roots(RingElement) -- compute the roots of a polynomial

Synopsis

Function: roots
Usage:

roots p
Inputs:
- p, a ring element, a univariate polynomial over ZZ, QQ, RR or CC.
Optional inputs:
- Precision => ..., default value -1, the number of precision bits used to compute the roots.The default precision is 53 bits for polynomials over ZZ or QQ and the same as the coefficient ring for RR[x] or CC[x].
- Unique => a Boolean value, default value false, whether to return multiple roots one or multiple times.
Outputs:
- a list, The roots of p each one represented as an elements of CC.

Description

i1 : RR_100[x]

o1 = RR   [x]
       100

o1 : PolynomialRing

i2 : p = x^13 + 5*x^9 + 7*x^4 + x +1

      13     9     4
o2 = x   + 5x  + 7x  + x + 1

o2 : RR   [x]
       100

i3 : roots p

o3 = {-.436092+.332356*ii, -.436092-.332356*ii, .425684-.538177*ii,
     ------------------------------------------------------------------------
     .425684+.538177*ii, -.285748-1.02908*ii, .937486-.552608*ii,
     ------------------------------------------------------------------------
     .937486+.552608*ii, -.285748+1.02908*ii, -1.02681, .98591+1.08117*ii,
     ------------------------------------------------------------------------
     -1.11384+1.06693*ii, -1.11384-1.06693*ii, .98591-1.08117*ii}

o3 : List

i4 : o3#0

o4 = -.436092315961826808232340406742+.332355643931146767051821089403*ii

o4 : CC (of precision 100)

i5 : ZZ[x]

o5 = ZZ[x]

o5 : PolynomialRing

i6 : p = x^13 + 5*x^9 + 7*x^4 + x +1

      13     9     4
o6 = x   + 5x  + 7x  + x + 1

o6 : ZZ[x]

i7 : roots(p^2, Precision=>150, Unique=>true)

o7 = {.425684+.538177*ii, -.436092+.332356*ii, -.436092-.332356*ii,
     ------------------------------------------------------------------------
     .425684-.538177*ii, -1.02681, -.436092-.332356*ii, -.285748-1.02908*ii,
     ------------------------------------------------------------------------
     .425684-.538177*ii, .937486-.552608*ii, .937486+.552608*ii,
     ------------------------------------------------------------------------
     .425684+.538177*ii, -.285748+1.02908*ii, -.436092+.332356*ii,
     ------------------------------------------------------------------------
     .98591-1.08117*ii, .937486-.552608*ii, .937486+.552608*ii,
     ------------------------------------------------------------------------
     .98591+1.08117*ii, -.285748+1.02908*ii, -1.11384+1.06693*ii, -1.02681,
     ------------------------------------------------------------------------
     -1.11384-1.06693*ii, -.285748-1.02908*ii, .98591+1.08117*ii,
     ------------------------------------------------------------------------
     -1.11384+1.06693*ii, -1.11384-1.06693*ii, .98591-1.08117*ii}

o7 : List

i8 : o7#0

o8 = .425684046781973469631387293327682995356019688
     +.538176972606594386584638016363221699619680744*ii

o8 : CC (of precision 150)

The roots are computed using MPSolve.

Ways to use this method: