Sturm's theorem

In mathematics, the Sturm sequence of a univariate polynomial $p$ is a sequence of polynomials associated with $p$ and its derivative by a variant of Euclid's algorithm for polynomials. Sturm's theorem expresses the number of distinct real roots of $p$ located in an interval in terms of the number of changes of signs of the values of the Sturm's sequence at the bounds of the interval. Applied to the interval of all the real numbers, it gives the total number of real roots of $p$ .

Whereas the fundamental theorem of algebra readily yields the overall number of complex roots, counted with multiplicity, it does not provide a procedure for calculating them. Sturm's theorem counts the number of distinct real roots and locates them in intervals. By subdividing the intervals containing some roots, it can isolate the roots into arbitrary small intervals, each containing exactly one root. This yields an arbitrary-precision numeric root finding algorithm for univariate polynomials.

Sturm's sequence and Sturm's theorems are named after Jacques Charles François Sturm.

Sturm chains

A Sturm chain or Sturm sequence is a finite sequence of polynomials

p_{0},p_{1},\dots ,p_{m}

of decreasing degree with these following properties:

$p 0 = p$ is square free (no square factors, i.e., no repeated roots);
if $p (ξ) = 0$ , then $sign(p 1 (ξ)) = sign(p' (ξ))$ ;
if $p i (ξ) = 0$ for $0 < i < m$ then $sign(p i -1 (ξ)) = -sign(p i +1 (ξ))$ ;
$p m$ does not change its sign.

Sturm's sequence is a modification of Fourier's sequence.

To obtain a Sturm chain, Sturm himself proposed to choose the intermediary results when applying Euclid's algorithm to $p$ and its derivative:

{\begin{aligned}p_{0}(x)&:=p(x),\\p_{1}(x)&:=p'(x),\\p_{2}(x)&:=-{{\rm {rem}}}(p_{0},p_{1})=p_{1}(x)q_{0}(x)-p_{0}(x),\\p_{3}(x)&:=-{{\rm {rem}}}(p_{1},p_{2})=p_{2}(x)q_{1}(x)-p_{1}(x),\\&{}\ \ \vdots \\0&=-{\text{rem}}(p_{{m-1}},p_{m}),\end{aligned}}

where $rem(p i, p j)$ and $q i$ are the remainder and the quotient of the polynomial long division of $p i$ by $p j$ , and where $m$ is the minimal number of polynomial divisions (never greater than $deg(p)$ ) needed to obtain a zero remainder. That is, successively take the remainders with polynomial division and change their signs. Since $deg(p i +1) < deg(p i)$ for $0 \leq i < m$ , the algorithm terminates. The final polynomial, $p m$ , is the greatest common divisor of $p$ and its derivative. If $p$ is square free, it shares no roots with its derivative, hence $p m$ will be a non-zero constant polynomial. The Sturm chain, called the canonical Sturm chain, then is

p_{0},p_{1},p_{2},\ldots ,p_{m}.

If $p$ is not square-free, this sequence does not formally satisfy the definition of a Sturm chain above; nevertheless it still satisfies the conclusion of Sturm's theorem (below).

Statement

Let $p 0, ..., p m$ be the Sturm chain of a square-free polynomial $p$ , and let $σ (ξ)$ denote the number of sign changes (ignoring zeroes) in the sequence

p_{0}(\xi ),p_{1}(\xi ),p_{2}(\xi ),\ldots ,p_{m}(\xi ).

Sturm's theorem then states that for two real numbers $a < b$ , the number of distinct roots of $p$ in the half-open interval $(a, b]$ is $σ (a) - σ (b)$ .

The non-square-free case

Let $p 0, ..., p m$ be the canonical Sturm sequence of a polynomial $p$ , not necessarily square-free. Then $σ (a) - σ (b)$ is the number of distinct roots of $p$ in $(a, b]$ whenever $a < b$ are real numbers such that neither $a$ nor $b$ is a multiple root of $p$ .

Example

Suppose we wish to find the number of roots in some range for the polynomial $p(x)=x^{4}+x^{3}-x-1$ . So

{\begin{aligned}p_{0}(x)&=p(x)=x^{4}+x^{3}-x-1\\p_{1}(x)&=p'(x)=4x^{3}+3x^{2}-1\end{aligned}}

Using polynomial long division to divide $p 0$ by $p 1$ gives the remainder $-{\tfrac {3}{16}}x^{2}-{\tfrac {3}{4}}x-{\tfrac {15}{16}}$ , and upon multiplying this remainder by $-1$ we obtain $p_{2}(x)={\tfrac {3}{16}}x^{2}+{\tfrac {3}{4}}x+{\tfrac {15}{16}}$ . Next dividing $p 1$ by $p 2$ and multiplying the remainder by $-1$ , we obtain $p_{3}(x)=-32x-64$ . And dividing $p 2$ by $p 3$ and multiplying the remainder by $-1$ , we obtain $p_{4}(x)=-{\tfrac {3}{16}}$ .

Thus the complete chain of Sturm polynomials is:

p_{0}(x)=x^{4}+x^{3}-x-1

p_{1}(x)=4x^{3}+3x^{2}-1

p_{2}(x)={\tfrac {3}{16}}x^{2}+{\tfrac {3}{4}}x+{\tfrac {15}{16}}

p_{3}(x)=-32x-64

p_{4}(x)=-{\tfrac {3}{16}}

To find the number of roots between $-\infty$ and $\infty$ , first evaluate $p 0, p 1, p 2, p 3,$ and $p 4$ at $-\infty$ and note the sequence of signs of the results: $+ - + + -$ , which contains three sign changes ( $+$ to $-$ , then $-$ to $+$ , then $+$ to $-$ ). The same procedure for $\infty$ gives the sign sequence $+ + + - -$ , which contains just one sign change. Hence the number of roots of the original polynomial between $-\infty$ and $\infty$ is $3 - 1 = 2.$ That this is correct can be seen by noting that $p (x) = x 4 + x 3 - x - 1$ can be factored as $(x 2 - 1)(x 2 + x + 1)$ , where it is readily verifiable that $x 2 - 1$ has the two roots $-1$ and $1$ while $x 2 + x + 1$ has no real roots. In more complicated examples in which there is no advance knowledge of the roots because factoring is either impossible or impractical, one can experiment with various finite bounds for the range to be considered, thus narrowing down the locations of the roots.

Proof

Polynomials are continuous functions, and any sign change must occur at a root, so consider the behavior of a Sturm chain around the roots of its constituent polynomials.

First, note that two adjacent polynomials can share a common root only when it is a multiple root of $p$ (in which case it is a root of every $p i$ ). Indeed, if $p i (ξ) = p i -1 (ξ) = 0$ , then $p i +1 (ξ) = 0$ , since $sign(p i -1 (ξ)) = -sign(p i +1 (ξ))$ . The zero then propagates recursively up and down the chain, so that $ξ$ is a root of all the polynomials $p 0, ..., p m$ .

Next, consider roots of polynomials in the interior (i.e., $0 < i < m$ ) of the Sturm chain that are not multiple roots of $p$ . If $p i (ξ) = 0$ , then from the previous paragraph it is true that $p i -1 (ξ) \neq 0$ and $p i +1 (ξ) \neq 0$ . Furthermore, $sign(p i -1 (ξ)) = -sign(p i +1 (ξ))$ . Since $p i -1$ and $p i +1$ are continuous, $sign(p i +1 (x)) = -sign(p i -1 (x))$ for every $x$ in the vicinity of $ξ$ . Similarly, the sign of $p i (x)$ is constant before and after $ξ$ , and changes as $x$ is crossing $ξ$ . Thus, whenever $x$ is crossing $ξ$ , moving from left to right, the part $p i -1, p i, p i +1$ of the Sturm chain loses a sign change at one side, and acquires a new sign change at the other side. Consequently, the total number of sign changes is never affected by the polynomial variations in the interior of the chain, and only roots of the original polynomial, at the top of the chain, can affect the total number of sign changes.

Consider a root $ξ$ , so $p (ξ)$ , and assume first that it is a simple root. Then the derivative of $p$ , which is $p 1$ , must be non-zero at $ξ$ , so $p$ must be either increasing or decreasing at $ξ$ . If it's increasing, then its sign is changing from negative to positive when moving from left to right while its derivative (again, $p 1$ ) is positive, so the total number of sign changes decreases by one. Conversely, if it's decreasing, then its sign changes from positive to negative while its derivative is negative, so again the total number of sign changes decreases by one.

Finally, let $ξ$ be a multiple root of $p$ , and let $p 0, ..., p m$ be the canonical Sturm chain. Let d = gcd(p,p′), $q 0$ = p/d and $q 1$ = p′/d, and let $q 0, ..., q m$ be the Sturm chain given by dividing the $p i$ by d. Then $σ (x)$ is the same for both chains whenever $x$ is not a root of $d$ . And the number of sign changes (in either chain) around $ξ$ decreases by one, since $ξ$ is a simple root of $q 0$ and is not a root of $q 1$ .

In summary, only sign changes at roots of the original polynomial affect the total number of sign changes across the chain, and the total number of sign changes always decreases by one as roots are passed. The theorem follows directly.

History and related methods

For counting and isolating the real roots, other methods are usually preferred, because they are computationally more efficient; these methods all use Descartes' rule of signs (just like Fourier^[1] did back in 1820) and Vincent's theorem. The very first one of those methods was initially called "modified Uspensky's algorithm" by its inventors,^[2] but it was later shown that there is no Uspensky's method;^[3] afterwards, people started calling it either "Collins-Akritas method"^[4] or "Descartes' method" only to be shown that there is no Descartes' method^[4] either. Finally, François Boulier, of the University of Lille,^[5] p. 24, gave it the name "Vincent-Collins-Akritas" (VCA for short) to also give credit to Vincent. VCA is a bisection method; there exists also a continued fractions method based on Vincent's theorem namely, the Vincent-Akritas-Strzeboński (VAS) method.^[6]

VAS is based on Budan's theorem whereas Sturm's method was inspired by Fourier's theorem. In fact Sturm himself,^[7] p. 108, acknowledges the great influence Fourier's theorem had on him: « C'est en m'appuyant sur les principes qu'il a posés, et en imitant ses démonstrations, que j'ai trouvé les nouveaux théorèmes que je vais énoncer. » which translates to "It is by relying upon the principles he has laid out and by imitating his proofs that I have found the new theorems which I am about to announce."

Applications

Number of real roots

Sturm's theorem can be used to compute the total number of real roots of a polynomial.

This may be done by choosing $- a = b = M$ where $M$ is larger than the absolute value of every root. For example, a bound due to Cauchy says that all real roots of a polynomial with coefficients $a i$ are in the interval $[- M, M]$ , where

M=1+{\frac {\max _{{i=0}}^{{n-1}}|a_{i}|}{|a_{n}|}}.

Although theoretically the above approach is the simplest, in practice bounds on the positive (only) roots of polynomials are used and the positive roots are isolated and evaluated first; the negative roots are treated by first substituting $x$ by $-x$ , then compute a new (positive root) bound to isolate and evaluate the negative roots. Sturm's method and VCA need to compute only one bound to isolate the positive roots. In contrast, to isolate the positive roots VAS needs to compute various positive bounds, for the various polynomials that appear in the process.^[8]^[9]

Another method is computationally simpler. One can use the fact that for large $x$ , the sign of

p(x)=a_{n}x^{n}+\cdots

is $sign(a n)$ , whereas $sign(p (- x))$ is $(-1) n a n$ . In this way, simply counting the sign changes in the leading coefficients in the Sturm chain readily gives the number of distinct real roots of a polynomial.

Sturm's theorem allows also to determine the multiplicity of a given root, say $ξ$ . Indeed, suppose that $a < ξ < b$ , with $σ (a) - σ (b) = 1$ . Then, $ξ$ has multiplicity $k$ precisely when $ξ$ is a root with multiplicity $k - 1$ of $p m$ (since it is the GCD of $p$ and its derivative). Thus the multiplicity of $ξ$ may be computed by recursively applying Sturm's theorem to $p m$ . However, this method is rarely used, as square-free factorization is computationally more efficient for this purpose.

Quotient

The remainder is needed to compute the chain using Euclid's algorithm. For two polynomials

p(x)=\sum _{{k=0}}^{i}p_{k}x^{k},\qquad \widetilde {p}(x)=\sum _{{k=0}}^{{i-1}}\widetilde {p}_{k}x^{k},\quad \widetilde {p}_{{i-1}}\neq 0,

this is accomplished by

{\text{rem}}\left(p,\widetilde {p}\right)=\widetilde {p}_{{i-1}}\cdot p(x)-p_{i}\cdot \widetilde {p}(x)\left(x+{\frac {p_{{i-1}}}{p_{i}}}-{\frac {\widetilde {p}_{{i-2}}}{\widetilde {p}_{{i-1}}}}\right),

where the quotient is built solely of the first two leading coefficients.

Generalized Sturm chains

Let $ξ$ be in the compact interval $[a, b]$ . A generalized Sturm chain over $[a, b]$ is a finite sequence of real polynomials $(X 0, X 1, ..., X r)$ such that:

$X (a) X (b) \neq 0$
$sign(X r)$ is constant on $[a, b]$
If $X i (ξ) = 0$ for $1 \leq i \leq r - 1$ , then $X i -1 (ξ) X i +1 (ξ) < 0$ .

One can check that each Sturm chain is indeed a generalized Sturm chain.

References

↑ Fourier, Jean Baptiste Joseph (1820). "Sur l'usage du théorème de Descartes dans la recherche des limites des racines" (PDF). Bulletin des Sciences, par la Société Philomatique de Paris: 156–165.
↑ Collins, George E.; Alkiviadis G. Akritas (1976). Polynomial Real Root Isolation Using Descartes' Rule of Signs. SYMSAC '76, Proceedings of the third ACM symposium on Symbolic and algebraic computation. Yorktown Heights, NY, USA: ACM. pp. 272–275.
↑ Akritas, Alkiviadis G. (1986). There is no "Uspensky's Method". In: Proceedings of the fifth ACM Symposium on Symbolic and Algebraic Computation (SYMSAC '86, Waterloo, Ontario, Canada), pp. 88–90.
1 2 Akritas, Alkiviadis G. (2008). There is no "Descartes' method" (PDF). In: M.J.Wester and M. Beaudin (Eds), Computer Algebra in Education, AullonaPress, USA, pp. 19-35.
↑ Boulier, François (2010). Systèmes polynomiaux : que signifie " résoudre " ? (PDF). Université Lille 1.
↑ Akritas, Alkiviadis G.; A.W. Strzeboński; P.S. Vigklas (2008). "Improving the performance of the continued fractions method using new bounds of positive roots" (PDF). Nonlinear Analysis: Modelling and Control. 13: 265–279.
↑ Hourya, Benis-Sinaceur (1988). "Deux moments dans l'histoire du Théorème d'algèbre de Ch. F. Sturm". In: Revue d'histoire des sciences. 41 (2): 99–132.
↑ Vigklas, Panagiotis, S. (2010). Upper bounds on the values of the positive roots of polynomials (PDF). Ph.D. Thesis, University of Thessaly, Greece.
↑ Akritas, Alkiviadis, G. (2009). "Linear and Quadratic Complexity Bounds on the Values of the Positive Roots of Polynomials". Journal of Universal Computer Science. 15 (3): 523–537.

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External links

C code from Graphic Gems by D.G. Hook and P.R. McAree.

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