Structured Matrix Methods Computing the Greatest Common Divisor of Polynomials

Abstract This paper revisits the Bézout, Sylvester, and power-basis matrix representations of the greatest common divisor (GCD) of sets of several polynomials. Furthermore, the present work introduces the application of the QR decomposition with column pivoting to a Bézout matrix achieving the computation of the degree and the coeffcients of the GCD through the range of the Bézout matrix. A comparison in terms of computational complexity and numerical effciency of the Bézout-QR, Sylvester-QR, and subspace-SVD methods for the computation of theGCDof sets of several polynomials with real coeffcients is provided.Useful remarks about the performance of the methods based on computational simulations of sets of several polynomials are also presented.

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A lower bound on the computational complexity of the QR decomposition on a shared memory SIMD computer

Parallel Computing ◽

10.1016/0167-8191(92)90102-d ◽

1992 ◽

Vol 18 (3) ◽

pp. 345-354

Author(s):

Eric Goles ◽

Marcos Kiwi

Keyword(s):

Computational Complexity ◽

Lower Bound ◽

Shared Memory ◽

Qr Decomposition

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Parallelization of the QR Decomposition with Column Pivoting Using Column Cyclic Distribution on Multicore and GPU Processors

Lecture Notes in Computer Science - High Performance Computing for Computational Science - VECPAR 2012 ◽

10.1007/978-3-642-38718-0_8 ◽

2013 ◽

pp. 50-58 ◽

Cited By ~ 4

Author(s):

Andrés Tomás ◽

Zhaojun Bai ◽

Vicente Hernández

Keyword(s):

Qr Decomposition ◽

Column Pivoting

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Structured matrix methods for the computation of multiple roots of a polynomial

Journal of Computational and Applied Mathematics ◽

10.1016/j.cam.2013.08.032 ◽

2014 ◽

Vol 272 ◽

pp. 449-467 ◽

Cited By ~ 6

Author(s):

Joab R. Winkler

Keyword(s):

Multiple Roots ◽

Matrix Methods ◽

Structured Matrix ◽

Roots Of A Polynomial

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Applications of matrix methods to the theory of lower bounds in computational complexity

COMBINATORICA ◽

10.1007/bf02122698 ◽

1990 ◽

Vol 10 (1) ◽

pp. 81-93 ◽

Cited By ~ 63

Author(s):

A. A. Razborov

Keyword(s):

Computational Complexity ◽

Lower Bounds ◽

Matrix Methods

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Zerofinding of analytic functions by structured matrix methods

A Panorama of Mathematics: Pure and Applied - Contemporary Mathematics ◽

10.1090/conm/658/13129 ◽

2016 ◽

pp. 47-66

Author(s):

Luca Gemignani

Keyword(s):

Analytic Functions ◽

Matrix Methods ◽

Structured Matrix

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Structured matrix methods for polynomial root-finding

Proceedings of the 2007 international symposium on Symbolic and algebraic computation - ISSAC '07 ◽

10.1145/1277548.1277573 ◽

2007 ◽

Cited By ~ 4

Author(s):

Luca Gemignani

Keyword(s):

Matrix Methods ◽

Root Finding ◽

Structured Matrix ◽

Polynomial Root Finding

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A sliding window update for the basis matrix of the QR decomposition

IEEE Transactions on Signal Processing ◽

10.1109/78.215313 ◽

1993 ◽

Vol 41 (5) ◽

pp. 1951-1953 ◽

Cited By ~ 9

Author(s):

M.P. Mahon ◽

L.H. Sibul ◽

H.M. Valenzuela

Keyword(s):

Sliding Window ◽

Qr Decomposition ◽

Basis Matrix

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Early Neighbor Rejection-Aided Tabu Search Detection for Large MIMO Systems

Applied Sciences ◽

10.3390/app11167305 ◽

2021 ◽

Vol 11 (16) ◽

pp. 7305

Author(s):

Uzokboy Ummatov ◽

Jin-Sil Park ◽

Gwang-Jae Jang ◽

Ju-Dong Lee

Keyword(s):

Computational Complexity ◽

Tabu Search ◽

Mimo Systems ◽

Multiple Input Multiple Output ◽

Low Complexity ◽

Qr Decomposition ◽

Efficient Computation ◽

Multiple Input ◽

Input Multiple Output ◽

Ordering Schemes

In this study, a low complexity tabu search (TS) algorithm for multiple-input multiple-output (MIMO) systems is proposed. To reduce the computational complexity of the TS algorithm, early neighbor rejection (ENR) and layer ordering schemes are employed. In the proposed ENR-aided TS (ENR-TS) algorithm, the least promising k neighbors are excluded from the neighbor set in each layer, which reduces the computational complexity of neighbor examination in each TS iteration. For efficient computation of the neighbors’ metrics, the ENR scheme can be incorporated into QR decomposition-aided TS (ENR-QR-TS). To further reduce the complexity and improve the performance of the ENR-QR-TS scheme, a layer ordering scheme is employed. The layer ordering scheme determines the order in which layers are detected based on their expected metrics, which reduces the risk of excluding likely neighbors in early layers. The simulation results show that the ENR-TS achieves nearly the same performance as the conventional TS while providing up to 82% complexity reduction.

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Numerical matrix methods in the computation of the greatest common divisor (GCD) of polynomials

10.1063/1.4965184 ◽

2016 ◽

Author(s):

Siti Nor Asiah binti Isa ◽

Nor’aini Aris ◽

Shazirawati Mohd Puzi

Keyword(s):

Greatest Common Divisor ◽

Matrix Methods

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Basis Matrix Representation of Morphological Filters with N-Dimensional Structuring Elements

Journal of Circuits System and Computers ◽

10.1142/s0218126697000255 ◽

1997 ◽

Vol 07 (04) ◽

pp. 345-352

Author(s):

Byung-Tae Choi ◽

Kyung-Hoon Lee ◽

Sung-Jea Ko ◽

Aldo Morales

Keyword(s):

Matrix Representation ◽

Efficient Implementation ◽

Morphological Filters ◽

Matrix Representations ◽

Matrix Operation ◽

Basis Matrix ◽

Block Basis ◽

Structuring Element ◽

Opening And Closing

In this paper, we present a basis matrix representation of grayscale morphological filters in N-dimensions. A procedure is proposed to derive the basis matrix and the block basis matrix (BBM) from an N-dimensional grayscale structuring element (GSE). It is shown that both opening and closing with arbitrary N-dimensional GSE can be accomplished by a local matrix operation using the basis matrix. Furthermore, these basis matrix representations are extended to the efficient implementation of open-closing (OC) and close-opening (CO) using the BBM.

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