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Approximate greatest common divisors of several polynomials with linearly constrained coefficients and singular polynomials. (English) Zbl 1356.12011

Dumas, Jean-Guillaume (ed.), Proceedings of the 2006 international symposium on symbolic and algebraic computation, ISSAC 06, Genova, Italy, July 9–12, 2006. New York, NY: ACM Press (ISBN 1-59593-276-3). 169-176 (2006).

MSC:

12Y05 Computational aspects of field theory and polynomials (MSC2010)
12D05 Polynomials in real and complex fields: factorization
68W30 Symbolic computation and algebraic computation
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[1] Anda, A. A., and Park, H. Fast plane with dynamic scaling. SIAM J. Matrix Anal. Applic. 15 (1994), 162-174. 10.1137/S0895479890193017 · Zbl 0799.65047
[2] Anda, A. A., and Park, H. Self-scaling fast rotations for stiff and equality-constrained linear least squares problems. Linear Algebra and Applications 234 (1996), 137-161. · Zbl 0842.65025
[3] Beckermann, B., and Labahn, G. A fast and numerically stable Euclidean-like algorithm for detecting relative prime numerical polynomials. J. Symbolic Comput. 26 (1998), 691-714. 10.1006/jsco.1998.0235 · Zbl 0935.65051
[4] Botting, B., Giesbrecht, M., and May, J. Using Riemannian SVD for problems in approximate algebra. In Wang and Zhi {33}, pp. 209-219.
[5] Chu, M. T., Funderlic, R. E., and Plemmons, R. J. Structured low rank approximation. Linear Algebra and Applications 366 (2003), 157-172. · Zbl 1018.65057
[6] Corless, R. M., Gianni, P. M., Trager, B. M., and Watt, S. M. The singular value decomposition for polynomial systems. In Proc. 1995 Internat. Symp. Symbolic Algebraic Comput. ISSAC’95 (New York, N. Y., 1995), A. H. M. Levelt, Ed., ACM Press, pp. 96-103. 10.1145/220346.220371
[7] Corless, R. M., Watt, S. M., and Zhi, L. QR factoring to compute the GCD of univariate approximate polynomials. IEEE Transactions on Signal Processing 52 (Dec. 2004), 3394-3402. 10.1109/TSP.2004.837413 · Zbl 1372.65120
[8] Dunaway, D. K. Calculation of zeros of a real polynomial through factorization using Euclid’s algorithm. SIAM J. Numer. Anal. 11, 6 (1974), 1087-1104. · Zbl 0292.65024
[9] Eckart, C., and Young, G. The approximation of one matrix by another of lower rank. Psychometrika 1, 3 (Sept. 1936), 211-218. · JFM 62.1075.02
[10] Gao, S., Kaltofen, E., May, J. P., Yang, Z., and Zhi, L. Approximate factorization of multivariate polynomials via differential equations. In Gutierrez {12}, pp. 167-174. 10.1145/1005285.1005311 · Zbl 1134.65346
[11] von zur Gathen, J., and Gerhard, J. Modern Computer Algebra. Cambridge University Press, Cambridge, New York, Melbourne, 1999. Second edition 2003. · Zbl 0936.11069
[12] Gutierrez, J., Ed. ISSAC 2004 Proc. 2004 Internat. Symp. Symbolic Algebraic Comput. (New York, N. Y., 2004), ACM Press. · Zbl 1050.68002
[13] Hitz, M. A., and Kaltofen, E. Efficient algorithms for computing the nearest polynomial with constrained roots. In Proc. 1998 Internat. Symp. Symbolic Algebraic Comput. (ISSAC’98) (New York, N. Y., 1998), O. Gloor, Ed., ACM Press, pp. 236-243. 10.1145/281508.281624 · Zbl 0917.65045
[14] Kahan, W. Numerical linear algebra. Canadian Math. Bull. 9 (1966), 757-801. · Zbl 0236.65025
[15] Kaltofen, E., and May, J. On approximate irreducibility of polynomials in several variables. In ISSAC 2003 Proc. 2003 Internat. Symp. Symbolic Algebraic Comput. (New York, N. Y., 2003), J. R. Sendra, Ed., ACM Press, pp. 161-168. 10.1145/860854.860893 · Zbl 1072.68676
[16] Kaltofen, E., May, J., Yang, Z., and Zhi, L. Approximate factorization of multivariate polynomials using singular value decomposition. Manuscript, 22 pages. Submitted, Jan. 2006.
[17] Kaltofen, E., Yang, Z., and Zhi, L. Structured low rank approximation of a Sylvester matrix. Manuscript, 15 pages, Oct. 2005. Preliminary version in SNC 2005 Proceedings, Dongming Wang and Lihong Zhi eds., pp. 188-201, distributed at the International Workshop on Symbolic-Numeric Computation in Xi’an, China, July 19-21, 2005.
[18] Kaltofen, E., Yang, Z., and Zhi, L. Structured low rank approximation of a generalized Sylvester matrix. In Proc. of the Seventh Asian Symposium on Computer Mathematics (Seoul, South Korea, 2005), S. Pae and H. Park, Eds., Korea Institute for Advanced Study, pp. 219-222. Extended abstract.
[19] Kaltofen, E., Yang, Z., and Zhi, L. Approximate greatest common divisors of several polynomials with linearly constrained coefficients and singular polynomials. Manuscript, 16 pages, Apr. 2006.
[20] Karmarkar, N. K., and Lakshman Y. N. On approximate GCDs of univariate polynomials. J. Symbolic Comput. 26, 6 (1998), 653-666. 10.1006/jsco.1998.0232 · Zbl 0967.12007
[21] Lemmerling, P. Structured total least squares: analysis, algorithms and applications. Dissertation, Katholieke Universiteit Leuven, Belgium, 1999.
[22] Lemmerling, P., Mastronardi, N., and Van Huffel, S. Fast algorithm for solving the Hankel/Toeplitz Structured Total Least Squares problem. Numerical Algorithms 23 (2000), 371-392. · Zbl 0957.65029
[23] Li, B., Liu, Z., and Zhi, L. Fast low rank approximation of a Sylvester matrix. In Wang and Zhi {33}, pp. 202-208.
[24] Li, B., Yang, Z., and Zhi, L. Fast low rank approximation of a Sylvester matrix by structured total least norm. J. JSSAC (Japan Society for Symbolic and Algebraic Computation) 11, 3,4 (2005), 165-174.
[25] Mastronardi, N., Lemmerling, P., and Van Huffel, S. Fast structured total least squares algorithm for solving the basic deconvolution problem. SIAM J. Matrix Anal. Applic. 22, 2 (2000), 533-553. 10.1137/S0895479898345813 · Zbl 0973.65030
[26] May, J. P. Approximate factorization of polynomials in many variables and other problems in approximate algebra via singular value decomposition methods. PhD thesis, North Carolina State Univ., Raleigh, North Carolina, Aug. 2005.
[27] Park, H., Zhang, L., and Rosen, J. B. Low rank approximation of a Hankel matrix by structured total least norm. BIT 39, 4 (1999), 757-779. · Zbl 0944.65043
[28] Pope, S., and Szanto, A. Nearest multivariate system with given root multiplicities. Manuscript available at http://www.math.ncsu.edu/ aszanto/papers.html, 2005.
[29] Rump, S. M. Structured perturbations part I: Normwise distances. SIAM J. Matrix Anal. Applic. 25, 1 (2003), 1-30. 10.1137/S0895479802405732 · Zbl 1061.15004
[30] Sasasaki, T., and Noda, M. T. Approximate square-free decomposition and root-finding of ill-conditioned algebraic equations. J. Inf. Process. 12 (1989), 159-168. · Zbl 0755.65046
[31] Schönhage, A. Quasi-gcd computations. Journal of Complexity 1 (1985), 118-137. · Zbl 0586.68031
[32] Stewart, G. W. Introduction to Matrix Computations. Academic Press, Inc., New York, 1973. · Zbl 0302.65021
[33] Wang, D., and Zhi, L., Eds. Proc. 2005 International Workshop on Symbolic-Numeric (July 2005). Distributed at the Workshop in Xi’an, China.
[34] Zeng, Z., and Dayton, B. H. The approximate GCD of inexact polynomials part II: a multivariate algorithm. In Gutierrez {12}, pp. 320-327. 10.1145/1005285.1005331 · Zbl 1134.13313
[35] Zhi, L. Displacement structure in computing approximate GCD of univariate polynomials. In Proc. Sixth Asian Symposium on Computer Mathematics (ASCM 2003) (Singapore, 2003), Z. Li and W. Sit, Eds., vol. 10 of Lecture Notes Series on Computing, World Scientific, pp. 288-298. · Zbl 1092.12003
[36] Zhi, L., Noda, M.-T., Kai, H., and Wu, W. Hybrid method for computing the nearest singular polynomials. Japan J. Industrial and Applied Math. 21, 2 (June 2004), 149-162. · Zbl 1138.65049
[37] Zhi, L., and Wu, W. Nearest singular polynomial. J. Symbolic Comput. 26, 6 (1998), 667-675. 10.1006/jsco.1998.0233 · Zbl 0917.65055
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