“Sturm Function”的概念、定义、习题解题方法及翻译-数学辞典

单词

Sturm Function

释义

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 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.

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.

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