int();
261 epvector::const_iterator it = seq.begin(), itend = seq.end();
264 int max_pow = INT_MIN;
265 while (it != itend) {
266 int pow = it->rest.degree(s);
275 /** Return degree of lowest power of the series. This is usually the exponent
276 * of the leading term. If s is not the expansion variable of the series, the
277 * series is examined termwise. If s is the expansion variable but the
278 * expansion point is not zero the series is not expanded to find the degree.
279 * I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
280 int pseries::ldegree(const ex &s) const
282 if (var.is_equal(s)) {
283 // Return first exponent
285 return ex_to_numeric((*(seq.begin())).coeff).to_int();
289 epvector::const_iterator it = seq.begin(), itend = seq.end();
292 int min_pow = INT_MAX;
293 while (it != itend) {
294 int pow = it->rest.ldegree(s);
303 /** Return coefficient of degree n in power series if s is the expansion
304 * variable. If the expansion point is nonzero, by definition the n=1
305 * coefficient in s of a+b*(s-z)+c*(s-z)^2+Order((s-z)^3) is b (assuming
306 * the expansion took place in the s in the first place).
307 * If s is not the expansion variable, an attempt is made to convert the
308 * series to a polynomial and return the corresponding coefficient from
310 ex pseries::coeff(const ex &s, int n) const
312 if (var.is_equal(s)) {
316 // Binary search in sequence for given power
317 numeric looking_for = numeric(n);
318 int lo = 0, hi = seq.size() - 1;
320 int mid = (lo + hi) / 2;
321 GINAC_ASSERT(is_ex_exactly_of_type(seq[mid].coeff, numeric));
322 int cmp = ex_to_numeric(seq[mid].coeff).compare(looking_for);
328 return seq[mid].rest;
333 throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
338 return convert_to_poly().coeff(s, n);
342 ex pseries::collect(const ex &s) const
348 /** Evaluate coefficients. */
349 ex pseries::eval(int level) const
354 if (level == -max_recursion_level)
355 throw (std::runtime_error("pseries::eval(): recursion limit exceeded"));
357 // Construct a new series with evaluated coefficients
359 new_seq.reserve(seq.size());
360 epvector::const_iterator it = seq.begin(), itend = seq.end();
361 while (it != itend) {
362 new_seq.push_back(expair(it->rest.eval(level-1), it->coeff));
365 return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
369 /** Evaluate coefficients numerically. */
370 ex pseries::evalf(int level) const
375 if (level == -max_recursion_level)
376 throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
378 // Construct a new series with evaluated coefficients
380 new_seq.reserve(seq.size());
381 epvector::const_iterator it = seq.begin(), itend = seq.end();
382 while (it != itend) {
383 new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff));
386 return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
390 ex pseries::subs(const lst & ls, const lst & lr) const
392 // If expansion variable is being substituted, convert the series to a
393 // polynomial and do the substitution there because the result might
394 // no longer be a power series
396 return convert_to_poly(true).subs(ls, lr);
398 // Otherwise construct a new series with substituted coefficients and
401 newseq.reserve(seq.size());
402 epvector::const_iterator it = seq.begin(), itend = seq.end();
403 while (it != itend) {
404 newseq.push_back(expair(it->rest.subs(ls, lr), it->coeff));
407 return (new pseries(relational(var,point.subs(ls, lr)), newseq))->setflag(status_flags::dynallocated);
411 /** Implementation of ex::expand() for a power series. It expands all the
412 * terms individually and returns the resulting series as a new pseries. */
413 ex pseries::expand(unsigned options) const
416 for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
417 ex restexp = i->rest.expand();
418 if (!restexp.is_zero())
419 newseq.push_back(expair(restexp, i->coeff));
421 return (new pseries(relational(var,point), newseq))
422 ->setflag(status_flags::dynallocated | status_flags::expanded);
426 /** Implementation of ex::diff() for a power series. It treats the series as a
429 ex pseries::derivative(const symbol & s) const
433 epvector::const_iterator it = seq.begin(), itend = seq.end();
435 // FIXME: coeff might depend on var
436 while (it != itend) {
437 if (is_order_function(it->rest)) {
438 new_seq.push_back(expair(it->rest, it->coeff - 1));
440 ex c = it->rest * it->coeff;
442 new_seq.push_back(expair(c, it->coeff - 1));
446 return pseries(relational(var,point), new_seq);
453 /** Convert a pseries object to an ordinary polynomial.
455 * @param no_order flag: discard higher order terms */
456 ex pseries::convert_to_poly(bool no_order) const
459 epvector::const_iterator it = seq.begin(), itend = seq.end();
461 while (it != itend) {
462 if (is_order_function(it->rest)) {
464 e += Order(power(var - point, it->coeff));
466 e += it->rest * power(var - point, it->coeff);
473 /** Returns true if there is no order term, i.e. the series terminates and
474 * false otherwise. */
475 bool pseries::is_terminating(void) const
477 return seq.size() == 0 || !is_order_function((seq.end()-1)->rest);
482 * Implementations of series expansion
485 /** Default implementation of ex::series(). This performs Taylor expansion.
487 ex basic::series(const relational & r, int order, unsigned options) const
492 ex coeff = deriv.subs(r);
493 const symbol &s = static_cast<symbol &>(*r.lhs().bp);
495 if (!coeff.is_zero())
496 seq.push_back(expair(coeff, numeric(0)));
499 for (n=1; n<order; ++n) {
500 fac = fac.mul(numeric(n));
501 deriv = deriv.diff(s).expand();
502 if (deriv.is_zero()) {
504 return pseries(r, seq);
506 coeff = deriv.subs(r);
507 if (!coeff.is_zero())
508 seq.push_back(expair(fac.inverse() * coeff, numeric(n)));
511 // Higher-order terms, if present
512 deriv = deriv.diff(s);
513 if (!deriv.expand().is_zero())
514 seq.push_back(expair(Order(_ex1()), numeric(n)));
515 return pseries(r, seq);
519 /** Implementation of ex::series() for symbols.
521 ex symbol::series(const relational & r, int order, unsigned options) const
524 const ex point = r.rhs();
525 GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
528 if (this->is_equal(*s.bp)) {
529 if (order > 0 && !point.is_zero())
530 seq.push_back(expair(point, _ex0()));
532 seq.push_back(expair(_ex1(), _ex1()));
534 seq.push_back(expair(Order(_ex1()), numeric(order)));
536 seq.push_back(expair(*this, _ex0()));
537 return pseries(r, seq);
541 /** Add one series object to another, producing a pseries object that
542 * represents the sum.
544 * @param other pseries object to add with
545 * @return the sum as a pseries */
546 ex pseries::add_series(const pseries &other) const
548 // Adding two series with different variables or expansion points
549 // results in an empty (constant) series
550 if (!is_compatible_to(other)) {
552 nul.push_back(expair(Order(_ex1()), _ex0()));
553 return pseries(relational(var,point), nul);
558 epvector::const_iterator a = seq.begin();
559 epvector::const_iterator b = other.seq.begin();
560 epvector::const_iterator a_end = seq.end();
561 epvector::const_iterator b_end = other.seq.end();
562 int pow_a = INT_MAX, pow_b = INT_MAX;
564 // If a is empty, fill up with elements from b and stop
567 new_seq.push_back(*b);
572 pow_a = ex_to_numeric((*a).coeff).to_int();
574 // If b is empty, fill up with elements from a and stop
577 new_seq.push_back(*a);
582 pow_b = ex_to_numeric((*b).coeff).to_int();
584 // a and b are non-empty, compare powers
586 // a has lesser power, get coefficient from a
587 new_seq.push_back(*a);
588 if (is_order_function((*a).rest))
591 } else if (pow_b < pow_a) {
592 // b has lesser power, get coefficient from b
593 new_seq.push_back(*b);
594 if (is_order_function((*b).rest))
598 // Add coefficient of a and b
599 if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
600 new_seq.push_back(expair(Order(_ex1()), (*a).coeff));
601 break; // Order term ends the sequence
603 ex sum = (*a).rest + (*b).rest;
604 if (!(sum.is_zero()))
605 new_seq.push_back(expair(sum, numeric(pow_a)));
611 return pseries(relational(var,point), new_seq);
615 /** Implementation of ex::series() for sums. This performs series addition when
616 * adding pseries objects.
618 ex add::series(const relational & r, int order, unsigned options) const
620 ex acc; // Series accumulator
622 // Get first term from overall_coeff
623 acc = overall_coeff.series(r, order, options);
625 // Add remaining terms
626 epvector::const_iterator it = seq.begin();
627 epvector::const_iterator itend = seq.end();
628 for (; it!=itend; ++it) {
630 if (is_ex_exactly_of_type(it->rest, pseries))
633 op = it->rest.series(r, order, options);
634 if (!it->coeff.is_equal(_ex1()))
635 op = ex_to_pseries(op).mul_const(ex_to_numeric(it->coeff));
638 acc = ex_to_pseries(acc).add_series(ex_to_pseries(op));
644 /** Multiply a pseries object with a numeric constant, producing a pseries
645 * object that represents the product.
647 * @param other constant to multiply with
648 * @return the product as a pseries */
649 ex pseries::mul_const(const numeric &other) const
652 new_seq.reserve(seq.size());
654 epvector::const_iterator it = seq.begin(), itend = seq.end();
655 while (it != itend) {
656 if (!is_order_function(it->rest))
657 new_seq.push_back(expair(it->rest * other, it->coeff));
659 new_seq.push_back(*it);
662 return pseries(relational(var,point), new_seq);
666 /** Multiply one pseries object to another, producing a pseries object that
667 * represents the product.
669 * @param other pseries object to multiply with
670 * @return the product as a pseries */
671 ex pseries::mul_series(const pseries &other) const
673 // Multiplying two series with different variables or expansion points
674 // results in an empty (constant) series
675 if (!is_compatible_to(other)) {
677 nul.push_back(expair(Order(_ex1()), _ex0()));
678 return pseries(relational(var,point), nul);
681 // Series multiplication
684 int a_max = degree(var);
685 int b_max = other.degree(var);
686 int a_min = ldegree(var);
687 int b_min = other.ldegree(var);
688 int cdeg_min = a_min + b_min;
689 int cdeg_max = a_max + b_max;
691 int higher_order_a = INT_MAX;
692 int higher_order_b = INT_MAX;
693 if (is_order_function(coeff(var, a_max)))
694 higher_order_a = a_max + b_min;
695 if (is_order_function(other.coeff(var, b_max)))
696 higher_order_b = b_max + a_min;
697 int higher_order_c = std::min(higher_order_a, higher_order_b);
698 if (cdeg_max >= higher_order_c)
699 cdeg_max = higher_order_c - 1;
701 for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
703 // c(i)=a(0)b(i)+...+a(i)b(0)
704 for (int i=a_min; cdeg-i>=b_min; ++i) {
705 ex a_coeff = coeff(var, i);
706 ex b_coeff = other.coeff(var, cdeg-i);
707 if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
708 co += a_coeff * b_coeff;
711 new_seq.push_back(expair(co, numeric(cdeg)));
713 if (higher_order_c < INT_MAX)
714 new_seq.push_back(expair(Order(_ex1()), numeric(higher_order_c)));
715 return pseries(relational(var, point), new_seq);
719 /** Implementation of ex::series() for product. This performs series
720 * multiplication when multiplying series.
722 ex mul::series(const relational & r, int order, unsigned options) const
724 ex acc; // Series accumulator
726 // Get first term from overall_coeff
727 acc = overall_coeff.series(r, order, options);
729 // Multiply with remaining terms
730 epvector::const_iterator it = seq.begin();
731 epvector::const_iterator itend = seq.end();
732 for (; it!=itend; ++it) {
734 if (op.info(info_flags::numeric)) {
735 // series * const (special case, faster)
736 ex f = power(op, it->coeff);
737 acc = ex_to_pseries(acc).mul_const(ex_to_numeric(f));
739 } else if (!is_ex_exactly_of_type(op, pseries))
740 op = op.series(r, order, options);
741 if (!it->coeff.is_equal(_ex1()))
742 op = ex_to_pseries(op).power_const(ex_to_numeric(it->coeff), order);
744 // Series multiplication
745 acc = ex_to_pseries(acc).mul_series(ex_to_pseries(op));
751 /** Compute the p-th power of a series.
753 * @param p power to compute
754 * @param deg truncation order of series calculation */
755 ex pseries::power_const(const numeric &p, int deg) const
758 // let A(x) be this series and for the time being let it start with a
759 // constant (later we'll generalize):
760 // A(x) = a_0 + a_1*x + a_2*x^2 + ...
761 // We want to compute
763 // C(x) = c_0 + c_1*x + c_2*x^2 + ...
764 // Taking the derivative on both sides and multiplying with A(x) one
765 // immediately arrives at
766 // C'(x)*A(x) = p*C(x)*A'(x)
767 // Multiplying this out and comparing coefficients we get the recurrence
769 // c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
770 // ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
771 // which can easily be solved given the starting value c_0 = (a_0)^p.
772 // For the more general case where the leading coefficient of A(x) is not
773 // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
774 // repeat the above derivation. The leading power of C2(x) = A2(x)^2 is
775 // then of course x^(p*m) but the recurrence formula still holds.
778 // as a spacial case, handle the empty (zero) series honoring the
779 // usual power laws such as implemented in power::eval()
780 if (p.real().is_zero())
781 throw (std::domain_error("pseries::power_const(): pow(0,I) is undefined"));
782 else if (p.real().is_negative())
783 throw (pole_error("pseries::power_const(): division by zero",1));
788 int ldeg = ldegree(var);
790 // Compute coefficients of the powered series
793 co.push_back(power(coeff(var, ldeg), p));
794 bool all_sums_zero = true;
795 for (int i=1; i<deg; ++i) {
797 for (int j=1; j<=i; ++j) {
798 ex c = coeff(var, j + ldeg);
799 if (is_order_function(c)) {
800 co.push_back(Order(_ex1()));
803 sum += (p * j - (i - j)) * co[i - j] * c;
806 all_sums_zero = false;
807 co.push_back(sum / coeff(var, ldeg) / numeric(i));
810 // Construct new series (of non-zero coefficients)
812 bool higher_order = false;
813 for (int i=0; i<deg; ++i) {
814 if (!co[i].is_zero())
815 new_seq.push_back(expair(co[i], numeric(i) + p * ldeg));
816 if (is_order_function(co[i])) {
821 if (!higher_order && !all_sums_zero)
822 new_seq.push_back(expair(Order(_ex1()), numeric(deg) + p * ldeg));
823 return pseries(relational(var,point), new_seq);
827 /** Return a new pseries object with the powers shifted by deg. */
828 pseries pseries::shift_exponents(int deg) const
830 epvector newseq(seq);
831 for (epvector::iterator i=newseq.begin(); i!=newseq.end(); ++i)
832 i->coeff = i->coeff + deg;
833 return pseries(relational(var, point), newseq);
837 /** Implementation of ex::series() for powers. This performs Laurent expansion
838 * of reciprocals of series at singularities.
840 ex power::series(const relational & r, int order, unsigned options) const
843 if (!is_ex_exactly_of_type(basis, pseries)) {
844 // Basis is not a series, may there be a singularity?
845 bool must_expand_basis = false;
848 } catch (pole_error) {
849 must_expand_basis = true;
852 // Is the expression of type something^(-int)?
853 if (!must_expand_basis && !exponent.info(info_flags::negint))
854 return basic::series(r, order, options);
856 // Is the expression of type 0^something?
857 if (!must_expand_basis && !basis.subs(r).is_zero())
858 return basic::series(r, order, options);
860 // Singularity encountered, expand basis into series
861 e = basis.series(r, order, options);
868 return ex_to_pseries(e).power_const(ex_to_numeric(exponent), order);
872 /** Re-expansion of a pseries object. */
873 ex pseries::series(const relational & r, int order, unsigned options) const
875 const ex p = r.rhs();
876 GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
877 const symbol &s = static_cast<symbol &>(*r.lhs().bp);
879 if (var.is_equal(s) && point.is_equal(p)) {
880 if (order > degree(s))
884 epvector::const_iterator it = seq.begin(), itend = seq.end();
885 while (it != itend) {
886 int o = ex_to_numeric(it->coeff).to_int();
888 new_seq.push_back(expair(Order(_ex1()), o));
891 new_seq.push_back(*it);
894 return pseries(r, new_seq);
897 return convert_to_poly().series(r, order, options);
901 /** Compute the truncated series expansion of an expression.
902 * This function returns an expression containing an object of class pseries
903 * to represent the series. If the series does not terminate within the given
904 * truncation order, the last term of the series will be an order term.
906 * @param r expansion relation, lhs holds variable and rhs holds point
907 * @param order truncation order of series calculations
908 * @param options of class series_options
909 * @return an expression holding a pseries object */
910 ex ex::series(const ex & r, int order, unsigned options) const
916 if (is_ex_exactly_of_type(r,relational))
917 rel_ = ex_to_relational(r);
918 else if (is_ex_exactly_of_type(r,symbol))
919 rel_ = relational(r,_ex0());
921 throw (std::logic_error("ex::series(): expansion point has unknown type"));
924 e = bp->series(rel_, order, options);
925 } catch (std::exception &x) {
926 throw (std::logic_error(std::string("unable to compute series (") + x.what() + ")"));
932 // static member variables
937 unsigned pseries::precedence = 38; // for clarity just below add::precedence