- remembering in divide_in_z() turned off (will eventually have to be replaced
authorChristian Bauer <Christian.Bauer@uni-mainz.de>
Tue, 28 Mar 2000 18:58:57 +0000 (18:58 +0000)
committerChristian Bauer <Christian.Bauer@uni-mainz.de>
Tue, 28 Mar 2000 18:58:57 +0000 (18:58 +0000)
  by the new remembering mechanism in remember.cpp)
- new #define options at top of source:
    USE_TRIAL_DIVISION makes divide_in_z() use trial division and Newton
      interpolation instead of recursive polynomial division (only useful
      for very large problems)
    STATISTICS enables some statistical output at the end of the program
- gcd(numeric, numeric) is now handled earlier; this should speed up normal()
- added new heuristic: gcd(p^n, p^m) with p a polynomial now immediately
  returns p^abs(n-m)
- if GCD is 1, cofactors are returned unexpanded

ginac/normal.cpp

index a8f64be82a988a330c6cb70997b60b86ae858af2..7d6ddca7cf3f4da9f0d212aa3dc3ebcda1621d3f 100644 (file)
@@ -57,7 +57,34 @@ namespace GiNaC {
 #define FAST_COMPARE 1
 
 // Set this if you want divide_in_z() to use remembering
-#define USE_REMEMBER 1
+#define USE_REMEMBER 0
+
+// Set this if you want divide_in_z() to use trial division followed by
+// polynomial interpolation (usually slower except for very large problems)
+#define USE_TRIAL_DIVISION 0
+
+// Set this to enable some statistical output for the GCD routines
+#define STATISTICS 0
+
+
+#if STATISTICS
+// Statistics variables
+static int gcd_called = 0;
+static int sr_gcd_called = 0;
+static int heur_gcd_called = 0;
+static int heur_gcd_failed = 0;
+
+// Print statistics at end of program
+static struct _stat_print {
+       _stat_print() {}
+       ~_stat_print() {
+               cout << "gcd() called " << gcd_called << " times\n";
+               cout << "sr_gcd() called " << sr_gcd_called << " times\n";
+               cout << "heur_gcd() called " << heur_gcd_called << " times\n";
+               cout << "heur_gcd() failed " << heur_gcd_failed << " times\n";
+       }
+} stat_print;
+#endif
 
 
 /** Return pointer to first symbol found in expression.  Due to GiNaCĀ“s
@@ -598,38 +625,9 @@ static bool divide_in_z(const ex &a, const ex &b, ex &q, sym_desc_vec::const_ite
     if (bdeg > adeg)
         return false;
 
-#if 1
+#if USE_TRIAL_DIVISION
 
-    // Polynomial long division (recursive)
-    ex r = a.expand();
-    if (r.is_zero())
-        return true;
-    int rdeg = adeg;
-    ex eb = b.expand();
-    ex blcoeff = eb.coeff(*x, bdeg);
-    while (rdeg >= bdeg) {
-        ex term, rcoeff = r.coeff(*x, rdeg);
-        if (!divide_in_z(rcoeff, blcoeff, term, var+1))
-            break;
-        term = (term * power(*x, rdeg - bdeg)).expand();
-        q += term;
-        r -= (term * eb).expand();
-        if (r.is_zero()) {
-#if USE_REMEMBER
-            dr_remember[ex2(a, b)] = exbool(q, true);
-#endif
-            return true;
-        }
-        rdeg = r.degree(*x);
-    }
-#if USE_REMEMBER
-    dr_remember[ex2(a, b)] = exbool(q, false);
-#endif
-    return false;
-
-#else
-
-    // Trial division using polynomial interpolation
+    // Trial division with polynomial interpolation
     int i, k;
 
     // Compute values at evaluation points 0..adeg
@@ -652,7 +650,7 @@ static bool divide_in_z(const ex &a, const ex &b, ex &q, sym_desc_vec::const_ite
 
     // Compute inverses
     vector<numeric> rcp; rcp.reserve(adeg + 1);
-    rcp.push_back(0);
+    rcp.push_back(_num0());
     for (k=1; k<=adeg; k++) {
         numeric product = alpha[k] - alpha[0];
         for (i=1; i<k; i++)
@@ -680,6 +678,36 @@ static bool divide_in_z(const ex &a, const ex &b, ex &q, sym_desc_vec::const_ite
         return true;
     } else
         return false;
+
+#else
+
+    // Polynomial long division (recursive)
+    ex r = a.expand();
+    if (r.is_zero())
+        return true;
+    int rdeg = adeg;
+    ex eb = b.expand();
+    ex blcoeff = eb.coeff(*x, bdeg);
+    while (rdeg >= bdeg) {
+        ex term, rcoeff = r.coeff(*x, rdeg);
+        if (!divide_in_z(rcoeff, blcoeff, term, var+1))
+            break;
+        term = (term * power(*x, rdeg - bdeg)).expand();
+        q += term;
+        r -= (term * eb).expand();
+        if (r.is_zero()) {
+#if USE_REMEMBER
+            dr_remember[ex2(a, b)] = exbool(q, true);
+#endif
+            return true;
+        }
+        rdeg = r.degree(*x);
+    }
+#if USE_REMEMBER
+    dr_remember[ex2(a, b)] = exbool(q, false);
+#endif
+    return false;
+
 #endif
 }
 
@@ -812,6 +840,9 @@ ex ex::primpart(const symbol &x, const ex &c) const
 static ex sr_gcd(const ex &a, const ex &b, const symbol *x)
 {
 //clog << "sr_gcd(" << a << "," << b << ")\n";
+#if STATISTICS
+       sr_gcd_called++;
+#endif
 
     // Sort c and d so that c has higher degree
     ex c, d;
@@ -837,6 +868,7 @@ static ex sr_gcd(const ex &a, const ex &b, const symbol *x)
         return gamma;
     c = c.primpart(*x, cont_c);
     d = d.primpart(*x, cont_d);
+//clog << " content " << gamma << " removed, continuing with sr_gcd(" << c << "," << d << ")\n";
 
     // First element of subresultant sequence
     ex r = _ex0(), ri = _ex1(), psi = _ex1();
@@ -844,11 +876,13 @@ static ex sr_gcd(const ex &a, const ex &b, const symbol *x)
 
     for (;;) {
         // Calculate polynomial pseudo-remainder
+//clog << " start of loop, psi = " << psi << ", calculating pseudo-remainder...\n";
         r = prem(c, d, *x, false);
         if (r.is_zero())
             return gamma * d.primpart(*x);
         c = d;
         cdeg = ddeg;
+//clog << " dividing...\n";
         if (!divide(r, ri * power(psi, delta), d, false))
             throw(std::runtime_error("invalid expression in sr_gcd(), division failed"));
         ddeg = d.degree(*x);
@@ -860,6 +894,7 @@ static ex sr_gcd(const ex &a, const ex &b, const symbol *x)
         }
 
         // Next element of subresultant sequence
+//clog << " calculating next subresultant...\n";
         ri = c.expand().lcoeff(*x);
         if (delta == 1)
             psi = ri;
@@ -1024,6 +1059,9 @@ class gcdheu_failed {};
 static ex heur_gcd(const ex &a, const ex &b, ex *ca, ex *cb, sym_desc_vec::const_iterator var)
 {
 //clog << "heur_gcd(" << a << "," << b << ")\n";
+#if STATISTICS
+       heur_gcd_called++;
+#endif
 
        // GCD of two numeric values -> CLN
     if (is_ex_exactly_of_type(a, numeric) && is_ex_exactly_of_type(b, numeric)) {
@@ -1057,9 +1095,9 @@ static ex heur_gcd(const ex &a, const ex &b, ex *ca, ex *cb, sym_desc_vec::const
         xi = mp * _num2() + _num2();
 
     // 6 tries maximum
-    for (int t=0; t<6; t++) {  // MAGIC
-        if (xi.int_length() * maxdeg > 100000) {  // MAGIC
-// clog << "giving up heur_gcd, xi.int_length = " << xi.int_length() << ", maxdeg = " << maxdeg << endl;
+    for (int t=0; t<6; t++) {
+        if (xi.int_length() * maxdeg > 100000) {
+//clog << "giving up heur_gcd, xi.int_length = " << xi.int_length() << ", maxdeg = " << maxdeg << endl;
             throw gcdheu_failed();
                }
 
@@ -1109,6 +1147,24 @@ static ex heur_gcd(const ex &a, const ex &b, ex *ca, ex *cb, sym_desc_vec::const
 ex gcd(const ex &a, const ex &b, ex *ca, ex *cb, bool check_args)
 {
 //clog << "gcd(" << a << "," << b << ")\n";
+#if STATISTICS
+       gcd_called++;
+#endif
+
+       // GCD of numerics -> CLN
+    if (is_ex_exactly_of_type(a, numeric) && is_ex_exactly_of_type(b, numeric)) {
+        numeric g = gcd(ex_to_numeric(a), ex_to_numeric(b));
+        if (ca)
+            *ca = ex_to_numeric(a) / g;
+        if (cb)
+            *cb = ex_to_numeric(b) / g;
+        return g;
+    }
+
+       // Check arguments
+    if (check_args && !a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial)) {
+        throw(std::invalid_argument("gcd: arguments must be polynomials over the rationals"));
+    }
 
        // Partially factored cases (to avoid expanding large expressions)
        if (is_ex_exactly_of_type(a, mul)) {
@@ -1149,6 +1205,51 @@ factored_b:
                return g;
        }
 
+#if FAST_COMPARE
+       // Input polynomials of the form poly^n are sometimes also trivial
+       if (is_ex_exactly_of_type(a, power)) {
+               ex p = a.op(0);
+               if (is_ex_exactly_of_type(b, power)) {
+                       if (p.is_equal(b.op(0))) {
+                               // a = p^n, b = p^m, gcd = p^min(n, m)
+                               ex exp_a = a.op(1), exp_b = b.op(1);
+                               if (exp_a < exp_b) {
+                                       if (ca)
+                                               *ca = _ex1();
+                                       if (cb)
+                                               *cb = power(p, exp_b - exp_a);
+                                       return power(p, exp_a);
+                               } else {
+                                       if (ca)
+                                               *ca = power(p, exp_a - exp_b);
+                                       if (cb)
+                                               *cb = _ex1();
+                                       return power(p, exp_b);
+                               }
+                       }
+               } else {
+                       if (p.is_equal(b)) {
+                               // a = p^n, b = p, gcd = p
+                               if (ca)
+                                       *ca = power(p, a.op(1) - 1);
+                               if (cb)
+                                       *cb = _ex1();
+                               return p;
+                       }
+               }
+       } else if (is_ex_exactly_of_type(b, power)) {
+               ex p = b.op(0);
+               if (p.is_equal(a)) {
+                       // a = p, b = p^n, gcd = p
+                       if (ca)
+                               *ca = _ex1();
+                       if (cb)
+                               *cb = power(p, b.op(1) - 1);
+                       return p;
+               }
+       }
+#endif
+
     // Some trivial cases
        ex aex = a.expand(), bex = b.expand();
     if (aex.is_zero()) {
@@ -1181,17 +1282,6 @@ factored_b:
         return a;
     }
 #endif
-    if (is_ex_exactly_of_type(aex, numeric) && is_ex_exactly_of_type(bex, numeric)) {
-        numeric g = gcd(ex_to_numeric(aex), ex_to_numeric(bex));
-        if (ca)
-            *ca = ex_to_numeric(aex) / g;
-        if (cb)
-            *cb = ex_to_numeric(bex) / g;
-        return g;
-    }
-    if (check_args && !a.info(info_flags::rational_polynomial) || !b.info(info_flags::rational_polynomial)) {
-        throw(std::invalid_argument("gcd: arguments must be polynomials over the rationals"));
-    }
 
     // Gather symbol statistics
     sym_desc_vec sym_stats;
@@ -1237,13 +1327,32 @@ factored_b:
     }
     if (is_ex_exactly_of_type(g, fail)) {
 //clog << "heuristics failed" << endl;
+#if STATISTICS
+               heur_gcd_failed++;
+#endif
         g = sr_gcd(aex, bex, x);
-        if (ca)
-            divide(aex, g, *ca, false);
-        if (cb)
-            divide(bex, g, *cb, false);
-    }
-    return g;
+               if (g.is_equal(_ex1())) {
+                       // Keep cofactors factored if possible
+                       if (ca)
+                               *ca = a;
+                       if (cb)
+                               *cb = b;
+               } else {
+               if (ca)
+                   divide(aex, g, *ca, false);
+               if (cb)
+                   divide(bex, g, *cb, false);
+               }
+    } else {
+               if (g.is_equal(_ex1())) {
+                       // Keep cofactors factored if possible
+                       if (ca)
+                               *ca = a;
+                       if (cb)
+                               *cb = b;
+               }
+           return g;
+       }
 }