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
28 #include "inifcns.h" // for Order function
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_a<print_tree>(c)) {
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 std::string par_open = is_a<print_latex>(c) ? "{(" : "(";
149 std::string par_close = is_a<print_latex>(c) ? ")}" : ")";
151 // objects of type pseries must not have any zero entries, so the
152 // trivial (zero) pseries needs a special treatment here:
155 for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
156 // print a sign, if needed
157 if (i != seq.begin())
159 if (!is_order_function(i->rest)) {
160 // print 'rest', i.e. the expansion coefficient
161 if (i->rest.info(info_flags::numeric) &&
162 i->rest.info(info_flags::positive)) {
169 // print 'coeff', something like (x-1)^42
170 if (!i->coeff.is_zero()) {
171 if (is_a<print_latex>(c))
175 if (!point.is_zero()) {
177 (var-point).print(c);
181 if (i->coeff.compare(_ex1())) {
183 if (i->coeff.info(info_flags::negative)) {
188 if (is_a<print_latex>(c)) {
198 Order(power(var-point,i->coeff)).print(c);
201 if (precedence() <= level)
206 int pseries::compare_same_type(const basic & other) const
208 GINAC_ASSERT(is_of_type(other, pseries));
209 const pseries &o = static_cast<const pseries &>(other);
211 // first compare the lengths of the series...
212 if (seq.size()>o.seq.size())
214 if (seq.size()<o.seq.size())
217 // ...then the expansion point...
218 int cmpval = var.compare(o.var);
221 cmpval = point.compare(o.point);
225 // ...and if that failed the individual elements
226 epvector::const_iterator it = seq.begin(), o_it = o.seq.begin();
227 while (it!=seq.end() && o_it!=o.seq.end()) {
228 cmpval = it->compare(*o_it);
235 // so they are equal.
239 /** Return the number of operands including a possible order term. */
240 unsigned pseries::nops(void) const
245 /** Return the ith term in the series when represented as a sum. */
246 ex pseries::op(int i) const
248 if (i < 0 || unsigned(i) >= seq.size())
249 throw (std::out_of_range("op() out of range"));
250 return seq[i].rest * power(var - point, seq[i].coeff);
253 ex &pseries::let_op(int i)
255 throw (std::logic_error("let_op not defined for pseries"));
258 /** Return degree of highest power of the series. This is usually the exponent
259 * of the Order term. If s is not the expansion variable of the series, the
260 * series is examined termwise. */
261 int pseries::degree(const ex &s) const
263 if (var.is_equal(s)) {
264 // Return last exponent
266 return ex_to<numeric>((*(seq.end() - 1)).coeff).to_int();
270 epvector::const_iterator it = seq.begin(), itend = seq.end();
273 int max_pow = INT_MIN;
274 while (it != itend) {
275 int pow = it->rest.degree(s);
284 /** Return degree of lowest power of the series. This is usually the exponent
285 * of the leading term. If s is not the expansion variable of the series, the
286 * series is examined termwise. If s is the expansion variable but the
287 * expansion point is not zero the series is not expanded to find the degree.
288 * I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
289 int pseries::ldegree(const ex &s) const
291 if (var.is_equal(s)) {
292 // Return first exponent
294 return ex_to<numeric>((*(seq.begin())).coeff).to_int();
298 epvector::const_iterator it = seq.begin(), itend = seq.end();
301 int min_pow = INT_MAX;
302 while (it != itend) {
303 int pow = it->rest.ldegree(s);
312 /** Return coefficient of degree n in power series if s is the expansion
313 * variable. If the expansion point is nonzero, by definition the n=1
314 * coefficient in s of a+b*(s-z)+c*(s-z)^2+Order((s-z)^3) is b (assuming
315 * the expansion took place in the s in the first place).
316 * If s is not the expansion variable, an attempt is made to convert the
317 * series to a polynomial and return the corresponding coefficient from
319 ex pseries::coeff(const ex &s, int n) const
321 if (var.is_equal(s)) {
325 // Binary search in sequence for given power
326 numeric looking_for = numeric(n);
327 int lo = 0, hi = seq.size() - 1;
329 int mid = (lo + hi) / 2;
330 GINAC_ASSERT(is_ex_exactly_of_type(seq[mid].coeff, numeric));
331 int cmp = ex_to<numeric>(seq[mid].coeff).compare(looking_for);
337 return seq[mid].rest;
342 throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
347 return convert_to_poly().coeff(s, n);
351 ex pseries::collect(const ex &s, bool distributed) const
356 /** Evaluate coefficients. */
357 ex pseries::eval(int level) const
362 if (level == -max_recursion_level)
363 throw (std::runtime_error("pseries::eval(): recursion limit exceeded"));
365 // Construct a new series with evaluated coefficients
367 new_seq.reserve(seq.size());
368 epvector::const_iterator it = seq.begin(), itend = seq.end();
369 while (it != itend) {
370 new_seq.push_back(expair(it->rest.eval(level-1), it->coeff));
373 return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
376 /** Evaluate coefficients numerically. */
377 ex pseries::evalf(int level) const
382 if (level == -max_recursion_level)
383 throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
385 // Construct a new series with evaluated coefficients
387 new_seq.reserve(seq.size());
388 epvector::const_iterator it = seq.begin(), itend = seq.end();
389 while (it != itend) {
390 new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff));
393 return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
396 ex pseries::subs(const lst & ls, const lst & lr, bool no_pattern) const
398 // If expansion variable is being substituted, convert the series to a
399 // polynomial and do the substitution there because the result might
400 // no longer be a power series
402 return convert_to_poly(true).subs(ls, lr, no_pattern);
404 // Otherwise construct a new series with substituted coefficients and
407 newseq.reserve(seq.size());
408 epvector::const_iterator it = seq.begin(), itend = seq.end();
409 while (it != itend) {
410 newseq.push_back(expair(it->rest.subs(ls, lr, no_pattern), it->coeff));
413 return (new pseries(relational(var,point.subs(ls, lr, no_pattern)), newseq))->setflag(status_flags::dynallocated);
416 /** Implementation of ex::expand() for a power series. It expands all the
417 * terms individually and returns the resulting series as a new pseries. */
418 ex pseries::expand(unsigned options) const
421 for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
422 ex restexp = i->rest.expand();
423 if (!restexp.is_zero())
424 newseq.push_back(expair(restexp, i->coeff));
426 return (new pseries(relational(var,point), newseq))
427 ->setflag(status_flags::dynallocated | status_flags::expanded);
430 /** Implementation of ex::diff() for a power series. It treats the series as a
433 ex pseries::derivative(const symbol & s) const
437 epvector::const_iterator it = seq.begin(), itend = seq.end();
439 // FIXME: coeff might depend on var
440 while (it != itend) {
441 if (is_order_function(it->rest)) {
442 new_seq.push_back(expair(it->rest, it->coeff - 1));
444 ex c = it->rest * it->coeff;
446 new_seq.push_back(expair(c, it->coeff - 1));
450 return pseries(relational(var,point), new_seq);
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);
472 bool pseries::is_terminating(void) const
474 return seq.size() == 0 || !is_order_function((seq.end()-1)->rest);
479 * Implementations of series expansion
482 /** Default implementation of ex::series(). This performs Taylor expansion.
484 ex basic::series(const relational & r, int order, unsigned options) const
489 ex coeff = deriv.subs(r);
490 const symbol &s = static_cast<symbol &>(*r.lhs().bp);
492 if (!coeff.is_zero())
493 seq.push_back(expair(coeff, _ex0()));
496 for (n=1; n<order; ++n) {
497 fac = fac.mul(numeric(n));
498 deriv = deriv.diff(s).expand();
499 if (deriv.is_zero()) {
501 return pseries(r, seq);
503 coeff = deriv.subs(r);
504 if (!coeff.is_zero())
505 seq.push_back(expair(fac.inverse() * coeff, numeric(n)));
508 // Higher-order terms, if present
509 deriv = deriv.diff(s);
510 if (!deriv.expand().is_zero())
511 seq.push_back(expair(Order(_ex1()), numeric(n)));
512 return pseries(r, seq);
516 /** Implementation of ex::series() for symbols.
518 ex symbol::series(const relational & r, int order, unsigned options) const
521 const ex point = r.rhs();
522 GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
525 if (this->is_equal(*s.bp)) {
526 if (order > 0 && !point.is_zero())
527 seq.push_back(expair(point, _ex0()));
529 seq.push_back(expair(_ex1(), _ex1()));
531 seq.push_back(expair(Order(_ex1()), numeric(order)));
533 seq.push_back(expair(*this, _ex0()));
534 return pseries(r, seq);
538 /** Add one series object to another, producing a pseries object that
539 * represents the sum.
541 * @param other pseries object to add with
542 * @return the sum as a pseries */
543 ex pseries::add_series(const pseries &other) const
545 // Adding two series with different variables or expansion points
546 // results in an empty (constant) series
547 if (!is_compatible_to(other)) {
549 nul.push_back(expair(Order(_ex1()), _ex0()));
550 return pseries(relational(var,point), nul);
555 epvector::const_iterator a = seq.begin();
556 epvector::const_iterator b = other.seq.begin();
557 epvector::const_iterator a_end = seq.end();
558 epvector::const_iterator b_end = other.seq.end();
559 int pow_a = INT_MAX, pow_b = INT_MAX;
561 // If a is empty, fill up with elements from b and stop
564 new_seq.push_back(*b);
569 pow_a = ex_to<numeric>((*a).coeff).to_int();
571 // If b is empty, fill up with elements from a and stop
574 new_seq.push_back(*a);
579 pow_b = ex_to<numeric>((*b).coeff).to_int();
581 // a and b are non-empty, compare powers
583 // a has lesser power, get coefficient from a
584 new_seq.push_back(*a);
585 if (is_order_function((*a).rest))
588 } else if (pow_b < pow_a) {
589 // b has lesser power, get coefficient from b
590 new_seq.push_back(*b);
591 if (is_order_function((*b).rest))
595 // Add coefficient of a and b
596 if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
597 new_seq.push_back(expair(Order(_ex1()), (*a).coeff));
598 break; // Order term ends the sequence
600 ex sum = (*a).rest + (*b).rest;
601 if (!(sum.is_zero()))
602 new_seq.push_back(expair(sum, numeric(pow_a)));
608 return pseries(relational(var,point), new_seq);
612 /** Implementation of ex::series() for sums. This performs series addition when
613 * adding pseries objects.
615 ex add::series(const relational & r, int order, unsigned options) const
617 ex acc; // Series accumulator
619 // Get first term from overall_coeff
620 acc = overall_coeff.series(r, order, options);
622 // Add remaining terms
623 epvector::const_iterator it = seq.begin();
624 epvector::const_iterator itend = seq.end();
625 for (; it!=itend; ++it) {
627 if (is_ex_exactly_of_type(it->rest, pseries))
630 op = it->rest.series(r, order, options);
631 if (!it->coeff.is_equal(_ex1()))
632 op = ex_to<pseries>(op).mul_const(ex_to<numeric>(it->coeff));
635 acc = ex_to<pseries>(acc).add_series(ex_to<pseries>(op));
641 /** Multiply a pseries object with a numeric constant, producing a pseries
642 * object that represents the product.
644 * @param other constant to multiply with
645 * @return the product as a pseries */
646 ex pseries::mul_const(const numeric &other) const
649 new_seq.reserve(seq.size());
651 epvector::const_iterator it = seq.begin(), itend = seq.end();
652 while (it != itend) {
653 if (!is_order_function(it->rest))
654 new_seq.push_back(expair(it->rest * other, it->coeff));
656 new_seq.push_back(*it);
659 return pseries(relational(var,point), new_seq);
663 /** Multiply one pseries object to another, producing a pseries object that
664 * represents the product.
666 * @param other pseries object to multiply with
667 * @return the product as a pseries */
668 ex pseries::mul_series(const pseries &other) const
670 // Multiplying two series with different variables or expansion points
671 // results in an empty (constant) series
672 if (!is_compatible_to(other)) {
674 nul.push_back(expair(Order(_ex1()), _ex0()));
675 return pseries(relational(var,point), nul);
678 // Series multiplication
681 int a_max = degree(var);
682 int b_max = other.degree(var);
683 int a_min = ldegree(var);
684 int b_min = other.ldegree(var);
685 int cdeg_min = a_min + b_min;
686 int cdeg_max = a_max + b_max;
688 int higher_order_a = INT_MAX;
689 int higher_order_b = INT_MAX;
690 if (is_order_function(coeff(var, a_max)))
691 higher_order_a = a_max + b_min;
692 if (is_order_function(other.coeff(var, b_max)))
693 higher_order_b = b_max + a_min;
694 int higher_order_c = std::min(higher_order_a, higher_order_b);
695 if (cdeg_max >= higher_order_c)
696 cdeg_max = higher_order_c - 1;
698 for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
700 // c(i)=a(0)b(i)+...+a(i)b(0)
701 for (int i=a_min; cdeg-i>=b_min; ++i) {
702 ex a_coeff = coeff(var, i);
703 ex b_coeff = other.coeff(var, cdeg-i);
704 if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
705 co += a_coeff * b_coeff;
708 new_seq.push_back(expair(co, numeric(cdeg)));
710 if (higher_order_c < INT_MAX)
711 new_seq.push_back(expair(Order(_ex1()), numeric(higher_order_c)));
712 return pseries(relational(var, point), new_seq);
716 /** Implementation of ex::series() for product. This performs series
717 * multiplication when multiplying series.
719 ex mul::series(const relational & r, int order, unsigned options) const
721 ex acc; // Series accumulator
723 // Get first term from overall_coeff
724 acc = overall_coeff.series(r, order, options);
726 // Multiply with remaining terms
727 epvector::const_iterator it = seq.begin();
728 epvector::const_iterator itend = seq.end();
729 for (; it!=itend; ++it) {
731 if (op.info(info_flags::numeric)) {
732 // series * const (special case, faster)
733 ex f = power(op, it->coeff);
734 acc = ex_to<pseries>(acc).mul_const(ex_to<numeric>(f));
736 } else if (!is_ex_exactly_of_type(op, pseries))
737 op = op.series(r, order, options);
738 if (!it->coeff.is_equal(_ex1()))
739 op = ex_to<pseries>(op).power_const(ex_to<numeric>(it->coeff), order);
741 // Series multiplication
742 acc = ex_to<pseries>(acc).mul_series(ex_to<pseries>(op));
748 /** Compute the p-th power of a series.
750 * @param p power to compute
751 * @param deg truncation order of series calculation */
752 ex pseries::power_const(const numeric &p, int deg) const
755 // let A(x) be this series and for the time being let it start with a
756 // constant (later we'll generalize):
757 // A(x) = a_0 + a_1*x + a_2*x^2 + ...
758 // We want to compute
760 // C(x) = c_0 + c_1*x + c_2*x^2 + ...
761 // Taking the derivative on both sides and multiplying with A(x) one
762 // immediately arrives at
763 // C'(x)*A(x) = p*C(x)*A'(x)
764 // Multiplying this out and comparing coefficients we get the recurrence
766 // c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
767 // ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
768 // which can easily be solved given the starting value c_0 = (a_0)^p.
769 // For the more general case where the leading coefficient of A(x) is not
770 // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
771 // repeat the above derivation. The leading power of C2(x) = A2(x)^2 is
772 // then of course x^(p*m) but the recurrence formula still holds.
775 // as a spacial case, handle the empty (zero) series honoring the
776 // usual power laws such as implemented in power::eval()
777 if (p.real().is_zero())
778 throw (std::domain_error("pseries::power_const(): pow(0,I) is undefined"));
779 else if (p.real().is_negative())
780 throw (pole_error("pseries::power_const(): division by zero",1));
785 int ldeg = ldegree(var);
787 // Compute coefficients of the powered series
790 co.push_back(power(coeff(var, ldeg), p));
791 bool all_sums_zero = true;
792 for (int i=1; i<deg; ++i) {
794 for (int j=1; j<=i; ++j) {
795 ex c = coeff(var, j + ldeg);
796 if (is_order_function(c)) {
797 co.push_back(Order(_ex1()));
800 sum += (p * j - (i - j)) * co[i - j] * c;
803 all_sums_zero = false;
804 co.push_back(sum / coeff(var, ldeg) / numeric(i));
807 // Construct new series (of non-zero coefficients)
809 bool higher_order = false;
810 for (int i=0; i<deg; ++i) {
811 if (!co[i].is_zero())
812 new_seq.push_back(expair(co[i], numeric(i) + p * ldeg));
813 if (is_order_function(co[i])) {
818 if (!higher_order && !all_sums_zero)
819 new_seq.push_back(expair(Order(_ex1()), numeric(deg) + p * ldeg));
820 return pseries(relational(var,point), new_seq);
824 /** Return a new pseries object with the powers shifted by deg. */
825 pseries pseries::shift_exponents(int deg) const
827 epvector newseq(seq);
828 for (epvector::iterator i=newseq.begin(); i!=newseq.end(); ++i)
829 i->coeff = i->coeff + deg;
830 return pseries(relational(var, point), newseq);
834 /** Implementation of ex::series() for powers. This performs Laurent expansion
835 * of reciprocals of series at singularities.
837 ex power::series(const relational & r, int order, unsigned options) const
840 if (!is_ex_exactly_of_type(basis, pseries)) {
841 // Basis is not a series, may there be a singularity?
842 bool must_expand_basis = false;
845 } catch (pole_error) {
846 must_expand_basis = true;
849 // Is the expression of type something^(-int)?
850 if (!must_expand_basis && !exponent.info(info_flags::negint))
851 return basic::series(r, order, options);
853 // Is the expression of type 0^something?
854 if (!must_expand_basis && !basis.subs(r).is_zero())
855 return basic::series(r, order, options);
857 // Singularity encountered, expand basis into series
858 e = basis.series(r, order, options);
865 return ex_to<pseries>(e).power_const(ex_to<numeric>(exponent), order);
869 /** Re-expansion of a pseries object. */
870 ex pseries::series(const relational & r, int order, unsigned options) const
872 const ex p = r.rhs();
873 GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
874 const symbol &s = static_cast<symbol &>(*r.lhs().bp);
876 if (var.is_equal(s) && point.is_equal(p)) {
877 if (order > degree(s))
881 epvector::const_iterator it = seq.begin(), itend = seq.end();
882 while (it != itend) {
883 int o = ex_to<numeric>(it->coeff).to_int();
885 new_seq.push_back(expair(Order(_ex1()), o));
888 new_seq.push_back(*it);
891 return pseries(r, new_seq);
894 return convert_to_poly().series(r, order, options);
898 /** Compute the truncated series expansion of an expression.
899 * This function returns an expression containing an object of class pseries
900 * to represent the series. If the series does not terminate within the given
901 * truncation order, the last term of the series will be an order term.
903 * @param r expansion relation, lhs holds variable and rhs holds point
904 * @param order truncation order of series calculations
905 * @param options of class series_options
906 * @return an expression holding a pseries object */
907 ex ex::series(const ex & r, int order, unsigned options) const
913 if (is_ex_exactly_of_type(r,relational))
914 rel_ = ex_to<relational>(r);
915 else if (is_ex_exactly_of_type(r,symbol))
916 rel_ = relational(r,_ex0());
918 throw (std::logic_error("ex::series(): expansion point has unknown type"));
921 e = bp->series(rel_, order, options);
922 } catch (std::exception &x) {
923 throw (std::logic_error(std::string("unable to compute series (") + x.what() + ")"));