PolyCoeff Routines

Summary

Converts polynomial roots to coefficients.

Declaration

Procedure PolyCoeff(Roots, Coeff: TVec);

Description

The PolyCoeff procedure converts polynomial roots to coefficients. The following scheme is used to construct a coefficients from the roots:


  P(x) = coeff[0]*x^n + coeff[1]*x^(n-1) + .. + coeff[n]
       = (x - roots[0])* .. *(x - roots[n-1])

  n .. order of the polynomial

The length and complex property of the Coeff object are set automatically. The inverse to this routine is PolyRoots

Declaration

Procedure PolyCoeff(Mtx: TMtx; Coeff: TVec);

Description

Returns coefficients of the characteristic polynomial: det( Mtx - Lambda * I )

See Also
PolyRoots

Example 1

Find coefficients of a polynomial with roots at 1,1 and 2.


    uses MtxExpr, Math387, MtxVec, MtxVecEdit, Polynoms;

    procedure TForm1.Button1Click(Sender: TObject);
    var Roots, Coeff: Vector;
    begin
    // roots of the polynomial: (x - 1)*(x - 1)*(x - 2)

        Roots.SetIt(false,[1, 1, 2]);
        PolyCoeff(Roots,Coeff);

    // Coeff returns the coefficients of the polynomial [ 1,-4, 5,-2]
    // which can be written as:  1*x^3 + -4*x^2 + 5*x - 2

        ViewValues(Coeff,'Coefficients');
    end;

    #include "MtxVecCPP.h" //MtxVecCPP.cpp must be included in the project
    #include "Polynoms.hpp"
    #include "MtxVecTee.hpp"

    void __fastcall TForm1::BitBtn1Click(TObject *Sender)
    {
        Vector Roots, Coeff;
        // roots of the polynomial: (x - 1)*(x - 1)*(x - 2)
        Roots->SetIt(false,OPENARRAY(double,(1, 1, 2)));
        PolyCoeff(Roots,Coeff);

        // Coeff returns the coefficients of the polynomial [ 1,-4, 5,-2]
        // which can be written as:  1*x^3 + -4*x^2 + 5*x - 2
        ViewValues(Coeff,"Coefficients");
    }

  using Dew.Math;
  using Dew.Math.Units;
  using Dew.Math.Editors;

  namespace Dew.Examples
  {
    private void Example()
    {
      Vector Roots = new Vector(0);
      Vector Coeff = new Vector(0);
      // roots of the polynomial: (x - 1)*(x - 1)*(x - 2)
      Roots.SetIt(false, new double[] {1,1,2});
      Polynoms.PolyCoeff(Roots,Coeff);
      // Coeff returns the coefficients of the polynomial [ 1,-4, 5,-2]
      // which can be written as:  1*x^3 + -4*x^2 + 5*x - 2
      MtxVecEdit.ViewValues(Coeff,"Coefficients");
    }
  }

Example 2

Find coefficients of a characteristic polynomial

  uses MtxExpr, Math387, MtxVec, MtxVecEdit, Polynoms;

  var Coeff: Vector;
      A: Matrix;
  begin
      A.SetIt(3,3,false,[1,  4, -1,
                         2,  1,  5,
                         3, -2,  0]);
      PolyCoeff(A,Coeff);
      ViewValues(Coeff,'Coefficients');
  end;


  #include "MtxVecCPP.h" //MtxVecCPP.cpp must be included in the project
  #include "Polynoms.hpp"
  #include "MtxVecTee.hpp"

  void __fastcall TForm1::BitBtn1Click(TObject *Sender)
  {
      Vector Coeff;
      Matrix A;

      A->SetIt(3,3,false,OPENARRAY(double,(1,  4, -1,
                                           2,  1,  5,
                                           3, -2,  0)));
      PolyCoeff(A,Coeff);
      ViewValues(Coeff,"Coefficients");
  }

  using Dew.Math;
  using Dew.Math.Units;
  using Dew.Math.Editors;

  namespace Dew.Examples
  {
    private void Example()
    {
      Vector Coeff = new Vector(0);
      Matrix A = new Matrix(0,0);
      A.SetIt(3,3,false, new double[] {1, 4, -1,
                                      2, 1, 5,
                                      3, -2, 0});
      Polynoms.PolyCoeff(A,Coeff);
      MtxVecEdit.ViewValues(Coeff,"Coefficients");
    }
  }