Abstract
We analyse the performance of five numerical methods for factoring a Laurent polynomial, which is positive on the unit circle, as the modulus squared of a real algebraic polynomial. It is found that there is a wide disparity between the methods, and all but one of the methods are significantly influenced by the variation in magnitude of the coefficients of the Laurent polynomial, by the closeness of the zeros of this polynomial to the unit circle, and by the spacing of these zeros.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
K. Amaratunga, A fast Matlab routine for calculating Daubechies filters, Wavelet Digest 4(4) (1995).
F.L. Bauer, Ein direktes Iterations Verfahren zur Hurwitz-zerlegung eines Polynoms, Arch. Elektr. Uebertragung 9 (1955) 285–290.
F.L. Bauer, Beiträge zur Entwicklung numerischer Verfahren für programmgesteuerte Rechenanlagen, ii. Direkte Faktorisierung eines Polynoms, Sitz. Ber. Bayer. Akad. Wiss. (1956) 163–203.
B.P. Bogert, M.J.R. Healy and J.W. Tukey, The quefrency alanysis of time serier for echoes: cepstrum pseudo-autocovariance, cross-cepstrum and saphe cracking, in: Proc. Symposium Time Series Analysis, ed. M. Rosenblatt (Wiley, New York, 1963) pp. 209–243.
A. Calderón, F. Spitzer and H. Widom, Inversion of Toeplitz matrices, Illinois J. Math. 3 (1959) 490–498.
A.S. Cavaretta, W. Dahmen and C.A. Micchelli, Stationary subdivision, Mem. Amer. Math. Soc. 93 (1991) 1–186.
W. Dahmen and C.A. Micchelli, Using the refinement equation for evaluating integrals of wavelets, SIAM J. Numer. Anal. 30(2) (1993) 507–537.
I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics 61 (SIAM, Philadelphia, PA, 1992).
C. de Boor, A Practical Guide to Splines, Applied Mathematical Sciences 27 (Springer, New York, 1978).
G.H. Golub and C.F. Van Loan, Matrix Computations (The Johns Hopkins University Press, Baltimore, 2nd ed., 1989).
T.N.T. Goodman, C.A. Micchelli, G. Rodriguez and S. Seatzu, On the Cholesky factorization of the Gram matrix of locally supported functions, BIT 35(2) (1995) 233–257.
T.N.T. Goodman, C.A. Micchelli, G. Rodriguez and S. Seatzu, On the limiting profile of exponentially decaying functions, Preprint (1996).
M.G. Krein, Integral equations on the half-line with kernel depending upon the difference of the arguments, Uspekhi Mat. Nauk 13(5) (1958) 3–120 (in Russian). Translation: AMS Translations 22 (1962) 163–288.
MATLAB Version 4.2c, The MathWorks Inc., South Natick, MA (1994).
C.A. Micchelli, Mathematical Aspects of Geometric Modeling, CBMS-NSF Regional Conference Series in Applied Mathematics 65 (SIAM, Philadelphia, PA, 1995).
A.V. Oppenheim and R.W. Schafer, Discrete-Time Signal Processing, Prentice-Hall Signal Processing Series (Prentice-Hall, Englewood Cliffs, NJ, 1989).
F. Riesz and B.Sz. Nagy, Functional Analysis (Frederick Ungar, New York, 1955).
I.J. Schoenberg, Cardinal Spline Interpolation, CBMS-NSF Regional Conference Series in Applied Mathematics 12 (SIAM, Philadelphia, PA, 1973).
J. Shen and G. Strang, The zeros of the Daubechies polynomials, Proc. Amer. Math. Soc. (1996).
G. Strang and T. Nguyen, Wavelets and Filter Banks (Wellesley-Cambridge Press, Wellesley, MA, 1996).
G. Wilson, Factorization of the covariance generating function of a pure moving average process, SIAM J. Numer. Anal. 6(1) (1969) 1–7.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Goodman, T.N., Micchelli, C.A., Rodriguez, G. et al. Spectral factorization of Laurent polynomials. Advances in Computational Mathematics 7, 429–454 (1997). https://doi.org/10.1023/A:1018915407202
Issue Date:
DOI: https://doi.org/10.1023/A:1018915407202