Added bisection variant for integers instead of intervals

UBigInt.pas

@@ -91,6 +91,8 @@ type
     class function FromBinaryString(const AValue: string): TBigInt; static;
   end;
 
+  TBigIntArray = array of TBigInt;
+
   { Operators }
 
   operator := (const A: Int64): TBigInt;

UPolynomialRoots.pas

@@ -52,8 +52,11 @@ type
     // Returns root-isolating intervals for non-negative, non-multiple roots.
     class function BisectIsolation(constref APolynomial: TBigIntPolynomial): TIsolatingIntervalArray;
     // Returns root-isolating intervals for non-multiple roots in the interval [0, 2^boundexp].
-    class function BisectIsolation(constref APolynomial: TBigIntPolynomial; constref ABoundExp: Cardinal):
-      TIsolatingIntervalArray;
+    class function BisectIsolation(constref APolynomial: TBigIntPolynomial; constref ABoundExp: Cardinal;
+      const AFindIntegers: Boolean = False): TIsolatingIntervalArray;
+    // Returns non-negative, non-multiple, integer roots in the interval [0, 2^boundexp].
+    class function BisectInteger(constref APolynomial: TBigIntPolynomial; constref ABoundExp: Cardinal):
+      TBigIntArray;
   end;
 
 implementation
@@ -112,8 +115,8 @@ end;
 
 // This is adapted from
 // https://en.wikipedia.org/wiki/Real-root_isolation#Bisection_method
-class function TPolynomialRoots.BisectIsolation(constref APolynomial: TBigIntPolynomial; constref ABoundExp: Cardinal):
-  TIsolatingIntervalArray;
+class function TPolynomialRoots.BisectIsolation(constref APolynomial: TBigIntPolynomial; constref ABoundExp: Cardinal;
+  const AFindIntegers: Boolean): TIsolatingIntervalArray;
 type
   TWorkItem = record
     C, K: Cardinal;
@@ -149,12 +152,11 @@ begin
 
     varq := item.P.RevertOrderOfCoefficients.TranslateVariableByOne;
     v := varq.CalcSignVariations;
-    if v = 1 then
-    begin
-      // Found isolating interval.
-      iso.Add(CreateIsolatingInterval(item.C, item.K, 1, ABoundExp));
-    end
-    else if v > 1 then
+    //WriteLn;
+    //WriteLn('var((x+1)^n*q(1/(x+1))): ', v);
+
+    if (v > 1)
+    or ((v = 1) and AFindIntegers and (item.K < ABoundExp)) then
     begin
       // Bisects, first new work item is (2c, k + 1, 2^n * q(x/2)).
       item.C := item.C << 1;
@@ -165,6 +167,12 @@ begin
       item.C := item.C + 1;
       item.P := item.P.TranslateVariableByOne;
       stack.Push(item);
+    end
+    else if v = 1 then
+    begin
+      // Found isolating interval.
+      //WriteLn('Found isolating interval (' + IntToStr(item.C) + ', ' + IntToStr(item.K) + ', 1).');
+      iso.Add(CreateIsolatingInterval(item.C, item.K, 1, ABoundExp));
     end;
   end;
   Result := iso.ToArray;
@@ -172,5 +180,29 @@ begin
   stack.Free;
 end;
 
+class function TPolynomialRoots.BisectInteger(constref APolynomial: TBigIntPolynomial; constref ABoundExp: Cardinal):
+  TBigIntArray;
+var
+  intervals: TIsolatingIntervalArray;
+  i: TIsolatingInterval;
+  r: specialize TList<TBigInt>;
+  value: Int64;
+begin
+  // Calculates isolating intervals.
+  intervals := BisectIsolation(APolynomial, ABoundExp, True);
+  r := specialize TList<TBigInt>.Create;
+
+  for i in intervals do
+    if i.H = 0 then
+      r.Add(i.A)
+    else if i.A.TryToInt64(value) and (APolynomial.CalcValueAt(value) = 0) then
+      r.Add(value)
+    else if i.B.TryToInt64(value) and (APolynomial.CalcValueAt(value) = 0) then
+      r.Add(value);
+
+  Result := r.ToArray;
+  r.Free;
+end;
+
 end.
 

tests/UPolynomialRootsTestCases.pas

@@ -32,9 +32,11 @@ type
   private
     procedure AssertBisectIntervals(AIsolatingIntervals: TIsolatingIntervalArray; constref AExpectedRoots:
       array of Cardinal);
+    procedure AssertBisectIntegers(ARoots: TBigIntArray; constref AExpectedRoots: array of Cardinal);
   published
     procedure TestBisectNoBound;
     procedure TestBisectWithBound;
+    procedure TestBisectInteger;
   end;
 
 implementation
@@ -64,6 +66,29 @@ begin
   end;
 end;
 
+procedure TPolynomialRootsTestCase.AssertBisectIntegers(ARoots: TBigIntArray; constref AExpectedRoots:
+  array of Cardinal);
+var
+  exp: Cardinal;
+  found: Boolean;
+  i, foundIndex: Integer;
+begin
+  AssertEquals('Unexpected number of integer roots.', Length(AExpectedRoots), Length(ARoots));
+  for exp in AExpectedRoots do
+  begin
+    found := False;
+    for i := 0 to Length(ARoots) - 1 do
+      if ARoots[i] = exp then
+      begin
+        found := True;
+        foundIndex := i;
+        Break;
+      end;
+    AssertTrue('Expected root ' + IntToStr(exp) + ' not found.', found);
+    Delete(ARoots, foundIndex, 1);
+  end;
+end;
+
 procedure TPolynomialRootsTestCase.TestBisectNoBound;
 const
   expRoots: array of Cardinal = (34000, 23017, 5);
@@ -92,6 +117,20 @@ begin
   AssertBisectIntervals(r, expRoots);
 end;
 
+procedure TPolynomialRootsTestCase.TestBisectInteger;
+const
+  expRoots: array of Cardinal = (23017, 5);
+var
+  a: TBigIntPolynomial;
+  r: TBigIntArray;
+begin
+  // y = 3 * (x - 34000) * (x - 23017) * (x - 5) * (x^2 - 19) * (x + 112)
+  //   = 3 * x^6 - 170730 * x^5 + 2329429920 * x^4 + 251300082690 * x^3 - 1270471872603 * x^2 + 4774763204640 * x - 24979889760000
+  a := TBigIntPolynomial.Create([-24979889760000, 4774763204640, -1270471872603, 251300082690, 2329429920, -170730, 3]);
+  r := TPolynomialRoots.BisectInteger(a, 15);
+  AssertBisectIntegers(r, expRoots);
+end;
+
 initialization
 
   RegisterTest(TPolynomialRootsTestCase);