Optimized look-ahead recurrences for adjacent rows in the Padé table
- 15 Downloads
A new look-ahead algorithm for recursively computing Padé approximants is introduced. It generates a subsequence of the Padé approximants on two adjacent rows (defined by fixed numerator degree) of the Padé table. Its two basic versions reduce to the classical Levinson and Schur algorithms if no look-ahead is required. The new algorithm can be viewed as a combination of the look-ahead sawtooth and the look-ahead Levinson and Schur algorithms that we proposed before, but now the look-ahead step size is minimal (as in the sawtooth version) and the computational costs are as low as in the least expensive competing algorithms (including our look-ahead Levinson and Schur algorithms). The underlying recurrences link well-conditioned basic pairs,i.e., pairs of sufficiently different neighboring Padé forms.
The algorithm can be used to solve Toeplitz systems of equationsTx = b. In this application it comes in several versions: anO(N2) Levinson-type form, anO(N2) Schur-type form, and a superfastO(N log2N) Schur-type version. As an option of the first two versions, the corresponding block LDU decompositions ofT−1 orT, respectively, can be found.
Key wordsPadé approximation row recurrence fast algorithm sawtooth recurrence look-ahead Toeplitz matrix Levinson algorithm Schur algorithm biorthogonal polynomials
Unable to display preview. Download preview PDF.
- 1.A. Bultheel,Laurent Series and their Padé Approximations, Birkhäuser, Basel/Boston, 1987.Google Scholar
- 2.A. Bultheel and M. Van Barel,Formal orthogonal polynomials and rational approximation for arbitrary bilinear forms, Tech. Report TW 163, Department of Computer Science, Katholieke Universiteit Leuven (Belgium), 1991.Google Scholar
- 3.T. F. Chan and P. C. Hansen,A look-ahead Levinson algorithm for general Toeplitz systems, IEEE Trans. Signal Processing, 40 (1992), pp. 1079–1090.Google Scholar
- 4.S. C. Cooper, A. Magnus, and J. H. McCabe,On the non-normal two-point Padé table, J. Comput. Appl. Math., 16 (1986), pp. 371–380.Google Scholar
- 5.A. Draux,Polynômes orthogonaux formels—applications, vol. 974 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1983.Google Scholar
- 6.R. Freund and H. Zha,Formally biorthogonal polynomials and a look-ahead Levinson algorithm for general Toeplitz systems, Linear Algebra Appl., 188/189 (1993), pp. 255–304.Google Scholar
- 7.R. W. Freund,A look-ahead Bareiss algorithm for general Toeplitz matrices, Numer. Math., 68 (1994), pp. 35–69.Google Scholar
- 8.W. B. Gragg,The Padé table and its relation to certain algorithms of numerical analysis, SIAM Rev., 14 (1972), pp. 1–62.Google Scholar
- 9.M. H. Gutknecht,Stable row recurrences in the Padé table and generically superfast lookahead solvers for non-Hermitian Toeplitz systems, Linear Algebra Appl., 188/189 (1993), pp. 351–421.Google Scholar
- 10.M. H. Gutknecht and M. Hochbruck,Look-ahead Levinson- and Schur-type recurrences in the Padé table, Electronic Trans. Numer. Anal., 2 (1994), pp. 104–129.Google Scholar
- 11.M. H. Gutknecht and M. Hochbruck,Look-ahead Levinson and Schur algorithms for non-Hermitian Toeplitz systems, Numer. Math., 70 (1995), pp. 181–227.Google Scholar
- 12.T. Huckle,Superfast solution of linear equations with low displacement rank, Tech. Report SCCM-93-15, Computer Science Department, Stanford University, December 1993.Google Scholar
- 13.N. Levinson,The Wiener rms (root-mean-square) error criterion in filter design and prediction, J. Math. Phys., 25 (1947), pp. 261–278.Google Scholar
- 14.I. Schur,Ueber Potenzreihen, die im Innern des Einheitskreises beschränkt sind, I, Crelle's J. (J. Reine Angew. Math.), 147 (1917), pp. 205–232.Google Scholar
- 15.L. N. Trefethen and M. H. Gutknecht,Padé, stable Padé, and Chebyshev-Padé approximation, in Algorithms for Approximation, J. Mason and M. Cox, eds., IMA Conference Series, new series, Vol. 10, Clarendon Press, Oxford, 1987, pp. 227–264.Google Scholar