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Taylor polynomials as an estimator for certain toeplitz matrices

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Abstract

We present an inherently parallel method using a Taylor polynomial as an estimator for several numerical examples of strictly diagonally dominant and banded triangular Toeplitz systems. We demonstrate the effectiveness of the method using multiple precisions for each of four test cases, and compare the results versus classical Jacobi and Newton methods. We also show that the residual norms can be driven down to any arbitrary precision with less hardware requirements than existing methods. Finally, we use truncated Taylor series as initial estimates for the Newton, and in each case, we obtain equal or better accuracy than the Newton method alone, while using less vector computations.

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Notes

  1. An FFT is not required. The result of this operation is a column of 1's.

  2. An FFT is not required. The result of this operation is a column of 1's.

  3. An FFT is not required. The result of this operation is a column of 1's.

  4. An FFT is not required. The result of this operation is a column of 1's.

  5. An FFT is not required. The result of this operation is a column of 1's.

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Authors and Affiliations

  1. Department of Electrical and Computer Engineering, University of Connecticut, Storrs, CT, 06269, USA

    Michael Kapralos

  2. Department of Electrical Engineering and Computer Science, US Military Academy, West Point, NY, 10996, USA

    Michael Kapralos

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  1. Michael Kapralos
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Correspondence to Michael Kapralos.

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Kapralos, M. Taylor polynomials as an estimator for certain toeplitz matrices. J Supercomput 78, 13245–13275 (2022). https://doi.org/10.1007/s11227-022-04373-y

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