realRootIsolation -- a list that isolates the real roots of a rational univariate polynomial

Synopsis

• Usage:

  realRootIsolation(f,r)

• Inputs:
  • f, a ring element, a rational univariate polynomial
  • r, a real number, a positive rational number
• Outputs:
  • a list, of intervals that contain all the real roots of f

Description

This method uses a Sturm sequence and a bisection method to isolate real solutions of f in intervals of length at most r.

i1 : R = QQ[t]

o1 = R

o1 : PolynomialRing

i2 : f = 45 - 39*t - 34*t^2 + 38*t^3 - 11*t^4 + t^5

      5      4      3      2
o2 = t  - 11t  + 38t  - 34t  - 39t + 45

o2 : R

i3 : realRootIsolation(f,1/2)

         3        1       5       9
o3 = {{- -, -1}, {-, 1}, {-, 3}, {-, 5}}
         2        2       2       2

o3 : List

i4 : realRootIsolation(f,.2)

         9        7       23       39
o4 = {{- -, -1}, {-, 1}, {--, 3}, {--, 5}}
         8        8        8        8

o4 : List

See also

• SturmSequence -- the Sturm sequence of a rational univariate polynomial