numSturm(RingElement,Number,Number)

Synopsis

• Function: numSturm

• Usage:

  numSturm(f,a,b)

• Inputs:
  • a ring element, f, a univariate polynomial
  • a number, a, a lower bound of the interval
  • a number, b, an upper bound of the interval
• Optional inputs:
  • Multiplicity => ..., default value false,
• Outputs:
  • an integer, the number of real roots of a univariate polynomial f not counting multiplicity in the interval (a,b]

Description

This computes the difference in variation of the Sturm sequence of f. If a and b are not specified, the interval will be taken from negative infinity to infinity.

i1 : R = QQ[t]

o1 = R

o1 : PolynomialRing

i2 : f = (t-5)*(t-3)^2*(t-1)*(t+1)

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

o2 : R

i3 : roots f

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

o3 : List

i4 : numSturm(f)

o4 = 4

i5 : numSturm(f,0,5)

o5 = 3

i6 : numSturm(f,-2,2)

o6 = 2

i7 : numSturm(f,-1,5)

o7 = 3

In the above example, multiplicity is not included so to include this we can make the multiplicity option true in the below example.

i8 : numSturm(f,Multiplicity=>true)

o8 = 5

i9 : numSturm(f,0,5,Multiplicity=>true)

o9 = 4

i10 : numSturm(f,0,3,Multiplicity=>true)

o10 = 3

If a is an InfiniteNumber, then the lower bound will be negative infinity and if b is an InfiniteNumber, then the upper bound is infinity.

i11 : numSturm(f,-infinity, 0)

o11 = 1

i12 : numSturm(f,0,infinity)

o12 = 3

i13 : numSturm(f,-infinity,infinity)

o13 = 4

See also

• SturmSequence