Next: Non-polynomial systems Up: benches Previous: benches

• La page de J-P. Merlet
• La page Présentation de COPRIN
• La page "Présentation" de l'INRIA

Subsections

# Polynomial systems

## Appolonius

Origin: D. Cox, Ideals, Varieties and Algorithms, Springer-Verlag, 1992
10 equations for 8 unknowns defined by:


2*x1-2=0
2*x2-1=0
2*x3-2=0
2*x4-1=0
2*x5-x6=0
x5+2*x6-2=0
(x1-x7)^2+x8^2-x7^2-(x8-x2)^2=0
(x1-x7)^2+x8^2-(x3-x7)^2-(x4-x8)^2=0
x1^2+x2^2-2*x1*x7+2*x2*x8=0
x1^2-x3^3-x4^2-2*x7*(x1-x3)+2*x4*x8=0

Ranges: for all unknowns [-1e8,1e8]

Solutions:: 0
Computation time (May 2004):

 DELL D400 (1.7Ghz) 0s

## Bellido

Origin: [1]
9 equations with 9 variables defined by:


(z1-6)^2+z2^2+z3^2-104=0
z4^2+(z5-6)^2+z6^2-104=0
z7^2+(z8-12)^2+(z9-6)^2-80=0
z1*(z4-6)+z5*(z2-6)+z3*z6-52=0
z1*(z7-6)+z8*(z2-12)+z9*(z3-6)+64=0
z4*z7+z8*(z5-12)+z9*(z6-6)-6*z5+32=0
2*z2+2*z3-2*z6-z4-z5-z7-z9+18=0
z1+z2+2*z3+2*z4+2*z6-2*z7+z8-z9-38=0
z1+z3+z5-z6+2*z7-2*z8-2*z4+8=0

Ranges: for all unknowns [-1e8,1e8]

Solutions:: 8 (exact)
Computation time (May 2004):

 Sun Blade 195s DELL D400 (1.7Ghz) 21s

## Bronstein

Origin: COCONUT
3 equations defined by:


x^2 + y^2 + z^2 - 36=0
x+y - z=0
x*y + z^2 - 1=0

Ranges: for all unknowns [-1e8,1e8]

Solutions:: 4 (exact)
Computation time (May 2004)

 DELL D400 (1.7 GHz) 0.01s

## Brown

Origin: [9]
n equations defined by:

solved here for n=5,30

Ranges: for all unknowns [-1e8,1e8]

Solutions:: 3 for n=5, 2 for n=30 (exact)
Computation time (April 2003):

 Sun Blade 1.23s (n=5), 9mn (n=30)

## Broyden banded

Origin: [9]
n equations defined by:

with ml=5, mu = 1 and

solved here for n=10,30

Ranges: for all unknowns [-100,100]

Solving method: HessianSolve+HullConsistencyStrong+3B
Solutions:: 1 for n=10, n=30 (exact)
Computation time:

 Evo 410C (1.2GHz) 0.76s (n=10), 11mn (n=30) (April 2003) DELL D400 (1.7Ghz) 0.76s (n=10), 130s (n=30) (May 2004)

## Broyden tridiagonal

Origin: [9]
n equations defined by:

with ,solved here for n=10,30

Ranges: for all unknowns [-100,100]

Solutions:: 2 for n=10, 2 for n=30 (exact)
Computation time:

 Evo 410C (1.2GHz) 0.55s (n=10), 112.98s (n=30) (April 2003) DELL D400 (1.7GHz) 0.24s (n=10), 26.14s (n=30) (May 2004)

## Caprasse

Origin: FRISCO
4 equations

y^2 z+2 x y t-2 x-z=0
-x^3 z+4 x y^2 z+4 x^2 y t+2 y^3 t+4 x^2-10 y^2+4 x z-10 y t+2=0
&2 y z t+x t^2-x-2 z=0
-x z^3+4 y z^2 t+4 x z t^2+2 y t^3+4 x z+4 z^2-10 y t-10 t^2+2=0

Ranges: for all unknowns [-10,10]
Solving method: HessianSolve+HullConsistency+3B
Solutions:: 18 (exact)
Computation time (April 2003):
 Sun Blade 30.15s EVO 410C (1.2Ghz) 6.71s

## Celestial

Origin: FRISCO
3 equations

-6 p^3+4 p^3 phi^3+15 phi^3 s^3 p-3 phi^3 s^5-12 phi^3 s p^2
-3 phi^3 s p+phi^3 s^3=0
-9 phi^3 s^2 p-5 phi^3 s^2-6 s p^3+3 phi^3 s^4+5 phi^3 p=0
-12 s^2 p-6 s^2+3 s^4+4 phi^2+3+12 p=0

 Unknown Range p [0.001,1000] s [0.001,1000] phi [0.001,1000]
Solutions:: 2 (exact)
Computation time (April 2003):
 Sun Blade 9.69s (April 2003) EVO 410C (1.2Ghz) 2.43s (April 2003) DELL D400 (1.7Ghz) 0.75s (May 2004)

## Chem

Origin: COCONUT
5 equations defined by:


3*y5 - y1*(y2 + 1)=0
y2*(2*y1 + y3^2 + r8 + 2*r10*y2 + r7*y3 + r9*y4) +y1 - r*y5=0
y3*(2*y2*y3 + 2*r5*y3 + r6 + r7*y2) - 8*y5=0
y4*(r9*y2 + 2*y4) - 4*r*y5=0
y2*(y1+r10*y2+y3^2+r8+r7*y3+r9*y4)+y1+r5*y3^2+y4^2-1+r6*y3=0

with

r = 10
r5 = 0.193
r6 = 0.002597/sqrt(40.)
r7 = 0.003448/sqrt(40.)
r8 = 0.00001799/40.
r9 = 0.0002155/sqrt(40.)
r10= 0.00003846/40.

Ranges: for all unknowns [0,1e8] or [-1e8,1e8]

Solutions:: 1 for range [0,1e8], 4 for range [-1e8,1e8] (exact)
Computation time (May 2004):

 DELL D400 (1.7 GHz) 0.05s (range =[0,1e8]), 0.5s (range =[-1e8,1e8])

## Chemk

Origin: Kearfott, COCONUT
4 equations defined by:


x1^2 - x2 =0
x4^2 - x3 =0
2.177e7*x2 - 1.697e7*x2*x4 + 0.55*x1*x4 + 0.45*x1 - x4=0

1.585e14*x2*x4 + 4.126e7*x1*x3 - 8.282e6*x1*x4 +
2.284e7*x3*x4 - 1.918e7*x3 + 48.4*x4 - 27.73=0

Ranges: for all unknowns [0,1]

Solutions:: 1 (exact)
Computation time (May 2004): will almost not change if the search space is extended to [-1e8,1e8]:

 DELL D400 (1.7 GHz) 0.01s

## Countercurrent reactors

Origin: [8]
n equations, a = 1/2

Ranges: [-1e8,1e8] for all unknowns

Solving method: HessianSolve+HullConsistency+ 3B
Solutions:: 18 (n=10), 38 (n=20) (exact)
Computation time:

 Evo 410C (1.2GHz) 6.52s (n=10) 1039.75s (n=20)(April 2003) DELL D400 (1.7Ghz) (1.2GHz) 3.22s (n=10) 570s (n=20)(May 2004)

## Cyclo

Origin: FRISCO
3 equations defined by:

-y^2*z^2-y^2+24*y*z-z^2-13=0
-x^2*z^2-x^2+24*x*z-z^2-13=0
-x^2*y^2-x^2+24*x*y-y^2-13=0

Ranges: for all unknowns [0,100000]
Solving method: HessianSolve+HullConsistency+ SimplexConsistency
Solutions:: 8 (exact)
Computation time (April 2003):
 Sun Blade 2.11s EVO 410C (1.2Ghz) 0.38s

## Dietmaier

Origin: Dietmaier  [4]
Physical meaning: forward kinematics of a parallel robot. Six points in a known position are chosen on two rigid bodies M1, M2. The set of points on M1 is A1,A2,..A6 and B1,B2,..B6 on M2. The distance between the 6 pairs (Ai,Bi) is known and the purpose of the system is to determine the possible location of M2 with respect to M2 (it can be shown that there will be at most 40 solutions)
12 equations:

xb_1^2+yb_1^2+zb_1^2- 1.0=0
(xb_2- 1.107915)^2+yb_2^2+zb_2^2- 0.4163798256=0
(xb_3- 0.549094)^2+(yb_3-0.756063)^2+zb_3^2- 1.180012929=0
(xb_4- 0.735077)^2+(yb_4+0.223935)^2+(zb_4- 0.525991)^2- 2.260328827=0
(xb_1-xb_2)^2+(yb_1-yb_2)^2+(zb_1-zb_2)^2- 0.2946372680=0
(xb_1-xb_3)^2+(yb_1-yb_3)^2+(zb_1-zb_3)^2- 1.195445050=0
(xb_1-xb_4)^2+(yb_1-yb_4)^2+(zb_1-zb_4)^2- 2.535465199=0
(xb_2-xb_3)^2+(yb_2-yb_3)^2+(zb_2-zb_3)^2- 0.4512414822=0
(xb_2-xb_4)^2+(yb_2-yb_4)^2+(zb_2-zb_4)^2- 2.107211052=0
(xb_3-xb_4)^2+(yb_3-yb_4)^2+(zb_3-zb_4)^2- 2.082590368=0
(- 1.574393890 xb_1+ 4.739721238 xb_2- 2.242096748 xb_3+ 0.07676939964 xb_4- 0.514188)^2+(- 1.574393890 yb_1+ 4.739721238 yb_2- 2.242096748 yb_3+0.07676939964 yb_4+ 0.526063)^2+(- 1.574393890 zb_1+ 4.739721238 zb_2- 2.242096748 zb_3+ 0.07676939964 zb_4+ 0.368418)^2- 1.643352216=0
( 1.520253264 xb_1- 1.048484327 xb_2+ 0.2758935157 xb_3+ 0.2523375480 xb_4- 0.590473)^{2}+( 1.520253264 yb_1- 1.048484327 yb_2+ 0.2758935157 yb_3+ 0.2523375480 yb_4- 0.094733)^2+( 1.520253264 zb_1- 1.048484327 zb_2+ 0.2758935157 zb_3+0.2523375480 zb_4+ 0.205018)^2- 0.5945504870=0

Unknowns:

Ranges: may be chosen arbitrary large without any change in the computation time

Solving method: SolveDistance
Solutions:: 40 (exact)
Computation time (April 2003):

 EVO 410C (1.2Ghz) 22s

## Dipole2

Origin: COCONUT, modified by COPRIN (original problem: see Dipole)
4 equations defined by:


a+b=0.63254
c+d =-1.34534
t*a+u*b-v*c-w*d=-0.8365348
v*a+w*b+t*c+u*d=1.7345334
a*t**2-a*v**2-2*c*t*v+b*u**2-b*w**2-2*d*u*w=1.352352
c*t**2-c*v**2+2*a*t*v+d*u**2-d*w**2+2*b*u*w=-0.843453
a*t**3-3*a*t*v**2+c*v**3-3*c*v*t**2+b*u**3-3*b*u*w**2+d*w**3-3*d*w*u**2=
-0.9563453
c*t**3-3*c*t*v**2-a*v**3+3*a*v*t**2+ d*u**3-3*d*u*w**2-b*w**3+3*b*w*u**2=1.2342523

The 4 first equations are linear in . They are solved to get a system in Ranges: for all unknowns [-1000,1000]
Solving method: HessianSolve+HullConsistency+3B+SimplexConsistency
Solutions:: 2 (exact)
Computation time (April 2007):
 Dell D620 (1.7Ghz) 69.72s

## Discrete boundary value function

Origin: [9]
n equations defined with h=1/(n+1), ti = ih

with ,solved here for n=20, 40

Ranges: for all unknowns [-100,100]

Solving method: HessianSolve+HullConsistency+3B
Solutions:: 1 for n=20, 1 for n=40 (exact)
Computation time (April 2003):

 Evo 410C (1.2Ghz) 1.13s (n=20), 22.92s (n=40)

## Discrete integral

Origin: [9]
n equations defined with h=1/(n+1), ti = ih

with

Ranges: for all unknowns [-100,100]
Solving method: HessianSolve+HullConsistency+3B (until May 2004), GlobalConsistencyTaylor, IntervalNewton and special 2B (2007)
Solutions:: 3 for n=5, 8,9,20, 1 for n=1,7,10,11 (exact)
Computation time:

 Evo 410C (1.2GHz) 40.46s (n=5), 915.97s (n=6) (April 2003) DELL D400 (1.7 GHz) 5.63s (n=5), 105.47s (n=6) (May 2004) Cluster (11 PC's) 3776s (n=7) (April 2003) DELL D620 (1.7Ghz) 0.027s (n=5), 0.032s(n=6), 0.045s(n=7), 1.15s (n=8), 1.39s (n=9), 2.58s (n=10), 2.98s (n=11), 1mn42s (n=20)

Note that until 2007 this system was considered as a difficult one

This system may be changed into another system of 2n equations in the unknowns y1..yn,x1..xn defined by:

Ranges: for all yj [-100,100], for all xj [-1000000,1000000]
Solving method: HessianSolve+HullConsistency+ SimplexConsistency+3B
Solutions:: 1 for n=6, 1 for n=20 (exact)
Computation time (April 2003):

 Evo 410C (1.2Ghz) 7.7s (n=6), 375.8s (n=20)

## Eco9

Origin: COCONUT [13]
8 equations with 8 unknowns:


x1 + x2*(x1 + x3) + x4*(x3 + x5) + x6*(x5 + x7) -( x8*((1/8) - x7))=0
x2 + x3*(x1 + x5) + x4*(x2 + x6) + x5*x7 -( x8*((2/8) - x6))=0
x3*(1 + x6) + x4*(x1 + x7) + x2*x5 -( x8*((3/8) - x5))=0
x4 + x1*x5 + x2*x6 + x3*x7 -( x8*((4/8) - x4))=0
x5 + x1*x6 + x2*x7 -( x8*((5/8) - x3))=0
x6 + x1*x7 -( x8*((6/8) - x2))=0
x7 -( x8*((7/8) - x1))=0
x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 +1=0

Ranges: [-100,100] for all unknowns

Solutions:: 16 (exact)
Computation time (May 2004):

 DELL D400 (1.7 Ghz) 208s

## Equilibrium Combustion

Origin: [6]
6 equations


y_1-3 y_5+y_1 y_2
y_1+0.00000044975 y_2-10 y_5+0.0000028845 y_2^2+2 y_1y_2+
0.0005451766686 y_2 y_3+0.00003407354179 y_2 y_4+y_2y_3^2

0.0004106217542 y_3-8 y_5+0.386 y_3^2+0.0005451766686 y_2y_3+2 y_2y_3^2
-40 y_5+2 y_4^2+0.00003407354179 y_2y_4

y_1+0.00000044975 y_2+0.0004106217542 y_3+0.0000009615y_2^2+
0.193 y_3^2+y_4^2+y_1y_2+0.0005451766686 y_2 y_3+0.00003407354179 y_2y_4+y_2y_3^2-1

Ranges: [-1e8,1e8] for all unknowns

Solving method: HessianSolve+HullConsistency+ SimplexConsistency+3B
Solutions:: 4 (exact)
Computation time (April 2003):

 Evo 410C (1.2Ghz) 2s

## Extended Freudenstein

Origin: [8]
Physical meaning: none
n equations:

Ranges: [-1e8,1e8] for all unknowns
Solving method: HessianSolve+HullConsistency+3B
Solutions:: 1 (exact)
Computation time (April 2003):

 EVO 410C (1.2Ghz) 5.2s (n=20)

## Extended Powell1

Origin: [8]
n=4m equations:

Ranges: [-1e8,1e8]

Solving method: HessianSolve+HullConsistency+3B
Solutions:: 4 (n=16) (approximate)
Computation time (April 2003):

 Evo 410C (1.2Ghz) 3.33s (n=16)

## Extended Wood

Origin: [8]
n=4m equations:

Ranges: [-1e8,1e8]

Solving method: HessianSolve+HullConsistency+ 3B + IntervalNewton (since May 2007)
Solutions:: 3 (n=4), 9 (n=8), 27 (n=12), 81 (n=16) (exact)
Computation time

 Evo 410C (April 2003): (1.2Ghz) 4.41s (n=4), 269.29s (n=8) DELL 620 (May 2007) (1.7GHz) 21.16s (n=8), 3mn5s (n=12), 38mn8s (n=16)
Note that this problem was intially classified as a difficult problem.

## Fourbar

Origin: COCONUT
4 equations defined by:

0.01692601*X1^2*Y1^2 - 0.888509280014*X1^2*Y2^2
+ 0.0411717692438*X2^2*Y1^2 - 0.00437457395884*X2^2*Y2^2
+ 0.331480641249*X1*X2*Y1^2 - 1.38036964668*X1*X2*Y2^2
- 0.270492270191*X1^2*Y1*Y2 + 1.44135801774*X2^2*Y1*Y2
+ 0.859888946812*X1*X2*Y1*Y2
+ 0.0791489659197*X1^2*Y1 - 0.00336032777032*X1^2*Y2
- 0.0620826738427*X1*Y1^2 + 0.501879647495*X1*Y2^2
+ 0.647156236961*X2^2*Y1 + 0.0926311741907*X2^2*Y2
- 0.255000006226*X2*Y1^2 - 0.0896892386081*X2*Y2^2
- 0.568007271041*X1*X2*Y2 + 0.095991501961*X1*X2*Y1
+ 0.165310767618*X1*Y1*Y2 - 0.563962321337*X2*Y1*Y2
- 0.0784871167595*X1*Y1 - 0.0784871167595*X2*Y2
+ 0.011807283256*X1*Y2  - 0.011807283256*X2*Y1
+ 0.0422876985355*X1^2 + 0.0422876985355*X2^2
+ 0.0372427422943*Y1^2 + 0.0372427422943*Y2^2=0

0.518178672335*X1^2*Y1^2 - 0.0414464807343*X1^2*Y2^2
+ 2.63600135179*X2^2*Y1^2 - 0.799490472298*X2^2*Y2^2
+ 0.29442805494*X1^2*Y1*Y2 + 1.46551534655*X2^2*Y1*Y2
- 0.631878110759*X1*X2*Y1^2 - 1.80296540237*X1*X2*Y2^2
- 2.87586667102*X1*X2*Y1*Y2
- 0.987856648177*X1^2*Y1 - 0.530579106676*X1^2*Y2
- 0.0397576281649*X1*Y1^2 + 0.317719102869*X1*Y2^2
- 1.93710490787*X2^2*Y1  + 0.00127693327315*X2^2*Y2
- 0.581380074072*X2*Y1^2 - 0.0672137066743*X2*Y2^2
+ 0.531856039949*X1*X2*Y1 + 0.949248259696*X1*X2*Y2
+ 0.514166367398*X1*Y1*Y2 - 0.357476731033*X2*Y1*Y2
+ 0.140965913657*X1*Y1  + 0.140965913657*X2*Y2
- 0.153347218606*X1*Y2  + 0.153347218606*X2*Y1
+ 0.283274882058*X1^2  + 0.283274882058*X2^2
+ 0.0382903330079*Y1^2 + 0.0382903330079*Y2^2 =0

0.0233560008057*X1^2*Y1^2 - 0.00428427501149*X1^2*Y1*Y2
- 0.792756311827*X1^2*Y2^2 + 0.0492185850289*X2^2*Y2^2
+ 0.0759264856293*X1*X2*Y1^2 + 1.14839711492*X1*X2*Y1*Y2
- 0.283066217262*X2^2*Y1^2 + 0.460041521291*X2^2*Y1*Y2
- 0.388399310674*X1*X2*Y2^2
- 0.0561169736293*X1*Y1^2 + 0.485064247792*X1*Y2^2
+ 0.0689567235492*X1^2*Y1 - 0.115620658768*X1^2*Y2
- 0.13286905328*X2*Y1^2 - 0.084375901147*X2*Y2^2
+ 0.639964831612*X2^2*Y1 + 0.101386684276*X2^2*Y2
+ 0.217007343044*X1*X2*Y1 - 0.571008108063*X1*X2*Y2
+ 0.0484931521334*X1*Y1*Y2 - 0.541181221422*X2*Y1*Y2
- 0.00363197918253*X1*Y2 + 0.00363197918253*X2*Y1
- 0.0781302968652*X1*Y1 - 0.0781302968652*X2*Y2
+ 0.0471311092612*X1^2 + 0.0471311092612*X2^2
+ 0.0324495575052*Y1^2 + 0.0324495575052*Y2^2 ;

0.393707415641*X1^2*Y1^2 + 0.59841456862*X1^2*Y2^2
+ 0.0735854940135*X2^2*Y1^2 + 0.0548997238169*X2^2*Y2^2
+ 0.0116156836985*X1^2*Y1*Y2 + 0.0699694273575*X2^2*Y1*Y2
- 0.305757340849*X1*X2*Y1^2 - 0.364111084508*X1*X2*Y2^2
- 0.223392923175*X1*X2*Y1*Y2
+ 0.0996725944534*X1^2*Y1 + 0.0113936468426*X1^2*Y2
- 0.381205205249*X1*Y1^2 - 0.473402150235*X1*Y2^2
- 0.0213613191759*X2^2*Y1 - 0.0372595571271*X2^2*Y2
+ 0.148904552394*X2*Y1^2 + 0.142408744984*X2*Y2^2
- 0.0486532039697*X1*X2*Y1 + 0.121033913629*X1*X2*Y2
- 0.00649580741066*X1*Y1*Y2 + 0.092196944986*X2*Y1*Y2
- 0.0483106652705*X1*Y1  - 0.0483106652705*X2*Y2
- 0.00316794272326*X1*Y2 + 0.00316794272326*X2*Y1
+ 0.00634952598374*X1^2 + 0.00634952598374*X2^2
+ 0.0922886309144*Y1^2  + 0.0922886309144*Y2^2 =0

Ranges: for all unknowns [-10,10]

Solutions:: 3 (2 exact, 1 singular)
Computation time (September 2004):

 DELL D400 (1.7Ghz) 364s

## Geneig

Origin: COCONUT
Physical meaning: generalized eigenvalue problem

6 equations:


-10*x1*x6^2+ 2*x2*x6^2-x3*x6^2+x4*x6^2+ 3*x5*x6^2+x1*x6+ 2*x2*x6+x3*x6+ 2*x4*
x6+x5*x6+ 10*x1+ 2*x2-x3+ 2*x4-2*x5 =0

2*x1*x6^2-11*x2*x6^2+ 2*x3*x6^2-2*x4*x6^2+x5*x6^2+ 2*x1*x6+x2*x6+ 2*x3*x6+x4*
x6+ 3*x5*x6+ 2*x1+ 9*x2+ 3*x3-x4-2*x5 =0

-x1*x6^2+ 2*x2*x6^2-12*x3*x6^2-x4*x6^2+x5*x6^2+x1*x6+ 2*x2*x6-2*x4*x6-2*x5*x6-
x1+ 3*x2+ 10*x3+ 2*x4-x5 =0

x1*x6^2-2*x2*x6^2-x3*x6^2-10*x4*x6^2+ 2*x5*x6^2+ 2*x1*x6+x2*x6-2*x3*x6+ 2*x4*
x6+ 3*x5*x6+ 2*x1-x2+ 2*x3+ 12*x4+x5 =0

3*x1*x6^2+x2*x6^2+x3*x6^2+ 2*x4*x6^2-11*x5*x6^2+x1*x6+ 3*x2*x6-2*x3*x6+ 3*x4*
x6+ 3*x5*x6-2*x1-2*x2-x3+x4+ 10*x5 =0

x1+x2+x3+x4+x5-1 :

Ranges: [-1e8,1e8] for all unknowns
Solutions:: 10 (exact)
Computation time (May 2004):
 DELL D400 (1.7Ghz) 108s

## Eiger-Sikorski-Stenger

Origin: [7]
n equations:

Ranges: [-1e8,1e8]

Solving method: HessianSolve+HullConsistency+ 3B
Solutions:: 2 (exact)
Computation time

 DELL 620 (May 2007) (1.7GHz) 0.012s (n=9), 1.49s (n=20), 9.96s (n=30)

## Freudenstein

Origin: [9]
Physical meaning: none
2 equations

-13+x1+((5-x2)*x2-2)*x2
-29+x1+((x2+1)*x2-14)*x2

Ranges: [-1e8,1e8] for x1, x2
Solving method: HessianSolve+HullConsistency+3B
Note that we don't use the difference between the two equations
Solutions:: 1 (exact)
Computation time (April 2003):
 EVO 410C (1.2Ghz) 0.04s

## Himmelblau

Origin: [9]
2 equations

Ranges: [-1e8,1e8] for x1, x2

Solving method: HessianSolve+HullConsistency+3B

Solutions:: 9 (exact)
Computation time (April 2003):

 EVO 410C (1.2Ghz) 0.11s

## I5

Origin: COCONUT [13]
10 equations with 10 unknowns:


x1  - 0.18324757*(x4*x3*x9)^3  + x3^4*x9^7  - 0.25428722=0
x2  - 0.16275449*(x1*x10*x6)^3 + x10^4*x6^7 - 0.37842197=0
x3  - 0.16955071*(x1*x2*x10)^3 + x2^4*x10^7 - 0.27162577=0
x4  - 0.15585316*(x7*x1*x6)^3  + x1^4*x6^7  - 0.19807914=0
x5  - 0.19950920*(x7*x6*x3)^3  + x6^4*x3^7  - 0.44166728=0
x6  - 0.18922793*(x8*x5*x10)^3 + x5^4*x10^7 - 0.14654113=0
x7  - 0.21180486*(x2*x5*x8)^3  + x5^4*x8^7  - 0.42937161=0
x8  - 0.17081208*(x1*x7*x6)^3  + x7^4*x6^7  - 0.07056438=0
x9  - 0.19612740*(x10*x6*x8)^3 + x6^4*x8^7  - 0.34504906=0
x10 - 0.21466544*(x4*x8*x1)^3  + x8^4*x1^7  - 0.42651102=0

Ranges: [-1,1] or [-100,100] for all unknowns

Solutions:: 1 for range [-1,1], 30 for range [-100,100] (exact)
Computation time (May 2004):

 DELL D400 (1.7GHz) 0.02s (range [-1,1]), 397s (range [-100,100])

## Kapur

Origin: COCONUT
9 equations defined by:


y*u1 + u3 - a=0
x*u1 + u3 - b=0
w*u2 + u4 - c=0
z*u2 + u4 - d=0
2*u4*u1 + 2*u3*u2 - 9*t=0
u1 - (z+w)=0
u2 - (x+y)=0
u3 - w*z=0
u4 - x*y=0

This is a modified version of the original system

y*z + z*w + w*y - a=0
z*x + x*w + w*z - b=0
w*x + x*y + y*w - c=0
x*y + y*z + z*x - d=0
2*x*y*z + 2*y*z*w + 2*z*w*x + 2*w*x*y - 9*t=0


Ranges: [-1e8,1e8] for u1,u2,u3,u4, [0,1000] for x,y,z,w,t

Solutions:: 1 (exact)
Computation time (May 2004)

 DELL D400 (1.7 GHz) 0.02s (original) 0.03s (modified)

## Katsura n

Origin: Faugère, http://www-calfor.lip6.fr
Physical meaning: the unknowns is the value of a distribution function of a field created by a mixture of a ferro antiferro magnetic bond at some some points. Hence the unknowns are probabilities and should lie in [0,1]

To respect the classical denomination in the bench of algebraic systems Katsura n has n+1 equations.

Unknowns: u(0),u(1),...u(n)
Ranges: [0,1]
Solving method: HessianSolve+HullConsistency+ SimplexConsistency+3B
Solutions:: 4 (n=9), 7 (n=12), 3 (n=13), 5 (n=14), 5 (n=15), 7 (n=20) (exact)
Computation time (April 2003):

 EVO 410C (1.2GHz) 21.61s (n=9), 514.86s (n=12), 1077s (n=13), 2948s (n=14) Cluster (12 PC's) 37mn (n=15), 6h55mn (n=20)

## Kearl11

Origin: COCONUT
8 equations with 8 unknowns:


x1^2 + x2^2 -1=0
x3^2 + x4^2 -1=0
x5^2 + x6^2 -1=0
x7^2 + x8^2 -1=0
0.004731*x1*x3-0.3578*x2*x3-0.1238*x1-0.001637*x2-0.9338*x4+x7 - 0.3571=0
0.2238*x1*x3+0.7623*x2*x3+0.2638*x1-0.07745*x2-0.6734*x4-0.6022 =0
x6*x8+0.3578*x1+0.004731*x2 =0
0.2238*x2-0.7623*x1+0.3461 =0

Ranges: [-1,1] for all unknowns

Solutions:: 16 (exact)
Computation time (May 2004):

 DELL D400 (1.7GHz) 0.23s

## Kincox

Origin: COCONUT
4 equations with 4 unknowns with a =1, b=4, l1=10, l2=6


-a + l2*(c1*c2 - s1*s2) + l1*c1=0
-b + l2*(c1*s2 + c2*s1) + l1*s1=0
c1^2 + s1^2 - 1=0
c2^2 + s2^2 - 1=0

Ranges: [-1,1] for all unknowns

Solutions:: 2 (exact)
Computation time (May 2004):

 DELL D400 (1.7GHz) 0.01s

## More 10/80

Origin: COCONUT
10 equations with 10 unknowns (or 80 equations with 80 unknowns)

X1+.3756574005e-2*(X1+1.090909091)^3+.3380916604e-2*(X2+1.181818182)^3
+.3005259204e-2*(X3+1.272727273)^3+.2629601803e-2*(X4+1.363636364)^3
+.2253944403e-2*(X5+1.454545455)^3+.1878287002e-2*(X6+1.545454545)^3
+.1502629602e-2*(X7+1.636363636)^3+.1126972201e-2*(X8+1.727272727)^3
+.7513148009e-3*(X9+1.818181818)^3+.3756574005e-3*(X10+1.909090909)^3=0

X2+.3380916604e-2*(X1+1.090909091)^3+.6761833208e-2*(X2+1.181818182)^3
+.6010518407e-2*(X3+1.272727273)^3+.5259203606e-2*(X4+1.363636364)^3
+.4507888805e-2*(X5+1.454545455)^3+.3756574005e-2*(X6+1.545454545)^3
+.3005259204e-2*(X7+1.636363636)^3+.2253944403e-2*(X8+1.727272727)^3
+.1502629602e-2*(X9+1.818181818)^3+.7513148009e-3*(X10+1.909090909)^3=0

X3+.3005259204e-2*(X1+1.090909091)^3+.6010518407e-2*(X2+1.181818182)^3
+.9015777611e-2*(X3+1.272727273)^3+.7888805409e-2*(X4+1.363636364)^3
+.6761833208e-2*(X5+1.454545455)^3+.5634861007e-2*(X6+1.545454545)^3
+.4507888805e-2*(X7+1.636363636)^3+.3380916604e-2*(X8+1.727272727)^3
+.2253944403e-2*(X9+1.818181818)^3+.1126972201e-2*(X10+1.909090909)^3=0

X4+.2629601803e-2*(X1+1.090909091)^3+.5259203606e-2*(X2+1.181818182)^3
+.7888805409e-2*(X3+1.272727273)^3+.1051840721e-1*(X4+1.363636364)^3
+.9015777611e-2*(X5+1.454545455)^3+.7513148009e-2*(X6+1.545454545)^3
+.6010518407e-2*(X7+1.636363636)^3+.4507888805e-2*(X8+1.727272727)^3
+.3005259204e-2*(X9+1.818181818)^3+.1502629602e-2*(X10+1.909090909)^3=0

X5+.2253944403e-2*(X1+1.090909091)^3+.4507888805e-2*(X2+1.181818182)^3
+.6761833208e-2*(X3+1.272727273)^3+.9015777611e-2*(X4+1.363636364)^3
+.1126972201e-1*(X5+1.454545455)^3+.9391435011e-2*(X6+1.545454545)^3
+.7513148009e-2*(X7+1.636363636)^3+.5634861007e-2*(X8+1.727272727)^3
+.3756574005e-2*(X9+1.818181818)^3+.1878287002e-2*(X10+1.909090909)^3=0

X6+.1878287002e-2*(X1+1.090909091)^3+.3756574005e-2*(X2+1.181818182)^3
+.5634861007e-2*(X3+1.272727273)^3+.7513148009e-2*(X4+1.363636364)^3
+.9391435011e-2*(X5+1.454545455)^3+.1126972201e-1*(X6+1.545454545)^3
+.9015777611e-2*(X7+1.636363636)^3+.6761833208e-2*(X8+1.727272727)^3
+.4507888805e-2*(X9+1.818181818)^3+.2253944403e-2*(X10+1.909090909)^3=0

X7+.1502629602e-2*(X1+1.090909091)^3+.3005259204e-2*(X2+1.181818182)^3
+.4507888805e-2*(X3+1.272727273)^3+.6010518407e-2*(X4+1.363636364)^3
+.7513148009e-2*(X5+1.454545455)^3+.9015777611e-2*(X6+1.545454545)^3
+.1051840721e-1*(X7+1.636363636)^3+.7888805409e-2*(X8+1.727272727)^3
+.5259203606e-2*(X9+1.818181818)^3+.2629601803e-2*(X10+1.909090909)^3=0

X8+.1126972201e-2*(X1+1.090909091)^3+.2253944403e-2*(X2+1.181818182)^3
+.3380916604e-2*(X3+1.272727273)^3+.4507888805e-2*(X4+1.363636364)^3
+.5634861007e-2*(X5+1.454545455)^3+.6761833208e-2*(X6+1.545454545)^3
+.7888805409e-2*(X7+1.636363636)^3+.9015777611e-2*(X8+1.727272727)^3
+.6010518407e-2*(X9+1.818181818)^3+.3005259204e-2*(X10+1.909090909)^3=0

X9+.7513148009e-3*(X1+1.090909091)^3+.1502629602e-2*(X2+1.181818182)^3
+.2253944403e-2*(X3+1.272727273)^3+.3005259204e-2*(X4+1.363636364)^3
+.3756574005e-2*(X5+1.454545455)^3+.4507888805e-2*(X6+1.545454545)^3
+.5259203606e-2*(X7+1.636363636)^3+.6010518407e-2*(X8+1.727272727)^3
+.6761833208e-2*(X9+1.818181818)^3+.3380916604e-2*(X10+1.909090909)^3=0

X10+.3756574005e-3*(X1+1.090909091)^3+.7513148009e-3*(X2+1.181818182)^3
+.1126972201e-2*(X3+1.272727273)^3+.1502629602e-2*(X4+1.363636364)^3
+.1878287002e-2*(X5+1.454545455)^3+.2253944403e-2*(X6+1.545454545)^3
+.2629601803e-2*(X7+1.636363636)^3+.3005259204e-2*(X8+1.727272727)^3
+.3380916604e-2*(X9+1.818181818)^3+.3756574005e-2*(X10+1.909090909)^3=0

Ranges: [-1e8,0] for all unknowns

Solutions:: 1 (exact)
Computation time (May 2004):

 DELL D400 (1.7GHz) 0.02s (10 equations), 437s (80 equations)

## Nauheim

Origin: COCONUT
8 equations with 8 unknowns


e*g + 2*d*h =0
9*e + 4*b =0
4*c*h + 2*e*f + 3*d*g =0
7*c - 9*a + 8*f =0
4*d*f + 5*c*g + 6*h + 3*e =0
5*d + 6*c*f + 7*g - 9*b =0
9*d + 6*a - 5*b=0
-9*c + 7*a - 8=0

Ranges: [-1e8,1e8] for all unknowns

Solutions:: 4 (exact)
Computation time (May 2004):

 DELL D400 (1.7GHz) 0.62s

## Neveu1

Origin: COPRIN
Physical meaning: we consider a set of points in a plane. Among these points the distance between some pairs is known
26 equations

(x1-2)^{2}+{y1}^{2}-16=0
(x4-x1)^{2}+(y4-y1)^{2}-16=0
(x7-x8)^{2}+(y7-y8)^{2}-16=0
(x10-x8)^{2}+(y10-y8)^{2}-16=0
(x13-x15)^{2}+(y13-y15)^{2}-16=0
(x14-x15)^{2}+(y14-y15)^{2}-16=0
(x4-2)^{2}+{y4}^{2}-25=0
(x5-2)^{2}+{y5}^{2}-9=0
(x4-x5)^{2}+(y4-y5)^{2}-16=0
(x5-5)^{2}+{y5}^{2}-18=0
(x5-x6)^{2}+(y5-y6)^{2}-25=0
(x4-x6)^{2}+(y4-y6)^{2}-9=0
(x7-5)^{2}+{y6}^{2}- 0.0625=0
(x9-x7)^{2}+(y9-y7)^{2}- 0.0625=0
(x10-x9)^{2}+(y10-y9)^{2}- 0.0625=0
(x10-x7)^{2}+(y10-y7)^{2}- 0.0625=0
(x11-x9)^{2}+(y11-y9)^{2}- 0.09=0
(x14-x11)^{2}+(y14-y11)^{2}- 0.0625=0
(x12-x11)^{2}+(y12-y11)^{2}- 0.0625=0
(x14-x12)^{2}+(y14-y12)^{2}- 0.0625=0
(x12-x6)^{2}+(y12-y6)^{2}- 0.0625=0
(x13-x6)^{2}+(y13-y6)^{2}- 0.0625=0
(x13-x14)^{2}+(y13-y14)^{2}- 0.0625=0
(x12-x13)^{2}+(y12-y13)^{2}- 0.0625=0
(x9-5)^{2}+{y9}^{2}- 0.0625=0
(x10-x11)^{2}+(y10-y11)^{2}- 0.0625=0

Unknowns:
x1,y1,x4,y4,x5,y5,x6,y6,x7,y7,x8,y8,x9,y9,x10,y10,x11,y11,x12,y12, x13,y13,x14,y14,x15,y15

Range: may be chosen arbitrary large without any change in the computation time
Solving method: SolveDistance
Solutions:: 128 (exact)
Computation time (April 2003):

 EVO 410C (1.2Ghz) 437.8s

## Piano

Origin: COCONUT
9 equations with 9 unknowns


x - l*t^3 - L*w =0
y - L*t - l*w^3 =0
L - 1           =0
l - 2           =0
x - a           =0
2*a - 3         =0
y - b           =0
b - r*t         =0
w^2 - 1 + t^2   =0

Ranges: [-100000,100000] for all unknowns except w [0,100000]

Solutions:: 1 (exact)
Computation time (May 2004):

 DELL D400 (1.7GHz) 0.01s

## Pramanik

Origin: Pramanik [10]
Physical meaning: the unknowns are geometrical parameters of a car steering mechanism. The purpose of the system is to determine the possible mechanisms such that the trajectory followed by the mechanism goes through pre-defined points

Let be defined as:

and the equation:

Let:

and

A system of 3 equations is obtained with for i in [1,3]

Unknowns:k1,k2,k3

Ranges: [0.06,1] for all unknowns

Solving method: HessianSolve
Solutions:: 2 (exact)
Computation time (April 2003):

 Sun Blade 31.31s EVO 410C (1.2Ghz) 12.58s

## Prolog

Origin: Norton, R D, and Scandizzo, P L, Market Equilibrium Computations in Activity Analysis Models. Operations Research 29, 2 (1981)
23 equations in 21 unknowns, x1,..x21
Ranges for the unknowns: [-1e8,1e8] for x1, [0.2,1e8] for x2,x3,x4, [0,1e8] for the other variables

x5 + x6 - 0.94*x11 - 0.94*x12 - 0.94*x13 + 0.244*x17 + 0.244*x18+ 0.244*x19 =0

0.064*x11 + 0.064*x12 + 0.064*x13 - 0.58*x14 - 0.58*x15 - 0.58*x16
+ 0.172*x17 + 0.172*x18 + 0.172*x19 =0

x7 + x8 + 0.048*x11 + 0.048*x12 + 0.048*x13 + 0.247*x14 + 0.247*x15
+ 0.247*x16 - 0.916*x17 - 0.916*x18 - 0.916*x19 =0

x11 + 1.2*x12 + 0.8*x13 + 2*x14 + 1.8*x15 + 2.4*x16 + 3*x17 + 2.7*x18
+ 3.2*x19 - 3712=0

2*x11 + 1.8*x12 + 2.2*x13 + 3*x14 + 3.5*x15 + 2.3*x16 + 3*x17 + 3.2*x18
+ 2.7*x19 - 5000=0

356.474947137148*x2 + 53.7083537310174*x4 + x5 - 0.564264890180399*x20- 352=0

339.983422262764*x2 + 43.5418249774113*x4 + x6 - 0.405939876920766*x21- 430=0

106.946746119538*x2 + 145.018955433089*x4 + x7 - 0.507117039797071*x20 - 222=0

173.929713444361*x2 + 203.031384299627*x4 + x8 - 0.578889145413521*x21 - 292=0

x5*x2 + x7*x4 - x20 =0

x6*x2 + x8*x4 - x21 =0

- 3340.8*x9 - 500*x10 + x20 =0

- 371.2*x9 - 4500*x10 + x21 =0

0.94*x2 - 0.064*x3 - 0.048*x4 - x9 - 2*x10 =0

0.94*x2 - 0.064*x3 - 0.048*x4 - 1.2*x9 - 1.8*x10 =0

0.94*x2 - 0.064*x3 - 0.048*x4 - 0.8*x9 - 2.2*x10 =0

0.58*x3 - 0.247*x4 - 2*x9 - 3*x10 =0

0.58*x3 - 0.247*x4 - 1.8*x9 - 3.5*x10 =0

0.58*x3 - 0.247*x4 - 2.4*x9 - 2.3*x10 =0

- 0.244*x2 - 0.172*x3 + 0.916*x4 - 3*x9 - 3*x10 =0

- 0.244*x2 - 0.172*x3 + 0.916*x4 - 2.7*x9 - 3.2*x10 =0

- 0.244*x2 - 0.172*x3 + 0.916*x4 - 3.2*x9 - 2.7*x10 =0

- (x5*x2 + x6*x2 + x7*x4 + x8*x4) - x1 + 3712*x9 + 5000*x10 =0

Solutions:: 0
Computation time (September 2004):
 DELL D400 (1.7GHz) 16.73s

## Puma

Origin: COCONUT
Physical meaning: inverse kinematics of a 3R robot
8 equations with 8 unknowns


x1^2 + x2^2 - 1 =0
x3^2 + x4^2 - 1 =0
x5^2 + x6^2 - 1 =0
x7^2 + x8^2 - 1 =0
0.004731*x1*x3 - 0.3578*x2*x3 - 0.1238*x1 - 0.001637*x2 - 0.9338*x4 + x7=0
1 =0
0.2238*x1*x3 + 0.7623*x2*x3 + 0.2638*x1 - 0.07745*x2 -0.6734*x4 -0.6022 =0
x6*x8 + 0.3578*x1 + 0.004731*x2 =0
-0.7623*x1 + 0.2238*x2 + 0.3461 =0

Ranges: [-1,1] for all unknowns

Solutions:: 16 (exact)
Computation time (May 2004):

 DELL D400 (1.7GHz) 0.23s

## Redeco8

Origin: COCONUT
8 equations with 8 unknowns


x1 + x1*x2 + x2*x3 + x3*x4 + x4*x5 + x5*x6 + x6*x7 - 1*u8 =0
x2 + x1*x3 + x2*x4 + x3*x5 + x4*x6 + x5*x7 - 2*u8 =0
x3 + x1*x4 + x2*x5 + x3*x6 + x4*x7 - 3*u8 =0
x4 + x1*x5 + x2*x6 + x3*x7 - 4*u8 =0
x5 + x1*x6 + x2*x7 - 5*u8 =0
x6 + x1*x7 - 6*u8 =0
x7 - 7*u8 =0
x1 + x2 + x3 + x4 + x5 + x6 + x7 + 1 =0

Ranges: [-1e8,1e8] for all unknowns

Solutions:: 8 (exact)
Computation time (May 2004):

 DELL D400 (1.7GHz) 39.46s

## Robot kinematics

Origin: [6]
Physical meaning: find the joint angles of a 4 degrees-of-freedom robot for reaching a given pose with the end-effector
8 equations


- 0.1238 x_1+x_7- 0.001637 x_2- 0.9338 x_4+ 0.004731 x_1 x_3- 0.3578 x_2 x_3- 0.3571
0.2638 x_1-x_7- 0.07745 x_2- 0.6734 x_4+ 0.2238 x_1 x_3+ 0.7623 x_2 x_3- 0.6022
0.3578 x_1+ 0.004731 x_2+x_6 x_8- 0.7623 x_1+ 0.2238 x_2+ 0.3461
x_1^2+x_2^2-1
x_3^2+x_4^2-1
x_5^2+x_6^2-1
x_7^2+x_8^2-1

Ranges: for all unknowns [-1,1]

Solving method: HessianSolve+HullConsistency+ SimplexConsistency+3B
Solutions:: 16 (exact)
Computation time (April 2003):

 Evo 410C (1.2Ghz) 0.38s

## Rose

Origin: COCONUT, Shoven :"Applied general equilibrium modelling", IMF Staff Papers, pages 394-419, 1983
Physical meaning: a general economic equilibrium model

3 equations defined by:


y^4-20/7*x^2 =0

x^2*z^4+7/10*x*z^4+7/48*z^4-50/27*x^2-35/27*x-49/216 =0

3/5*x^6*y^2*z+x^5*y^3+3/7*x^5*y^2*z+7/5*x^4*y^3-7/20*x^4*y*z^2-
3/20*x^4*z^3+609/1000*x^3*y^3+63/200*x^3*y^2*z-77/125*x^3*y*z^2-
21/50*x^3*z^3+49/1250*x^2*y^3+147/2000*x^2*y^2*z-
23863/60000*x^2*y*z^2-91/400*x^2*z^3-27391/800000*x*y^3+
4137/800000*x*y^2*z-1078/9375*x*y*z^2-5887/200000*x*z^3-1029/160000*y^3-
24353/1920000*y*z^2-343/128000*z^3 =0

Ranges: for all unknowns [-1e8,1e8]

Solutions:: 16 (exact) +2 singular
Computation time (May 2004)

 DELL D400 (1.7 GHz) 2.87s

## 6body

Origin: FRISCO
Physical meaning: Polynomial systems arising in the study of central configurations in the N-body problem of Celestial Mechanics
6 equations

3*(b-d)*(B-D)+B+D-6*F+4=0
3*(b-d)*(B+D-2*F)+5*(B-D)=0
3*(b-d)^2-6*(b+d)+4*f+3=0
B^2*b^3-1=0
D^2*d^3-1=0
F^2*f^3-1=0

Ranges: for all unknowns [0,1e5]
Solving method: HessianSolve+HullConsistencyStrong+0.95
Solutions:: 5 (exact)
Computation time:
 Sun Blade 28.69s (April 2003) EVO 410C (1.2Ghz) 4.68s (April 2003) DELL D400 (1.7Ghz) 0.95s (May 2004)

## Stenger

Origin: [7]
2 equations:

Ranges: [-1e8,1e8]

Solving method: HessianSolve+HullConsistency+ 3B
Solutions:: 2 (exact)
Computation time

 DELL 620 (May 2007) (1.7GHz) 0.01s

## Yamamura1

Origin: Yamamura [15]
n equations defined by:

solved here for n=16,32

Ranges: for all unknowns [-1e8,1e8]

Solutions:: 9 for n=16, 7 for n=32 (exact)
Computation time (April 2003):

 Sun Blade 2.11s EVO 410C (1.2Ghz) 39.25s (n=16), 975s (n=32)

Next: Non-polynomial systems Up: benches Previous: benches
• La page de J-P. Merlet