## Sturm Function

Given a function , write and define the Sturm functions by (1)

where is a polynomial quotient. Then construct the following chain of Sturm functions,         (2)    known as a Sturm Chain. The chain is terminated when a constant is obtained.

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)

The following table shows the signs of and the number of sign changes obtained for points separated by .        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:

1. Two neighboring functions do not vanish simultaneously at any point in the interval.

2. At a null point of a Sturm function, its two neighboring functions are of different signs.

3. Within a sufficiently small Area surrounding a zero point of , is everywhere greater than zero or everywhere smaller than zero.

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.