1. Vincent’s Theorem of 1836: Overview and Recent Developments
Alkiviadis G. Akritas
Department of Computer & Communication Engineering
University of Thessaly
Volos, Greece

2. Outline of the talk:
We will present Vincent’s theorem and review the various real root isolation methods derived from it (2 bisection methods and 1 Continued Fractions method).
We will concentrate on the Continued Fractions method (the fastest of them all) and show how it was recently speeded up by 40% over its initial implementation.
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3. Descartes’ Rule of Signs --- in the open interval ]0, +∞[
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4. Special Cases of Descartes’ Rule of Signs
0 sign variations positive roots
1 sign variation positive root
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5. Vincent’s Theorem (1836) – the Continued Fractions version
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6. Vincent’s Theorem …
was kept “alive” by Uspensky --- in his book “Theory of Equations” (1949), and
was rediscovered by Akritas in 1976; was the subject of his Ph.D. Thesis.
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7. Historical Note on Uspensky (1883-1947):
Uspensky never read Vincent’s original paper of Instead,
Uspesky read Vincent’s theorem (only) in a Russian translation of Serret’s (French) book on Algebra (1866).
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8. Vincent’s Theorem – the Bisection version by Alesina & Galuzzi (2000)
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9. Alesina-Galuzzi’s proof of Vincent’s theorem …
is the most recent one; it uses Obreschkoff’s theorem of which gives the necessary condition for a polynomial with one positive root to have one sign variation!
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10. Obreschkoff’s Cone or Sector Theorem (V=1)
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11. Obreschkoff’s Cone & Circles in picture …
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12. Real Root Isolation Using Vincent’s Theorem
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13. Different Ways to Isolate the Real Roots:
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14. Why bother with these and not use Numeric Methods?
Numeric Methods CANNOT isolate just the positive roots! They isolate ALL roots (real and complex).
They can give wrong answers as the following example demonstrates.
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15. Consider the polynomial: 10999(x-1)50 – 1 (2 positive roots ≠1)
Numeric Method using 1010 digits takes 56 ms and fails (finds all 50 roots =1)!
Numeric Method using 1020 digits successfully isolates the roots but takes ms!
The VAS Continued Fractions method, discussed below, takes 4 ms!
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16. Therefore …
We DO need the real root isolation methods derived from Vincent’s Theorem of 1836.
Especially so since the one developed by Vincent in 1836 is exponential!
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17. Vincent’s Exponential Method of 1836 …
… uses continued fractions and is described in his original paper of 1836!, and
… was erroneously and unintentionally attributed by Uspensky to himself (1947).
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18. Methods derived from Vincent’s Theorem …
… were developed to overcome the exponential behavior of Vincent’s method.
We begin with the bisection methods
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19. The Bisection Methods Derived from Vincent’s Theorem differ in:
the termination test (criterion) used, and
the open interval, ]a, b[, they bisect.
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20. The first bisection method, VCA, …
was developed in 1976,
uses Uspensky’s termination test, and
bisects the open interval ]0, 1[.
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21. The termination test used in VCA is named after Uspensky because …
Uspensky was the one to use it as a test, in an isolation method that he erroneously attributed to himself;
Uspensky developed the test since he was not aware of Budan’s theorem of 1807 (eclipsed by Fourier’s theorem).
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22. Budan’s theorem – to be found in Vincent’s paper of 1836 (and my work).
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23. Vincent vs Uspensky
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24. Uspensky’s Termination Test, for the interval ]a, b[ = ]0,1[
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25. VCA method: The 1st Bisection Method Derived from Vincent’s Theorem (1976)
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26. VCA is implemented in maple and its other names were:
“Modified Uspensky’s method” ( )
1986 : “There is no Uspensky’s method”
“Descartes’ method” OR “Collins-Akritas” ( )
2007 : “There is no Descartes’ method”
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27. Method’s correct name:
Since it does NOT use Descrates’ test
Methode de Vincent-Collins-Akritas
Coined by Francois Boulier, University of Lille, France, 2007
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28. What’s in a name?
It makes clear the existing relation with Vincent’s theorem.
Can use the results by Alesina & Galuzzi (2000) to estimate the computing time (termination) of all methods derived from Vincent’s theorem.
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29. The Computing Time of the VCA Bisection Method …
is O( n4 τ2 ), where n is the degree of the polynomial, and τ bounds the coefficient bitsize.
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30. … and …
the fastest implementation of the VCA method was developed by Rouillier & Zimmermann in 2004 (REL)!
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31. The second bisection method …
… uses Vincent’s termination test, and
… bisects the interval ]a, b[ = ]0, ub[.
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32. Vincent’s Termination Test, for the interval ]a, b[ = ]0, ub[
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33. The 2nd Bisection Method Derived from Vincent’s Theorem (A+G:2000)
ACA 2007, Oakland University

34. Comparison of the Two Bisection Methods
VCA, the method using the simpler termination test (Uspensky’s) is faster than the one using Vincent’s (more complex) termination test!
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35. The Continued Fractions Method Derived from Vincent’s Theorem (1978) …
… uses Descartes’ Rule of Signs as the termination test, and
… relies, heavily, on the (REPEATED) estimation of lower bounds on the values of the positive roots of polynomials.
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36. VAS – Continued Fractions Method (VAS-CF)
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37. Comment on the VAS-CF Real Root Isolation Method:
Without steps 4 and 5 it is simply Vincent’s original exponential method.
Strzebonski’s contribution is ommitted for simplicity.
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38. VAS-CF is implemented in Mma. Over the past 30 years …
it has been using Cauchy’s bound on the values of the positive roots,
it has been several thousand times faster than the VCA bisection method (for random polys) --- even up to times faster than VCA, for Mignotte polys (and nobody believed it until 2005!),
only in the case of very many (>50) very large roots (~ 10100) has it been up to 4 times slower than VCA.
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39. Computing Time of the VAS Continued Fractions Method
Using a plausible hypothesis and the fast translation algorithm by von zu Gathen, its computing time is O( n4 τ2 ), where n is the degree of the polynomial, and τ bounds the coefficient bitsize. (Akritas 1978, Tsigaridas-Emiris 2005)
Without any hypotheses its computing time is O( n8 τ3 ) (Sharma 2007). HOWEVER, this bound does NOT match its performance.
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40. To improve the performance of VAS-CF even further …
… new bounds on the values of the positive roots of polynomials were needed.
To understand the nature of these bounds we used Doru Stefanescu’s inspirational work!
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41. Stefanescu’s Theorem (2005) : it matches +ive with –ive coefficients when the # of sign variations is even
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42. Our Theorem (2006): it breakes up & matches +ive with –ive coefficients, (a).
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43. Our Theorem (2006): it breakes up & matches +ive with –ive coefficients , (b).
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44. On our Theorem (Akritas, Strzebonski and Vigklas, 2006):
It is a general theorem from which ALL methods for computing positive bounds on the values of positive roots are derived!
This generality is achieved by breaking up and pairing (unmatched) positive coefficients with negative ones in various ways!
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45. For the estimation of the bounds …
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46. Bounds in the Literature are, in general, of Linear Complexity!
That is, EACH negative coefficient of the polynomial is paired with ONE of the preceding (unmatched) positive coefficients and the maximum of all the computed values is taken as the estimate of the bound.
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47. Linear Complexity Bounds discussed:
We present four of them.
Of those four bounds, the last two were developed by Akritas, Strzebonski and Vigklas in 2007.
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48. 1st Linear Complexity Bound Derived from Our Theorem
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49. 2nd Linear Complexity Bound Derived from Our Theorem
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50. 3rd Linear Complexity Bound Derived from Our Theorem
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51. 4th Linear Complexity Bound Derived from Our Theorem
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52. Problems with a single method of computing bounds
There does NOT exist one which ALWAYS works better.
Therefore, to compute the bound use two methods & pick the minimum
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53. Example: the single positive root of the poly = 1.
x x2 – x - 1
The bound computed by the first three methods is UNACCEPTABLE (>= 1050)!
However, with LM, we pick the max from the coefficient pairs
{10100 x2 / 2, x} and
{{10100 x2 / 22, -1}
and obtain : 2
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54. Min(FL, LM), the Best Linear Complexity bound!
Experimental results showed that min( FL, LM) gives the best estimates.
Using min( FL, LM) we obtain a 15% average overall speed-up over the VAS-CF implementation using Cauchy’s rule.
VAS-CF using min( FL, LM) is ALWAYS faster than the VCA-bisection method!
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55. To Further Improve the Performance of VAS-CF …
… we decided to use quadratic complexity bounds because their improved estimates SHOULD compensate for the extra time needed to compute them.
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56. GI: The General Idea of the Quadratic Complexity Bounds:
EACH negative coefficient of the polynomial is paired with ALL the preceding positive coefficients and the minimum of the computed values is taken;
the maximum of all those minimums is taken as the estimate of the bound.
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57. Quadratic Complexity Bounds discussed:
We will present four Quadratic Complexity Bounds derived by applying the general idea (GI) to the four linear complexity bounds we discussed before.
Three out of those four bounds, were developed by Akritas, Argyris, Strzebonski and Vigklas in 2008.
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58. 1st Quadratic Complexity Bound Derived from Our Theorem
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59. 2nd Quadratic Complexity Bound Derived from Our Theorem (developed by Hong)
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60. 3rd Quadratic Complexity Bound Derived from Our Theorem, (a)
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61. 3rd Quadratic Complexity Bound Derived from Our Theorem, (b)
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62. FLQ is the Fastest Quadratic Complexity method.
EACH negative coefficient of the polynomial is paired only with one of the first λ preceding positive coefficients and the minimum is taken.
Each of the first λ preceding positive coefficients is used once.
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63. 4th Quadratic Complexity Bound Derived from Our Theorem
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64. Revisited Example: the single positive root of the poly = 1.
x x2 – x - 1
The bound computed by three quadratic complexity bounds is 2!
However, with FLQ we pick the Max of the
Min{{x3, x},
{10100 x2, x}}=1, &
Min{{x3, -1} , {10100 x2, -1}}=1
to obtain : 1
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65. Linear vs Quadratic Complexity Bounds
The quadratic complexity bounds CANNOT perform worse than the linear complexity ones; most of the times they perform (a lot) better!
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66. LMQ, the best quadratic complexity bound!
LMQ always gives a better (<=) estimate than KQ.
Experimental results indicate that FLQ, LMQ and min( FLQ, LMQ ) behave equally well!
Therefore, pick LMQ.
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67. VAS-CF using Linear and Quadratic Complexity Bounds: a
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68. VAS-CF using Linear and Quadratic Complexity Bounds: b
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69. Conclusions:
Linear Complexity Bounds: VAS-CF using min ( FL, LM ), the best linear bound, was speeded up 15% over VAS-CF using Cauchy (AND became always faster than VCA).
Quadratic Complexity Bounds: VAS-CF using LMQ, one of the three best quadratic bounds, was speeded up 40% over VAS-CF using Cauchy.
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70. References I
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71. References II
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72. References III
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73. References IV
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