Given a function
, write
and define the Sturm functions by
(1) |
(2) | |||
Sturm functions provide a convenient way for finding the number of real roots of an algebraic equation with real coefficients over a given interval. Specifically, the difference in the number of sign changes between the Sturm functions evaluated at two points and gives the number of real roots in the interval . This powerful result is known as the Sturm Theorem.
As a specific application of Sturm functions toward finding Polynomial Roots, consider the function
,
plotted above, which has roots , ,
, and 1.38879 (three of which are real). The
Derivative is given by , and the Sturm Chain is then given by
(3) | |||
(4) | |||
(5) | |||
(6) |
1 | 1 | 3 | |||
0 | 1 | 1 | 1 | ||
2 | 1 | 1 | 1 | 1 | 0 |
This shows that real roots lie in , and real root lies in . Reducing the spacing to gives the following table.
1 | 1 | 3 | |||
1 | 1 | 3 | |||
1 | 1 | 1 | 2 | ||
1 | 1 | 2 | |||
0.0 | 1 | 1 | 1 | ||
0.5 | 1 | 1 | 1 | ||
1.0 | 1 | 1 | 1 | 1 | |
1.5 | 1 | 1 | 1 | 1 | 0 |
2.0 | 1 | 1 | 1 | 1 | 0 |
This table isolates the three real roots and shows that they lie in the intervals , , and . If desired, the intervals in which the roots fall could be further reduced.
The Sturm functions satisfy the following conditions:
See also Descartes' Sign Rule, Sturm Chain, Sturm Theorem
References
Acton, F. S. Numerical Methods That Work, 2nd printing. Washington, DC: Math. Assoc. Amer., p. 334, 1990.
Dörrie, H. ``Sturm's Problem of the Number of Roots.'' §24 in
100 Great Problems of Elementary Mathematics: Their History and Solutions. New York: Dover, pp. 112-116, 1965.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T.
Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England:
Cambridge University Press, p. 469, 1992.
Rusin, D. ``Known Math.'' http://www.math.niu.edu./~rusin/known-math/polynomials/sturm.
Sturm, C. ``Mémoire sur la résolution des équations numériques.'' Bull. des sciences de Férussac 11, 1929.
© 1996-9 Eric W. Weisstein