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SUPERFAST SOLUTION OF REAL POSITIVE DEFINITE TOEPLITZ GREGORY S. AMMAR + AND WILLIAM B. GRAGG #
 

Summary: SUPERFAST SOLUTION OF REAL POSITIVE DEFINITE TOEPLITZ
SYSTEMS #
GREGORY S. AMMAR + AND WILLIAM B. GRAGG #
Abstract. We describe an implementation of the generalized Schur algorithm for the superfast
solution of real positive definite Toeplitz systems of order n + 1, where n = 2 # . Our implementation
uses the split­radix fast Fourier transform algorithms for real data of Duhamel. We are able to obtain
the nth Szego polynomial using fewer than 8n log 2
2 n real arithmetic operations without explicit use of
the bit­reversal permutation. Since Levinson's algorithm requires slightly more than 2n 2 operations
to obtain this polynomial, we achieve crossover with Levinson's algorithm at n = 256.
Key words. Toeplitz matrix, Schur's algorithm, split­radix Fast Fourier Transform
AMS subject classifications. 65F05, 65E05
1. Introduction. Consider the linear system of equations Mx = b, where
M = Mn+1 =
# # # # # # # #
” 0 ” 1 ” 2 · · · ”n
” 1 ” 0 ” 1 · · · ”n-1
” 2 ” 1 ” 0
. . . . . .
. . .

  

Source: Ammar, Greg - Department of Mathematical Sciences, Northern Illinois University

 

Collections: Mathematics