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
29 #include "inifcns.h" // for Order function
33 #include "relational.h"
41 GINAC_IMPLEMENT_REGISTERED_CLASS(pseries, basic)
45 * Default ctor, dtor, copy ctor, assignment operator and helpers
48 pseries::pseries() : inherited(TINFO_pseries) { }
50 void pseries::copy(const pseries &other)
52 inherited::copy(other);
58 DEFAULT_DESTROY(pseries)
65 /** Construct pseries from a vector of coefficients and powers.
66 * expair.rest holds the coefficient, expair.coeff holds the power.
67 * The powers must be integers (positive or negative) and in ascending order;
68 * the last coefficient can be Order(_ex1) to represent a truncated,
69 * non-terminating series.
71 * @param rel_ expansion variable and point (must hold a relational)
72 * @param ops_ vector of {coefficient, power} pairs (coefficient must not be zero)
73 * @return newly constructed pseries */
74 pseries::pseries(const ex &rel_, const epvector &ops_) : basic(TINFO_pseries), seq(ops_)
76 GINAC_ASSERT(is_exactly_a<relational>(rel_));
77 GINAC_ASSERT(is_exactly_a<symbol>(rel_.lhs()));
87 pseries::pseries(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
89 for (unsigned int i=0; true; ++i) {
92 if (n.find_ex("coeff", rest, sym_lst, i) && n.find_ex("power", coeff, sym_lst, i))
93 seq.push_back(expair(rest, coeff));
97 n.find_ex("var", var, sym_lst);
98 n.find_ex("point", point, sym_lst);
101 void pseries::archive(archive_node &n) const
103 inherited::archive(n);
104 epvector::const_iterator i = seq.begin(), iend = seq.end();
106 n.add_ex("coeff", i->rest);
107 n.add_ex("power", i->coeff);
110 n.add_ex("var", var);
111 n.add_ex("point", point);
114 DEFAULT_UNARCHIVE(pseries)
117 // functions overriding virtual functions from base classes
120 void pseries::print(const print_context & c, unsigned level) const
122 if (is_a<print_tree>(c)) {
124 c.s << std::string(level, ' ') << class_name()
125 << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
127 unsigned delta_indent = static_cast<const print_tree &>(c).delta_indent;
128 unsigned num = seq.size();
129 for (unsigned i=0; i<num; ++i) {
130 seq[i].rest.print(c, level + delta_indent);
131 seq[i].coeff.print(c, level + delta_indent);
132 c.s << std::string(level + delta_indent, ' ') << "-----" << std::endl;
134 var.print(c, level + delta_indent);
135 point.print(c, level + delta_indent);
139 if (precedence() <= level)
142 std::string par_open = is_a<print_latex>(c) ? "{(" : "(";
143 std::string par_close = is_a<print_latex>(c) ? ")}" : ")";
145 // objects of type pseries must not have any zero entries, so the
146 // trivial (zero) pseries needs a special treatment here:
149 epvector::const_iterator i = seq.begin(), end = seq.end();
151 // print a sign, if needed
152 if (i != seq.begin())
154 if (!is_order_function(i->rest)) {
155 // print 'rest', i.e. the expansion coefficient
156 if (i->rest.info(info_flags::numeric) &&
157 i->rest.info(info_flags::positive)) {
164 // print 'coeff', something like (x-1)^42
165 if (!i->coeff.is_zero()) {
166 if (is_a<print_latex>(c))
170 if (!point.is_zero()) {
172 (var-point).print(c);
176 if (i->coeff.compare(_ex1)) {
178 if (i->coeff.info(info_flags::negative)) {
183 if (is_a<print_latex>(c)) {
193 Order(power(var-point,i->coeff)).print(c);
197 if (precedence() <= level)
202 int pseries::compare_same_type(const basic & other) const
204 GINAC_ASSERT(is_a<pseries>(other));
205 const pseries &o = static_cast<const pseries &>(other);
207 // first compare the lengths of the series...
208 if (seq.size()>o.seq.size())
210 if (seq.size()<o.seq.size())
213 // ...then the expansion point...
214 int cmpval = var.compare(o.var);
217 cmpval = point.compare(o.point);
221 // ...and if that failed the individual elements
222 epvector::const_iterator it = seq.begin(), o_it = o.seq.begin();
223 while (it!=seq.end() && o_it!=o.seq.end()) {
224 cmpval = it->compare(*o_it);
231 // so they are equal.
235 /** Return the number of operands including a possible order term. */
236 unsigned pseries::nops(void) const
241 /** Return the ith term in the series when represented as a sum. */
242 ex pseries::op(int i) const
244 if (i < 0 || unsigned(i) >= seq.size())
245 throw (std::out_of_range("op() out of range"));
246 return seq[i].rest * power(var - point, seq[i].coeff);
249 ex &pseries::let_op(int i)
251 throw (std::logic_error("let_op not defined for pseries"));
254 /** Return degree of highest power of the series. This is usually the exponent
255 * of the Order term. If s is not the expansion variable of the series, the
256 * series is examined termwise. */
257 int pseries::degree(const ex &s) const
259 if (var.is_equal(s)) {
260 // Return last exponent
262 return ex_to<numeric>((seq.end()-1)->coeff).to_int();
266 epvector::const_iterator it = seq.begin(), itend = seq.end();
269 int max_pow = INT_MIN;
270 while (it != itend) {
271 int pow = it->rest.degree(s);
280 /** Return degree of lowest power of the series. This is usually the exponent
281 * of the leading term. If s is not the expansion variable of the series, the
282 * series is examined termwise. If s is the expansion variable but the
283 * expansion point is not zero the series is not expanded to find the degree.
284 * I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
285 int pseries::ldegree(const ex &s) const
287 if (var.is_equal(s)) {
288 // Return first exponent
290 return ex_to<numeric>((seq.begin())->coeff).to_int();
294 epvector::const_iterator it = seq.begin(), itend = seq.end();
297 int min_pow = INT_MAX;
298 while (it != itend) {
299 int pow = it->rest.ldegree(s);
308 /** Return coefficient of degree n in power series if s is the expansion
309 * variable. If the expansion point is nonzero, by definition the n=1
310 * coefficient in s of a+b*(s-z)+c*(s-z)^2+Order((s-z)^3) is b (assuming
311 * the expansion took place in the s in the first place).
312 * If s is not the expansion variable, an attempt is made to convert the
313 * series to a polynomial and return the corresponding coefficient from
315 ex pseries::coeff(const ex &s, int n) const
317 if (var.is_equal(s)) {
321 // Binary search in sequence for given power
322 numeric looking_for = numeric(n);
323 int lo = 0, hi = seq.size() - 1;
325 int mid = (lo + hi) / 2;
326 GINAC_ASSERT(is_exactly_a<numeric>(seq[mid].coeff));
327 int cmp = ex_to<numeric>(seq[mid].coeff).compare(looking_for);
333 return seq[mid].rest;
338 throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
343 return convert_to_poly().coeff(s, n);
347 ex pseries::collect(const ex &s, bool distributed) const
352 /** Perform coefficient-wise automatic term rewriting rules in this class. */
353 ex pseries::eval(int level) const
358 if (level == -max_recursion_level)
359 throw (std::runtime_error("pseries::eval(): recursion limit exceeded"));
361 // Construct a new series with evaluated coefficients
363 new_seq.reserve(seq.size());
364 epvector::const_iterator it = seq.begin(), itend = seq.end();
365 while (it != itend) {
366 new_seq.push_back(expair(it->rest.eval(level-1), it->coeff));
369 return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
372 /** Evaluate coefficients numerically. */
373 ex pseries::evalf(int level) const
378 if (level == -max_recursion_level)
379 throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
381 // Construct a new series with evaluated coefficients
383 new_seq.reserve(seq.size());
384 epvector::const_iterator it = seq.begin(), itend = seq.end();
385 while (it != itend) {
386 new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff));
389 return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
392 ex pseries::subs(const lst & ls, const lst & lr, bool no_pattern) const
394 // If expansion variable is being substituted, convert the series to a
395 // polynomial and do the substitution there because the result might
396 // no longer be a power series
398 return convert_to_poly(true).subs(ls, lr, no_pattern);
400 // Otherwise construct a new series with substituted coefficients and
403 newseq.reserve(seq.size());
404 epvector::const_iterator it = seq.begin(), itend = seq.end();
405 while (it != itend) {
406 newseq.push_back(expair(it->rest.subs(ls, lr, no_pattern), it->coeff));
409 return (new pseries(relational(var,point.subs(ls, lr, no_pattern)), newseq))->setflag(status_flags::dynallocated);
412 /** Implementation of ex::expand() for a power series. It expands all the
413 * terms individually and returns the resulting series as a new pseries. */
414 ex pseries::expand(unsigned options) const
417 epvector::const_iterator i = seq.begin(), end = seq.end();
419 ex restexp = i->rest.expand();
420 if (!restexp.is_zero())
421 newseq.push_back(expair(restexp, i->coeff));
424 return (new pseries(relational(var,point), newseq))
425 ->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
428 /** Implementation of ex::diff() for a power series. It treats the series as a
431 ex pseries::derivative(const symbol & s) const
435 epvector::const_iterator it = seq.begin(), itend = seq.end();
437 // FIXME: coeff might depend on var
438 while (it != itend) {
439 if (is_order_function(it->rest)) {
440 new_seq.push_back(expair(it->rest, it->coeff - 1));
442 ex c = it->rest * it->coeff;
444 new_seq.push_back(expair(c, it->coeff - 1));
448 return pseries(relational(var,point), new_seq);
454 ex pseries::convert_to_poly(bool no_order) const
457 epvector::const_iterator it = seq.begin(), itend = seq.end();
459 while (it != itend) {
460 if (is_order_function(it->rest)) {
462 e += Order(power(var - point, it->coeff));
464 e += it->rest * power(var - point, it->coeff);
470 bool pseries::is_terminating(void) const
472 return seq.empty() || !is_order_function((seq.end()-1)->rest);
477 * Implementations of series expansion
480 /** Default implementation of ex::series(). This performs Taylor expansion.
482 ex basic::series(const relational & r, int order, unsigned options) const
487 ex coeff = deriv.subs(r);
488 const symbol &s = ex_to<symbol>(r.lhs());
490 if (!coeff.is_zero())
491 seq.push_back(expair(coeff, _ex0));
494 for (n=1; n<order; ++n) {
496 // We need to test for zero in order to see if the series terminates.
497 // The problem is that there is no such thing as a perfect test for
498 // zero. Expanding the term occasionally helps a little...
499 deriv = deriv.diff(s).expand();
500 if (deriv.is_zero()) // Series terminates
501 return pseries(r, seq);
503 coeff = deriv.subs(r);
504 if (!coeff.is_zero())
505 seq.push_back(expair(fac.inverse() * coeff, 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), 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_exactly_a<symbol>(r.lhs()));
524 if (this->is_equal_same_type(ex_to<symbol>(r.lhs()))) {
525 if (order > 0 && !point.is_zero())
526 seq.push_back(expair(point, _ex0));
528 seq.push_back(expair(_ex1, _ex1));
530 seq.push_back(expair(Order(_ex1), numeric(order)));
532 seq.push_back(expair(*this, _ex0));
533 return pseries(r, seq);
537 /** Add one series object to another, producing a pseries object that
538 * represents the sum.
540 * @param other pseries object to add with
541 * @return the sum as a pseries */
542 ex pseries::add_series(const pseries &other) const
544 // Adding two series with different variables or expansion points
545 // results in an empty (constant) series
546 if (!is_compatible_to(other)) {
548 nul.push_back(expair(Order(_ex1), _ex0));
549 return pseries(relational(var,point), nul);
554 epvector::const_iterator a = seq.begin();
555 epvector::const_iterator b = other.seq.begin();
556 epvector::const_iterator a_end = seq.end();
557 epvector::const_iterator b_end = other.seq.end();
558 int pow_a = INT_MAX, pow_b = INT_MAX;
560 // If a is empty, fill up with elements from b and stop
563 new_seq.push_back(*b);
568 pow_a = ex_to<numeric>((*a).coeff).to_int();
570 // If b is empty, fill up with elements from a and stop
573 new_seq.push_back(*a);
578 pow_b = ex_to<numeric>((*b).coeff).to_int();
580 // a and b are non-empty, compare powers
582 // a has lesser power, get coefficient from a
583 new_seq.push_back(*a);
584 if (is_order_function((*a).rest))
587 } else if (pow_b < pow_a) {
588 // b has lesser power, get coefficient from b
589 new_seq.push_back(*b);
590 if (is_order_function((*b).rest))
594 // Add coefficient of a and b
595 if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
596 new_seq.push_back(expair(Order(_ex1), (*a).coeff));
597 break; // Order term ends the sequence
599 ex sum = (*a).rest + (*b).rest;
600 if (!(sum.is_zero()))
601 new_seq.push_back(expair(sum, numeric(pow_a)));
607 return pseries(relational(var,point), new_seq);
611 /** Implementation of ex::series() for sums. This performs series addition when
612 * adding pseries objects.
614 ex add::series(const relational & r, int order, unsigned options) const
616 ex acc; // Series accumulator
618 // Get first term from overall_coeff
619 acc = overall_coeff.series(r, order, options);
621 // Add remaining terms
622 epvector::const_iterator it = seq.begin();
623 epvector::const_iterator itend = seq.end();
624 for (; it!=itend; ++it) {
626 if (is_ex_exactly_of_type(it->rest, pseries))
629 op = it->rest.series(r, order, options);
630 if (!it->coeff.is_equal(_ex1))
631 op = ex_to<pseries>(op).mul_const(ex_to<numeric>(it->coeff));
634 acc = ex_to<pseries>(acc).add_series(ex_to<pseries>(op));
640 /** Multiply a pseries object with a numeric constant, producing a pseries
641 * object that represents the product.
643 * @param other constant to multiply with
644 * @return the product as a pseries */
645 ex pseries::mul_const(const numeric &other) const
648 new_seq.reserve(seq.size());
650 epvector::const_iterator it = seq.begin(), itend = seq.end();
651 while (it != itend) {
652 if (!is_order_function(it->rest))
653 new_seq.push_back(expair(it->rest * other, it->coeff));
655 new_seq.push_back(*it);
658 return pseries(relational(var,point), new_seq);
662 /** Multiply one pseries object to another, producing a pseries object that
663 * represents the product.
665 * @param other pseries object to multiply with
666 * @return the product as a pseries */
667 ex pseries::mul_series(const pseries &other) const
669 // Multiplying two series with different variables or expansion points
670 // results in an empty (constant) series
671 if (!is_compatible_to(other)) {
673 nul.push_back(expair(Order(_ex1), _ex0));
674 return pseries(relational(var,point), nul);
677 // Series multiplication
679 int a_max = degree(var);
680 int b_max = other.degree(var);
681 int a_min = ldegree(var);
682 int b_min = other.ldegree(var);
683 int cdeg_min = a_min + b_min;
684 int cdeg_max = a_max + b_max;
686 int higher_order_a = INT_MAX;
687 int higher_order_b = INT_MAX;
688 if (is_order_function(coeff(var, a_max)))
689 higher_order_a = a_max + b_min;
690 if (is_order_function(other.coeff(var, b_max)))
691 higher_order_b = b_max + a_min;
692 int higher_order_c = std::min(higher_order_a, higher_order_b);
693 if (cdeg_max >= higher_order_c)
694 cdeg_max = higher_order_c - 1;
696 for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
698 // c(i)=a(0)b(i)+...+a(i)b(0)
699 for (int i=a_min; cdeg-i>=b_min; ++i) {
700 ex a_coeff = coeff(var, i);
701 ex b_coeff = other.coeff(var, cdeg-i);
702 if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
703 co += a_coeff * b_coeff;
706 new_seq.push_back(expair(co, numeric(cdeg)));
708 if (higher_order_c < INT_MAX)
709 new_seq.push_back(expair(Order(_ex1), numeric(higher_order_c)));
710 return pseries(relational(var, point), new_seq);
714 /** Implementation of ex::series() for product. This performs series
715 * multiplication when multiplying series.
717 ex mul::series(const relational & r, int order, unsigned options) const
719 pseries acc; // Series accumulator
721 // Multiply with remaining terms
722 const epvector::const_iterator itbeg = seq.begin();
723 const epvector::const_iterator itend = seq.end();
724 for (epvector::const_iterator it=itbeg; it!=itend; ++it) {
725 ex op = recombine_pair_to_ex(*it).series(r, order, options);
727 // Series multiplication
729 acc = ex_to<pseries>(op);
731 acc = ex_to<pseries>(acc.mul_series(ex_to<pseries>(op)));
733 return acc.mul_const(ex_to<numeric>(overall_coeff));
737 /** Compute the p-th power of a series.
739 * @param p power to compute
740 * @param deg truncation order of series calculation */
741 ex pseries::power_const(const numeric &p, int deg) const
744 // (due to Leonhard Euler)
745 // let A(x) be this series and for the time being let it start with a
746 // constant (later we'll generalize):
747 // A(x) = a_0 + a_1*x + a_2*x^2 + ...
748 // We want to compute
750 // C(x) = c_0 + c_1*x + c_2*x^2 + ...
751 // Taking the derivative on both sides and multiplying with A(x) one
752 // immediately arrives at
753 // C'(x)*A(x) = p*C(x)*A'(x)
754 // Multiplying this out and comparing coefficients we get the recurrence
756 // c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
757 // ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
758 // which can easily be solved given the starting value c_0 = (a_0)^p.
759 // For the more general case where the leading coefficient of A(x) is not
760 // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
761 // repeat the above derivation. The leading power of C2(x) = A2(x)^2 is
762 // then of course x^(p*m) but the recurrence formula still holds.
765 // as a special case, handle the empty (zero) series honoring the
766 // usual power laws such as implemented in power::eval()
767 if (p.real().is_zero())
768 throw std::domain_error("pseries::power_const(): pow(0,I) is undefined");
769 else if (p.real().is_negative())
770 throw pole_error("pseries::power_const(): division by zero",1);
775 const int ldeg = ldegree(var);
776 if (!(p*ldeg).is_integer())
777 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
779 // O(x^n)^(-m) is undefined
780 if (seq.size() == 1 && is_order_function(seq[0].rest) && p.real().is_negative())
781 throw pole_error("pseries::power_const(): division by zero",1);
783 // Compute coefficients of the powered series
786 co.push_back(power(coeff(var, ldeg), p));
787 bool all_sums_zero = true;
788 for (int i=1; i<deg; ++i) {
790 for (int j=1; j<=i; ++j) {
791 ex c = coeff(var, j + ldeg);
792 if (is_order_function(c)) {
793 co.push_back(Order(_ex1));
796 sum += (p * j - (i - j)) * co[i - j] * c;
799 all_sums_zero = false;
800 co.push_back(sum / coeff(var, ldeg) / i);
803 // Construct new series (of non-zero coefficients)
805 bool higher_order = false;
806 for (int i=0; i<deg; ++i) {
807 if (!co[i].is_zero())
808 new_seq.push_back(expair(co[i], p * ldeg + i));
809 if (is_order_function(co[i])) {
814 if (!higher_order && !all_sums_zero)
815 new_seq.push_back(expair(Order(_ex1), p * ldeg + deg));
816 return pseries(relational(var,point), new_seq);
820 /** Return a new pseries object with the powers shifted by deg. */
821 pseries pseries::shift_exponents(int deg) const
823 epvector newseq = seq;
824 epvector::iterator i = newseq.begin(), end = newseq.end();
829 return pseries(relational(var, point), newseq);
833 /** Implementation of ex::series() for powers. This performs Laurent expansion
834 * of reciprocals of series at singularities.
836 ex power::series(const relational & r, int order, unsigned options) const
838 // If basis is already a series, just power it
839 if (is_ex_exactly_of_type(basis, pseries))
840 return ex_to<pseries>(basis).power_const(ex_to<numeric>(exponent), order);
842 // Basis is not a series, may there be a singularity?
843 bool must_expand_basis = false;
846 } catch (pole_error) {
847 must_expand_basis = true;
850 // Is the expression of type something^(-int)?
851 if (!must_expand_basis && !exponent.info(info_flags::negint))
852 return basic::series(r, order, options);
854 // Is the expression of type 0^something?
855 if (!must_expand_basis && !basis.subs(r).is_zero())
856 return basic::series(r, order, options);
858 // Singularity encountered, is the basis equal to (var - point)?
859 if (basis.is_equal(r.lhs() - r.rhs())) {
861 if (ex_to<numeric>(exponent).to_int() < order)
862 new_seq.push_back(expair(_ex1, exponent));
864 new_seq.push_back(expair(Order(_ex1), exponent));
865 return pseries(r, new_seq);
868 // No, expand basis into series
869 ex e = basis.series(r, order, options);
870 return ex_to<pseries>(e).power_const(ex_to<numeric>(exponent), order);
874 /** Re-expansion of a pseries object. */
875 ex pseries::series(const relational & r, int order, unsigned options) const
877 const ex p = r.rhs();
878 GINAC_ASSERT(is_exactly_a<symbol>(r.lhs()));
879 const symbol &s = ex_to<symbol>(r.lhs());
881 if (var.is_equal(s) && point.is_equal(p)) {
882 if (order > degree(s))
886 epvector::const_iterator it = seq.begin(), itend = seq.end();
887 while (it != itend) {
888 int o = ex_to<numeric>(it->coeff).to_int();
890 new_seq.push_back(expair(Order(_ex1), o));
893 new_seq.push_back(*it);
896 return pseries(r, new_seq);
899 return convert_to_poly().series(r, order, options);
903 /** Compute the truncated series expansion of an expression.
904 * This function returns an expression containing an object of class pseries
905 * to represent the series. If the series does not terminate within the given
906 * truncation order, the last term of the series will be an order term.
908 * @param r expansion relation, lhs holds variable and rhs holds point
909 * @param order truncation order of series calculations
910 * @param options of class series_options
911 * @return an expression holding a pseries object */
912 ex ex::series(const ex & r, int order, unsigned options) const
918 if (is_ex_exactly_of_type(r,relational))
919 rel_ = ex_to<relational>(r);
920 else if (is_ex_exactly_of_type(r,symbol))
921 rel_ = relational(r,_ex0);
923 throw (std::logic_error("ex::series(): expansion point has unknown type"));
926 e = bp->series(rel_, order, options);
927 } catch (std::exception &x) {
928 throw (std::logic_error(std::string("unable to compute series (") + x.what() + ")"));