Efficient parallel rooting of complex polynomials on the unit circle

An efficient method is presented for rooting complex polynomial equations of any order where the root space is restricted to the unit circle. The method restates the evaluation of polynomials as a recursive algorithm involving only additions. By means of the bilinear transformation, the straight line, uniformly spaced, recursive polynomials evaluation method of A.H. Nuttall (1987) is extended to the unit circle where the root positions are determined by thresholding. General coefficient transformations are provided along with a comparison to the Horner method. >

[1]  René Alt,et al.  Computing roots of polynomials on vector processing machines , 1985 .

[2]  B. Anderson,et al.  A simplified Schur-Cohn test , 1973 .

[3]  Rolf Unbehauen,et al.  Efficient evaluation of 1-D and 2-D polynomials at equispaced points , 1989, IEEE Trans. Acoust. Speech Signal Process..

[4]  A. Davies,et al.  Bilinear transformation of polynomials , 1974 .

[5]  Kevin C. McGill,et al.  A Storage-Efficient Method for Solving Banded Toeplitz Systems , 1991 .

[6]  L. Volfson,et al.  S-plane to Z-plane mapping using a simultaneous equation algorithm based on the bilinear transformation , 1987 .

[7]  James Hardy Wilkinson,et al.  Rounding errors in algebraic processes , 1964, IFIP Congress.

[8]  Nathaniel Macon,et al.  Numerical analysis , 1963 .

[9]  H. M. Paynter,et al.  Inners and Stability of Dynamic Systems , 1975 .

[10]  Albert H. Nuttall,et al.  Efficient evaluation of polynomials and exponentials of polynomials for equispaced arguments , 1987, IEEE Trans. Acoust. Speech Signal Process..

[11]  James A. Heinen,et al.  A simple algorithm for arbitrary polynomial transformation , 1988, IEEE Trans. Acoust. Speech Signal Process..

[12]  Nirmal K. Bose.  Efficient evaluation of multivariate polynomials over multidimensional grids , 1987, IEEE Trans. Acoust. Speech Signal Process..

[13]  N. K. Bose,et al.  Digital filters: Theory and applications , 1985 .

[14]  P. Henrici.  Elements of numerical analysis , 1966 .

[15]  L. Jackson.  Digital filters and signal processing , 1985 .