Abstract
This paper gives a numerically stable method to compute H-basis which is based on the computing a minimal basis for the module of syzygies using singular value decomposition. We illustrate the performance of this method by means of various examples.
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Batselier, K., Dreesen, P., De Moor, B.: The canonical decomposition of \({C}_d^n\) and numerical Gröbner and border bases. SIAM J. Matrix Anal. Appl. 35, 1242–1264 (2014)
Batselier, K., Dreesen, P., De Moor, B.: A fast recursive orthogonalization scheme for the Macaulay matrix. J. Comput. Appl. Math. 267, 20–32 (2014)
Batselier, K., Dreesen, P., De Moor, B.: On the null spaces of the Macaulay matrix. Linear Algebra Appl. 460, 259–289 (2014)
Boor, C.: Computational aspects of multivariate polynomial interpolation: indexing the coefficients. Adv. Comput. Math. 12, 289–301 (2000)
Buchberger, B.: Gröbner bases: an algorithmic method in polynomial ideal theory. In: Bose, N.K. (ed.) Multidimensional Systems Theory, pp. 184–232. D. Reidel Publishing Company, Dordrecht (1985)
Cox, D., Little, J., O’Shea, D.: Ideals, Varieties and Algorithms. Undergraduate Texts in Mathematics, 2nd edn. Springer, Berlin (1996)
Czekansky, J., Sauer, T.: The multivariate Horner scheme revisited. BIT Numer. Math. 55, 1043–1056 (2015). https://doi.org/10.1007/s10543-014-0533-x
Decker, W., Greuel, G.M., Pfister, G.: Primary decomposition: Algorithms and comparisons. In: Matzaat, B.H., Greuel, G.-M., Hiss, G. (eds.) Algorithmic Algebra and Number Theory. Springer, Berlin (1999)
Demmel, J.W., Grigori, L., Gu, M., Xiang, H.: Communication avoiding rank revealing QR factorization with column pivoting. SIAM J. Matrix Anal. Appl. 36, 55–89 (2015)
Foster, L.V., Liu, X.: Comparison of rank revealing algorithms applied to matrices with well defined numerical rank (2006) (Preprint unpublished)
Golub, G., van Loan, C.F.: Matrix Computations, 3rd edn. The Johns Hopkins University Press, Baltimore (1996)
Gröbner, W.: Über das Macaulaysche inverse System und dessen Bedeutung für die Theorie der linearen Differentialgleichungen mit konstanten Koeffizienten. Abh. Math. Sem. Hamburg 12, 127–132 (1937)
Hansen, P.: Rank-deficient and discrete ill-posed problems. SIAM, Philadelphia (1998)
Hashemi, A., Javanbakht, M.: Computing H-bases via minimal bases for syzygy modules (2017) (Submitted for publication)
Heldt, D., Kreuzer, M., Pokutta, S., Poulisse, H.: Approximate computation of zero-dimensional polynomial ideals. J. Symb. Comput. 44, 1566–1591 (2009)
Higham, N.J.: Accuracy and Stability of Numerical Algorithms, 2nd edn. SIAM, Philadelphia (2002)
Kreuzer, M., Robbiano, L.: Computational Commutative Algebra 1. Springer, Berlin (2000)
Lazard, D.: Gröbner bases, Gaussian elimination, and resolution of systems of algebraic equations. In: van Hulzen, J.A. (ed.) EUROCAL. Springer, Berlin (1983)
Li, T.Y., Zeng, Z.: A rank-revealing method with updating, downdating, and applications. SIAM J. Matrix Anal. Appl. 26, 918–946 (2009)
Macaulay, F.S.: The Algebraic Theory of Modular Systems. No. 19 in Cambridge Tracts in Math. and Math. Physics. Cambridge University Press (1916)
Möller, H.M., Sauer, T.: H-bases for polynomial interpolation and system solving. Adv. Comput. Math. 12(4), 335–362 (2000)
Möller, H.M., Sauer, T.: H-bases II: Applications to numerical problems. In: Cohen, A., Rabut, C., Schumaker, L.L. (eds.) Curve and Surface fitting: Saint-Malo 1999, pp. 333–342. Vanderbilt University Press, Nashville (2000)
Möller, H.M., Tenberg, R.: Multivariate polynomial system solving using intersections of eigenspaces. J. Symb. Comput. 32, 513–531 (2001)
Peeva, I.: Graded Syzygies. Springer, Berlin (2011)
Sauer, T.: Gröbner bases, H-bases and interpolation. Trans. Am. Math. Soc. 353, 2293–2308 (2001)
Sauer, T.: Approximate varieties, approximate ideals and dimension reduction. Numer. Algorithms 45, 295–313 (2007)
Sauer, T.: Prony’s method in several variables. Numer. Math. 136, 411–438 (2017). https://doi.org/10.1007/s00211-016-0844-8; ArXiv:1602.02352
Schreyer, F.O.: Die Berechnung von Syzygien mit dem verallgemeinerten Weierstrassschen Divisionssatz. Diplomarbeit, University of Hamburg (1980)
Stetter, H.J.: Numerical Polynomial Algebra. SIAM, Philadelphia (2005)
Acknowledgements
This research was performed during the first authors stay at the University of Passau. We are grateful for the support provided by both institutions. In addition, we thank the referees for their careful reading and their valuable suggestions.
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Communicated by Daniel Kressner.
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Javanbakht, M., Sauer, T. Numerical computation of H-bases. Bit Numer Math 59, 417–442 (2019). https://doi.org/10.1007/s10543-018-0733-x
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DOI: https://doi.org/10.1007/s10543-018-0733-x