Further improved bound and skipped bisection for isolated roots withi…

…n interval size

Combined the Lagrange and Cauchy bounds to return the smallest. realRootIsolation no longer bisects intervals when the root is isolated and the interval is below the required size (e.g. when two roots are very close together, the interval widths with be smaller, but the interval widths do not need to be smaller for the other roots). Final list is also sorted in case bisection stops in an interval after the first.

Test also corrected (the new isolating intervals only look 'less nice' because the previous initial bounds happened to be divisible by 8).

M2/Macaulay2/packages/RealRoots.m2

@@ -398,26 +398,31 @@ realRootIsolation (RingElement,A) := List => (f,r)->(
 
 	--bound for real roots
 	C := (listForm ((f-leadTerm(f))/leadCoefficient(f)))/last; --make the polynomial monic, and obtain list of coefficients of non-lead monomials.
-    	M := max(1,sum(C,abs)); --obtains Lagrange bound.
+    	M := min(1+max(apply(C,abs)),max(1,sum(C,abs))); --obtains Cauchy or Lagrange bound.
 
 	L := {{-M,M}};
 	midp := 0;
 	v := new MutableHashTable from {M=>variations apply(l,g->signAt(g,M)),-M=>variations apply(l,g->signAt(g,-M))};
 
 	while (max apply(L,I-> I#1-I#0) > r) or (max apply(L,I-> v#(I#0)-v#(I#1)) > 1) do (
 	    for I in L do (
-		midp = (sum I)/2;
-		v#midp = variations apply(l,g->signAt(g,midp));
-		L = drop(L,1);
-		if (v#(I#0)-v#midp > 0) then (
-		    L = append(L,{I#0,midp})
-		    );
-		if (v#midp-v#(I#1) > 0) then (
-		    L = append(L,{midp,I#1})
+		if ((v#(I#0)-v#(I#1) == 1) and (I#1-I#0 <= r)) then (
+	            L = take(L,{1,#L})|{L#0}; -- skip bisection if root is identified and bound is within interval size.
+		)
+		else (
+		    midp = (sum I)/2;
+		    v#midp = variations apply(l,g->signAt(g,midp));
+		    L = drop(L,1);
+		    if (v#(I#0)-v#midp > 0) then (
+		        L = append(L,{I#0,midp})
+		        );
+	  	    if (v#midp-v#(I#1) > 0) then (
+		        L = append(L,{midp,I#1})
+		        );
 		    )
 		)
 	    );
-	L
+	sort L --list is ordered from smallest to largest interval, in case the while look stops after a larger interval
 	) else (
 	{}
 	)
@@ -1033,7 +1038,7 @@ TEST ///
 TEST ///
     R = QQ[t];
     f = (t-1)^2*(t+3)*(t+5)*(t-6);
-    assert(realRootIsolation(f,1/2) == {{-161/32, -299/64}, {-207/64, -23/8}, {23/32, 69/64}, {23/4, 391/64}});
+    assert(realRootIsolation(f,1/2) == {{-1365/256,-637/128},{-819/256,-91/32},{91/128,273/256},{91/16,1547/256}});
     ///    
 
 TEST ///
@@ -1073,10 +1078,4 @@ TEST ///
     -- assert(rationalUnivariateRepresentationresentation(I) == {Z^2 - (3/2)*Z - 9, x + y});
     -- ///	 
 
-end
-
-
-
-
-
-
+end