Algebraic Number and their Practical Applications

Ron Wein January 2002

Algebraic NumbersA real number x is algebraic if for some d there exist d+1 integers ad,…,a0 such that x is a root of the polynomial:

01)( axaxaxp dd

Two libraries support exact computations with algebraic numbers:LEDA – developed in Max Planck Institut, Saarbrücken, Gremany.

CORE – developed in Courant Institute, New-York, USA.

Separation BoundsFor any given expression E a separation bound is an easily computable function sep: R such that the value val(E) of any non-zero expression E is lower bounded by sep(E)

(that is |val(E)| < sep(E)).

Using the separation bound, we can compute an approximation app(E) such that the approximation error is guaranteed not to exceed : | app(E) - val(E) | <

• If |app(E)| > then sign(val(E)) = sign(app(E).• Otherwise, since |app(E)| – then |val(E)| < 2 : If 2 sep(E) – the whole expression equals 0. Otherwise, repeat the process process with /2.

Tests with a Linear Eqation  Checks Correct Wrong Time (sec.)

double 100,000 89,864 10,136 less than 0.01

Double 100,000 100,000 0 0.03

leda_real 100,000 100,000 0 22.33

         

leda_real 100,000 100,000 0 0.21

0

1000,1000,

?

bax

a

bx

ba

1?bax

Tests with a Quadratic Eqation  Checks Correct Wrong Time (sec.)

double 100,000 45,108 54,892 0.02

Double 100,000 99,962 308 0.07

leda_real 100,000 100,000 0 60.6

         

leda_real 100,000 100,000 0 0.39

0

2

4±

1000,1000,,

?2

2

cbxax

a

acbbx

cba

1?2 cbxax

Tests with a Quartic Eqation  Checks Correct Wrong Time (sec.)

double 100,000 36,267 63,733 0.02

Double 100,000 99,710 290 0.09

leda_real 100,000 100,000 0 282.62

         

leda_real 100,000 100,000 0 0.52

0

2

4±

1000,1000,,

?24

2

cbxax

a

acbbx

cba

1?24 cbxax

LEDA vs. COREAll tests consist of 1000 equation (both libraries were 100% correct in all cases). Running times are given in seconds:

Number Type Core’s EXPR Leda’s leda_real

Query E = 0 (?) E = 1 (?) E = 0 (?) E = 1 (?)

Linear Test 0.21 0.24 0.22 0.002

Quadratic Test 2.57 2.68 0.61 0.004

Quartic Test 25.85 27.14 2.83 0.005

Expression DAGs

a b c

42

-

-

x

a b c

+

+

a

acbbxcbxaxE

2

422

Evaluating Expressions (LEDA)E - Expression u(E) l(E)

integer n |n| 1

E1E2 u(E1)l(E2)+u(E2)l(E1) l(E1)l(E2)

E1E2 u(E1)u(E2) l(E1)l(E2)

E1E2 u(E1)l(E2) l(E1)u(E2)

kE1ku(E1)

kl(E1)

kE1 and u(E1)l(E1)k[u(E1)l(E1)

k-1] l(E1)

kE1 and u(E1)<l(E1) u(E1)k[u(E1)

k-1l(E1)]

At each stage, ifE=U(E)/l(E), we keep an upper bound u(E) on U(E) and a lower bound l(E) on L(E).

E - Expression u’(E) l’(E)

integer n log2 n 0

E1E2 1+max[u’(E1)+l’(E2),u’(E2)+l’(E1)

]

l’(E1)+l’(E2)

E1E2 u’(E1)+u’(E2) l’(E1)+l’(E2)

E1E2 u’(E1)+l’(E2) l’(E1)+u’(E2)

kE1 u’(E1)/k l’(E1)/kkE1 and u(E1)l(E1) [u’(E1)+(k-1)l’(E1)]/k l’(E1)

kE1 and u(E1)<l(E1) u’(E1) [(k-1)u’(E1)+ l’(E1)]/k

For practical purposes it is easier to keep u’(E) and l’(E) that estimate the exponents of the bound.

The Separation Bound

Let D(E) be the weight of the expression E, i.e. the product of the degrees of all radical nodes in the expression DAG.

Then either val(E) = 0, or:

1)(11)( )()()(val)()( EDED ElEuEEuEl

The old bound (LEDA):

1)(1

1)( 22

)()()(val)()(

EDED ElEuEEuEl

Separation Bound (Linear)

E u(E) l(E)

a, b M = max(|a|,|b|) 1

x=-b/a u(a)l(b)=M u(b)l(a)=M

ax u(a)u(x)=M 2 l(a)l(x)=M

E’=ax+b u(ax)l(b)+l(ax)u(b)=2M 2 l(ax)l(b)=M

D(E’)=1

(l(E’)u(E’)D(E’)-1)-1=M -1

u(E’)l(E’)D(E’)-1=2M 2

Separation Bound (Quadratic)E u(E) l(E)

a, b, c M = max(|a|,|b|,|c|) 1

4ac 4M 2 1

b2 M 2 1

b2-4ac 5M 2 1

b2-4ac u(b2-4ac)l(b2-4ac)=2.2M 1

b±b2-4ac 3.2M 1

x=(b±b2-4ac)/2a 3.2M 2M

x2 10.5M 2 4M 2

ax2 4M 3+10.5M 2 4M 2

bx 3.2M 2+2M 2M

ax2+bx 21M 4+29M 3 8M 3

E’=ax2+bx+c 29M 4+29M 330M 4 8M 3

D(E’)=2

(l(E’)u(E’)D(E’)-1)-1 = (8M 3 30M 4)-1 = (240M 7)-1

u(E’)l(E’)D(E’)-1=240M 7

Planar Arrangements

Given a collection C of curves in the plane, the arrangement of C is the subdivision of the plane into vertices, edges and faces induced by the curves in C.

Arrangements in CGAL

CGAL separates the topological handling of arrangements from the geometry, supplying a uniform construction method for any type of planar curve. The users should supply a traits class that should handle the geometry of the curves.

Existing Traits in CGAL

Line segments.

Poly-lines.

Circles.

Canonic parabolas.

Our Goal

Providing a unified approach for handling arrangements for all these curve types (and more).

Conic Curves

A conic curve is the set of points (x,y) satisfying the equation:

022 wvyuxtxysyrx

Ellipse

04 2 trs

Hyperbola

04 2 trs

Parabola

04 2 trs

Conic Arcs

A conic arc is defined by: - its base conic, - its source, - its target.

In case of an ellipse, we can define the full conic as the arcs (not specifying the source and the target).

Representation of Conic Arcs

The point p = (x,y) is on the conic arc a = (C,s,t) in iff it satisfies the equation of C and spt is a left turn (or a right turn if the orientation of C is negative).

Representation of Conic Arcs

Given x0 solve the quadratic equation:

002

002 wuxrxyvtxsy

Vertical ray-shooting:

04 22 wuxrxsvtx

Find all xs for which the equation above have exactly one solution – i..e. its discriminant is 0:

Vertical tangency points:

Representation of Conic ArcsIntersection points:

Given two conic curves:0:

0:22

2

221

wyvxuxytysxrC

wvyuxtxysyrxC

Let us denote:

vsvsEtstsD

wswsCususBrsrsA

Applying this to C1 leads us to the quartic equation:

022

22

22

222

2222

32422

vCEsCwExvBECvDtEsBCwDEuE

xvAEBvDtEtCDsBsACwDuDErE

xAvDtEtBDsABuDrDExtADsArD

EDx

CBxAxy

2

The Problem (1)In order to give an exact analytic solution to a quartic equation, one needs to use trigonometric functions.

The number type we use, leda_real, does not support trigonometric functions.

In the future leda_real will be extended to include the diamond operator, where diamond(p(x),d) means the dth root of the polynomial p(x).

The Solution (1)

Use an approximation method to solve quartic equations.

Bairstow’s method: Given a polynomial p(x) with real-valued coefficients, extract a quadratic factor (x2-rx-q) from it, using the following iterated process:

11

221

002

1

)()(

)()(

BxAqrxxxpxp

BxAqrxxxpxp

0

0

1

11

111

B

A

BqA

ABrA

q

r

q

r

The Problem (2)

Cascaded computations may lead to very complicated expression trees, resulting in a high time consumption.

The Solution (2)

We can define the following types of points:

Original end-points.

Vertical tangency points. (conic, index)

Intersection points. (conic #1, conic #2, index)

Vertical ray-shooting points. (point, conic)

When creating a new point, we can compute only an approximation for this point. If necessary, we can use the attached information to refine the approximation (or even use an exact computation).

Experimental ResultsArrangement Naïve conic

traitsFiltered

conic traitsFiltered

and cachedUsing epsilon-

tweaking

rot_ellipses_2 12.73 0.75 0.64 0.01

tng_circles_8 3.19 2.67 2.67 0.06

tng_ellipses_9 14.55 11.11 2.03 0.14

rose_14 7.44 4.12 2.19 0.17

mix_20 N/A 234.44 5.28 0.28

mix_50 56.01 29.24 15.50 0.91

The Segment/Circle Traits

As an intermediate by product, we have released an implementation of a traits class for arrangement of segments and circular arcs, based on the conic traits class.

Handling segments and circles is much simpler, since all the algebra needed is solutions to linear and quadratic equations.

Disc Robot Motion Planning

Example of usage:

Motion planning for a disc robot in a room with polygonal obstacles.

References

1. A Seperation Bound for Real Algebraic Expressions,C. Burnikel, S. Funke, K. Mehlhorn, S. Schirra, S. Schmitt (2001).

2. A New Constrcutive Root-Bound for Algebraic Expressions,C. Li, C. Yap (2000).

3. The LEDA homepage: http:/www.mpi-sb.mpg.de/LEDA/leda.html

4. The CORE hompage: http:/www.cs.nyu.edu/exact/core/