Can We Optimize Toeplitz/Hankel Computations?
نویسنده
چکیده
The classical and intensively studied problem of solving a Toeplitz/Hankel linear system of equations is omnipresent in computations in sciences, engineering and communication. Its equivalent formulations include computing polynomial gcd and lcm, Padé approximation, and BerlekampMassey’s problem of recovering the linear recurrence coefficients. To improve the current record asymptotic bit operation cost of the solution, we rely on Hensel’s p-adic lifting. We accelerate its recovery stage by exploiting randomization and the correlation between lifting and the computation of Smith’s invariant factors of the input matrix. Furthermore, for the average input, the 2-adic version of lifting is sufficient, allowing entire computation in binary form, which promises to be valuable for practical computations. Our resulting algorithms solve a nonsingular Toeplitz/Hankel linear system of n equations by using O(m(n)nμ(log n)) bit operations (versus the information lower bound of the order of n log n), where m(n) and μ(d) bound the arithmetic and Boolean cost of multiplying polynomials of degree n and integers modulo 2 +1, respectively, and where the input coefficients are in n. Our algorithms can be applied to a larger class of Toeplitz/Hankel-like linear systems.
منابع مشابه
TR-2002002: Can We Optimize Toeplitz/Hankel Computations? II. Singular Toeplitz/Hankel-like Case
In Part I, under the bit operation cost model we achieved nearly optimal randomized solution of nonsingular Toeplitz/Hankel linear system of equations based on Hensel's lifting. In Part II, we extend these results to the singular Toeplitz/Hankel-like case based on the MBA divide-and conquer algorithm and its combination with Hensel's lifting. We specify randomization and estimate the error/fail...
متن کاملMean value theorem for integrals and its application on numerically solving of Fredholm integral equation of second kind with Toeplitz plus Hankel Kernel
The subject of this paper is the solution of the Fredholm integral equation with Toeplitz, Hankel and the Toeplitz plus Hankel kernel. The mean value theorem for integrals is applied and then extended for solving high dimensional problems and finally, some example and graph of error function are presented to show the ability and simplicity of the method.
متن کاملToeplitz transforms of Fibonacci sequences
We introduce a matricial Toeplitz transform and prove that the Toeplitz transform of a second order recurrence sequence is another second order recurrence sequence. We investigate the injectivity of this transform and show how this distinguishes the Fibonacci sequence among other recurrence sequences. We then obtain new Fibonacci identities as an application of our transform.
متن کاملMultidimensional structured matrices and polynomial systems
We apply and extend some well known and some recent techniques from algebraic residue theory in order to relate to each other two major subjects of algebraic and numerical computing that is computations with structured matrices and solving a system of polynomial equations In the rst part of our paper we extend the Toeplitz and Hankel structures of matrices and some of their known properties to ...
متن کامل