sturm - Maple Programming Help

sturm

number of real roots of a polynomial in an interval

sturmseq

Sturm sequence of a polynomial

 Calling Sequence sturmseq(p, x) sturm(s, x, a, b)

Parameters

 p - polynomial in x with rational or float coefficients x - variable in polynomial p a, b - rationals or floats such that $a\le b$; a can be $-\mathrm{\infty }$ and b can be $\mathrm{\infty }$ s - Sturm sequence for polynomial p

Description

 • The procedure sturmseq computes a Sturm sequence for the polynomial p in x.  It returns the Sturm sequence as a list of polynomials and replaces multiple roots with single roots. It uses the procedures sturmrem and sturmquo instead of quo and rem to avoid the fuzzy zero.  This can be reinstated by defining

$\mathrm{sturmrem}≔\mathrm{rem}$

$\mathrm{sturmquo}≔\mathrm{quo}$

 • The procedure sturm uses Sturm's theorem to return the number of real roots in the interval (a,b] of polynomial p in x. The first argument to sturm should be a Sturm sequence for p.  This may be computed by sturmseq.
 Note: The interval excludes the lower endpoint a and includes the upper endpoint b (unless it is $\mathrm{\infty }$).

Examples

 > $s≔\mathrm{sturmseq}\left(\mathrm{expand}\left(\left(x-1\right)\left(x-2\right)\left(x-3\right)\right),x\right)$
 ${s}{≔}\left[{{x}}^{{3}}{-}{6}{}{{x}}^{{2}}{+}{11}{}{x}{-}{6}{,}{{x}}^{{2}}{-}{4}{}{x}{+}\frac{{11}}{{3}}{,}{x}{-}{2}{,}{1}\right]$ (1)
 > $\mathrm{sturm}\left(s,x,\frac{3}{2},4\right)$
 ${2}$ (2)
 > $\mathrm{sturm}\left(s,x,1,2\right)$
 ${1}$ (3)
 > $\mathrm{sturm}\left(s,x,-\mathrm{∞},\mathrm{∞}\right)$
 ${3}$ (4)