A stable test for strict sign regularity
HTML articles powered by AMS MathViewer
- by V. Cortés and J. M. Peña;
- Math. Comp. 77 (2008), 2155-2171
- DOI: https://doi.org/10.1090/S0025-5718-08-02107-8
- Published electronically: March 19, 2008
- PDF | Request permission
Abstract:
A stable test to check if a given matrix is strictly sign regular is provided. Among other nice properties, we prove that it has an optimal growth factor. The test is compared with other alternative tests appearing in the literature, and its advantages are shown.References
- T. Ando, Totally positive matrices, Linear Algebra Appl. 90 (1987), 165–219. MR 884118, DOI 10.1016/0024-3795(87)90313-2
- Lawrence D. Brown, Iain M. Johnstone, and K. Brenda MacGibbon, Variation diminishing transformations: a direct approach to total positivity and its statistical applications, J. Amer. Statist. Assoc. 76 (1981), no. 376, 824–832. MR 650893
- J. M. Carnicer and J. M. Peña, On transforming a Tchebycheff system into a strictly totally positive system, J. Approx. Theory 81 (1995), no. 2, 274–295. MR 1327172, DOI 10.1006/jath.1995.1050
- V. Cortes and J. M. Peña, Sign regular matrices and Neville elimination, Linear Algebra Appl. 421 (2007), no. 1, 53–62. MR 2290685, DOI 10.1016/j.laa.2006.03.040
- Shaun M. Fallat and P. van den Driessche, On matrices with all minors negative, Electron. J. Linear Algebra 7 (2000), 92–99. MR 1779434, DOI 10.13001/1081-3810.1049
- M. Gasca and J. M. Peña, Total positivity and Neville elimination, Linear Algebra Appl. 165 (1992), 25–44. MR 1149743, DOI 10.1016/0024-3795(92)90226-Z
- M. Gasca and J. M. Peña, A test for strict sign-regularity, Linear Algebra Appl. 197/198 (1994), 133–142. Second Conference of the International Linear Algebra Society (ILAS) (Lisbon, 1992). MR 1275612, DOI 10.1016/0024-3795(94)90485-5
- M. Gasca and J. M. Peña, On factorizations of totally positive matrices, Total positivity and its applications (Jaca, 1994) Math. Appl., vol. 359, Kluwer Acad. Publ., Dordrecht, 1996, pp. 109–130. MR 1421600
- Samuel Karlin, Total positivity. Vol. I, Stanford University Press, Stanford, CA, 1968. MR 230102
- Plamen Koev and Froilán Dopico, Accurate eigenvalues of certain sign regular matrices, Linear Algebra Appl. 424 (2007), no. 2-3, 435–447. MR 2329485, DOI 10.1016/j.laa.2007.02.012
- G. Mühlbach and M. Gasca, A test for strict total positivity via Neville elimination, Current trends in matrix theory (Auburn, Ala., 1986) North-Holland, New York, 1987, pp. 225–232. MR 898910
- J.M. Peña (Ed.), Shape Preserving Representations in Computer-Aided Geometric Design, Nova Science Publishers, 1999.
- J. M. Peña, On nonsingular sign regular matrices, Linear Algebra Appl. 359 (2003), 91–100. MR 1948436, DOI 10.1016/S0024-3795(02)00437-8
- J. M. Peña, Characterizations and stable tests for the Routh-Hurwitz conditions and for total positivity, Linear Algebra Appl. 393 (2004), 319–332. MR 2098621, DOI 10.1016/j.laa.2003.11.013
- J. M. Peña, A stable test to check if a matrix is a nonsingular $M$-matrix, Math. Comp. 73 (2004), no. 247, 1385–1392. MR 2047092, DOI 10.1090/S0025-5718-04-01639-4
- Isac Schoenberg, Über variationsvermindernde lineare Transformationen, Math. Z. 32 (1930), no. 1, 321–328 (German). MR 1545169, DOI 10.1007/BF01194637
Bibliographic Information
- V. Cortés
- Affiliation: Departamento de Matemática Aplicada, Universidad de Zaragoza, 50009, Zaragoza, Spain
- Email: vcortes@unizar.es
- J. M. Peña
- Affiliation: Departamento de Matemática Aplicada, Universidad de Zaragoza, 50009, Zaragoza, Spain
- Email: jmpena@posta.unizar.es
- Received by editor(s): November 17, 2007
- Published electronically: March 19, 2008
- Additional Notes: This research has been partially supported by the Spanish Research Grant MTM2006-03388 and by Gobierno de Aragón and Fondo Social Europeo.
- © Copyright 2008 American Mathematical Society
- Journal: Math. Comp. 77 (2008), 2155-2171
- MSC (2000): Primary 65F05, 65F40, 15A48
- DOI: https://doi.org/10.1090/S0025-5718-08-02107-8
- MathSciNet review: 2429879