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Regularization and Matrix Computation in Numerical Polynomial Algebra

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Approximate Commutative Algebra

Part of the book series: Texts and Monographs in Symbolic Computation ((TEXTSMONOGR))

Abstract

Numerical polynomial algebra emerges as a growing field of study in recent years with a broad spectrum of applications and many robust algorithms. Among the challenges we must face when solving polynomial algebra problems with floating-point arithmetic, the most frequently encountered difficulties include the regularization of ill-posedness and the handling of large matrices. We develop regularization principles for reformulating the ill-posed algebraic problems, derive matrix computations arising in numerical polynomial algebra, as well as subspace strategies that substantially improve computational efficiency by reducing matrix sizes. Those strategies have been successfully applied to numerical polynomial algebra problems such as GCD, factorization, multiplicity structure and elimination.

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Acknowledgements

The author is supported in part by the National Science Foundation of U.S. under Grants DMS-0412003 and DMS-0715127. The author thanks Hans Stetter for providing his student’s thesis [88], and Erich Kaltofen for his comment as well as several references in §5.4.2.

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  1. Department of Mathematics, Northeastern Illinois University, 60625, Chicago, IL, USA

    Zhonggang Zeng

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  1. Dipto. Matematica, Università Genova, Via Dodecaneso 35, Genova, 16146, Italy

    Lorenzo Robbiano

  2. Dipto. Matematica, Università Genova, Via Dodecaneso 35, Genova, 16146, Italy

    John Abbott

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Zeng, Z. (2009). Regularization and Matrix Computation in Numerical Polynomial Algebra. In: Robbiano, L., Abbott, J. (eds) Approximate Commutative Algebra. Texts and Monographs in Symbolic Computation. Springer, Vienna. https://doi.org/10.1007/978-3-211-99314-9_5

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