Speed up special cases of square-free factorization.
authorRichard Kreckel <kreckel@ginac.de>
Wed, 24 Jan 2018 21:37:24 +0000 (22:37 +0100)
committerRichard Kreckel <kreckel@ginac.de>
Wed, 24 Jan 2018 21:37:24 +0000 (22:37 +0100)
Square-free factorization of polynomials containing a factor which is
a high power P of a symbol x did scale like O(P) in space and time.
This patch introduces a shortcut in Yun's algorithm, such that the
computation is only O(1) in space and time.

This makes it possible to compute, say sqrfree(x^P + x^(P+1)) =>
(1+x)*x^P with P=123456789.

Found this to be a bottleneck while debugging one of Vitaly Magerya's
examples.

ginac/normal.cpp

index 231ef7e..9f8b7b4 100644 (file)
@@ -1785,32 +1785,39 @@ ex lcm(const ex &a, const ex &b, bool check_args)
  *  @param a  multivariate polynomial over Z[X], treated here as univariate
  *            polynomial in x (needs not be expanded).
  *  @param x  variable to factor in
- *  @return   vector of factors sorted in ascending degree */
-static exvector sqrfree_yun(const ex &a, const symbol &x)
+ *  @return   vector of expairs (factor, exponent), sorted by exponents */
+static epvector sqrfree_yun(const ex &a, const symbol &x)
 {
-       exvector res;
        ex w = a;
        ex z = w.diff(x);
        ex g = gcd(w, z);
        if (g.is_zero()) {
-               return res;
+               return epvector{};
        }
        if (g.is_equal(_ex1)) {
-               res.push_back(a);
-               return res;
+               return epvector{expair(a, _ex1)};
        }
-       ex y;
+       epvector results;
+       ex exponent = _ex0;
        do {
                w = quo(w, g, x);
                if (w.is_zero()) {
                        return res;
                }
-               y = quo(z, g, x);
-               z = y - w.diff(x);
+               z = quo(z, g, x) - w.diff(x);
+               exponent = exponent + 1;
+               if (w.is_equal(x)) {
+                       // shortcut for x^n with n ∈ ℕ
+                       exponent += quo(z, w.diff(x), x);
+                       results.push_back(expair(w, exponent));
+                       break;
+               }
                g = gcd(w, z);
-               res.push_back(g);
+               if (!g.is_equal(_ex1)) {
+                       results.push_back(expair(g, exponent));
+               }
        } while (!z.is_zero());
-       return res;
+       return results;
 }
 
 
@@ -1878,30 +1885,28 @@ ex sqrfree(const ex &a, const lst &l)
        const ex tmp = multiply_lcm(a,lcm);
 
        // find the factors
-       exvector factors = sqrfree_yun(tmp, x);
+       epvector factors = sqrfree_yun(tmp, x);
 
-       // construct the next list of symbols with the first element popped
-       lst newargs = args;
-       newargs.remove_first();
+       // remove symbol x and proceed recursively with the remaining symbols
+       args.remove_first();
 
        // recurse down the factors in remaining variables
-       if (newargs.nops()>0) {
+       if (args.nops()>0) {
                for (auto & it : factors)
-                       it = sqrfree(it, newargs);
+                       it.rest = sqrfree(it.rest, args);
        }
 
        // Done with recursion, now construct the final result
        ex result = _ex1;
-       int p = 1;
        for (auto & it : factors)
-               result *= pow(it, p++);
+               result *= pow(it.rest, it.coeff);
 
        // Yun's algorithm does not account for constant factors.  (For univariate
        // polynomials it works only in the monic case.)  We can correct this by
        // inserting what has been lost back into the result.  For completeness
        // we'll also have to recurse down that factor in the remaining variables.
-       if (newargs.nops()>0)
-               result *= sqrfree(quo(tmp, result, x), newargs);
+       if (args.nops()>0)
+               result *= sqrfree(quo(tmp, result, x), args);
        else
                result *= quo(tmp, result, x);
 
@@ -1929,24 +1934,21 @@ ex sqrfree_parfrac(const ex & a, const symbol & x)
 //clog << "red_poly = " << red_poly << ", red_numer = " << red_numer << endl;
 
        // Factorize denominator and compute cofactors
-       exvector yun = sqrfree_yun(denom, x);
-//clog << "yun factors: " << exprseq(yun) << endl;
-       size_t num_yun = yun.size();
-       exvector factor; factor.reserve(num_yun);
-       exvector cofac; cofac.reserve(num_yun);
-       for (size_t i=0; i<num_yun; i++) {
-               if (!yun[i].is_equal(_ex1)) {
-                       for (size_t j=0; j<=i; j++) {
-                               factor.push_back(pow(yun[i], j+1));
-                               ex prod = _ex1;
-                               for (size_t k=0; k<num_yun; k++) {
-                                       if (k == i)
-                                               prod *= pow(yun[k], i-j);
-                                       else
-                                               prod *= pow(yun[k], k+1);
-                               }
-                               cofac.push_back(prod.expand());
+       epvector yun = sqrfree_yun(denom, x);
+       size_t yun_max_exponent = yun.empty() ? 0 : ex_to<numeric>(yun.back().coeff).to_int();
+       exvector factor, cofac;
+       for (size_t i=0; i<yun.size(); i++) {
+               numeric i_exponent = ex_to<numeric>(yun[i].coeff);
+               for (size_t j=0; j<i_exponent; j++) {
+                       factor.push_back(pow(yun[i].rest, j+1));
+                       ex prod = _ex1;
+                       for (size_t k=0; k<yun.size(); k++) {
+                               if (yun[k].coeff == i_exponent)
+                                       prod *= pow(yun[k].rest, i_exponent-1-j);
+                               else
+                                       prod *= pow(yun[k].rest, yun[k].coeff);
                        }
+                       cofac.push_back(prod.expand());
                }
        }
        size_t num_factors = factor.size();