Updated bisection root finding algorithm and test case

This commit is contained in:
Stefan Müller 2024-05-24 20:20:28 +02:00
parent cbaffbf55e
commit 53e3922654
2 changed files with 109 additions and 15 deletions

View File

@ -22,17 +22,29 @@ unit UPolynomialRoots;
interface
uses
Classes, SysUtils, UPolynomial, UBigInt;
Classes, SysUtils, Generics.Collections, UPolynomial, UBigInt;
type
{ TIsolatingInterval }
// 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
// respect to [0, bound], with A = C * bound and B = (C + H) * bound.
TIsolatingInterval = record
C, K, H: Cardinal;
Bound, A, B: TBigInt;
end;
TIsolatingIntervals = specialize TList<TIsolatingInterval>;
{ TRootIsolation }
TRootIsolation = class
private
function CalcSimpleRootBound(constref APolynomial: TBigIntPolynomial): TBigInt;
function GetIsolatingInterval(const AC, AK, AH: Cardinal; constref ABound: TBigInt): TIsolatingInterval;
public
function Bisect(constref APolynomial: TBigIntPolynomial): Int64;
function Bisect(constref APolynomial: TBigIntPolynomial): TIsolatingIntervals;
end;
implementation
@ -42,30 +54,96 @@ implementation
function TRootIsolation.CalcSimpleRootBound(constref APolynomial: TBigIntPolynomial): TBigInt;
var
i, sign: Integer;
a: TBigInt;
an, ai, max: TBigInt;
numeratorBit, denominatorBit: Int64;
begin
// We need a_n > 0 here, so we use -sign(a_n) instead of actually flipping the polynomial.
// Sign is not 0 because a_n is not 0.
sign := -APolynomial.Coefficient[APolynomial.Degree].Sign;
an := APolynomial.Coefficient[APolynomial.Degree];
sign := -an.Sign;
// This is a simplification of Cauchy's bound to avoid division.
// This is a simplification of Cauchy's bound to avoid division and make it a power of two.
// https://en.wikipedia.org/wiki/Geometrical_properties_of_polynomial_roots#Bounds_of_positive_real_roots
Result := TBigInt.Zero;
max := TBigInt.Zero;
for i := 0 to APolynomial.Degree - 1 do begin
a := sign * APolynomial.Coefficient[i];
if Result < a then
Result := a;
ai := sign * APolynomial.Coefficient[i];
if max < ai then
max := ai;
end;
Result := Result + 1;
numeratorBit := max.GetMostSignificantBitIndex + 1;
denominatorBit := an.GetMostSignificantBitIndex;
Result := TBigInt.One << (numeratorBit - denominatorBit);
end;
function TRootIsolation.Bisect(constref APolynomial: TBigIntPolynomial): Int64;
function TRootIsolation.GetIsolatingInterval(const AC, AK, AH: Cardinal; constref ABound: TBigInt): TIsolatingInterval;
begin
Result.C := AC;
Result.K := AK;
Result.H := AH;
Result.Bound := ABound;
Result.A := AC * ABound;
Result.B := (AC + AH) * ABound;
end;
// This is adapted from
// https://en.wikipedia.org/wiki/Real-root_isolation#Bisection_method
function TRootIsolation.Bisect(constref APolynomial: TBigIntPolynomial): TIsolatingIntervals;
type
TWorkItem = record
C, K: Cardinal;
P: TBigIntPolynomial;
end;
TWorkStack = specialize TStack<TWorkItem>;
var
bound: TBigInt;
p: TBigIntPolynomial;
item: TWorkItem;
stack: TWorkStack;
n, v: Integer;
varq: TBigIntPolynomial;
begin
Result := TIsolatingIntervals.Create;
stack := TWorkStack.Create;
bound := CalcSimpleRootBound(APolynomial);
p := APolynomial.ScaleVariable(bound);
n := item.P.Degree;
item.C := 0;
item.K := 0;
item.P := APolynomial.ScaleVariable(bound);
stack.Push(item);
while stack.Count > 0 do
begin
item := stack.Pop;
if item.P.Coefficient[0] = TBigInt.Zero then
begin
// Found an integer root at 0.
item.P := item.P.DivideByVariable;
Dec(n);
Result.Add(GetIsolatingInterval(item.C, item.K, 0, bound));
end;
varq := item.P.RevertOrderOfCoefficients.TranslateVariableByOne;
v := varq.CalcSignVariations;
if v = 1 then
begin
// Found isolating interval.
Result.Add(GetIsolatingInterval(item.C, item.K, 1, bound));
end
else if v > 1 then
begin
// Bisects, first new work item is (2c, k + 1, 2^n * q(x/2)).
item.C := item.C << 1;
Inc(item.K);
item.P := item.P.ScaleVariableByHalf.ScaleByPowerOfTwo(n);
stack.Push(item);
// ... second new work item is (2c + 1, k + 1, 2^n * q((x+1)/2)).
item.C := item.C + 1;
item.P := item.P.TranslateVariableByOne;
stack.Push(item);
end;
end;
stack.Free;
end;
end.

View File

@ -54,15 +54,31 @@ begin
end;
procedure TPolynomialRootsTestCase.TestBisectionRootIsolation;
const
expRoots: array of Cardinal = (34000, 23017, 5);
var
exp: Cardinal;
a: TBigIntPolynomial;
r: Int64;
r: TIsolatingIntervals;
ri: TIsolatingInterval;
found: Boolean;
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 := FRootIsolation.Bisect(a);
AssertEquals(0, r);
AssertEquals(Length(expRoots), r.Count);
for exp in expRoots do
begin
found := False;
for ri in r do
if (ri.A <= exp) and (exp <= ri.B) then
begin
found := True;
Break;
end;
AssertTrue('No isolating interval for expected root ' + IntToStr(exp) + ' found.', found);
end;
end;
initialization