کد مقاله کد نشریه سال انتشار مقاله انگلیسی نسخه تمام متن
4598766 1631104 2016 16 صفحه PDF دانلود رایگان
عنوان انگلیسی مقاله ISI
Essential spectral equivalence via multiple step preconditioning and applications to ill conditioned Toeplitz matrices
ترجمه فارسی عنوان
معادلات طیفی ضروری از طریق پیشگیری و برنامهریزی چند گام به ماتریس های تئپلیتس متعلق به بیمار
موضوعات مرتبط
مهندسی و علوم پایه ریاضیات اعداد جبر و تئوری
چکیده انگلیسی

In this note, we study the fast solution of Toeplitz linear systems with coefficient matrix Tn(f)Tn(f), where the generating function f is nonnegative and has a unique zero at zero of any real positive order θ  . As preconditioner we choose a matrix τn(f)τn(f) belonging to the so-called τ   algebra, which is diagonalized by the sine transform associated to the discrete Laplacian. In previous works, the spectral equivalence of the matrix sequences {τn(f)}n{τn(f)}n and {Tn(f)}n{Tn(f)}n was proven under the assumption that the order of the zero is equal to 2: in other words the preconditioned matrix sequence {τn−1(f)Tn(f)}n has eigenvalues, which are uniformly away from zero and from infinity. Here we prove a partial generalization of the above result when θ<2θ<2. Furthermore, by making use of multiple step preconditioning, we show that the matrix sequences {τn(f)}n{τn(f)}n and {Tn(f)}n{Tn(f)}n are essentially spectrally equivalent for every θ>2θ>2, i.e., for every θ>2θ>2, there exist mθmθ and a positive interval [αθ,βθ][αθ,βθ] such that all the eigenvalues of {τn−1(f)Tn(f)}n belong to this interval, except at most mθmθ outliers larger than βθβθ: while the essential bound from above is proven, the bound from below is only observed numerically. Such a nice property, already known only when θ is an even positive integer greater than 2, is coupled with the fact that the preconditioned sequence has an eigenvalue cluster at one, so that the convergence rate of the associated preconditioned conjugate gradient method is optimal. As a conclusion we discuss possible generalizations and we present selected numerical experiments.

ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: Linear Algebra and its Applications - Volume 491, 15 February 2016, Pages 276–291
نویسندگان
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