73 lines
2.0 KiB
Plaintext
73 lines
2.0 KiB
Plaintext
{
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Solutions to the Advent Of Code.
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Copyright (C) 2024 Stefan Müller
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This program is free software: you can redistribute it and/or modify it under
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the terms of the GNU General Public License as published by the Free Software
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Foundation, either version 3 of the License, or (at your option) any later
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version.
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This program is distributed in the hope that it will be useful, but WITHOUT
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ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
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FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.
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You should have received a copy of the GNU General Public License along with
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this program. If not, see <http://www.gnu.org/licenses/>.
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}
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unit UPolynomialRoots;
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{$mode ObjFPC}{$H+}
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interface
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uses
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Classes, SysUtils, UPolynomial, UBigInt;
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type
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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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public
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function Bisect(constref APolynomial: TBigIntPolynomial): Int64;
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end;
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implementation
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{ TRootIsolation }
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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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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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// This is a simplification of Cauchy's bound to avoid division.
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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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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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end;
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Result := Result + 1;
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end;
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function TRootIsolation.Bisect(constref APolynomial: TBigIntPolynomial): Int64;
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var
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bound: TBigInt;
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p: TBigIntPolynomial;
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begin
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bound := CalcSimpleRootBound(APolynomial);
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p := APolynomial.ScaleVariable(bound);
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end;
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end.
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