Evaluation of All Complex Poles and Zeros of a Transfer Function Expressed as an Arbitrary Rational Polynomial


  • Automatic parsing of user defined polynomials for both numerator and denominator
  • Capability of computing all complex roots of any given polynomial
  • Optional refinement of roots' precision with criterion set by the user
  • Graphical display of all zeros and poles

Rules and Theories

Roots of a polynomial are evaluated by the eigenvalues of its companion matrix. Root refinement are carried out using Newton's method.

The Applet and User's Guide

  • Choose a character, such as 's' or 'x', for use as the independent variable in defining the polynomials
  • Enter polynomials for either or both numerator and denominator
    • Simple polynomial term can be entered with free-hand written forms, e.g., s^6-7s^2+2
    • Factored terms can be entered using parantheses, e.g., (s)(s+1)(s^2+3s-6)
  • An empty entry is interpreted as unity, a constant: 1
  • Click the "Find" button to trigger the calculations
  • All zeros and poles will be plotted, a click on the "Clear" button will remove the plots
  • Set the precision criterion and the maximum number of iterations, click the "Refine Roots" button cause the refinement to take place for all zeros and poles


[1] Alain Reverchon and Marc Ducamp, "Mathematical Software Tools in C++", John Wiley, New York, 1993.

[2] Gene Golob and James M. Ortega, "Scientific Computing, An Introduction with Parallel Computing", Academic Press, Boston, 1993.

[3] Robert J. Schilling and Sandra L. Harris, "Applied Numerical Methods for Engineers using MATLAB and C", Brooks/Cole, Pacific Grove, 1999.

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