Computing the nearest singular univariate polynomials with given root multiplicities. (English) Zbl 1291.65082
Summary: We derive explicit expressions for the nearest singular polynomials with given root multiplicities and its distance to the given polynomial. These expressions can be computed recursively. These results extend previous results [L. Zhi et al., Japan J. Ind. Appl. Math. 21, No. 2, 149–162 (2004; Zbl 1138.65049); L. Zhi and W. Wu, J. Symb. Comput. 26, No. 6, 667–675 (1998; Zbl 0917.65055)].
MSC:
| 65D99 | Numerical approximation and computational geometry (primarily algorithms) |
| 68W30 | Symbolic computation and algebraic computation |
| 12D10 | Polynomials in real and complex fields: location of zeros (algebraic theorems) |
| 26C10 | Real polynomials: location of zeros |
| 41A10 | Approximation by polynomials |
Software:
MultRootReferences:
| [1] | Corless, R. M.; Rezvani, N., The nearest polynomial of lower degree, (Proceedings of the 2007 International Workshop on Symbolic-Numeric Computation, SNC ’07 (2007), ACM: ACM New York, NY, USA), 199-200 |
| [2] | Hitz, M. A.; Kaltofen, E., Efficient algorithms for computing the nearest polynomial with constrained roots, (Proceedings of the 1998 International Symposium on Symbolic and Algebraic Computation, ISSAC ’98 (1998), ACM: ACM New York, NY, USA), 236-243 · Zbl 0917.65045 |
| [3] | Hitz, M. A.; Kaltofen, E.; Lakshman, Y. N., Efficient algorithms for computing the nearest polynomial with a real root and related problems, (Proceedings of the 1999 International Symposium on Symbolic and Algebraic Computation, ISSAC ’99 (1999), ACM: ACM New York, NY, USA), 205-212 |
| [4] | Karmarkar, N.; Lakshman, Y. N., Approximate polynomial greatest common divisors and nearest singular polynomials, (Proceedings of the 1996 International Symposium on Symbolic and Algebraic Computation, ISSAC ’96 (1996), ACM: ACM New York, NY, USA), 35-39 · Zbl 0928.13017 |
| [5] | Pope, S. R.; Szanto, A., Nearest multivariate system with given root multiplicities, J. Symbolic. Comput., 44, June, 606-625 (2009) · Zbl 1168.65347 |
| [6] | Rezvani, N.; Corless, R. M., The nearest polynomial with a given zero, revisited, SIGSAM Bull., 39, September, 73-79 (2005) · Zbl 1341.12002 |
| [7] | Stetter, H. J., The nearest polynomial with a given zero, and similar problems, SIGSAM Bull., 33, December, 2-4 (1999) · Zbl 1097.65524 |
| [8] | Stetter, H. J., Numerical Polynomial Algebra (2004), Society for Industrial and Applied Mathematics: Society for Industrial and Applied Mathematics Philadelphia, PA, USA · Zbl 1058.65054 |
| [9] | Zeng, Z., Computing multiple roots of inexact polynomials, Math. Comp, 74, 869-903 (2005) · Zbl 1079.12007 |
| [10] | Zhi, L.; Noda, M.-T.; Kai, H.; Wu, W., Hybrid method for computing the nearest singular polynomials, Jpn. J. Ind. Appl. Math., 21, 149-162 (2004) · Zbl 1138.65049 |
| [11] | Zhi, L.; Wu, W., Nearest singular polynomials, J. Symbolic. Comput., 26, December, 667-675 (1998) · Zbl 0917.65055 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
