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Superfast solution of real positive definite Toeplitz systems
 SIAM J. MATRIX ANAL. APPL
, 1988
"... 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 = 2ν. Our implementation uses the splitradix fast Fourier transform algorithms for real data of Duhamel. We are able to obtain the nth Szeg ..."
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Cited by 72 (2 self)
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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 = 2ν. Our implementation uses the splitradix fast Fourier transform algorithms for real data of Duhamel. We are able to obtain the nth Szegő polynomial using fewer than 8n log2 2 n real arithmetic operations without explicit use of the bitreversal permutation. Since Levinson’s algorithm requires slightly more than 2n2 operations to obtain this polynomial, we achieve crossover with Levinson’s algorithm at n = 256.
A SUPERFAST ALGORITHM FOR TOEPLITZ SYSTEMS OF LINEAR EQUATIONS
"... In this paper we develop a new superfast solver for Toeplitz systems of linear equations. To solve Toeplitz systems many people use displacement equation methods. With displacement structures, Toeplitz matrices can be transformed into Cauchylike matrices using the FFT or other trigonometric transf ..."
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Cited by 24 (4 self)
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In this paper we develop a new superfast solver for Toeplitz systems of linear equations. To solve Toeplitz systems many people use displacement equation methods. With displacement structures, Toeplitz matrices can be transformed into Cauchylike matrices using the FFT or other trigonometric transformations. These Cauchylike matrices have a special property, that is, their offdiagonal blocks have small numerical ranks. This lowrank property plays a central role in our superfast Toeplitz solver. It enables us to quickly approximate the Cauchylike matrices by structured matrices called sequentially semiseparable (SSS) matrices. The major work of the constructions of these SSS forms can be done in precomputations (independent of the Toeplitz matrix entries). These SSS representations are compact because of the lowrank property. The SSS Cauchylike systems can be solved in linear time with linear storage. Excluding precomputations the main operations are the FFT and SSS system solvers, which are both very efficient. Our new Toeplitz solver is stable in practice. Numerical examples are presented to illustrate the efficiency and the practical stability.
Lookahead Levinson and Schur Algorithms For Nonhermitian Toeplitz Systems
, 1995
"... We present generalizations of the nonsymmetric Levinson and Schur algorithms for nonHermitian Toeplitz matrices with some singular or illconditioned leading principal submatrices. The underlying recurrences allow us to go from any pair of successive wellconditioned leading principal submatrices ..."
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Cited by 24 (3 self)
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We present generalizations of the nonsymmetric Levinson and Schur algorithms for nonHermitian Toeplitz matrices with some singular or illconditioned leading principal submatrices. The underlying recurrences allow us to go from any pair of successive wellconditioned leading principal submatrices to any such pair of larger order. If the lookahead step size between these pairs is bounded, our generalized Levinson and Schur recurrences require O(N 2) operations, and the Schur recurrences can be combined with recursive doubling so that an O(N log2N) algorithm results. The overhead (in operations and storage) of lookahead steps is very small. There are various options for applying these algorithms to solving linear systems with Toeplitz matrix.
The Generalized Schur Algorithm for the Superfast Solution of Toeplitz Systems
 IN RATIONAL APPROXIMATION AND ITS APPLICATIONS IN MATHEMATICS AND PHYSICS
, 1987
"... We review the connections between fast, O(n²), Toeplitz solvers and the classical theory of Szego polynomials and Schur's algorithm. We then give ..."
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Cited by 22 (5 self)
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We review the connections between fast, O(n²), Toeplitz solvers and the classical theory of Szego polynomials and Schur's algorithm. We then give
Parallel algorithms for Toeplitz systems
 Numerical Linear Algebra, Digital Signal Processing and Parallel Algorithms
, 1990
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Lookahead Levinson and Schurtype recurrences in the Pad'e table
 Electron. Trans. Numer. Anal
, 1994
"... Abstract. For computing Padé approximants, we present presumably stable recursive algorithms that follow two adjacent rows of the Padé table and generalize the wellknown classical Levinson and Schur recurrences to the case of a nonnormal Padé table. Singular blocks in the table are crossed by look ..."
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Abstract. For computing Padé approximants, we present presumably stable recursive algorithms that follow two adjacent rows of the Padé table and generalize the wellknown classical Levinson and Schur recurrences to the case of a nonnormal Padé table. Singular blocks in the table are crossed by lookahead steps. Illconditioned Padé approximants are skipped also. If the size of these lookahead steps is bounded, the recursive computation of an (m, n) Padé approximant with either the lookahead Levinson or the lookahead Schur algorithm requires O(n 2) operations. With recursive doubling and fast polynomial multiplication, the cost of the lookahead Schur algorithm can be reduced to O(n log 2 n).
The stability of inversion formulas for Toeplitz matrices,” Linear Algebra and its
 Applications,
, 1995
"... ..."
Parallel Complexity of Computations with General and Toeplitzlike Matrices Filled with Integers and Extensions
, 1999
"... 1 Computations with Toeplitz and Toeplitzlike matrices are fundamental for many areas of algebraic and numerical computing. The list of computational problems reducible to Toeplitz and Toeplitzlike computations includes, in particular, the evaluation of the gcd, the lcm, and the resultant of two p ..."
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1 Computations with Toeplitz and Toeplitzlike matrices are fundamental for many areas of algebraic and numerical computing. The list of computational problems reducible to Toeplitz and Toeplitzlike computations includes, in particular, the evaluation of the gcd, the lcm, and the resultant of two polynomials, computing Pad'e approximation and the BerlekampMassey recurrence coefficients, as well as numerous problems reducible to these. Transition to Toeplitz and Toeplitzlike computations is currently the basis for the design of the fastest known parallel (RNC) algorithms for these computational problems. Our main result is in contructing nearly optimal randomized parallel algorithms for Toeplitz and Toeplitzlike computations and, consequently, for numerous related computational problems (including the computational problems listed above), where all the input values are integers and all the output values are computed exactly. This includes randomized parallel algorithms for computing...
Old and new algorithms for Toeplitz systems
 PROCEEDINGS SPIE, VOLUME 975, ADVANCED ALGORITHMS AND ARCHITECTURES FOR SIGNAL PROCESSING III (EDITED BY FRANKLIN T. LUK), SPIE
, 1989
"... Toeplitz linear systems and Toeplitz least squares problems commonly arise in digital signal processing. In this paper we survey some old, “well known” algorithms and some recent algorithms for solving these problems. We concentrate our attention on algorithms which can be implemented efficiently on ..."
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Cited by 3 (3 self)
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Toeplitz linear systems and Toeplitz least squares problems commonly arise in digital signal processing. In this paper we survey some old, “well known” algorithms and some recent algorithms for solving these problems. We concentrate our attention on algorithms which can be implemented efficiently on a variety of parallel machines (including pipelined vector processors and systolic arrays). We distinguish between algorithms which require inner products, and algorithms which avoid inner products, and thus are better suited to parallel implementation on some parallel architectures. Finally, we mention some “asymptotically fast” O(n(log n)²) algorithms and compare them with O(n²) algorithms.
Parallel Output Sensitive Algorithms for Combinatorial and Linear Algebra Problems
, 2000
"... This paper gives output sensitive parallel algorithms whose performance ..."
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Cited by 2 (2 self)
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This paper gives output sensitive parallel algorithms whose performance