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"
38 #ifndef NO_NAMESPACE_GINAC
40 #endif // ndef NO_NAMESPACE_GINAC
42 GINAC_IMPLEMENT_REGISTERED_CLASS(pseries, basic)
45 * Default constructor, destructor, copy constructor, assignment operator and helpers
48 pseries::pseries() : basic(TINFO_pseries)
50 debugmsg("pseries default constructor", LOGLEVEL_CONSTRUCT);
53 void pseries::copy(const pseries &other)
55 inherited::copy(other);
61 void pseries::destroy(bool call_parent)
64 inherited::destroy(call_parent);
72 /** Construct pseries from a vector of coefficients and powers.
73 * expair.rest holds the coefficient, expair.coeff holds the power.
74 * The powers must be integers (positive or negative) and in ascending order;
75 * the last coefficient can be Order(_ex1()) to represent a truncated,
76 * non-terminating series.
78 * @param rel_ expansion variable and point (must hold a relational)
79 * @param ops_ vector of {coefficient, power} pairs (coefficient must not be zero)
80 * @return newly constructed pseries */
81 pseries::pseries(const ex &rel_, const epvector &ops_) : basic(TINFO_pseries), seq(ops_)
83 debugmsg("pseries constructor from ex,epvector", LOGLEVEL_CONSTRUCT);
84 GINAC_ASSERT(is_ex_exactly_of_type(rel_, relational));
85 GINAC_ASSERT(is_ex_exactly_of_type(rel_.lhs(),symbol));
87 var = *static_cast<symbol *>(rel_.lhs().bp);
95 /** Construct object from archive_node. */
96 pseries::pseries(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
98 debugmsg("pseries constructor from archive_node", LOGLEVEL_CONSTRUCT);
99 for (unsigned int i=0; true; ++i) {
102 if (n.find_ex("coeff", rest, sym_lst, i) && n.find_ex("power", coeff, sym_lst, i))
103 seq.push_back(expair(rest, coeff));
107 n.find_ex("var", var, sym_lst);
108 n.find_ex("point", point, sym_lst);
111 /** Unarchive the object. */
112 ex pseries::unarchive(const archive_node &n, const lst &sym_lst)
114 return (new pseries(n, sym_lst))->setflag(status_flags::dynallocated);
117 /** Archive the object. */
118 void pseries::archive(archive_node &n) const
120 inherited::archive(n);
121 epvector::const_iterator i = seq.begin(), iend = seq.end();
123 n.add_ex("coeff", i->rest);
124 n.add_ex("power", i->coeff);
127 n.add_ex("var", var);
128 n.add_ex("point", point);
132 // functions overriding virtual functions from bases classes
135 basic *pseries::duplicate() const
137 debugmsg("pseries duplicate", LOGLEVEL_DUPLICATE);
138 return new pseries(*this);
141 void pseries::print(std::ostream &os, unsigned upper_precedence) const
143 debugmsg("pseries print", LOGLEVEL_PRINT);
144 if (precedence<=upper_precedence) os << "(";
145 for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
147 if (i->rest.is_zero())
149 // print a sign, if needed
152 if (!is_order_function(i->rest)) {
153 // print 'rest', i.e. the expansion coefficient
154 if (i->rest.info(info_flags::numeric) &&
155 i->rest.info(info_flags::positive)) {
158 os << "(" << i->rest << ')';
159 // print 'coeff', something like (x-1)^42
160 if (!i->coeff.is_zero()) {
162 if (!point.is_zero())
163 os << '(' << var-point << ')';
166 if (i->coeff.compare(_ex1())) {
168 if (i->coeff.info(info_flags::negative))
169 os << '(' << i->coeff << ')';
175 os << Order(power(var-point,i->coeff));
178 if (precedence<=upper_precedence) os << ")";
182 void pseries::printraw(std::ostream &os) const
184 debugmsg("pseries printraw", LOGLEVEL_PRINT);
185 os << "pseries(" << var << ";" << point << ";";
186 for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
187 os << "(" << (*i).rest << "," << (*i).coeff << "),";
193 void pseries::printtree(std::ostream & os, unsigned indent) const
195 debugmsg("pseries printtree",LOGLEVEL_PRINT);
196 os << std::string(indent,' ') << "pseries "
197 << ", hash=" << hashvalue
198 << " (0x" << std::hex << hashvalue << std::dec << ")"
199 << ", flags=" << flags << std::endl;
200 for (unsigned i=0; i<seq.size(); ++i) {
201 seq[i].rest.printtree(os,indent+delta_indent);
202 seq[i].coeff.printtree(os,indent+delta_indent);
204 os << std::string(indent+delta_indent,' ') << "-----" << std::endl;
206 var.printtree(os, indent+delta_indent);
207 point.printtree(os, indent+delta_indent);
210 /** Return the number of operands including a possible order term. */
211 unsigned pseries::nops(void) const
217 /** Return the ith term in the series when represented as a sum. */
218 ex pseries::op(int i) const
220 if (i < 0 || unsigned(i) >= seq.size())
221 throw (std::out_of_range("op() out of range"));
222 return seq[i].rest * power(var - point, seq[i].coeff);
226 ex &pseries::let_op(int i)
228 throw (std::logic_error("let_op not defined for pseries"));
232 /** Return degree of highest power of the series. This is usually the exponent
233 * of the Order term. If s is not the expansion variable of the series, the
234 * series is examined termwise. */
235 int pseries::degree(const symbol &s) const
237 if (var.is_equal(s)) {
238 // Return last exponent
240 return ex_to_numeric((*(seq.end() - 1)).coeff).to_int();
244 epvector::const_iterator it = seq.begin(), itend = seq.end();
247 int max_pow = INT_MIN;
248 while (it != itend) {
249 int pow = it->rest.degree(s);
258 /** Return degree of lowest power of the series. This is usually the exponent
259 * of the leading term. If s is not the expansion variable of the series, the
260 * series is examined termwise. If s is the expansion variable but the
261 * expansion point is not zero the series is not expanded to find the degree.
262 * I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
263 int pseries::ldegree(const symbol &s) const
265 if (var.is_equal(s)) {
266 // Return first exponent
268 return ex_to_numeric((*(seq.begin())).coeff).to_int();
272 epvector::const_iterator it = seq.begin(), itend = seq.end();
275 int min_pow = INT_MAX;
276 while (it != itend) {
277 int pow = it->rest.ldegree(s);
286 ex pseries::coeff(const symbol &s, int n) const
288 if (var.is_equal(s)) {
292 // Binary search in sequence for given power
293 numeric looking_for = numeric(n);
294 int lo = 0, hi = seq.size() - 1;
296 int mid = (lo + hi) / 2;
297 GINAC_ASSERT(is_ex_exactly_of_type(seq[mid].coeff, numeric));
298 int cmp = ex_to_numeric(seq[mid].coeff).compare(looking_for);
304 return seq[mid].rest;
309 throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
314 return convert_to_poly().coeff(s, n);
318 ex pseries::collect(const symbol &s) const
324 /** Evaluate coefficients. */
325 ex pseries::eval(int level) const
330 if (level == -max_recursion_level)
331 throw (std::runtime_error("pseries::eval(): recursion limit exceeded"));
333 // Construct a new series with evaluated coefficients
335 new_seq.reserve(seq.size());
336 epvector::const_iterator it = seq.begin(), itend = seq.end();
337 while (it != itend) {
338 new_seq.push_back(expair(it->rest.eval(level-1), it->coeff));
341 return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
345 /** Evaluate coefficients numerically. */
346 ex pseries::evalf(int level) const
351 if (level == -max_recursion_level)
352 throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
354 // Construct a new series with evaluated coefficients
356 new_seq.reserve(seq.size());
357 epvector::const_iterator it = seq.begin(), itend = seq.end();
358 while (it != itend) {
359 new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff));
362 return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
366 ex pseries::subs(const lst & ls, const lst & lr) const
368 // If expansion variable is being substituted, convert the series to a
369 // polynomial and do the substitution there because the result might
370 // no longer be a power series
372 return convert_to_poly(true).subs(ls, lr);
374 // Otherwise construct a new series with substituted coefficients and
377 newseq.reserve(seq.size());
378 epvector::const_iterator it = seq.begin(), itend = seq.end();
379 while (it != itend) {
380 newseq.push_back(expair(it->rest.subs(ls, lr), it->coeff));
383 return (new pseries(relational(var,point.subs(ls, lr)), newseq))->setflag(status_flags::dynallocated);
387 /** Implementation of ex::expand() for a power series. It expands all the
388 * terms individually and returns the resulting series as a new pseries.
390 ex pseries::expand(unsigned options) const
393 newseq.reserve(seq.size());
394 for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i)
395 newseq.push_back(expair(i->rest.expand(), i->coeff));
396 return (new pseries(relational(var,point), newseq))
397 ->setflag(status_flags::dynallocated | status_flags::expanded);
401 /** Implementation of ex::diff() for a power series. It treats the series as a
404 ex pseries::derivative(const symbol & s) const
408 epvector::const_iterator it = seq.begin(), itend = seq.end();
410 // FIXME: coeff might depend on var
411 while (it != itend) {
412 if (is_order_function(it->rest)) {
413 new_seq.push_back(expair(it->rest, it->coeff - 1));
415 ex c = it->rest * it->coeff;
417 new_seq.push_back(expair(c, it->coeff - 1));
421 return pseries(relational(var,point), new_seq);
429 * Construct ordinary polynomial out of series
432 /** Convert a pseries object to an ordinary polynomial.
434 * @param no_order flag: discard higher order terms */
435 ex pseries::convert_to_poly(bool no_order) const
438 epvector::const_iterator it = seq.begin(), itend = seq.end();
440 while (it != itend) {
441 if (is_order_function(it->rest)) {
443 e += Order(power(var - point, it->coeff));
445 e += it->rest * power(var - point, it->coeff);
451 /** Returns true if there is no order term, i.e. the series terminates and
452 * false otherwise. */
453 bool pseries::is_terminating(void) const
455 return seq.size() == 0 || !is_order_function((seq.end()-1)->rest);
460 * Implementation of series expansion
463 /** Default implementation of ex::series(). This performs Taylor expansion.
465 ex basic::series(const relational & r, int order, unsigned options) const
470 ex coeff = deriv.subs(r);
471 const symbol *s = static_cast<symbol *>(r.lhs().bp);
473 if (!coeff.is_zero())
474 seq.push_back(expair(coeff, numeric(0)));
477 for (n=1; n<order; ++n) {
478 fac = fac.mul(numeric(n));
479 deriv = deriv.diff(*s).expand();
480 if (deriv.is_zero()) {
482 return pseries(r, seq);
484 coeff = deriv.subs(r);
485 if (!coeff.is_zero())
486 seq.push_back(expair(fac.inverse() * coeff, numeric(n)));
489 // Higher-order terms, if present
490 deriv = deriv.diff(*s);
491 if (!deriv.expand().is_zero())
492 seq.push_back(expair(Order(_ex1()), numeric(n)));
493 return pseries(r, seq);
497 /** Implementation of ex::series() for symbols.
499 ex symbol::series(const relational & r, int order, unsigned options) const
502 const ex point = r.rhs();
503 GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
504 const symbol *s = static_cast<symbol *>(r.lhs().bp);
506 if (this->is_equal(*s)) {
507 if (order > 0 && !point.is_zero())
508 seq.push_back(expair(point, _ex0()));
510 seq.push_back(expair(_ex1(), _ex1()));
512 seq.push_back(expair(Order(_ex1()), numeric(order)));
514 seq.push_back(expair(*this, _ex0()));
515 return pseries(r, seq);
519 /** Add one series object to another, producing a pseries object that
520 * represents the sum.
522 * @param other pseries object to add with
523 * @return the sum as a pseries */
524 ex pseries::add_series(const pseries &other) const
526 // Adding two series with different variables or expansion points
527 // results in an empty (constant) series
528 if (!is_compatible_to(other)) {
530 nul.push_back(expair(Order(_ex1()), _ex0()));
531 return pseries(relational(var,point), nul);
536 epvector::const_iterator a = seq.begin();
537 epvector::const_iterator b = other.seq.begin();
538 epvector::const_iterator a_end = seq.end();
539 epvector::const_iterator b_end = other.seq.end();
540 int pow_a = INT_MAX, pow_b = INT_MAX;
542 // If a is empty, fill up with elements from b and stop
545 new_seq.push_back(*b);
550 pow_a = ex_to_numeric((*a).coeff).to_int();
552 // If b is empty, fill up with elements from a and stop
555 new_seq.push_back(*a);
560 pow_b = ex_to_numeric((*b).coeff).to_int();
562 // a and b are non-empty, compare powers
564 // a has lesser power, get coefficient from a
565 new_seq.push_back(*a);
566 if (is_order_function((*a).rest))
569 } else if (pow_b < pow_a) {
570 // b has lesser power, get coefficient from b
571 new_seq.push_back(*b);
572 if (is_order_function((*b).rest))
576 // Add coefficient of a and b
577 if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
578 new_seq.push_back(expair(Order(_ex1()), (*a).coeff));
579 break; // Order term ends the sequence
581 ex sum = (*a).rest + (*b).rest;
582 if (!(sum.is_zero()))
583 new_seq.push_back(expair(sum, numeric(pow_a)));
589 return pseries(relational(var,point), new_seq);
593 /** Implementation of ex::series() for sums. This performs series addition when
594 * adding pseries objects.
596 ex add::series(const relational & r, int order, unsigned options) const
598 ex acc; // Series accumulator
600 // Get first term from overall_coeff
601 acc = overall_coeff.series(r, order, options);
603 // Add remaining terms
604 epvector::const_iterator it = seq.begin();
605 epvector::const_iterator itend = seq.end();
606 for (; it!=itend; ++it) {
608 if (is_ex_exactly_of_type(it->rest, pseries))
611 op = it->rest.series(r, order, options);
612 if (!it->coeff.is_equal(_ex1()))
613 op = ex_to_pseries(op).mul_const(ex_to_numeric(it->coeff));
616 acc = ex_to_pseries(acc).add_series(ex_to_pseries(op));
622 /** Multiply a pseries object with a numeric constant, producing a pseries
623 * object that represents the product.
625 * @param other constant to multiply with
626 * @return the product as a pseries */
627 ex pseries::mul_const(const numeric &other) const
630 new_seq.reserve(seq.size());
632 epvector::const_iterator it = seq.begin(), itend = seq.end();
633 while (it != itend) {
634 if (!is_order_function(it->rest))
635 new_seq.push_back(expair(it->rest * other, it->coeff));
637 new_seq.push_back(*it);
640 return pseries(relational(var,point), new_seq);
644 /** Multiply one pseries object to another, producing a pseries object that
645 * represents the product.
647 * @param other pseries object to multiply with
648 * @return the product as a pseries */
649 ex pseries::mul_series(const pseries &other) const
651 // Multiplying two series with different variables or expansion points
652 // results in an empty (constant) series
653 if (!is_compatible_to(other)) {
655 nul.push_back(expair(Order(_ex1()), _ex0()));
656 return pseries(relational(var,point), nul);
659 // Series multiplication
662 const symbol *s = static_cast<symbol *>(var.bp);
663 int a_max = degree(*s);
664 int b_max = other.degree(*s);
665 int a_min = ldegree(*s);
666 int b_min = other.ldegree(*s);
667 int cdeg_min = a_min + b_min;
668 int cdeg_max = a_max + b_max;
670 int higher_order_a = INT_MAX;
671 int higher_order_b = INT_MAX;
672 if (is_order_function(coeff(*s, a_max)))
673 higher_order_a = a_max + b_min;
674 if (is_order_function(other.coeff(*s, b_max)))
675 higher_order_b = b_max + a_min;
676 int higher_order_c = std::min(higher_order_a, higher_order_b);
677 if (cdeg_max >= higher_order_c)
678 cdeg_max = higher_order_c - 1;
680 for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
682 // c(i)=a(0)b(i)+...+a(i)b(0)
683 for (int i=a_min; cdeg-i>=b_min; ++i) {
684 ex a_coeff = coeff(*s, i);
685 ex b_coeff = other.coeff(*s, cdeg-i);
686 if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
687 co += a_coeff * b_coeff;
690 new_seq.push_back(expair(co, numeric(cdeg)));
692 if (higher_order_c < INT_MAX)
693 new_seq.push_back(expair(Order(_ex1()), numeric(higher_order_c)));
694 return pseries(relational(var,point), new_seq);
698 /** Implementation of ex::series() for product. This performs series
699 * multiplication when multiplying series.
701 ex mul::series(const relational & r, int order, unsigned options) const
703 ex acc; // Series accumulator
705 // Get first term from overall_coeff
706 acc = overall_coeff.series(r, order, options);
708 // Multiply with remaining terms
709 epvector::const_iterator it = seq.begin();
710 epvector::const_iterator itend = seq.end();
711 for (; it!=itend; ++it) {
713 if (op.info(info_flags::numeric)) {
714 // series * const (special case, faster)
715 ex f = power(op, it->coeff);
716 acc = ex_to_pseries(acc).mul_const(ex_to_numeric(f));
718 } else if (!is_ex_exactly_of_type(op, pseries))
719 op = op.series(r, order, options);
720 if (!it->coeff.is_equal(_ex1()))
721 op = ex_to_pseries(op).power_const(ex_to_numeric(it->coeff), order);
723 // Series multiplication
724 acc = ex_to_pseries(acc).mul_series(ex_to_pseries(op));
730 /** Compute the p-th power of a series.
732 * @param p power to compute
733 * @param deg truncation order of series calculation */
734 ex pseries::power_const(const numeric &p, int deg) const
737 // let A(x) be this series and for the time being let it start with a
738 // constant (later we'll generalize):
739 // A(x) = a_0 + a_1*x + a_2*x^2 + ...
740 // We want to compute
742 // C(x) = c_0 + c_1*x + c_2*x^2 + ...
743 // Taking the derivative on both sides and multiplying with A(x) one
744 // immediately arrives at
745 // C'(x)*A(x) = p*C(x)*A'(x)
746 // Multiplying this out and comparing coefficients we get the recurrence
748 // c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
749 // ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
750 // which can easily be solved given the starting value c_0 = (a_0)^p.
751 // For the more general case where the leading coefficient of A(x) is not
752 // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
753 // repeat the above derivation. The leading power of C2(x) = A2(x)^2 is
754 // then of course x^(p*m) but the recurrence formula still holds.
755 const symbol *s = static_cast<symbol *>(var.bp);
756 int ldeg = ldegree(*s);
758 // Compute coefficients of the powered series
761 co.push_back(power(coeff(*s, ldeg), p));
762 bool all_sums_zero = true;
763 for (int i=1; i<deg; ++i) {
765 for (int j=1; j<=i; ++j) {
766 ex c = coeff(*s, j + ldeg);
767 if (is_order_function(c)) {
768 co.push_back(Order(_ex1()));
771 sum += (p * j - (i - j)) * co[i - j] * c;
774 all_sums_zero = false;
775 co.push_back(sum / coeff(*s, ldeg) / numeric(i));
778 // Construct new series (of non-zero coefficients)
780 bool higher_order = false;
781 for (int i=0; i<deg; ++i) {
782 if (!co[i].is_zero())
783 new_seq.push_back(expair(co[i], numeric(i) + p * ldeg));
784 if (is_order_function(co[i])) {
789 if (!higher_order && !all_sums_zero)
790 new_seq.push_back(expair(Order(_ex1()), numeric(deg) + p * ldeg));
791 return pseries(relational(var,point), new_seq);
795 /** Return a new pseries object with the powers shifted by deg. */
796 pseries pseries::shift_exponents(int deg) const
798 epvector newseq(seq);
799 for (epvector::iterator i=newseq.begin(); i!=newseq.end(); ++i)
800 i->coeff = i->coeff + deg;
801 return pseries(relational(var, point), newseq);
805 /** Implementation of ex::series() for powers. This performs Laurent expansion
806 * of reciprocals of series at singularities.
808 ex power::series(const relational & r, int order, unsigned options) const
811 if (!is_ex_exactly_of_type(basis, pseries)) {
812 // Basis is not a series, may there be a singularity?
813 bool must_expand_basis = false;
816 } catch (pole_error) {
817 must_expand_basis = true;
820 // Is the expression of type something^(-int)?
821 if (!must_expand_basis && !exponent.info(info_flags::negint))
822 return basic::series(r, order, options);
824 // Is the expression of type 0^something?
825 if (!must_expand_basis && !basis.subs(r).is_zero())
826 return basic::series(r, order, options);
828 // Singularity encountered, expand basis into series
829 e = basis.series(r, order, options);
836 return ex_to_pseries(e).power_const(ex_to_numeric(exponent), order);
840 /** Re-expansion of a pseries object. */
841 ex pseries::series(const relational & r, int order, unsigned options) const
843 const ex p = r.rhs();
844 GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
845 const symbol *s = static_cast<symbol *>(r.lhs().bp);
847 if (var.is_equal(*s) && point.is_equal(p)) {
848 if (order > degree(*s))
852 epvector::const_iterator it = seq.begin(), itend = seq.end();
853 while (it != itend) {
854 int o = ex_to_numeric(it->coeff).to_int();
856 new_seq.push_back(expair(Order(_ex1()), o));
859 new_seq.push_back(*it);
862 return pseries(r, new_seq);
865 return convert_to_poly().series(r, order, options);
869 /** Compute the truncated series expansion of an expression.
870 * This function returns an expression containing an object of class pseries
871 * to represent the series. If the series does not terminate within the given
872 * truncation order, the last term of the series will be an order term.
874 * @param r expansion relation, lhs holds variable and rhs holds point
875 * @param order truncation order of series calculations
876 * @param options of class series_options
877 * @return an expression holding a pseries object */
878 ex ex::series(const ex & r, int order, unsigned options) const
884 if (is_ex_exactly_of_type(r,relational))
885 rel_ = ex_to_relational(r);
886 else if (is_ex_exactly_of_type(r,symbol))
887 rel_ = relational(r,_ex0());
889 throw (std::logic_error("ex::series(): expansion point has unknown type"));
892 e = bp->series(rel_, order, options);
893 } catch (std::exception &x) {
894 throw (std::logic_error(std::string("unable to compute series (") + x.what() + ")"));
900 // static member variables
905 unsigned pseries::precedence = 38; // for clarity just below add::precedence
907 #ifndef NO_NAMESPACE_GINAC
909 #endif // ndef NO_NAMESPACE_GINAC