Updated bisection root finding algorithm and test case
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@ -22,17 +22,29 @@ unit UPolynomialRoots;
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interface
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uses
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Classes, SysUtils, UPolynomial, UBigInt;
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Classes, SysUtils, Generics.Collections, UPolynomial, UBigInt;
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type
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{ TIsolatingInterval }
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// Represents an isolating interval of the form [C / 2^K, (C + H) / 2^K] in respect to [0, 1] or [A / 2^K, B / 2^K] in
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// respect to [0, bound], with A = C * bound and B = (C + H) * bound.
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TIsolatingInterval = record
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C, K, H: Cardinal;
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Bound, A, B: TBigInt;
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end;
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TIsolatingIntervals = specialize TList<TIsolatingInterval>;
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{ TRootIsolation }
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TRootIsolation = class
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private
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function CalcSimpleRootBound(constref APolynomial: TBigIntPolynomial): TBigInt;
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function GetIsolatingInterval(const AC, AK, AH: Cardinal; constref ABound: TBigInt): TIsolatingInterval;
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public
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function Bisect(constref APolynomial: TBigIntPolynomial): Int64;
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function Bisect(constref APolynomial: TBigIntPolynomial): TIsolatingIntervals;
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end;
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implementation
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@ -42,30 +54,96 @@ implementation
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function TRootIsolation.CalcSimpleRootBound(constref APolynomial: TBigIntPolynomial): TBigInt;
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var
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i, sign: Integer;
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a: TBigInt;
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an, ai, max: TBigInt;
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numeratorBit, denominatorBit: Int64;
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begin
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// We need a_n > 0 here, so we use -sign(a_n) instead of actually flipping the polynomial.
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// Sign is not 0 because a_n is not 0.
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sign := -APolynomial.Coefficient[APolynomial.Degree].Sign;
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an := APolynomial.Coefficient[APolynomial.Degree];
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sign := -an.Sign;
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// This is a simplification of Cauchy's bound to avoid division.
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// This is a simplification of Cauchy's bound to avoid division and make it a power of two.
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// https://en.wikipedia.org/wiki/Geometrical_properties_of_polynomial_roots#Bounds_of_positive_real_roots
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Result := TBigInt.Zero;
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max := TBigInt.Zero;
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for i := 0 to APolynomial.Degree - 1 do begin
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a := sign * APolynomial.Coefficient[i];
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if Result < a then
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Result := a;
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ai := sign * APolynomial.Coefficient[i];
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if max < ai then
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max := ai;
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end;
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Result := Result + 1;
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numeratorBit := max.GetMostSignificantBitIndex + 1;
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denominatorBit := an.GetMostSignificantBitIndex;
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Result := TBigInt.One << (numeratorBit - denominatorBit);
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end;
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function TRootIsolation.Bisect(constref APolynomial: TBigIntPolynomial): Int64;
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function TRootIsolation.GetIsolatingInterval(const AC, AK, AH: Cardinal; constref ABound: TBigInt): TIsolatingInterval;
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begin
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Result.C := AC;
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Result.K := AK;
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Result.H := AH;
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Result.Bound := ABound;
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Result.A := AC * ABound;
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Result.B := (AC + AH) * ABound;
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end;
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// This is adapted from
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// https://en.wikipedia.org/wiki/Real-root_isolation#Bisection_method
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function TRootIsolation.Bisect(constref APolynomial: TBigIntPolynomial): TIsolatingIntervals;
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type
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TWorkItem = record
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C, K: Cardinal;
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P: TBigIntPolynomial;
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end;
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TWorkStack = specialize TStack<TWorkItem>;
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var
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bound: TBigInt;
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p: TBigIntPolynomial;
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item: TWorkItem;
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stack: TWorkStack;
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n, v: Integer;
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varq: TBigIntPolynomial;
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begin
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Result := TIsolatingIntervals.Create;
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stack := TWorkStack.Create;
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bound := CalcSimpleRootBound(APolynomial);
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p := APolynomial.ScaleVariable(bound);
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n := item.P.Degree;
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item.C := 0;
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item.K := 0;
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item.P := APolynomial.ScaleVariable(bound);
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stack.Push(item);
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while stack.Count > 0 do
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begin
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item := stack.Pop;
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if item.P.Coefficient[0] = TBigInt.Zero then
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begin
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// Found an integer root at 0.
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item.P := item.P.DivideByVariable;
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Dec(n);
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Result.Add(GetIsolatingInterval(item.C, item.K, 0, bound));
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end;
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varq := item.P.RevertOrderOfCoefficients.TranslateVariableByOne;
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v := varq.CalcSignVariations;
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if v = 1 then
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begin
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// Found isolating interval.
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Result.Add(GetIsolatingInterval(item.C, item.K, 1, bound));
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end
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else if v > 1 then
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begin
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// Bisects, first new work item is (2c, k + 1, 2^n * q(x/2)).
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item.C := item.C << 1;
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Inc(item.K);
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item.P := item.P.ScaleVariableByHalf.ScaleByPowerOfTwo(n);
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stack.Push(item);
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// ... second new work item is (2c + 1, k + 1, 2^n * q((x+1)/2)).
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item.C := item.C + 1;
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item.P := item.P.TranslateVariableByOne;
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stack.Push(item);
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end;
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end;
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stack.Free;
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end;
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end.
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@ -54,15 +54,31 @@ begin
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end;
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procedure TPolynomialRootsTestCase.TestBisectionRootIsolation;
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const
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expRoots: array of Cardinal = (34000, 23017, 5);
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var
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exp: Cardinal;
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a: TBigIntPolynomial;
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r: Int64;
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r: TIsolatingIntervals;
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ri: TIsolatingInterval;
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found: Boolean;
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begin
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// y = 3 * (x - 34000) * (x - 23017) * (x - 5) * (x^2 - 19) * (x + 112)
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// = 3 * x^6 - 170730 * x^5 + 2329429920 * x^4 + 251300082690 * x^3 - 1270471872603 * x^2 + 4774763204640 * x - 24979889760000
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a := TBigIntPolynomial.Create([-24979889760000, 4774763204640, -1270471872603, 251300082690, 2329429920, -170730, 3]);
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r := FRootIsolation.Bisect(a);
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AssertEquals(0, r);
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AssertEquals(Length(expRoots), r.Count);
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for exp in expRoots do
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begin
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found := False;
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for ri in r do
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if (ri.A <= exp) and (exp <= ri.B) then
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begin
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found := True;
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Break;
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end;
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AssertTrue('No isolating interval for expected root ' + IntToStr(exp) + ' found.', found);
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end;
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end;
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initialization
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