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polylib::realroots -- isolate all real roots of a real univariate polynomial

Introduction

polylib::realroots(p) returns intervals isolating the real roots of the real univariate polynomial p.

polylib::realroots(p, eps) returns refined intervals approximating the real roots of p to the relative precision given by eps.

Call(s)

polylib::realroots(p)
polylib::realroots(p, eps)

Parameters

p - a univariate polynomial: either an expression or a polyomial of domain type DOM_POLY.
eps - a (small) positive real number determining the size of the returned intervals.

Returns

A list of lists [[a1,b1], [a2,b2], ...] with rational numbers a.i <= b.i is returned. Lists with a.i=b.i represent exact rational roots. Lists with a.i<b.i represent open intervals containing exactly one real root. If the polynomial has no real roots, then the empty list [ ] is returned.

Side Effects

The function is sensitive to the environment variable DIGITS, if there are non-integer or non-rational coefficients in the polynomial. Any such coefficient is replaced by a rational number approximating the coefficient to DIGITS significant decimal places.

Related Functions

numeric::fsolve, numeric::polyroots, numeric::realroot, numeric::realroots

Details

Example 1

We use a polynomial expression as input to polylib::realroots:

>> p := (x - 1/3)*(x - 1)*(x - 4/3)*(x - 2)*(x - 17):
>> polylib::realroots(p)
                [[0, 1], [1, 1], [1, 2], [2, 2], [16, 32]]

The roots 1 and 2 are found exactly: the corresponding intervals have length 0. The other isolating intervals are quite large. We refine the intervals such that they approximate the roots to 12 decimal places. Note that this is independent of the current value of DIGITS, because no floating point arithmetic is used:

>> polylib::realroots(p, 10^(-12))
      [[1466015503701/4398046511104, 733007751851/2199023255552],
      
         [1, 1], [1466015503701/1099511627776,
      
         733007751851/549755813888], [2, 2], [17, 17]]

We convert these exact bounds for the real roots to floating point approximations. Note that with the default value of DIGITS=10 we ignore 2 of the 12 correct digits the rational bounds could potentially give:

>> map(%, map, float)
      [[0.3333333333, 0.3333333333], [1.0, 1.0],
      
         [1.333333333, 1.333333333], [2.0, 2.0], [17.0, 17.0]]
>> delete p:

Example 2

Orthogonal polynomials of degree n have n simple real roots. We consider the Legendre polynomial of degree 5, available in the library orthpoly for orthogonal polynomials:

>> polylib::realroots(orthpoly::legendre(5, x), 10^(-DIGITS)):
>> map(%, float@op, 1)
      [-0.9061798459, -0.5384693101, 0.0, 0.5384693101, 0.9061798459]

Example 3

We consider a polynomial with a multiple root:

>> p := poly((x - 1/3)^3*(x - 1), [x])
                    4      3        2
              poly(x  - 2 x  + 4/3 x  - 10/27 x + 1/27, [x])

Note that only one isolating interval [0, 1] is returned for the triple root 1/3:

>> polylib::realroots(p)
                             [[0, 1], [1, 1]]
>> delete p:

Example 4

We consider a polynomial with non-rational roots:

>> p := (x - 3)^2*(x - PI)^2:

Converting the result of polylib::realroots to floating point numbers one sees that the exact roots 3, 3, PI, PI are approximated only to 3 decimal places:

>> map(polylib::realroots(p, 10^(-10)), map, float)
      [[2.998807805, 2.998807805], [3.001213582, 3.001213583],
      
         [3.140323518, 3.140323519], [3.142840401, 3.142840401]]

This is caused by the internal rationalization of the coefficients of p. Information on the rationalized polynomial may be optained by a builtin userinfo command:

>> setuserinfo(polylib::realroots, 1): 
>> polylib::realroots(p, 10^(-10))
      Info: polynomial rationalized to poly(x^4 - 12283185307/1000..)
      ...

The intervals returned by polylib::realroots(p, 10^(-10)) correctly locate the 4 exact roots of this rationalized polynomial to a precision of 10 digits. However, because all 4 roots are close, the small perturbations of the coefficients introduced by rationalization have a drastic effect on the location of the roots. In particular, rationalization splits the two original double roots into 4 simple roots.

>> setuserinfo(polylib::realroots, 0): delete p:

Example 5

We consider a further example involving non-exact coefficients. First we approximate the roots of a polynomial with exact coefficients:

>> p1 := (x - 1/3)^3*(x - 4/3):
>> map(polylib::realroots(p1, 10^(-10)), map, float)
        [[0.3333333333, 0.3333333333], [1.333333333, 1.333333333]]

Now we introduce roundoff errors by replacing one entry by a floating point approximation:

>> p2 := (x - 1.0/3)^3*(x - 4/3):
>> map(polylib::realroots(p2, 10^(-10)),map,float)
        [[0.3332481323, 0.3332481323], [1.333333333, 1.333333333]]

In this example rationalization caused the triple root 1/3 to split into one real root and two complex conjugate roots.

>> delete p1, p2:

Example 6

We want to approximate roots to a precision of 1000 digits:

>> p := x^5 - 129/20*x^4 + 69/5*x^3 - 14*x^2 + 12*x - 8:

We recommend not to obtain the result directly by polylib::realroots(p,10^(-1000)), because the internal bisectioning process for refining crude isolating intervals converges only linearly. Instead, we compute first approximations of the roots to a precision of 10 digits:

>> approx := map(polylib::realroots(p, 10^(-10)), float@op, 1)
                  [1.489177599, 1.752191733, 3.255184556]

These values are used as starting points for a numerical root finder. The internal Newton search in numeric::fsolve converges quadratically and yields the high precision results much faster than polylib::realroots:

>> DIGITS := 1000: Roots := []:
>> for x0 in approx do
       Roots := append(Roots, numeric::fsolve([p = 0], [x = x0]));
   end_for
      [[x = 1.489177598846870281338916114673844643894...],
      
          [x = 1.752191733304413195335101727880090131407...],
      
          [x = 3.255184555797733438479691333705558491124...]]
>> delete approx, Digits, Roots, x0:

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