3 * Implementation of class for extended truncated power series and
4 * methods for series expansion. */
7 * GiNaC Copyright (C) 1999-2001 Johannes Gutenberg University Mainz, Germany
9 * This program is free software; you can redistribute it and/or modify
10 * it under the terms of the GNU General Public License as published by
11 * the Free Software Foundation; either version 2 of the License, or
12 * (at your option) any later version.
14 * This program is distributed in the hope that it will be useful,
15 * but WITHOUT ANY WARRANTY; without even the implied warranty of
16 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
17 * GNU General Public License for more details.
19 * You should have received a copy of the GNU General Public License
20 * along with this program; if not, write to the Free Software
21 * Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
32 #include "relational.h"
41 GINAC_IMPLEMENT_REGISTERED_CLASS(pseries, basic)
45 * Default ctor, dtor, copy ctor, assignment operator and helpers
48 pseries::pseries() : basic(TINFO_pseries)
50 debugmsg("pseries default ctor", LOGLEVEL_CONSTRUCT);
53 void pseries::copy(const pseries &other)
55 inherited::copy(other);
61 DEFAULT_DESTROY(pseries)
68 /** Construct pseries from a vector of coefficients and powers.
69 * expair.rest holds the coefficient, expair.coeff holds the power.
70 * The powers must be integers (positive or negative) and in ascending order;
71 * the last coefficient can be Order(_ex1()) to represent a truncated,
72 * non-terminating series.
74 * @param rel_ expansion variable and point (must hold a relational)
75 * @param ops_ vector of {coefficient, power} pairs (coefficient must not be zero)
76 * @return newly constructed pseries */
77 pseries::pseries(const ex &rel_, const epvector &ops_) : basic(TINFO_pseries), seq(ops_)
79 debugmsg("pseries ctor from ex,epvector", LOGLEVEL_CONSTRUCT);
80 GINAC_ASSERT(is_ex_exactly_of_type(rel_, relational));
81 GINAC_ASSERT(is_ex_exactly_of_type(rel_.lhs(),symbol));
83 var = *static_cast<symbol *>(rel_.lhs().bp);
91 pseries::pseries(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
93 debugmsg("pseries ctor from archive_node", LOGLEVEL_CONSTRUCT);
94 for (unsigned int i=0; true; ++i) {
97 if (n.find_ex("coeff", rest, sym_lst, i) && n.find_ex("power", coeff, sym_lst, i))
98 seq.push_back(expair(rest, coeff));
102 n.find_ex("var", var, sym_lst);
103 n.find_ex("point", point, sym_lst);
106 void pseries::archive(archive_node &n) const
108 inherited::archive(n);
109 epvector::const_iterator i = seq.begin(), iend = seq.end();
111 n.add_ex("coeff", i->rest);
112 n.add_ex("power", i->coeff);
115 n.add_ex("var", var);
116 n.add_ex("point", point);
119 DEFAULT_UNARCHIVE(pseries)
122 // functions overriding virtual functions from bases classes
125 void pseries::print(const print_context & c, unsigned level) const
127 debugmsg("pseries print", LOGLEVEL_PRINT);
129 if (is_of_type(c, print_tree)) {
131 c.s << std::string(level, ' ') << class_name()
132 << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
134 unsigned delta_indent = static_cast<const print_tree &>(c).delta_indent;
135 for (unsigned i=0; i<seq.size(); ++i) {
136 seq[i].rest.print(c, level + delta_indent);
137 seq[i].coeff.print(c, level + delta_indent);
138 c.s << std::string(level + delta_indent, ' ') << "-----" << std::endl;
140 var.print(c, level + delta_indent);
141 point.print(c, level + delta_indent);
145 if (precedence <= level)
148 std::string par_open = is_of_type(c, print_latex) ? "{(" : "(";
149 std::string par_close = is_of_type(c, print_latex) ? ")}" : ")";
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()) {
172 if (!point.is_zero()) {
174 (var-point).print(c);
178 if (i->coeff.compare(_ex1())) {
180 if (i->coeff.info(info_flags::negative)) {
185 if (is_of_type(c, print_latex)) {
195 Order(power(var-point,i->coeff)).print(c);
198 if (precedence <= level)
203 int pseries::compare_same_type(const basic & other) const
205 GINAC_ASSERT(is_of_type(other, pseries));
206 const pseries &o = static_cast<const pseries &>(other);
208 // first compare the lengths of the series...
209 if (seq.size()>o.seq.size())
211 if (seq.size()<o.seq.size())
214 // ...then the expansion point...
215 int cmpval = var.compare(o.var);
218 cmpval = point.compare(o.point);
222 // ...and if that failed the individual elements
223 epvector::const_iterator it = seq.begin(), o_it = o.seq.begin();
224 while (it!=seq.end() && o_it!=o.seq.end()) {
225 cmpval = it->compare(*o_it);
232 // so they are equal.
236 /** Return the number of operands including a possible order term. */
237 unsigned pseries::nops(void) const
242 /** Return the ith term in the series when represented as a sum. */
243 ex pseries::op(int i) const
245 if (i < 0 || unsigned(i) >= seq.size())
246 throw (std::out_of_range("op() out of range"));
247 return seq[i].rest * power(var - point, seq[i].coeff);
250 ex &pseries::let_op(int i)
252 throw (std::logic_error("let_op not defined for pseries"));
255 /** Return degree of highest power of the series. This is usually the exponent
256 * of the Order term. If s is not the expansion variable of the series, the
257 * series is examined termwise. */
258 int pseries::degree(const ex &s) const
260 if (var.is_equal(s)) {
261 // Return last exponent
263 return ex_to_numeric((*(seq.end() - 1)).coeff).to_int();
267 epvector::const_iterator it = seq.begin(), itend = seq.end();
270 int max_pow = INT_MIN;
271 while (it != itend) {
272 int pow = it->rest.degree(s);
281 /** Return degree of lowest power of the series. This is usually the exponent
282 * of the leading term. If s is not the expansion variable of the series, the
283 * series is examined termwise. If s is the expansion variable but the
284 * expansion point is not zero the series is not expanded to find the degree.
285 * I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
286 int pseries::ldegree(const ex &s) const
288 if (var.is_equal(s)) {
289 // Return first exponent
291 return ex_to_numeric((*(seq.begin())).coeff).to_int();
295 epvector::const_iterator it = seq.begin(), itend = seq.end();
298 int min_pow = INT_MAX;
299 while (it != itend) {
300 int pow = it->rest.ldegree(s);
309 /** Return coefficient of degree n in power series if s is the expansion
310 * variable. If the expansion point is nonzero, by definition the n=1
311 * coefficient in s of a+b*(s-z)+c*(s-z)^2+Order((s-z)^3) is b (assuming
312 * the expansion took place in the s in the first place).
313 * If s is not the expansion variable, an attempt is made to convert the
314 * series to a polynomial and return the corresponding coefficient from
316 ex pseries::coeff(const ex &s, int n) const
318 if (var.is_equal(s)) {
322 // Binary search in sequence for given power
323 numeric looking_for = numeric(n);
324 int lo = 0, hi = seq.size() - 1;
326 int mid = (lo + hi) / 2;
327 GINAC_ASSERT(is_ex_exactly_of_type(seq[mid].coeff, numeric));
328 int cmp = ex_to_numeric(seq[mid].coeff).compare(looking_for);
334 return seq[mid].rest;
339 throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
344 return convert_to_poly().coeff(s, n);
348 ex pseries::collect(const ex &s) const
353 /** Evaluate coefficients. */
354 ex pseries::eval(int level) const
359 if (level == -max_recursion_level)
360 throw (std::runtime_error("pseries::eval(): recursion limit exceeded"));
362 // Construct a new series with evaluated coefficients
364 new_seq.reserve(seq.size());
365 epvector::const_iterator it = seq.begin(), itend = seq.end();
366 while (it != itend) {
367 new_seq.push_back(expair(it->rest.eval(level-1), it->coeff));
370 return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
373 /** Evaluate coefficients numerically. */
374 ex pseries::evalf(int level) const
379 if (level == -max_recursion_level)
380 throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
382 // Construct a new series with evaluated coefficients
384 new_seq.reserve(seq.size());
385 epvector::const_iterator it = seq.begin(), itend = seq.end();
386 while (it != itend) {
387 new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff));
390 return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
393 ex pseries::subs(const lst & ls, const lst & lr) const
395 // If expansion variable is being substituted, convert the series to a
396 // polynomial and do the substitution there because the result might
397 // no longer be a power series
399 return convert_to_poly(true).subs(ls, lr);
401 // Otherwise construct a new series with substituted coefficients and
404 newseq.reserve(seq.size());
405 epvector::const_iterator it = seq.begin(), itend = seq.end();
406 while (it != itend) {
407 newseq.push_back(expair(it->rest.subs(ls, lr), it->coeff));
410 return (new pseries(relational(var,point.subs(ls, lr)), newseq))->setflag(status_flags::dynallocated);
413 /** Implementation of ex::expand() for a power series. It expands all the
414 * terms individually and returns the resulting series as a new pseries. */
415 ex pseries::expand(unsigned options) const
418 for (epvector::const_iterator i=seq.begin(); i!=seq.end(); ++i) {
419 ex restexp = i->rest.expand();
420 if (!restexp.is_zero())
421 newseq.push_back(expair(restexp, i->coeff));
423 return (new pseries(relational(var,point), newseq))
424 ->setflag(status_flags::dynallocated | status_flags::expanded);
427 /** Implementation of ex::diff() for a power series. It treats the series as a
430 ex pseries::derivative(const symbol & s) const
434 epvector::const_iterator it = seq.begin(), itend = seq.end();
436 // FIXME: coeff might depend on var
437 while (it != itend) {
438 if (is_order_function(it->rest)) {
439 new_seq.push_back(expair(it->rest, it->coeff - 1));
441 ex c = it->rest * it->coeff;
443 new_seq.push_back(expair(c, it->coeff - 1));
447 return pseries(relational(var,point), new_seq);
453 ex pseries::convert_to_poly(bool no_order) const
456 epvector::const_iterator it = seq.begin(), itend = seq.end();
458 while (it != itend) {
459 if (is_order_function(it->rest)) {
461 e += Order(power(var - point, it->coeff));
463 e += it->rest * power(var - point, it->coeff);
469 bool pseries::is_terminating(void) const
471 return seq.size() == 0 || !is_order_function((seq.end()-1)->rest);
476 * Implementations of series expansion
479 /** Default implementation of ex::series(). This performs Taylor expansion.
481 ex basic::series(const relational & r, int order, unsigned options) const
486 ex coeff = deriv.subs(r);
487 const symbol &s = static_cast<symbol &>(*r.lhs().bp);
489 if (!coeff.is_zero())
490 seq.push_back(expair(coeff, numeric(0)));
493 for (n=1; n<order; ++n) {
494 fac = fac.mul(numeric(n));
495 deriv = deriv.diff(s).expand();
496 if (deriv.is_zero()) {
498 return pseries(r, seq);
500 coeff = deriv.subs(r);
501 if (!coeff.is_zero())
502 seq.push_back(expair(fac.inverse() * coeff, numeric(n)));
505 // Higher-order terms, if present
506 deriv = deriv.diff(s);
507 if (!deriv.expand().is_zero())
508 seq.push_back(expair(Order(_ex1()), numeric(n)));
509 return pseries(r, seq);
513 /** Implementation of ex::series() for symbols.
515 ex symbol::series(const relational & r, int order, unsigned options) const
518 const ex point = r.rhs();
519 GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
522 if (this->is_equal(*s.bp)) {
523 if (order > 0 && !point.is_zero())
524 seq.push_back(expair(point, _ex0()));
526 seq.push_back(expair(_ex1(), _ex1()));
528 seq.push_back(expair(Order(_ex1()), numeric(order)));
530 seq.push_back(expair(*this, _ex0()));
531 return pseries(r, seq);
535 /** Add one series object to another, producing a pseries object that
536 * represents the sum.
538 * @param other pseries object to add with
539 * @return the sum as a pseries */
540 ex pseries::add_series(const pseries &other) const
542 // Adding two series with different variables or expansion points
543 // results in an empty (constant) series
544 if (!is_compatible_to(other)) {
546 nul.push_back(expair(Order(_ex1()), _ex0()));
547 return pseries(relational(var,point), nul);
552 epvector::const_iterator a = seq.begin();
553 epvector::const_iterator b = other.seq.begin();
554 epvector::const_iterator a_end = seq.end();
555 epvector::const_iterator b_end = other.seq.end();
556 int pow_a = INT_MAX, pow_b = INT_MAX;
558 // If a is empty, fill up with elements from b and stop
561 new_seq.push_back(*b);
566 pow_a = ex_to_numeric((*a).coeff).to_int();
568 // If b is empty, fill up with elements from a and stop
571 new_seq.push_back(*a);
576 pow_b = ex_to_numeric((*b).coeff).to_int();
578 // a and b are non-empty, compare powers
580 // a has lesser power, get coefficient from a
581 new_seq.push_back(*a);
582 if (is_order_function((*a).rest))
585 } else if (pow_b < pow_a) {
586 // b has lesser power, get coefficient from b
587 new_seq.push_back(*b);
588 if (is_order_function((*b).rest))
592 // Add coefficient of a and b
593 if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
594 new_seq.push_back(expair(Order(_ex1()), (*a).coeff));
595 break; // Order term ends the sequence
597 ex sum = (*a).rest + (*b).rest;
598 if (!(sum.is_zero()))
599 new_seq.push_back(expair(sum, numeric(pow_a)));
605 return pseries(relational(var,point), new_seq);
609 /** Implementation of ex::series() for sums. This performs series addition when
610 * adding pseries objects.
612 ex add::series(const relational & r, int order, unsigned options) const
614 ex acc; // Series accumulator
616 // Get first term from overall_coeff
617 acc = overall_coeff.series(r, order, options);
619 // Add remaining terms
620 epvector::const_iterator it = seq.begin();
621 epvector::const_iterator itend = seq.end();
622 for (; it!=itend; ++it) {
624 if (is_ex_exactly_of_type(it->rest, pseries))
627 op = it->rest.series(r, order, options);
628 if (!it->coeff.is_equal(_ex1()))
629 op = ex_to_pseries(op).mul_const(ex_to_numeric(it->coeff));
632 acc = ex_to_pseries(acc).add_series(ex_to_pseries(op));
638 /** Multiply a pseries object with a numeric constant, producing a pseries
639 * object that represents the product.
641 * @param other constant to multiply with
642 * @return the product as a pseries */
643 ex pseries::mul_const(const numeric &other) const
646 new_seq.reserve(seq.size());
648 epvector::const_iterator it = seq.begin(), itend = seq.end();
649 while (it != itend) {
650 if (!is_order_function(it->rest))
651 new_seq.push_back(expair(it->rest * other, it->coeff));
653 new_seq.push_back(*it);
656 return pseries(relational(var,point), new_seq);
660 /** Multiply one pseries object to another, producing a pseries object that
661 * represents the product.
663 * @param other pseries object to multiply with
664 * @return the product as a pseries */
665 ex pseries::mul_series(const pseries &other) const
667 // Multiplying two series with different variables or expansion points
668 // results in an empty (constant) series
669 if (!is_compatible_to(other)) {
671 nul.push_back(expair(Order(_ex1()), _ex0()));
672 return pseries(relational(var,point), nul);
675 // Series multiplication
678 int a_max = degree(var);
679 int b_max = other.degree(var);
680 int a_min = ldegree(var);
681 int b_min = other.ldegree(var);
682 int cdeg_min = a_min + b_min;
683 int cdeg_max = a_max + b_max;
685 int higher_order_a = INT_MAX;
686 int higher_order_b = INT_MAX;
687 if (is_order_function(coeff(var, a_max)))
688 higher_order_a = a_max + b_min;
689 if (is_order_function(other.coeff(var, b_max)))
690 higher_order_b = b_max + a_min;
691 int higher_order_c = std::min(higher_order_a, higher_order_b);
692 if (cdeg_max >= higher_order_c)
693 cdeg_max = higher_order_c - 1;
695 for (int cdeg=cdeg_min; cdeg<=cdeg_max; ++cdeg) {
697 // c(i)=a(0)b(i)+...+a(i)b(0)
698 for (int i=a_min; cdeg-i>=b_min; ++i) {
699 ex a_coeff = coeff(var, i);
700 ex b_coeff = other.coeff(var, cdeg-i);
701 if (!is_order_function(a_coeff) && !is_order_function(b_coeff))
702 co += a_coeff * b_coeff;
705 new_seq.push_back(expair(co, numeric(cdeg)));
707 if (higher_order_c < INT_MAX)
708 new_seq.push_back(expair(Order(_ex1()), numeric(higher_order_c)));
709 return pseries(relational(var, point), new_seq);
713 /** Implementation of ex::series() for product. This performs series
714 * multiplication when multiplying series.
716 ex mul::series(const relational & r, int order, unsigned options) const
718 ex acc; // Series accumulator
720 // Get first term from overall_coeff
721 acc = overall_coeff.series(r, order, options);
723 // Multiply with remaining terms
724 epvector::const_iterator it = seq.begin();
725 epvector::const_iterator itend = seq.end();
726 for (; it!=itend; ++it) {
728 if (op.info(info_flags::numeric)) {
729 // series * const (special case, faster)
730 ex f = power(op, it->coeff);
731 acc = ex_to_pseries(acc).mul_const(ex_to_numeric(f));
733 } else if (!is_ex_exactly_of_type(op, pseries))
734 op = op.series(r, order, options);
735 if (!it->coeff.is_equal(_ex1()))
736 op = ex_to_pseries(op).power_const(ex_to_numeric(it->coeff), order);
738 // Series multiplication
739 acc = ex_to_pseries(acc).mul_series(ex_to_pseries(op));
745 /** Compute the p-th power of a series.
747 * @param p power to compute
748 * @param deg truncation order of series calculation */
749 ex pseries::power_const(const numeric &p, int deg) const
752 // let A(x) be this series and for the time being let it start with a
753 // constant (later we'll generalize):
754 // A(x) = a_0 + a_1*x + a_2*x^2 + ...
755 // We want to compute
757 // C(x) = c_0 + c_1*x + c_2*x^2 + ...
758 // Taking the derivative on both sides and multiplying with A(x) one
759 // immediately arrives at
760 // C'(x)*A(x) = p*C(x)*A'(x)
761 // Multiplying this out and comparing coefficients we get the recurrence
763 // c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
764 // ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
765 // which can easily be solved given the starting value c_0 = (a_0)^p.
766 // For the more general case where the leading coefficient of A(x) is not
767 // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
768 // repeat the above derivation. The leading power of C2(x) = A2(x)^2 is
769 // then of course x^(p*m) but the recurrence formula still holds.
772 // as a spacial case, handle the empty (zero) series honoring the
773 // usual power laws such as implemented in power::eval()
774 if (p.real().is_zero())
775 throw (std::domain_error("pseries::power_const(): pow(0,I) is undefined"));
776 else if (p.real().is_negative())
777 throw (pole_error("pseries::power_const(): division by zero",1));
782 int ldeg = ldegree(var);
784 // Compute coefficients of the powered series
787 co.push_back(power(coeff(var, ldeg), p));
788 bool all_sums_zero = true;
789 for (int i=1; i<deg; ++i) {
791 for (int j=1; j<=i; ++j) {
792 ex c = coeff(var, j + ldeg);
793 if (is_order_function(c)) {
794 co.push_back(Order(_ex1()));
797 sum += (p * j - (i - j)) * co[i - j] * c;
800 all_sums_zero = false;
801 co.push_back(sum / coeff(var, ldeg) / numeric(i));
804 // Construct new series (of non-zero coefficients)
806 bool higher_order = false;
807 for (int i=0; i<deg; ++i) {
808 if (!co[i].is_zero())
809 new_seq.push_back(expair(co[i], numeric(i) + p * ldeg));
810 if (is_order_function(co[i])) {
815 if (!higher_order && !all_sums_zero)
816 new_seq.push_back(expair(Order(_ex1()), numeric(deg) + p * ldeg));
817 return pseries(relational(var,point), new_seq);
821 /** Return a new pseries object with the powers shifted by deg. */
822 pseries pseries::shift_exponents(int deg) const
824 epvector newseq(seq);
825 for (epvector::iterator i=newseq.begin(); i!=newseq.end(); ++i)
826 i->coeff = i->coeff + deg;
827 return pseries(relational(var, point), newseq);
831 /** Implementation of ex::series() for powers. This performs Laurent expansion
832 * of reciprocals of series at singularities.
834 ex power::series(const relational & r, int order, unsigned options) const
837 if (!is_ex_exactly_of_type(basis, pseries)) {
838 // Basis is not a series, may there be a singularity?
839 bool must_expand_basis = false;
842 } catch (pole_error) {
843 must_expand_basis = true;
846 // Is the expression of type something^(-int)?
847 if (!must_expand_basis && !exponent.info(info_flags::negint))
848 return basic::series(r, order, options);
850 // Is the expression of type 0^something?
851 if (!must_expand_basis && !basis.subs(r).is_zero())
852 return basic::series(r, order, options);
854 // Singularity encountered, expand basis into series
855 e = basis.series(r, order, options);
862 return ex_to_pseries(e).power_const(ex_to_numeric(exponent), order);
866 /** Re-expansion of a pseries object. */
867 ex pseries::series(const relational & r, int order, unsigned options) const
869 const ex p = r.rhs();
870 GINAC_ASSERT(is_ex_exactly_of_type(r.lhs(),symbol));
871 const symbol &s = static_cast<symbol &>(*r.lhs().bp);
873 if (var.is_equal(s) && point.is_equal(p)) {
874 if (order > degree(s))
878 epvector::const_iterator it = seq.begin(), itend = seq.end();
879 while (it != itend) {
880 int o = ex_to_numeric(it->coeff).to_int();
882 new_seq.push_back(expair(Order(_ex1()), o));
885 new_seq.push_back(*it);
888 return pseries(r, new_seq);
891 return convert_to_poly().series(r, order, options);
895 /** Compute the truncated series expansion of an expression.
896 * This function returns an expression containing an object of class pseries
897 * to represent the series. If the series does not terminate within the given
898 * truncation order, the last term of the series will be an order term.
900 * @param r expansion relation, lhs holds variable and rhs holds point
901 * @param order truncation order of series calculations
902 * @param options of class series_options
903 * @return an expression holding a pseries object */
904 ex ex::series(const ex & r, int order, unsigned options) const
910 if (is_ex_exactly_of_type(r,relational))
911 rel_ = ex_to_relational(r);
912 else if (is_ex_exactly_of_type(r,symbol))
913 rel_ = relational(r,_ex0());
915 throw (std::logic_error("ex::series(): expansion point has unknown type"));
918 e = bp->series(rel_, order, options);
919 } catch (std::exception &x) {
920 throw (std::logic_error(std::string("unable to compute series (") + x.what() + ")"));
926 // static member variables
931 unsigned pseries::precedence = 38; // for clarity just below add::precedence