Sturm Function
Given a function 
, write 
and define the Sturm functions by
 | (1) |
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 coefficientsover a given interval. Specifically, the difference in the number of sign changes between the Sturm functions evaluated at twopoints 
and 
gives the number of real roots in the interval 
. This powerful result is known as the SturmTheorem.
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). TheDerivative 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 thanzero 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.
- Euler's Rule
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- Euler's Sum of Powers Conjecture
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- Eventually Periodic
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