SML000669 - c 2003 Society for Industrial and Applied...

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
A SUPERFAST TOEPLITZ SOLVER WITH IMPROVED NUMERICAL STABILITY MICHAEL STEWART SIAM J. M ATRIX A NAL. A PPL . c 2003 Society for Industrial and Applied Mathematics Vol. 25, No. 3, pp. 669–693 Abstract. This paper describes a new O ( n log 3 ( n )) solver for the positive definite Toeplitz system Tx = b . Instead of computing generators for the inverse of T , the new algorithm adjoins b to T and applies a superfast Schur algorithm to the resulting augmented matrix. The genera- tors of this augmented matrix and its Schur complements are used by a divide-and-conquer block back-substitution routine to complete the solution of the system. The goal is to avoid the well- known numerical instability inherent in explicit inversion. Experiments suggest that the algorithm is backward stable in most cases. Key words. Toeplitz matrix, Schur algorithm AMS subject classification. 65F05 DOI. 10.1137/S089547980241791X 1. Background. We start with the positive definite Toeplitz matrix T = t 0 t 1 · · · · · · t m 1 t 1 t 0 t 1 . . . . . . t 1 . . . . . . . . . . . . . . . . . . t 1 t m 1 · · · · · · t 1 t 0 C m × m and the system of equations Tx = b . There are several classes of algorithms for solving such systems: these include slow algorithms requiring O ( n 3 ) unstructured matrix computations, fast O ( n 2 ) algorithms that exploit the Toeplitz structure, and superfast algorithms that achieve a complexity strictly less than O ( n 2 ). Examples of fast algorithms include the Schur and Levinson algorithms. Superfast algorithms have been developed in [3, 5, 10, 1, 7, 2]. One way to view the related approaches of [10, 1, 7, 2] is as a divide-and-conquer variant of the O ( n 2 ) Schur algorithm with fast polynomial multiplication via the FFT used to extend computations from submatrices and Schur complements to the full matrix T . The underlying Schur algorithm, along with several generalizations, is numerically stable [4, 15, 6], but it has not been shown that this stability extends to the superfast Schur algorithm. In fact, the proposed application of the algorithm to linear systems involves computing generators of T 1 and then forming T 1 b using the FFT. Numerical methods based on explicit inversion are usually unstable [9]. Experiments presented in section 5 show that the superfast Schur algorithm is no exception; it is not a backward stable algorithm. We will propose an alternative method that parallels the conventional and stable method of triangular factorization and back-substitution. Instead of inverting T we Received by the editors November 13, 2002; accepted for publication (in revised form) by L. Reichel May 6, 2003; published electronically December 17, 2003. http://www.siam.org/journals/simax/25-3/41791.html Department of Mathematics and Statistics, Georgia State University, Atlanta, GA 30303 ([email protected]).
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern