3 * Implementation of class for extended truncated power series and
4 * methods for series expansion. */
7 * GiNaC Copyright (C) 1999-2003 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
29 #include "inifcns.h" // for Order function
33 #include "relational.h"
34 #include "operators.h"
42 GINAC_IMPLEMENT_REGISTERED_CLASS(pseries, basic)
49 pseries::pseries() : inherited(TINFO_pseries) { }
56 /** Construct pseries from a vector of coefficients and powers.
57 * expair.rest holds the coefficient, expair.coeff holds the power.
58 * The powers must be integers (positive or negative) and in ascending order;
59 * the last coefficient can be Order(_ex1) to represent a truncated,
60 * non-terminating series.
62 * @param rel_ expansion variable and point (must hold a relational)
63 * @param ops_ vector of {coefficient, power} pairs (coefficient must not be zero)
64 * @return newly constructed pseries */
65 pseries::pseries(const ex &rel_, const epvector &ops_) : basic(TINFO_pseries), seq(ops_)
67 GINAC_ASSERT(is_a<relational>(rel_));
68 GINAC_ASSERT(is_a<symbol>(rel_.lhs()));
78 pseries::pseries(const archive_node &n, lst &sym_lst) : inherited(n, sym_lst)
80 for (unsigned int i=0; true; ++i) {
83 if (n.find_ex("coeff", rest, sym_lst, i) && n.find_ex("power", coeff, sym_lst, i))
84 seq.push_back(expair(rest, coeff));
88 n.find_ex("var", var, sym_lst);
89 n.find_ex("point", point, sym_lst);
92 void pseries::archive(archive_node &n) const
94 inherited::archive(n);
95 epvector::const_iterator i = seq.begin(), iend = seq.end();
97 n.add_ex("coeff", i->rest);
98 n.add_ex("power", i->coeff);
101 n.add_ex("var", var);
102 n.add_ex("point", point);
105 DEFAULT_UNARCHIVE(pseries)
108 // functions overriding virtual functions from base classes
111 void pseries::print(const print_context & c, unsigned level) const
113 if (is_a<print_tree>(c)) {
115 c.s << std::string(level, ' ') << class_name()
116 << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
118 unsigned delta_indent = static_cast<const print_tree &>(c).delta_indent;
119 size_t num = seq.size();
120 for (size_t i=0; i<num; ++i) {
121 seq[i].rest.print(c, level + delta_indent);
122 seq[i].coeff.print(c, level + delta_indent);
123 c.s << std::string(level + delta_indent, ' ') << "-----" << std::endl;
125 var.print(c, level + delta_indent);
126 point.print(c, level + delta_indent);
128 } else if (is_a<print_python_repr>(c)) {
129 c.s << class_name() << "(relational(";
134 size_t num = seq.size();
135 for (size_t i=0; i<num; ++i) {
139 seq[i].rest.print(c);
141 seq[i].coeff.print(c);
147 if (precedence() <= level)
150 std::string par_open = is_a<print_latex>(c) ? "{(" : "(";
151 std::string par_close = is_a<print_latex>(c) ? ")}" : ")";
153 // objects of type pseries must not have any zero entries, so the
154 // trivial (zero) pseries needs a special treatment here:
157 epvector::const_iterator i = seq.begin(), end = seq.end();
159 // print a sign, if needed
160 if (i != seq.begin())
162 if (!is_order_function(i->rest)) {
163 // print 'rest', i.e. the expansion coefficient
164 if (i->rest.info(info_flags::numeric) &&
165 i->rest.info(info_flags::positive)) {
172 // print 'coeff', something like (x-1)^42
173 if (!i->coeff.is_zero()) {
174 if (is_a<print_latex>(c))
178 if (!point.is_zero()) {
180 (var-point).print(c);
184 if (i->coeff.compare(_ex1)) {
185 if (is_a<print_python>(c))
189 if (i->coeff.info(info_flags::negative)) {
194 if (is_a<print_latex>(c)) {
204 Order(power(var-point,i->coeff)).print(c);
208 if (precedence() <= level)
213 int pseries::compare_same_type(const basic & other) const
215 GINAC_ASSERT(is_a<pseries>(other));
216 const pseries &o = static_cast<const pseries &>(other);
218 // first compare the lengths of the series...
219 if (seq.size()>o.seq.size())
221 if (seq.size()<o.seq.size())
224 // ...then the expansion point...
225 int cmpval = var.compare(o.var);
228 cmpval = point.compare(o.point);
232 // ...and if that failed the individual elements
233 epvector::const_iterator it = seq.begin(), o_it = o.seq.begin();
234 while (it!=seq.end() && o_it!=o.seq.end()) {
235 cmpval = it->compare(*o_it);
242 // so they are equal.
246 /** Return the number of operands including a possible order term. */
247 size_t pseries::nops() const
252 /** Return the ith term in the series when represented as a sum. */
253 ex pseries::op(size_t i) const
256 throw (std::out_of_range("op() out of range"));
258 return seq[i].rest * power(var - point, seq[i].coeff);
261 /** Return degree of highest power of the series. This is usually the exponent
262 * of the Order term. If s is not the expansion variable of the series, the
263 * series is examined termwise. */
264 int pseries::degree(const ex &s) const
266 if (var.is_equal(s)) {
267 // Return last exponent
269 return ex_to<numeric>((seq.end()-1)->coeff).to_int();
273 epvector::const_iterator it = seq.begin(), itend = seq.end();
276 int max_pow = INT_MIN;
277 while (it != itend) {
278 int pow = it->rest.degree(s);
287 /** Return degree of lowest power of the series. This is usually the exponent
288 * of the leading term. If s is not the expansion variable of the series, the
289 * series is examined termwise. If s is the expansion variable but the
290 * expansion point is not zero the series is not expanded to find the degree.
291 * I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
292 int pseries::ldegree(const ex &s) const
294 if (var.is_equal(s)) {
295 // Return first exponent
297 return ex_to<numeric>((seq.begin())->coeff).to_int();
301 epvector::const_iterator it = seq.begin(), itend = seq.end();
304 int min_pow = INT_MAX;
305 while (it != itend) {
306 int pow = it->rest.ldegree(s);
315 /** Return coefficient of degree n in power series if s is the expansion
316 * variable. If the expansion point is nonzero, by definition the n=1
317 * coefficient in s of a+b*(s-z)+c*(s-z)^2+Order((s-z)^3) is b (assuming
318 * the expansion took place in the s in the first place).
319 * If s is not the expansion variable, an attempt is made to convert the
320 * series to a polynomial and return the corresponding coefficient from
322 ex pseries::coeff(const ex &s, int n) const
324 if (var.is_equal(s)) {
328 // Binary search in sequence for given power
329 numeric looking_for = numeric(n);
330 int lo = 0, hi = seq.size() - 1;
332 int mid = (lo + hi) / 2;
333 GINAC_ASSERT(is_exactly_a<numeric>(seq[mid].coeff));
334 int cmp = ex_to<numeric>(seq[mid].coeff).compare(looking_for);
340 return seq[mid].rest;
345 throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
350 return convert_to_poly().coeff(s, n);
354 ex pseries::collect(const ex &s, bool distributed) const
359 /** Perform coefficient-wise automatic term rewriting rules in this class. */
360 ex pseries::eval(int level) const
365 if (level == -max_recursion_level)
366 throw (std::runtime_error("pseries::eval(): recursion limit exceeded"));
368 // Construct a new series with evaluated coefficients
370 new_seq.reserve(seq.size());
371 epvector::const_iterator it = seq.begin(), itend = seq.end();
372 while (it != itend) {
373 new_seq.push_back(expair(it->rest.eval(level-1), it->coeff));
376 return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
379 /** Evaluate coefficients numerically. */
380 ex pseries::evalf(int level) const
385 if (level == -max_recursion_level)
386 throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
388 // Construct a new series with evaluated coefficients
390 new_seq.reserve(seq.size());
391 epvector::const_iterator it = seq.begin(), itend = seq.end();
392 while (it != itend) {
393 new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff));
396 return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
399 ex pseries::subs(const exmap & m, unsigned options) const
401 // If expansion variable is being substituted, convert the series to a
402 // polynomial and do the substitution there because the result might
403 // no longer be a power series
404 if (m.find(var) != m.end())
405 return convert_to_poly(true).subs(m, options);
407 // Otherwise construct a new series with substituted coefficients and
410 newseq.reserve(seq.size());
411 epvector::const_iterator it = seq.begin(), itend = seq.end();
412 while (it != itend) {
413 newseq.push_back(expair(it->rest.subs(m, options), it->coeff));
416 return (new pseries(relational(var,point.subs(m, options)), newseq))->setflag(status_flags::dynallocated);
419 /** Implementation of ex::expand() for a power series. It expands all the
420 * terms individually and returns the resulting series as a new pseries. */
421 ex pseries::expand(unsigned options) const
424 epvector::const_iterator i = seq.begin(), end = seq.end();
426 ex restexp = i->rest.expand();
427 if (!restexp.is_zero())
428 newseq.push_back(expair(restexp, i->coeff));
431 return (new pseries(relational(var,point), newseq))
432 ->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
435 /** Implementation of ex::diff() for a power series.
437 ex pseries::derivative(const symbol & s) const
440 epvector::const_iterator it = seq.begin(), itend = seq.end();
444 // FIXME: coeff might depend on var
445 while (it != itend) {
446 if (is_order_function(it->rest)) {
447 new_seq.push_back(expair(it->rest, it->coeff - 1));
449 ex c = it->rest * it->coeff;
451 new_seq.push_back(expair(c, it->coeff - 1));
458 while (it != itend) {
459 if (is_order_function(it->rest)) {
460 new_seq.push_back(*it);
462 ex c = it->rest.diff(s);
464 new_seq.push_back(expair(c, it->coeff));
470 return pseries(relational(var,point), new_seq);
473 ex pseries::convert_to_poly(bool no_order) const
476 epvector::const_iterator it = seq.begin(), itend = seq.end();
478 while (it != itend) {
479 if (is_order_function(it->rest)) {
481 e += Order(power(var - point, it->coeff));
483 e += it->rest * power(var - point, it->coeff);
489 bool pseries::is_terminating() const
491 return seq.empty() || !is_order_function((seq.end()-1)->rest);
496 * Implementations of series expansion
499 /** Default implementation of ex::series(). This performs Taylor expansion.
501 ex basic::series(const relational & r, int order, unsigned options) const
506 ex coeff = deriv.subs(r);
507 const symbol &s = ex_to<symbol>(r.lhs());
509 if (!coeff.is_zero())
510 seq.push_back(expair(coeff, _ex0));
513 for (n=1; n<order; ++n) {
515 // We need to test for zero in order to see if the series terminates.
516 // The problem is that there is no such thing as a perfect test for
517 // zero. Expanding the term occasionally helps a little...
518 deriv = deriv.diff(s).expand();
519 if (deriv.is_zero()) // Series terminates
520 return pseries(r, seq);
522 coeff = deriv.subs(r);
523 if (!coeff.is_zero())
524 seq.push_back(expair(fac.inverse() * coeff, n));
527 // Higher-order terms, if present
528 deriv = deriv.diff(s);
529 if (!deriv.expand().is_zero())
530 seq.push_back(expair(Order(_ex1), n));
531 return pseries(r, seq);
535 /** Implementation of ex::series() for symbols.
537 ex symbol::series(const relational & r, int order, unsigned options) const
540 const ex point = r.rhs();
541 GINAC_ASSERT(is_a<symbol>(r.lhs()));
543 if (this->is_equal_same_type(ex_to<symbol>(r.lhs()))) {
544 if (order > 0 && !point.is_zero())
545 seq.push_back(expair(point, _ex0));
547 seq.push_back(expair(_ex1, _ex1));
549 seq.push_back(expair(Order(_ex1), numeric(order)));
551 seq.push_back(expair(*this, _ex0));
552 return pseries(r, seq);
556 /** Add one series object to another, producing a pseries object that
557 * represents the sum.
559 * @param other pseries object to add with
560 * @return the sum as a pseries */
561 ex pseries::add_series(const pseries &other) const
563 // Adding two series with different variables or expansion points
564 // results in an empty (constant) series
565 if (!is_compatible_to(other)) {
567 nul.push_back(expair(Order(_ex1), _ex0));
568 return pseries(relational(var,point), nul);
573 epvector::const_iterator a = seq.begin();
574 epvector::const_iterator b = other.seq.begin();
575 epvector::const_iterator a_end = seq.end();
576 epvector::const_iterator b_end = other.seq.end();
577 int pow_a = INT_MAX, pow_b = INT_MAX;
579 // If a is empty, fill up with elements from b and stop
582 new_seq.push_back(*b);
587 pow_a = ex_to<numeric>((*a).coeff).to_int();
589 // If b is empty, fill up with elements from a and stop
592 new_seq.push_back(*a);
597 pow_b = ex_to<numeric>((*b).coeff).to_int();
599 // a and b are non-empty, compare powers
601 // a has lesser power, get coefficient from a
602 new_seq.push_back(*a);
603 if (is_order_function((*a).rest))
606 } else if (pow_b < pow_a) {
607 // b has lesser power, get coefficient from b
608 new_seq.push_back(*b);
609 if (is_order_function((*b).rest))
613 // Add coefficient of a and b
614 if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
615 new_seq.push_back(expair(Order(_ex1), (*a).coeff));
616 break; // Order term ends the sequence
618 ex sum = (*a).rest + (*b).rest;
619 if (!(sum.is_zero()))
620 new_seq.push_back(expair(sum, numeric(pow_a)));
626 return pseries(relational(var,point), new_seq);
630 /** Implementation of ex::series() for sums. This performs series addition when
631 * adding pseries objects.
633 ex add::series(const relational & r, int order, unsigned options) const
635 ex acc; // Series accumulator
637 // Get first term from overall_coeff
638 acc = overall_coeff.series(r, order, options);
640 // Add remaining terms
641 epvector::const_iterator it = seq.begin();
642 epvector::const_iterator itend = seq.end();
643 for (; it!=itend; ++it) {
645 if (is_exactly_a<pseries>(it->rest))
648 op = it->rest.series(r, order, options);
649 if (!it->coeff.is_equal(_ex1))
650 op = ex_to<pseries>(op).mul_const(ex_to<numeric>(it->coeff));
653 acc = ex_to<pseries>(acc).add_series(ex_to<pseries>(op));
659 /** Multiply a pseries object with a numeric constant, producing a pseries
660 * object that represents the product.
662 * @param other constant to multiply with
663 * @return the product as a pseries */
664 ex pseries::mul_const(const numeric &other) const
667 new_seq.reserve(seq.size());
669 epvector::const_iterator it = seq.begin(), itend = seq.end();
670 while (it != itend) {
671 if (!is_order_function(it->rest))
672 new_seq.push_back(expair(it->rest * other, it->coeff));
674 new_seq.push_back(*it);
677 return pseries(relational(var,point), new_seq);
681 /** Multiply one pseries object to another, producing a pseries object that
682 * represents the product.
684 * @param other pseries object to multiply with
685 * @return the product as a pseries */
686 ex pseries::mul_series(const pseries &other) const
688 // Multiplying two series with different variables or expansion points
689 // results in an empty (constant) series
690 if (!is_compatible_to(other)) {
692 nul.push_back(expair(Order(_ex1), _ex0));
693 return pseries(relational(var,point), nul);
696 // Series multiplication
698 int a_max = degree(var);
699 int b_max = other.degree(var);
700 int a_min = ldegree(var);
701 int b_min = other.ldegree(var);
702 int cdeg_min = a_min + b_min;
703 int cdeg_max = a_max + b_max;
705 int higher_order_a = INT_MAX;
706 int higher_order_b = INT_MAX;
707 if (is_order_function(coeff(var, a_max)))
708 higher_order_a = a_max + b_min;
709 if (is_order_function(other.coeff(var, b_max)))
710 higher_order_b = b_max + a_min;
711 int higher_order_c = std::min(higher_order_a, higher_order_b);
712 if (cdeg_max >= higher_order_c)
713 cdeg_max = higher_order_c - 1;
715 for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
717 // c(i)=a(0)b(i)+...+a(i)b(0)
718 for (int i=a_min; cdeg-i>=b_min; ++i) {
719 ex a_coeff = coeff(var, i);
720 ex b_coeff = other.coeff(var, cdeg-i);
721 if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
722 co += a_coeff * b_coeff;
725 new_seq.push_back(expair(co, numeric(cdeg)));
727 if (higher_order_c < INT_MAX)
728 new_seq.push_back(expair(Order(_ex1), numeric(higher_order_c)));
729 return pseries(relational(var, point), new_seq);
733 /** Implementation of ex::series() for product. This performs series
734 * multiplication when multiplying series.
736 ex mul::series(const relational & r, int order, unsigned options) const
738 pseries acc; // Series accumulator
740 // Multiply with remaining terms
741 const epvector::const_iterator itbeg = seq.begin();
742 const epvector::const_iterator itend = seq.end();
743 for (epvector::const_iterator it=itbeg; it!=itend; ++it) {
744 ex op = recombine_pair_to_ex(*it).series(r, order, options);
746 // Series multiplication
748 acc = ex_to<pseries>(op);
750 acc = ex_to<pseries>(acc.mul_series(ex_to<pseries>(op)));
752 return acc.mul_const(ex_to<numeric>(overall_coeff));
756 /** Compute the p-th power of a series.
758 * @param p power to compute
759 * @param deg truncation order of series calculation */
760 ex pseries::power_const(const numeric &p, int deg) const
763 // (due to Leonhard Euler)
764 // let A(x) be this series and for the time being let it start with a
765 // constant (later we'll generalize):
766 // A(x) = a_0 + a_1*x + a_2*x^2 + ...
767 // We want to compute
769 // C(x) = c_0 + c_1*x + c_2*x^2 + ...
770 // Taking the derivative on both sides and multiplying with A(x) one
771 // immediately arrives at
772 // C'(x)*A(x) = p*C(x)*A'(x)
773 // Multiplying this out and comparing coefficients we get the recurrence
775 // c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
776 // ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
777 // which can easily be solved given the starting value c_0 = (a_0)^p.
778 // For the more general case where the leading coefficient of A(x) is not
779 // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
780 // repeat the above derivation. The leading power of C2(x) = A2(x)^2 is
781 // then of course x^(p*m) but the recurrence formula still holds.
784 // as a special case, handle the empty (zero) series honoring the
785 // usual power laws such as implemented in power::eval()
786 if (p.real().is_zero())
787 throw std::domain_error("pseries::power_const(): pow(0,I) is undefined");
788 else if (p.real().is_negative())
789 throw pole_error("pseries::power_const(): division by zero",1);
794 const int ldeg = ldegree(var);
795 if (!(p*ldeg).is_integer())
796 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
798 // O(x^n)^(-m) is undefined
799 if (seq.size() == 1 && is_order_function(seq[0].rest) && p.real().is_negative())
800 throw pole_error("pseries::power_const(): division by zero",1);
802 // Compute coefficients of the powered series
805 co.push_back(power(coeff(var, ldeg), p));
806 bool all_sums_zero = true;
807 for (int i=1; i<deg; ++i) {
809 for (int j=1; j<=i; ++j) {
810 ex c = coeff(var, j + ldeg);
811 if (is_order_function(c)) {
812 co.push_back(Order(_ex1));
815 sum += (p * j - (i - j)) * co[i - j] * c;
818 all_sums_zero = false;
819 co.push_back(sum / coeff(var, ldeg) / i);
822 // Construct new series (of non-zero coefficients)
824 bool higher_order = false;
825 for (int i=0; i<deg; ++i) {
826 if (!co[i].is_zero())
827 new_seq.push_back(expair(co[i], p * ldeg + i));
828 if (is_order_function(co[i])) {
833 if (!higher_order && !all_sums_zero)
834 new_seq.push_back(expair(Order(_ex1), p * ldeg + deg));
835 return pseries(relational(var,point), new_seq);
839 /** Return a new pseries object with the powers shifted by deg. */
840 pseries pseries::shift_exponents(int deg) const
842 epvector newseq = seq;
843 epvector::iterator i = newseq.begin(), end = newseq.end();
848 return pseries(relational(var, point), newseq);
852 /** Implementation of ex::series() for powers. This performs Laurent expansion
853 * of reciprocals of series at singularities.
855 ex power::series(const relational & r, int order, unsigned options) const
857 // If basis is already a series, just power it
858 if (is_exactly_a<pseries>(basis))
859 return ex_to<pseries>(basis).power_const(ex_to<numeric>(exponent), order);
861 // Basis is not a series, may there be a singularity?
862 bool must_expand_basis = false;
865 } catch (pole_error) {
866 must_expand_basis = true;
869 // Is the expression of type something^(-int)?
870 if (!must_expand_basis && !exponent.info(info_flags::negint))
871 return basic::series(r, order, options);
873 // Is the expression of type 0^something?
874 if (!must_expand_basis && !basis.subs(r).is_zero())
875 return basic::series(r, order, options);
877 // Singularity encountered, is the basis equal to (var - point)?
878 if (basis.is_equal(r.lhs() - r.rhs())) {
880 if (ex_to<numeric>(exponent).to_int() < order)
881 new_seq.push_back(expair(_ex1, exponent));
883 new_seq.push_back(expair(Order(_ex1), exponent));
884 return pseries(r, new_seq);
887 // No, expand basis into series
888 ex e = basis.series(r, order, options);
889 return ex_to<pseries>(e).power_const(ex_to<numeric>(exponent), order);
893 /** Re-expansion of a pseries object. */
894 ex pseries::series(const relational & r, int order, unsigned options) const
896 const ex p = r.rhs();
897 GINAC_ASSERT(is_a<symbol>(r.lhs()));
898 const symbol &s = ex_to<symbol>(r.lhs());
900 if (var.is_equal(s) && point.is_equal(p)) {
901 if (order > degree(s))
905 epvector::const_iterator it = seq.begin(), itend = seq.end();
906 while (it != itend) {
907 int o = ex_to<numeric>(it->coeff).to_int();
909 new_seq.push_back(expair(Order(_ex1), o));
912 new_seq.push_back(*it);
915 return pseries(r, new_seq);
918 return convert_to_poly().series(r, order, options);
922 /** Compute the truncated series expansion of an expression.
923 * This function returns an expression containing an object of class pseries
924 * to represent the series. If the series does not terminate within the given
925 * truncation order, the last term of the series will be an order term.
927 * @param r expansion relation, lhs holds variable and rhs holds point
928 * @param order truncation order of series calculations
929 * @param options of class series_options
930 * @return an expression holding a pseries object */
931 ex ex::series(const ex & r, int order, unsigned options) const
936 if (is_a<relational>(r))
937 rel_ = ex_to<relational>(r);
938 else if (is_a<symbol>(r))
939 rel_ = relational(r,_ex0);
941 throw (std::logic_error("ex::series(): expansion point has unknown type"));
944 e = bp->series(rel_, order, options);
945 } catch (std::exception &x) {
946 throw (std::logic_error(std::string("unable to compute series (") + x.what() + ")"));