Changeset 7682:6747edea46a8

Timestamp:

11/20/07 02:42:48 (6 years ago)

Author:

Robert Bradshaw <robertwb@…>

Branch:

default

Message:

n-th root for complex numbers, integer divisors

Location:

sage/rings

Files:

• complex_double.pyx (modified) (2 diffs)
• complex_number.pyx (modified) (1 diff)
• integer.pyx (modified) (1 diff)
• real_double.pyx (modified) (1 diff)

sage/rings/complex_double.pyx

@@ -727 +727 @@
         """
         return RealDoubleElement(gsl_complex_abs(self._complex))
+        
+    def argument(self):
+        return RealDoubleElement(gsl_complex_arg(self._complex))
     
     def abs2(self):
@@ -848 +851 @@
                  return [z, -z]
         return z
+        
+    def nth_root(self, n, all=False):
+        """
+        The n-th root function.
+                
+        INPUT:
+            all -- bool (default: False); if True, return a list
+                of all n-th roots.
+                
+        EXAMPLES: 
+            sage: a = CDF(125)
+            sage: a.nth_root(3)
+            5.0
+            sage: a = CDF(10, 2)
+            sage: [r^5 for r in a.nth_root(5, all=True)]
+            [10.0 + 2.0*I, 10.0 + 2.0*I, 10.0 + 2.0*I, 10.0 + 2.0*I, 10.0 + 2.0*I]
+            sage: abs(sum(a.nth_root(111, all=True))) # random but close to zero
+            6.00659385991e-14
+        """
+        if not self:
+            return [self] if all else self
+        arg = self.argument() / n
+        abs = self.abs().nth_root(n)
+        z = ComplexDoubleElement(abs * arg.cos(), abs*arg.sin())
+        if all:
+            zeta = self._parent.zeta(n)
+            return [z * zeta**k for k in range(n)]
+        else:
+            return z
+
         
     def is_square(self):

sage/rings/complex_number.pyx

@@ -830 +830 @@
                 return [z, -z]
         return z
+        
+    def nth_root(self, n, all=False):
+        """
+        The n-th root function.
+                
+        INPUT:
+            all -- bool (default: False); if True, return a list
+                of all n-th roots.
+                
+        EXAMPLES: 
+            sage: a = CC(27)
+            sage: a.nth_root(3)
+            3.00000000000000
+            sage: a.nth_root(3, all=True)
+            [3.00000000000000, -1.50000000000000 + 2.59807621135332*I, -1.50000000000000 - 2.59807621135332*I]
+            sage: a = ComplexField(20)(2,1)
+            sage: [r^7 for r in a.nth_root(7, all=True)]
+            [2.0000 + 1.0000*I, 2.0000 + 1.0000*I, 2.0000 + 1.0000*I, 2.0000 + 1.0000*I, 2.0000 + 1.0000*I, 2.0000 + 1.0000*I, 2.0000 + 1.0000*I]
+        """
+        if not self:
+            return [self] if all else self
+        arg = self.argument() / n
+        abs = self.abs().nth_root(n)
+        z = ComplexNumber(self._parent, abs * arg.cos(), abs*arg.sin())
+        if all:
+            zeta = self._parent.zeta(n)
+            return [z * zeta**k for k in range(n)]
+        else:
+            return z
     
     def is_square(self):

sage/rings/integer.pyx

@@ -1471 +1471 @@
         import sage.rings.integer_ring
         return sage.rings.integer_ring.factor(self, algorithm=algorithm, proof=proof)
+        
+    def prime_divisors(self, **kwds):
+        return [p for p,e in self.factor(**kwds)]
         
     def coprime_integers(self, m):

sage/rings/real_double.pyx

@@ -891 +891 @@
             raise ValueError, "negative number %s does not have a square root in the real field"%self
         return self._complex_double_(sage.rings.complex_double.CDF).sqrt(all=all)
-        
-        
+                
     def is_square(self):
         """