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"
40 GINAC_IMPLEMENT_REGISTERED_CLASS(pseries, basic)
44 * Default ctor, dtor, copy ctor, assignment operator and helpers
47 pseries::pseries() : inherited(TINFO_pseries) { }
49 void pseries::copy(const pseries &other)
51 inherited::copy(other);
57 DEFAULT_DESTROY(pseries)
64 /** Construct pseries from a vector of coefficients and powers.
65 * expair.rest holds the coefficient, expair.coeff holds the power.
66 * The powers must be integers (positive or negative) and in ascending order;
67 * the last coefficient can be Order(_ex1) to represent a truncated,
68 * non-terminating series.
70 * @param rel_ expansion variable and point (must hold a relational)
71 * @param ops_ vector of {coefficient, power} pairs (coefficient must not be zero)
72 * @return newly constructed pseries */
73 pseries::pseries(const ex &rel_, const epvector &ops_) : basic(TINFO_pseries), seq(ops_)
75 GINAC_ASSERT(is_exactly_a<relational>(rel_));
76 GINAC_ASSERT(is_exactly_a<symbol>(rel_.lhs()));
86 pseries::pseries(const archive_node &n, const lst &sym_lst) : inherited(n, sym_lst)
88 for (unsigned int i=0; true; ++i) {
91 if (n.find_ex("coeff", rest, sym_lst, i) && n.find_ex("power", coeff, sym_lst, i))
92 seq.push_back(expair(rest, coeff));
96 n.find_ex("var", var, sym_lst);
97 n.find_ex("point", point, sym_lst);
100 void pseries::archive(archive_node &n) const
102 inherited::archive(n);
103 epvector::const_iterator i = seq.begin(), iend = seq.end();
105 n.add_ex("coeff", i->rest);
106 n.add_ex("power", i->coeff);
109 n.add_ex("var", var);
110 n.add_ex("point", point);
113 DEFAULT_UNARCHIVE(pseries)
116 // functions overriding virtual functions from base classes
119 void pseries::print(const print_context & c, unsigned level) const
121 if (is_a<print_tree>(c)) {
123 c.s << std::string(level, ' ') << class_name()
124 << std::hex << ", hash=0x" << hashvalue << ", flags=0x" << flags << std::dec
126 unsigned delta_indent = static_cast<const print_tree &>(c).delta_indent;
127 unsigned num = seq.size();
128 for (unsigned i=0; i<num; ++i) {
129 seq[i].rest.print(c, level + delta_indent);
130 seq[i].coeff.print(c, level + delta_indent);
131 c.s << std::string(level + delta_indent, ' ') << "-----" << std::endl;
133 var.print(c, level + delta_indent);
134 point.print(c, level + delta_indent);
138 if (precedence() <= level)
141 std::string par_open = is_a<print_latex>(c) ? "{(" : "(";
142 std::string par_close = is_a<print_latex>(c) ? ")}" : ")";
144 // objects of type pseries must not have any zero entries, so the
145 // trivial (zero) pseries needs a special treatment here:
148 epvector::const_iterator i = seq.begin(), end = seq.end();
150 // print a sign, if needed
151 if (i != seq.begin())
153 if (!is_order_function(i->rest)) {
154 // print 'rest', i.e. the expansion coefficient
155 if (i->rest.info(info_flags::numeric) &&
156 i->rest.info(info_flags::positive)) {
163 // print 'coeff', something like (x-1)^42
164 if (!i->coeff.is_zero()) {
165 if (is_a<print_latex>(c))
169 if (!point.is_zero()) {
171 (var-point).print(c);
175 if (i->coeff.compare(_ex1)) {
177 if (i->coeff.info(info_flags::negative)) {
182 if (is_a<print_latex>(c)) {
192 Order(power(var-point,i->coeff)).print(c);
196 if (precedence() <= level)
201 int pseries::compare_same_type(const basic & other) const
203 GINAC_ASSERT(is_a<pseries>(other));
204 const pseries &o = static_cast<const pseries &>(other);
206 // first compare the lengths of the series...
207 if (seq.size()>o.seq.size())
209 if (seq.size()<o.seq.size())
212 // ...then the expansion point...
213 int cmpval = var.compare(o.var);
216 cmpval = point.compare(o.point);
220 // ...and if that failed the individual elements
221 epvector::const_iterator it = seq.begin(), o_it = o.seq.begin();
222 while (it!=seq.end() && o_it!=o.seq.end()) {
223 cmpval = it->compare(*o_it);
230 // so they are equal.
234 /** Return the number of operands including a possible order term. */
235 unsigned pseries::nops(void) const
240 /** Return the ith term in the series when represented as a sum. */
241 ex pseries::op(int i) const
243 if (i < 0 || unsigned(i) >= seq.size())
244 throw (std::out_of_range("op() out of range"));
245 return seq[i].rest * power(var - point, seq[i].coeff);
248 ex &pseries::let_op(int i)
250 throw (std::logic_error("let_op not defined for pseries"));
253 /** Return degree of highest power of the series. This is usually the exponent
254 * of the Order term. If s is not the expansion variable of the series, the
255 * series is examined termwise. */
256 int pseries::degree(const ex &s) const
258 if (var.is_equal(s)) {
259 // Return last exponent
261 return ex_to<numeric>((seq.end()-1)->coeff).to_int();
265 epvector::const_iterator it = seq.begin(), itend = seq.end();
268 int max_pow = INT_MIN;
269 while (it != itend) {
270 int pow = it->rest.degree(s);
279 /** Return degree of lowest power of the series. This is usually the exponent
280 * of the leading term. If s is not the expansion variable of the series, the
281 * series is examined termwise. If s is the expansion variable but the
282 * expansion point is not zero the series is not expanded to find the degree.
283 * I.e.: (1-x) + (1-x)^2 + Order((1-x)^3) has ldegree(x) 1, not 0. */
284 int pseries::ldegree(const ex &s) const
286 if (var.is_equal(s)) {
287 // Return first exponent
289 return ex_to<numeric>((seq.begin())->coeff).to_int();
293 epvector::const_iterator it = seq.begin(), itend = seq.end();
296 int min_pow = INT_MAX;
297 while (it != itend) {
298 int pow = it->rest.ldegree(s);
307 /** Return coefficient of degree n in power series if s is the expansion
308 * variable. If the expansion point is nonzero, by definition the n=1
309 * coefficient in s of a+b*(s-z)+c*(s-z)^2+Order((s-z)^3) is b (assuming
310 * the expansion took place in the s in the first place).
311 * If s is not the expansion variable, an attempt is made to convert the
312 * series to a polynomial and return the corresponding coefficient from
314 ex pseries::coeff(const ex &s, int n) const
316 if (var.is_equal(s)) {
320 // Binary search in sequence for given power
321 numeric looking_for = numeric(n);
322 int lo = 0, hi = seq.size() - 1;
324 int mid = (lo + hi) / 2;
325 GINAC_ASSERT(is_exactly_a<numeric>(seq[mid].coeff));
326 int cmp = ex_to<numeric>(seq[mid].coeff).compare(looking_for);
332 return seq[mid].rest;
337 throw(std::logic_error("pseries::coeff: compare() didn't return -1, 0 or 1"));
342 return convert_to_poly().coeff(s, n);
346 ex pseries::collect(const ex &s, bool distributed) const
351 /** Perform coefficient-wise automatic term rewriting rules in this class. */
352 ex pseries::eval(int level) const
357 if (level == -max_recursion_level)
358 throw (std::runtime_error("pseries::eval(): recursion limit exceeded"));
360 // Construct a new series with evaluated coefficients
362 new_seq.reserve(seq.size());
363 epvector::const_iterator it = seq.begin(), itend = seq.end();
364 while (it != itend) {
365 new_seq.push_back(expair(it->rest.eval(level-1), it->coeff));
368 return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
371 /** Evaluate coefficients numerically. */
372 ex pseries::evalf(int level) const
377 if (level == -max_recursion_level)
378 throw (std::runtime_error("pseries::evalf(): recursion limit exceeded"));
380 // Construct a new series with evaluated coefficients
382 new_seq.reserve(seq.size());
383 epvector::const_iterator it = seq.begin(), itend = seq.end();
384 while (it != itend) {
385 new_seq.push_back(expair(it->rest.evalf(level-1), it->coeff));
388 return (new pseries(relational(var,point), new_seq))->setflag(status_flags::dynallocated | status_flags::evaluated);
391 ex pseries::subs(const lst & ls, const lst & lr, bool no_pattern) const
393 // If expansion variable is being substituted, convert the series to a
394 // polynomial and do the substitution there because the result might
395 // no longer be a power series
397 return convert_to_poly(true).subs(ls, lr, no_pattern);
399 // Otherwise construct a new series with substituted coefficients and
402 newseq.reserve(seq.size());
403 epvector::const_iterator it = seq.begin(), itend = seq.end();
404 while (it != itend) {
405 newseq.push_back(expair(it->rest.subs(ls, lr, no_pattern), it->coeff));
408 return (new pseries(relational(var,point.subs(ls, lr, no_pattern)), newseq))->setflag(status_flags::dynallocated);
411 /** Implementation of ex::expand() for a power series. It expands all the
412 * terms individually and returns the resulting series as a new pseries. */
413 ex pseries::expand(unsigned options) const
416 epvector::const_iterator i = seq.begin(), end = seq.end();
418 ex restexp = i->rest.expand();
419 if (!restexp.is_zero())
420 newseq.push_back(expair(restexp, i->coeff));
423 return (new pseries(relational(var,point), newseq))
424 ->setflag(status_flags::dynallocated | (options == 0 ? status_flags::expanded : 0));
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.empty() || !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 = ex_to<symbol>(r.lhs());
489 if (!coeff.is_zero())
490 seq.push_back(expair(coeff, _ex0));
493 for (n=1; n<order; ++n) {
495 // We need to test for zero in order to see if the series terminates.
496 // The problem is that there is no such thing as a perfect test for
497 // zero. Expanding the term occasionally helps a little...
498 deriv = deriv.diff(s).expand();
499 if (deriv.is_zero()) // Series terminates
500 return pseries(r, seq);
502 coeff = deriv.subs(r);
503 if (!coeff.is_zero())
504 seq.push_back(expair(fac.inverse() * coeff, n));
507 // Higher-order terms, if present
508 deriv = deriv.diff(s);
509 if (!deriv.expand().is_zero())
510 seq.push_back(expair(Order(_ex1), n));
511 return pseries(r, seq);
515 /** Implementation of ex::series() for symbols.
517 ex symbol::series(const relational & r, int order, unsigned options) const
520 const ex point = r.rhs();
521 GINAC_ASSERT(is_exactly_a<symbol>(r.lhs()));
523 if (this->is_equal_same_type(ex_to<symbol>(r.lhs()))) {
524 if (order > 0 && !point.is_zero())
525 seq.push_back(expair(point, _ex0));
527 seq.push_back(expair(_ex1, _ex1));
529 seq.push_back(expair(Order(_ex1), numeric(order)));
531 seq.push_back(expair(*this, _ex0));
532 return pseries(r, seq);
536 /** Add one series object to another, producing a pseries object that
537 * represents the sum.
539 * @param other pseries object to add with
540 * @return the sum as a pseries */
541 ex pseries::add_series(const pseries &other) const
543 // Adding two series with different variables or expansion points
544 // results in an empty (constant) series
545 if (!is_compatible_to(other)) {
547 nul.push_back(expair(Order(_ex1), _ex0));
548 return pseries(relational(var,point), nul);
553 epvector::const_iterator a = seq.begin();
554 epvector::const_iterator b = other.seq.begin();
555 epvector::const_iterator a_end = seq.end();
556 epvector::const_iterator b_end = other.seq.end();
557 int pow_a = INT_MAX, pow_b = INT_MAX;
559 // If a is empty, fill up with elements from b and stop
562 new_seq.push_back(*b);
567 pow_a = ex_to<numeric>((*a).coeff).to_int();
569 // If b is empty, fill up with elements from a and stop
572 new_seq.push_back(*a);
577 pow_b = ex_to<numeric>((*b).coeff).to_int();
579 // a and b are non-empty, compare powers
581 // a has lesser power, get coefficient from a
582 new_seq.push_back(*a);
583 if (is_order_function((*a).rest))
586 } else if (pow_b < pow_a) {
587 // b has lesser power, get coefficient from b
588 new_seq.push_back(*b);
589 if (is_order_function((*b).rest))
593 // Add coefficient of a and b
594 if (is_order_function((*a).rest) || is_order_function((*b).rest)) {
595 new_seq.push_back(expair(Order(_ex1), (*a).coeff));
596 break; // Order term ends the sequence
598 ex sum = (*a).rest + (*b).rest;
599 if (!(sum.is_zero()))
600 new_seq.push_back(expair(sum, numeric(pow_a)));
606 return pseries(relational(var,point), new_seq);
610 /** Implementation of ex::series() for sums. This performs series addition when
611 * adding pseries objects.
613 ex add::series(const relational & r, int order, unsigned options) const
615 ex acc; // Series accumulator
617 // Get first term from overall_coeff
618 acc = overall_coeff.series(r, order, options);
620 // Add remaining terms
621 epvector::const_iterator it = seq.begin();
622 epvector::const_iterator itend = seq.end();
623 for (; it!=itend; ++it) {
625 if (is_ex_exactly_of_type(it->rest, pseries))
628 op = it->rest.series(r, order, options);
629 if (!it->coeff.is_equal(_ex1))
630 op = ex_to<pseries>(op).mul_const(ex_to<numeric>(it->coeff));
633 acc = ex_to<pseries>(acc).add_series(ex_to<pseries>(op));
639 /** Multiply a pseries object with a numeric constant, producing a pseries
640 * object that represents the product.
642 * @param other constant to multiply with
643 * @return the product as a pseries */
644 ex pseries::mul_const(const numeric &other) const
647 new_seq.reserve(seq.size());
649 epvector::const_iterator it = seq.begin(), itend = seq.end();
650 while (it != itend) {
651 if (!is_order_function(it->rest))
652 new_seq.push_back(expair(it->rest * other, it->coeff));
654 new_seq.push_back(*it);
657 return pseries(relational(var,point), new_seq);
661 /** Multiply one pseries object to another, producing a pseries object that
662 * represents the product.
664 * @param other pseries object to multiply with
665 * @return the product as a pseries */
666 ex pseries::mul_series(const pseries &other) const
668 // Multiplying two series with different variables or expansion points
669 // results in an empty (constant) series
670 if (!is_compatible_to(other)) {
672 nul.push_back(expair(Order(_ex1), _ex0));
673 return pseries(relational(var,point), nul);
676 // 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 pseries acc; // Series accumulator
720 // Multiply with remaining terms
721 const epvector::const_iterator itbeg = seq.begin();
722 const epvector::const_iterator itend = seq.end();
723 for (epvector::const_iterator it=itbeg; it!=itend; ++it) {
724 ex op = recombine_pair_to_ex(*it).series(r, order, options);
726 // Series multiplication
728 acc = ex_to<pseries>(op);
730 acc = ex_to<pseries>(acc.mul_series(ex_to<pseries>(op)));
732 return acc.mul_const(ex_to<numeric>(overall_coeff));
736 /** Compute the p-th power of a series.
738 * @param p power to compute
739 * @param deg truncation order of series calculation */
740 ex pseries::power_const(const numeric &p, int deg) const
743 // (due to Leonhard Euler)
744 // let A(x) be this series and for the time being let it start with a
745 // constant (later we'll generalize):
746 // A(x) = a_0 + a_1*x + a_2*x^2 + ...
747 // We want to compute
749 // C(x) = c_0 + c_1*x + c_2*x^2 + ...
750 // Taking the derivative on both sides and multiplying with A(x) one
751 // immediately arrives at
752 // C'(x)*A(x) = p*C(x)*A'(x)
753 // Multiplying this out and comparing coefficients we get the recurrence
755 // c_i = (i*p*a_i*c_0 + ((i-1)*p-1)*a_{i-1}*c_1 + ...
756 // ... + (p-(i-1))*a_1*c_{i-1})/(a_0*i)
757 // which can easily be solved given the starting value c_0 = (a_0)^p.
758 // For the more general case where the leading coefficient of A(x) is not
759 // a constant, just consider A2(x) = A(x)*x^m, with some integer m and
760 // repeat the above derivation. The leading power of C2(x) = A2(x)^2 is
761 // then of course x^(p*m) but the recurrence formula still holds.
764 // as a special case, handle the empty (zero) series honoring the
765 // usual power laws such as implemented in power::eval()
766 if (p.real().is_zero())
767 throw std::domain_error("pseries::power_const(): pow(0,I) is undefined");
768 else if (p.real().is_negative())
769 throw pole_error("pseries::power_const(): division by zero",1);
774 const int ldeg = ldegree(var);
775 if (!(p*ldeg).is_integer())
776 throw std::runtime_error("pseries::power_const(): trying to assemble a Puiseux series");
778 // O(x^n)^(-m) is undefined
779 if (seq.size() == 1 && is_order_function(seq[0].rest) && p.real().is_negative())
780 throw pole_error("pseries::power_const(): division by zero",1);
782 // Compute coefficients of the powered series
785 co.push_back(power(coeff(var, ldeg), p));
786 bool all_sums_zero = true;
787 for (int i=1; i<deg; ++i) {
789 for (int j=1; j<=i; ++j) {
790 ex c = coeff(var, j + ldeg);
791 if (is_order_function(c)) {
792 co.push_back(Order(_ex1));
795 sum += (p * j - (i - j)) * co[i - j] * c;
798 all_sums_zero = false;
799 co.push_back(sum / coeff(var, ldeg) / i);
802 // Construct new series (of non-zero coefficients)
804 bool higher_order = false;
805 for (int i=0; i<deg; ++i) {
806 if (!co[i].is_zero())
807 new_seq.push_back(expair(co[i], p * ldeg + i));
808 if (is_order_function(co[i])) {
813 if (!higher_order && !all_sums_zero)
814 new_seq.push_back(expair(Order(_ex1), p * ldeg + deg));
815 return pseries(relational(var,point), new_seq);
819 /** Return a new pseries object with the powers shifted by deg. */
820 pseries pseries::shift_exponents(int deg) const
822 epvector newseq = seq;
823 epvector::iterator i = newseq.begin(), end = newseq.end();
828 return pseries(relational(var, point), newseq);
832 /** Implementation of ex::series() for powers. This performs Laurent expansion
833 * of reciprocals of series at singularities.
835 ex power::series(const relational & r, int order, unsigned options) const
837 // If basis is already a series, just power it
838 if (is_ex_exactly_of_type(basis, pseries))
839 return ex_to<pseries>(basis).power_const(ex_to<numeric>(exponent), order);
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, is the basis equal to (var - point)?
858 if (basis.is_equal(r.lhs() - r.rhs())) {
860 if (ex_to<numeric>(exponent).to_int() < order)
861 new_seq.push_back(expair(_ex1, exponent));
863 new_seq.push_back(expair(Order(_ex1), exponent));
864 return pseries(r, new_seq);
867 // No, expand basis into series
868 ex e = basis.series(r, order, options);
869 return ex_to<pseries>(e).power_const(ex_to<numeric>(exponent), order);
873 /** Re-expansion of a pseries object. */
874 ex pseries::series(const relational & r, int order, unsigned options) const
876 const ex p = r.rhs();
877 GINAC_ASSERT(is_exactly_a<symbol>(r.lhs()));
878 const symbol &s = ex_to<symbol>(r.lhs());
880 if (var.is_equal(s) && point.is_equal(p)) {
881 if (order > degree(s))
885 epvector::const_iterator it = seq.begin(), itend = seq.end();
886 while (it != itend) {
887 int o = ex_to<numeric>(it->coeff).to_int();
889 new_seq.push_back(expair(Order(_ex1), o));
892 new_seq.push_back(*it);
895 return pseries(r, new_seq);
898 return convert_to_poly().series(r, order, options);
902 /** Compute the truncated series expansion of an expression.
903 * This function returns an expression containing an object of class pseries
904 * to represent the series. If the series does not terminate within the given
905 * truncation order, the last term of the series will be an order term.
907 * @param r expansion relation, lhs holds variable and rhs holds point
908 * @param order truncation order of series calculations
909 * @param options of class series_options
910 * @return an expression holding a pseries object */
911 ex ex::series(const ex & r, int order, unsigned options) const
917 if (is_ex_exactly_of_type(r,relational))
918 rel_ = ex_to<relational>(r);
919 else if (is_ex_exactly_of_type(r,symbol))
920 rel_ = relational(r,_ex0);
922 throw (std::logic_error("ex::series(): expansion point has unknown type"));
925 e = bp->series(rel_, order, options);
926 } catch (std::exception &x) {
927 throw (std::logic_error(std::string("unable to compute series (") + x.what() + ")"));