Sturm Function

Given a function $f(x)\equiv f_0(x)$, write $f_1\equiv f'(x)$ and define the Sturm functions by

$$f_n(x)=-\left\{{f_{n-2}(x)-f_{n-1}(x) \left[{f_{n-2}(x)\over f_{n-1}(x)}\right]}\right\},$$ (1)

where $[P(x)/Q(x)]$ is a polynomial quotient. Then construct the following chain of Sturm functions,

$$f_0 = q_0f_1-f_2$$

$$f_1 = q_1f_2-f_3$$

$$f_2 = q_2f_3-f_4$$ (2)

$$\vdots$$

$$f_{s-2} = q_{s-2}f_{s-1}-f_s,$$

known as a Sturm Chain. The chain is terminated when a constant $-f_s(x)$ 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 $x=a$ and $x=b$ gives the number of real roots in the interval $(a,b)$. This powerful result is known as the Sturm Theorem.

As a specific application of Sturm functions toward finding Polynomial Roots, consider the function $f_0(x)=x^5-3x-1$, plotted above, which has roots $-1.21465$, $-0.334734$, $0.0802951\pm 1.32836i$, and 1.38879 (three of which are real). The Derivative is given by $f'(x)=5x^4-3$, and the Sturm Chain is then given by

$$f_0 = x^5-3x-1$$ (3)

$$f_1 = 5x^4-3$$ (4)

$$f_2 = {\textstyle{1\over 5}}(12x+5)$$ (5)

$$f_3 = {\textstyle{59083\over 20736}}.$$ (6)

The following table shows the signs of $f_i$ and the number of sign changes $\Delta$ obtained for points separated by $\Delta x=2$.

$x$	$f_0$	$f_1$	$f_2$	$f_3$	$\Delta$
$-2$	$-1$	1	$-1$	1	3
0	$-1$	$-1$	1	1	1
2	1	1	1	1	0

This shows that $3-1=2$ real roots lie in $(-2,0)$, and $1-0=1$ real root lies in $(0,2)$. Reducing the spacing to $\Delta x=0.5$ gives the following table.

$x$	$f_0$	$f_1$	$f_2$	$f_3$	$\Delta$
$-2.0$	$-1$	1	$-1$	1	3
$-1.5$	$-1$	1	$-1$	1	3
$-1.0$	1	1	$-1$	1	2
$-0.5$	1	$-1$	$-1$	1	2
0.0	$-1$	$-1$	1	1	1
0.5	$-1$	$-1$	1	1	1
1.0	$-1$	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 $(-1.5,-1.0)$, $(-0.5,0.0)$, and $(1.0,1.5)$. 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 $f_0(x)$, $f_1(x)$ is everywhere greater than zero or everywhere smaller than zero.

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.