* Implementation of GiNaC's symbolic exponentiation (basis^exponent). */
/*
- * GiNaC Copyright (C) 1999-2014 Johannes Gutenberg University Mainz, Germany
+ * GiNaC Copyright (C) 1999-2015 Johannes Gutenberg University Mainz, Germany
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
print_func<print_python_repr>(&power::do_print_python_repr).
print_func<print_csrc_cl_N>(&power::do_print_csrc_cl_N))
-typedef std::vector<int> intvector;
-
//////////
// default constructor
//////////
case info_flags::positive:
return basis.info(info_flags::positive) && exponent.info(info_flags::real);
case info_flags::nonnegative:
- return basis.info(info_flags::real) && exponent.info(info_flags::integer) && exponent.info(info_flags::even);
+ return (basis.info(info_flags::positive) && exponent.info(info_flags::real)) ||
+ (basis.info(info_flags::real) && exponent.info(info_flags::integer) && exponent.info(info_flags::even));
case info_flags::has_indices: {
if (flags & status_flags::has_indices)
return true;
// protected
-// protected
-
/** Implementation of ex::diff() for a power.
* @see ex::diff */
ex power::derivative(const symbol & s) const
return *this;
}
+ // (x*p)^c -> x^c * p^c, if p>0
+ // makes sense before expanding the basis
+ if (is_exactly_a<mul>(basis) && !basis.info(info_flags::indefinite)) {
+ const mul &m = ex_to<mul>(basis);
+ exvector prodseq;
+ epvector powseq;
+ prodseq.reserve(m.seq.size() + 1);
+ powseq.reserve(m.seq.size() + 1);
+ epvector::const_iterator last = m.seq.end();
+ epvector::const_iterator cit = m.seq.begin();
+ bool possign = true;
+
+ // search for positive/negative factors
+ while (cit!=last) {
+ ex e=m.recombine_pair_to_ex(*cit);
+ if (e.info(info_flags::positive))
+ prodseq.push_back(pow(e, exponent).expand(options));
+ else if (e.info(info_flags::negative)) {
+ prodseq.push_back(pow(-e, exponent).expand(options));
+ possign = !possign;
+ } else
+ powseq.push_back(*cit);
+ ++cit;
+ }
+
+ // take care on the numeric coefficient
+ ex coeff=(possign? _ex1 : _ex_1);
+ if (m.overall_coeff.info(info_flags::positive) && m.overall_coeff != _ex1)
+ prodseq.push_back(power(m.overall_coeff, exponent));
+ else if (m.overall_coeff.info(info_flags::negative) && m.overall_coeff != _ex_1)
+ prodseq.push_back(power(-m.overall_coeff, exponent));
+ else
+ coeff *= m.overall_coeff;
+
+ // If positive/negative factors are found, then extract them.
+ // In either case we set a flag to avoid the second run on a part
+ // which does not have positive/negative terms.
+ if (prodseq.size() > 0) {
+ ex newbasis = coeff*mul(powseq);
+ ex_to<basic>(newbasis).setflag(status_flags::purely_indefinite);
+ return ((new mul(prodseq))->setflag(status_flags::dynallocated)*(new power(newbasis, exponent))->setflag(status_flags::dynallocated).expand(options)).expand(options);
+ } else
+ ex_to<basic>(basis).setflag(status_flags::purely_indefinite);
+ }
+
const ex expanded_basis = basis.expand(options);
const ex expanded_exponent = exponent.expand(options);
// non-virtual functions in this class
//////////
+namespace { // anonymous namespace for power::expand_add() helpers
+
+/** Helper class to generate all bounded combinatorial partitions of an integer
+ * n with exactly m parts (including zero parts) in non-decreaing order.
+ */
+class partition_generator {
+private:
+ // Partitions n into m parts, not including zero parts.
+ // (Cf. OEIS sequence A008284; implementation adapted from Jörg Arndt's
+ // FXT library)
+ struct mpartition2
+ {
+ // partition: x[1] + x[2] + ... + x[m] = n and sentinel x[0] == 0
+ std::vector<int> x;
+ int n; // n>0
+ int m; // 0<m<=n
+ mpartition2(unsigned n_, unsigned m_)
+ : x(m_+1), n(n_), m(m_)
+ {
+ for (int k=1; k<m; ++k)
+ x[k] = 1;
+ x[m] = n - m + 1;
+ }
+ bool next_partition()
+ {
+ int u = x[m]; // last element
+ int k = m;
+ int s = u;
+ while (--k) {
+ s += x[k];
+ if (x[k] + 2 <= u)
+ break;
+ }
+ if (k==0)
+ return false; // current is last
+ int f = x[k] + 1;
+ while (k < m) {
+ x[k] = f;
+ s -= f;
+ ++k;
+ }
+ x[m] = s;
+ return true;
+ }
+ } mpgen;
+ int m; // number of parts 0<m<=n
+ mutable std::vector<int> partition; // current partition
+public:
+ partition_generator(unsigned n_, unsigned m_)
+ : mpgen(n_, 1), m(m_), partition(m_)
+ { }
+ // returns current partition in non-decreasing order, padded with zeros
+ const std::vector<int>& current() const
+ {
+ for (int i = 0; i < m - mpgen.m; ++i)
+ partition[i] = 0; // pad with zeros
+
+ for (int i = m - mpgen.m; i < m; ++i)
+ partition[i] = mpgen.x[i - m + mpgen.m + 1];
+
+ return partition;
+ }
+ bool next()
+ {
+ if (!mpgen.next_partition()) {
+ if (mpgen.m == m || mpgen.m == mpgen.n)
+ return false; // current is last
+ // increment number of parts
+ mpgen = mpartition2(mpgen.n, mpgen.m + 1);
+ }
+ return true;
+ }
+};
+
+/** Helper class to generate all compositions of a partition of an integer n,
+ * starting with the compositions which has non-decreasing order.
+ */
+class composition_generator {
+private:
+ // Generates all distinct permutations of a multiset.
+ // (Based on Aaron Williams' algorithm 1 from "Loopless Generation of
+ // Multiset Permutations using a Constant Number of Variables by Prefix
+ // Shifts." <http://webhome.csc.uvic.ca/~haron/CoolMulti.pdf>)
+ struct coolmulti {
+ // element of singly linked list
+ struct element {
+ int value;
+ element* next;
+ element(int val, element* n)
+ : value(val), next(n) {}
+ ~element()
+ { // recurses down to the end of the singly linked list
+ delete next;
+ }
+ };
+ element *head, *i, *after_i;
+ // NB: Partition must be sorted in non-decreasing order.
+ explicit coolmulti(const std::vector<int>& partition)
+ {
+ head = NULL;
+ for (unsigned n = 0; n < partition.size(); ++n) {
+ head = new element(partition[n], head);
+ if (n <= 1)
+ i = head;
+ }
+ after_i = i->next;
+ }
+ ~coolmulti()
+ { // deletes singly linked list
+ delete head;
+ }
+ void next_permutation()
+ {
+ element *before_k;
+ if (after_i->next != NULL && i->value >= after_i->next->value)
+ before_k = after_i;
+ else
+ before_k = i;
+ element *k = before_k->next;
+ before_k->next = k->next;
+ k->next = head;
+ if (k->value < head->value)
+ i = k;
+ after_i = i->next;
+ head = k;
+ }
+ bool finished() const
+ {
+ return after_i->next == NULL && after_i->value >= head->value;
+ }
+ } cmgen;
+ bool atend; // needed for simplifying iteration over permutations
+ bool trivial; // likewise, true if all elements are equal
+ mutable std::vector<int> composition; // current compositions
+public:
+ explicit composition_generator(const std::vector<int>& partition)
+ : cmgen(partition), atend(false), trivial(true), composition(partition.size())
+ {
+ for (unsigned i=1; i<partition.size(); ++i)
+ trivial = trivial && (partition[0] == partition[i]);
+ }
+ const std::vector<int>& current() const
+ {
+ coolmulti::element* it = cmgen.head;
+ size_t i = 0;
+ while (it != NULL) {
+ composition[i] = it->value;
+ it = it->next;
+ ++i;
+ }
+ return composition;
+ }
+ bool next()
+ {
+ // This ugly contortion is needed because the original coolmulti
+ // algorithm requires code duplication of the payload procedure,
+ // one before the loop and one inside it.
+ if (trivial || atend)
+ return false;
+ cmgen.next_permutation();
+ atend = cmgen.finished();
+ return true;
+ }
+};
+
+/** Helper function to compute the multinomial coefficient n!/(p1!*p2!*...*pk!)
+ * where n = p1+p2+...+pk, i.e. p is a partition of n.
+ */
+const numeric
+multinomial_coefficient(const std::vector<int> p)
+{
+ numeric n = 0, d = 1;
+ std::vector<int>::const_iterator it = p.begin(), itend = p.end();
+ while (it != itend) {
+ n += numeric(*it);
+ d *= factorial(numeric(*it));
+ ++it;
+ }
+ return factorial(numeric(n)) / d;
+}
+
+} // anonymous namespace
+
/** expand a^n where a is an add and n is a positive integer.
* @see power::expand */
ex power::expand_add(const add & a, int n, unsigned options) const
{
+ // The special case power(+(x,...y;x),2) can be optimized better.
if (n==2)
return expand_add_2(a, options);
- const size_t m = a.nops();
- exvector result;
+ // method:
+ //
+ // Consider base as the sum of all symbolic terms and the overall numeric
+ // coefficient and apply the binomial theorem:
+ // S = power(+(x,...,z;c),n)
+ // = power(+(+(x,...,z;0);c),n)
+ // = sum(binomial(n,k)*power(+(x,...,z;0),k)*c^(n-k), k=1..n) + c^n
+ // Then, apply the multinomial theorem to expand all power(+(x,...,z;0),k):
+ // The multinomial theorem is computed by an outer loop over all
+ // partitions of the exponent and an inner loop over all compositions of
+ // that partition. This method makes the expansion a combinatorial
+ // problem and allows us to directly construct the expanded sum and also
+ // to re-use the multinomial coefficients (since they depend only on the
+ // partition, not on the composition).
+ //
+ // multinomial power(+(x,y,z;0),3) example:
+ // partition : compositions : multinomial coefficient
+ // [0,0,3] : [3,0,0],[0,3,0],[0,0,3] : 3!/(3!*0!*0!) = 1
+ // [0,1,2] : [2,1,0],[1,2,0],[2,0,1],... : 3!/(2!*1!*0!) = 3
+ // [1,1,1] : [1,1,1] : 3!/(1!*1!*1!) = 6
+ // => (x + y + z)^3 =
+ // x^3 + y^3 + z^3
+ // + 3*x^2*y + 3*x*y^2 + 3*y^2*z + 3*y*z^2 + 3*x*z^2 + 3*x^2*z
+ // + 6*x*y*z
+ //
+ // multinomial power(+(x,y,z;0),4) example:
+ // partition : compositions : multinomial coefficient
+ // [0,0,4] : [4,0,0],[0,4,0],[0,0,4] : 4!/(4!*0!*0!) = 1
+ // [0,1,3] : [3,1,0],[1,3,0],[3,0,1],... : 4!/(3!*1!*0!) = 4
+ // [0,2,2] : [2,2,0],[2,0,2],[0,2,2] : 4!/(2!*2!*0!) = 6
+ // [1,1,2] : [2,1,1],[1,2,1],[1,1,2] : 4!/(2!*1!*1!) = 12
+ // (no [1,1,1,1] partition since it has too many parts)
+ // => (x + y + z)^4 =
+ // x^4 + y^4 + z^4
+ // + 4*x^3*y + 4*x*y^3 + 4*y^3*z + 4*y*z^3 + 4*x*z^3 + 4*x^3*z
+ // + 6*x^2*y^2 + 6*y^2*z^2 + 6*x^2*z^2
+ // + 12*x^2*y*z + 12*x*y^2*z + 12*x*y*z^2
+ //
+ // Summary:
+ // r = 0
+ // for k from 0 to n:
+ // f = c^(n-k)*binomial(n,k)
+ // for p in all partitions of n with m parts (including zero parts):
+ // h = f * multinomial coefficient of p
+ // for c in all compositions of p:
+ // t = 1
+ // for e in all elements of c:
+ // t = t * a[e]^e
+ // r = r + h*t
+ // return r
+
+ epvector result;
// The number of terms will be the number of combinatorial compositions,
// i.e. the number of unordered arrangements of m nonnegative integers
// which sum up to n. It is frequently written as C_n(m) and directly
- // related with binomial coefficients:
- result.reserve(binomial(numeric(n+m-1), numeric(m-1)).to_int());
- intvector k(m-1);
- intvector k_cum(m-1); // k_cum[l]:=sum(i=0,l,k[l]);
- intvector upper_limit(m-1);
-
- for (size_t l=0; l<m-1; ++l) {
- k[l] = 0;
- k_cum[l] = 0;
- upper_limit[l] = n;
+ // related with binomial coefficients: binomial(n+m-1,m-1).
+ size_t result_size = binomial(numeric(n+a.nops()-1), numeric(a.nops()-1)).to_int();
+ if (!a.overall_coeff.is_zero()) {
+ // the result's overall_coeff is one of the terms
+ --result_size;
}
-
- while (true) {
- exvector term;
- term.reserve(m+1);
- for (std::size_t l = 0; l < m - 1; ++l) {
- const ex & b = a.op(l);
- GINAC_ASSERT(!is_exactly_a<add>(b));
- GINAC_ASSERT(!is_exactly_a<power>(b) ||
- !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
- !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
- !is_exactly_a<add>(ex_to<power>(b).basis) ||
- !is_exactly_a<mul>(ex_to<power>(b).basis) ||
- !is_exactly_a<power>(ex_to<power>(b).basis));
- if (is_exactly_a<mul>(b))
- term.push_back(expand_mul(ex_to<mul>(b), numeric(k[l]), options, true));
- else
- term.push_back(power(b,k[l]));
- }
-
- const ex & b = a.op(m - 1);
- GINAC_ASSERT(!is_exactly_a<add>(b));
- GINAC_ASSERT(!is_exactly_a<power>(b) ||
- !is_exactly_a<numeric>(ex_to<power>(b).exponent) ||
- !ex_to<numeric>(ex_to<power>(b).exponent).is_pos_integer() ||
- !is_exactly_a<add>(ex_to<power>(b).basis) ||
- !is_exactly_a<mul>(ex_to<power>(b).basis) ||
- !is_exactly_a<power>(ex_to<power>(b).basis));
- if (is_exactly_a<mul>(b))
- term.push_back(expand_mul(ex_to<mul>(b), numeric(n-k_cum[m-2]), options, true));
- else
- term.push_back(power(b,n-k_cum[m-2]));
-
- numeric f = binomial(numeric(n),numeric(k[0]));
- for (std::size_t l = 1; l < m - 1; ++l)
- f *= binomial(numeric(n-k_cum[l-1]),numeric(k[l]));
-
- term.push_back(f);
-
- result.push_back(ex((new mul(term))->setflag(status_flags::dynallocated)).expand(options));
-
- // increment k[]
- bool done = false;
- std::size_t l = m - 2;
- while ((++k[l]) > upper_limit[l]) {
- k[l] = 0;
- if (l != 0)
- --l;
- else {
- done = true;
- break;
+ result.reserve(result_size);
+
+ // Iterate over all terms in binomial expansion of
+ // S = power(+(x,...,z;c),n)
+ // = sum(binomial(n,k)*power(+(x,...,z;0),k)*c^(n-k), k=1..n) + c^n
+ for (int k = 1; k <= n; ++k) {
+ numeric binomial_coefficient; // binomial(n,k)*c^(n-k)
+ if (a.overall_coeff.is_zero()) {
+ // degenerate case with zero overall_coeff:
+ // apply multinomial theorem directly to power(+(x,...z;0),n)
+ binomial_coefficient = 1;
+ if (k < n) {
+ continue;
}
+ } else {
+ binomial_coefficient = binomial(numeric(n), numeric(k)) * pow(ex_to<numeric>(a.overall_coeff), numeric(n-k));
}
- if (done)
- break;
- // recalc k_cum[] and upper_limit[]
- k_cum[l] = (l==0 ? k[0] : k_cum[l-1]+k[l]);
+ // Multinomial expansion of power(+(x,...,z;0),k)*c^(n-k):
+ // Iterate over all partitions of k with exactly as many parts as
+ // there are symbolic terms in the basis (including zero parts).
+ partition_generator partitions(k, a.seq.size());
+ do {
+ const std::vector<int>& partition = partitions.current();
+ const numeric coeff = multinomial_coefficient(partition) * binomial_coefficient;
+
+ // Iterate over all compositions of the current partition.
+ composition_generator compositions(partition);
+ do {
+ const std::vector<int>& exponent = compositions.current();
+ exvector term;
+ term.reserve(n);
+ numeric factor = coeff;
+ for (unsigned i = 0; i < exponent.size(); ++i) {
+ const ex & r = a.seq[i].rest;
+ const ex & c = a.seq[i].coeff;
+ GINAC_ASSERT(!is_exactly_a<add>(r));
+ GINAC_ASSERT(!is_exactly_a<power>(r) ||
+ !is_exactly_a<numeric>(ex_to<power>(r).exponent) ||
+ !ex_to<numeric>(ex_to<power>(r).exponent).is_pos_integer() ||
+ !is_exactly_a<add>(ex_to<power>(r).basis) ||
+ !is_exactly_a<mul>(ex_to<power>(r).basis) ||
+ !is_exactly_a<power>(ex_to<power>(r).basis));
+ if (exponent[i] == 0) {
+ // optimize away
+ } else if (exponent[i] == 1) {
+ // optimized
+ term.push_back(r);
+ factor = factor.mul(ex_to<numeric>(c));
+ } else { // general case exponent[i] > 1
+ term.push_back((new power(r, exponent[i]))->setflag(status_flags::dynallocated));
+ factor = factor.mul(ex_to<numeric>(c).power(exponent[i]));
+ }
+ }
+ result.push_back(a.combine_ex_with_coeff_to_pair(mul(term).expand(options), factor));
+ } while (compositions.next());
+ } while (partitions.next());
+ }
- for (size_t i=l+1; i<m-1; ++i)
- k_cum[i] = k_cum[i-1]+k[i];
+ GINAC_ASSERT(result.size() == result_size);
- for (size_t i=l+1; i<m-1; ++i)
- upper_limit[i] = n-k_cum[i-1];
+ if (a.overall_coeff.is_zero()) {
+ return (new add(result))->setflag(status_flags::dynallocated |
+ status_flags::expanded);
+ } else {
+ return (new add(result, ex_to<numeric>(a.overall_coeff).power(n)))->setflag(status_flags::dynallocated |
+ status_flags::expanded);
}
-
- return (new add(result))->setflag(status_flags::dynallocated |
- status_flags::expanded);
}
if (c.is_equal(_ex1)) {
if (is_exactly_a<mul>(r)) {
- sum.push_back(expair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
- _ex1));
+ sum.push_back(a.combine_ex_with_coeff_to_pair(expand_mul(ex_to<mul>(r), *_num2_p, options, true),
+ _ex1));
} else {
- sum.push_back(expair((new power(r,_ex2))->setflag(status_flags::dynallocated),
- _ex1));
+ sum.push_back(a.combine_ex_with_coeff_to_pair((new power(r,_ex2))->setflag(status_flags::dynallocated),
+ _ex1));
}
} else {
if (is_exactly_a<mul>(r)) {
for (epvector::const_iterator cit1=cit0+1; cit1!=last; ++cit1) {
const ex & r1 = cit1->rest;
const ex & c1 = cit1->coeff;
- sum.push_back(a.combine_ex_with_coeff_to_pair((new mul(r,r1))->setflag(status_flags::dynallocated),
+ sum.push_back(a.combine_ex_with_coeff_to_pair(mul(r,r1).expand(options),
_num2_p->mul(ex_to<numeric>(c)).mul_dyn(ex_to<numeric>(c1))));
}
}
GINAC_ASSERT(sum.size()==(a.seq.size()*(a.seq.size()+1))/2);
- // second part: add terms coming from overall_factor (if != 0)
+ // second part: add terms coming from overall_coeff (if != 0)
if (!a.overall_coeff.is_zero()) {
epvector::const_iterator i = a.seq.begin(), end = a.seq.end();
while (i != end) {
git://www.ginac.de / ginac.git / blobdiff