Positivity and Nonnegativity Conditions of a Quartic Equation and Related Problems

IEEE Transactions on Automatic Control

Volumn 26, Issue 2, 1981, Pages 444-451

  Jury, E I ^a   Mansour, M ^b  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b ETH ZURICH   (Switzerland)

Author keywords

[No Author keywords available]

Indexed keywords

MATHEMATICAL TECHNIQUES;

EID: 0019558395     PISSN: 00189286     EISSN: 15582523     Source Type: Journal    
DOI: 10.1109/TAC.1981.1102589     Document Type: Article

References (29)

1. O. Brune, “Synthesis of a finite two-terminal network whose driving point impedance is a prescribed function of frequency,” J. Math. Phys., vol. 10, pp. 191–236, Oct. 1931.
2. G. A. Guillemin, Synthesis of Passive Networks. New York: Wiley, 1957.
3. L. Weinberg, Network Analysis and Synthesis. New York: McGraw-Hill, 1962, pp. 247–269.
4. D. D. šiljak, Nonlinear Systems. New York: Wiley, 1964, pp. 352–367.
5. —, “New algebraic criteria for positive realness,” J. Franklin Inst., pp. 291-120, Feb. 1971.
6. M. V. Meerov, “Analytic conditions for positivity of a real function” (in Russian), Dokl. Akad. Nauk USSR, vol. 165, pp. 780–782, 1965.
7. D. D. šiljak. “Algebraic criteria for absolute stability, optimality and passivity of dynamic systems,” Proc. Inst. Elec. Eng., vol. 117, Oct. 1970.
8. R. E. Kalman, “When is linear control system optimal?” J. Basic Eng., vol. 186(D), pp. 51–60, Mar. 1964.
9. V. M. Popov, “Incompletely controllable positive system and applications to optimization and stability of automatic control systems,” Rev. Roum. Sci., Tech. Ser. Electrotech. Energ., 1967.
10. E. I. Jury, “Stability of multidimensional scalar and matrix polynomials,” Proc. IEEE, vol. 66, pp. 1018–1047, Sept. 1978.
11. A. L. Cauchy, “Calcul des indices des fonctions,” J. Ecole Polytechnique, vol. 15, pp. 175–230, 1837.
12. C. Hermite, “Sur le nombre des racines d'une equation algebrique comprise entre des limites donnees,” J. Reine Angew. Math, vol. 52. pp. 39–51, 1854.
13. C. Sturm, “Memoire sur la resolution des equations numeriques,” Memoirs des Savants Etrangers, vol. 6, pp. 273–318, 1835.
14. J. J. Sylvester, “Sur la theorie des racines réelles et imaginaires des equations du cinquieme degré,” C. R. Acad. Sci., vol. 59 pp. 749–753. 1864; also Math Papers by J. J. Sylvester. London: Cambridge University Press, 1908, pp. 371–375.
15. A. Cayley, “A discussion of the Sturmian constants for cubic and quartic equations,” Quart. J. Pure Appl. Math., vol. 4. pp. 7–12, 1861.
16. W. H. Ku, “Explicit criterion for the positive definiteness of a general quartic form,” IEEE Trans. Automat. Contr., vol. AC-10, no. 3, pp. 372–373. July 1965.
17. J. Marik, “On real biquadratic polynomials,” Casopis pro Pestoruni Matematiky, vol. 90, Praha. Czechoslovakia, pp. 33–43, 1965.
18. E I. Jury, Inners and Stability of Dynamic Systems. New York: Wiley, 1974, ch. 5.
19. R. N. Gadenz and C. C. Li, “On positive definiteness of quartic forms of two-variables,” IEEE Trans. Automat. Contr., vol. AC-9, pp. 187–188, Apr. 1964.
20. R. Ball. “Note on the algebraical solution of biquadratic equations,” Quart. J. Pure Appl. Math., vol. 7, pp. 6–9, 358–369, 1866.
21. H. W. Turnbull, Theory of Equations. London: Oliver and Boyd. 1957.
22. J. Nestmeier, “Circle positivity condition of low degree polynomials and conditions for the absence of real roots in an interval,” M.S. thesis. Dep. Elec. Eng. Comput. Sci., Univ. of California. Berkeley, June 1979.
23. A. T. Fuller, “On redundance in aperiodicity criteria,” J. Math. Anal. Appl., vol. 68, pp. 371–394, 1979.
24. R. Courant and D. Hilbert. Methods of Mathematical Physics, vol. II. New York: Wiley-Interscience, 1962.
25. W. S. Burnside and A. H. Panton, The Theory of Equations. Dublin: Hodges, Figgs and Co., 1892.
26. C. Hermite. “Sur L'equation du cinquieme degre,” oeuvres 2, pp. 347–424; (originally published serially in C. R. Acad. Sci., vol. 61, 1865, vol. 62, 1866.)
27. Masakazu Iwaki, “Positivity test algorithm for 8th degree even polynomials,” M.S. thesis, Dep. Elec. Eng, Comput. Sc., Univ. of California, Berkeley, CA. June 1979.
28. N. K. Bose and H. R. Modaressi, “General procedure to multivariable polynomial positivity test with control applications, IEEE Trans. Automat. Contr., pp. 696–701, Oct. 1976.
29. E. J. Routh, The Advanced Part of a Treatise on the Dynamics of a System of Rigid Bodies. London: Macmillan, ch. 6, p. 220.