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rootbound

compute bound on complex roots of a polynomial Calling Sequence rootbound(p, x) Parameters

 p - polynomial in x with numeric coefficients x - name Description

 • Returns a positive integer N such that $|r| for all complex roots r of p.  In general, this bound is better than Cauchy's bound of
 $⌈1+\frac{\mathrm{maxnorm}\left(p\right)}{\left|\mathrm{lcoeff}\left(p\right)\right|}⌉$. Examples

 > $p≔x↦{x}^{4}-10\cdot {x}^{2}+1:$
 > $\mathrm{rootbound}\left(p\left(x\right),x\right)$
 ${4}$ (1)
 > $\mathrm{ceil}\left(1+\frac{\mathrm{maxnorm}\left(p\left(x\right)\right)}{\mathrm{abs}\left(\mathrm{lcoeff}\left(p\left(x\right)\right)\right)}\right)$
 ${11}$ (2)
 > $\mathrm{fsolve}\left(p\left(x\right)=0,x\right)$
 ${-3.146264370}{,}{-0.3178372452}{,}{0.3178372452}{,}{3.146264370}$ (3)
 > $q≔x↦\left(x-3.2\right)\cdot \left(x-2.5\right)\cdot \left(x-0.3\right)\cdot \left(x+1.3\right)\cdot \left(x+2.5\right)\cdot \left(x+3.6\right):$
 > $\mathrm{rootbound}\left(q\left(x\right),x\right)$
 ${6}$ (4)
 > $\mathrm{ceil}\left(1+\frac{\mathrm{maxnorm}\left(\mathrm{expand}\left(q\left(x\right)\right)\right)}{\mathrm{abs}\left(\mathrm{lcoeff}\left(q\left(x\right)\right)\right)}\right)$
 ${78}$ (5) References

 Monagan, M.B. "A Heuristic Irreducibility Test for Univariate Polynomials." J. of Symbolic Comp. Vol. 13 No. 1. Academic Press, (1992): 47-57.