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Subsections

• AOL-cosh1
• Non algebraic: AOL-legentil
• AOL-log1
• Box3
• Branin system
• Bratu
• Bullard
• Collins
• comp.soft-sys.math.maple-14706
• Design problem
• DiGregorio
• Electrical circuit
• Extended Powell
• Ferraris
• Gaussian
• Geometrica1
• Helical
• Helical1
• Helical valley
• Jennrich1
• Kincox modified
• Kin1 modified
• Lambert
• Lambert1
• Mixed algebraic-trigonometric
• Powell
• Puma modified
• Reactor
• Rump univariate
• sci.math.num-analysis1
• sci.math.num-analysis-89741
• sci.math.num-analysis90897
• sci.math.num-analysis92191
• sci.math.num-analysis97384
• Semi-conductor (Molenaar)
• Sinxx
• Sinxx1
• Sjirk-Boon
• stoutemyer-eq-2007
• Trigexp1
• Trigexp2
• Trigexp3
• Trigo1
• Troesch
• Xu
• Zufiris

Non-polynomial systems

AOL-cosh1

Origin:AOL
3 equations

$$\begin{eqnarray*} x * cosh( ( 1 / x ) + y) + z - 1=0\\ x * cosh( ( 2 / x ) + y) + z - 4=0\\ x * cosh( ( 3 / x ) + y) + z - 9=0 \end{eqnarray*}$$

Ranges: [-100,100]

Solving method: GradientSolve+HullConsistency+3B
Solutions:: 1 (exact)
Computation time (October 2004):

DELL D400 (1.7GHz)

5.96s

Non algebraic: AOL-legentil

3 equations

 
10/3*cos(x)/sin(x)^2+4*(1+tan(x)^2)/cos(y)+z*(-50/3*sin(y)*cos(x)/(sin(x)^2*(3.5-5*sin(y)))-10/3*cos(x)/sin(x)^2-4*(1+tan(x)^2)/cos(y))=0

4*tan(x)*sin(y)/cos(y)^2+z*(50/3*cos(y)/(sin(x)*(3.5-5*sin(y)))+250/3*sin(y)*cos(y)/(sin(x)*(3.5-5*sin(y))^2)-4*tan(x)*sin(y)/cos(y)^2)=0

50/3*sin(y)/(sin(x)*(3.5-5*sin(y)))+20+10/3/sin(x)-4*tan(x)/cos(y)=0

Ranges: $]0,\pi/2[, [0,\pi/2[,[-1e8,1e8]$ for x,y,z
Solving method: GradientSolve+3B+Simp2B
Solutions:: 0
Computation time:

DELL D620 (1.7GHz), (May 2007):

9mn

Note that this problem may easily be solved by using the Weierstrass substitution to transform the equations into algebraic one and using resultants to obtain an univariate polynomial that leads to all solutions.

AOL-log1

Origin:AOL

$$n-8\frac{ln(n)}{ln(2)}=0$$

Range: [1,1000]

Solving method: GradientSolve
Solutions:: 2 (exact)
Computation time (February 2005):

Sun Blade

0.06s

Box3

Origin: [9]
Let

$$t_i =0.1 i$$

and

$$f_i = e^{-t_ix_1}-e^{-t_ix_2}-x_3(e^{-t_i}- e^{-10 t_i})$$

for i =1,2,3
Range: [-100,100],[-100,100],[0.1,100]

Solving method: HessianSolve,3B,2B
Solutions:: 1 (exact)
Computation time (April 2003):

EVO 410C (1.2GHz)

10.4s

Branin system

Origin: mentioned by Rump[11]

$$\begin{eqnarray*} &&2 \sin(2 \pi x1/5.) \sin(2 \pi x3/5.)-x2=0\\ &&2.5-x3+0.1~ x2~ \sin(2 \pi x3)-x1=0\\ &&1+0.1~ x2~ \sin(2 \pi x1)-x3=0\\ \end{eqnarray*}$$

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

Solving method: GradientSolve,3B,HullConsistency
Solutions:: 1 (exact)
Computation time (September 2006):

DELL D400 (1.7GHz)

0.01s

Bratu

Physical meaning: PDE in combustion theory
Origin: unknown

Let

$$x_0 = x_{n+1}=0~~~~~h=\frac{1}{(n+1)^2}$$

and

$$x_{k-1}-2 x_k+ x_{k+1} + h e^{x_k}~~{\rm for}~k \in [1,n]$$

with n=15 Unknowns: x,.., xn
Range: [-1e8,20]

Solving method: HessianSolve,HullConsistency
Solutions:: 2 pour n=15, 30 (exact)
Computation time:

Sun Blade	6.94s (April 2003)
DELL D400 (1.7Ghz)	0.59s (n=15), 9.1s (n=30 (May 2004))

Bullard

Origin: [6]

$$\begin{eqnarray*} &&10000 x_1 x_2-1\\ &&e^{-x_1}+e^{-x_2}-1.001\\ \end{eqnarray*}$$

Ranges: [-1e8,1e8]

Solving method: HessianSolve+HullConsistency+3B
Solutions:: 2 (exact)
Computation time (April 2003):

Evo 410C (1.2Ghz)

0.01s

Collins

Origin: [2]
Physical meaning: allows to determine the configuration of a parallel robot being given its joint coordinates

$$3.9852- 10.039\,{q_4}^{2}+ 7.2338\,{q_4}^{4}- 1.17775\,{q_4... ...0.091\,{q_4}^{3}- 11.177\,{q_4}^{5}\right )\sqrt {1-{q_4}^{2}}$$

Range: [-1,1]

Solving method: GradientSolve,HullConsistency
Solutions:: 6 (exact)
Computation time (April 2003):

EVO 410C (1.2GHz)

0.2s

comp.soft-sys.math.maple-14706

Origin: Article 14706 of comp.soft-sys.math.maple

$$\begin{eqnarray*} &&-sin(x)*cos(y)-2*cos(x)*sin(y)\\ &&-cos(x)*sin(y)-2*sin(x)*cos(y) \end{eqnarray*}$$

Ranges: [0,2Pi]

Solving method: HessianSolve+3B
Solutions:: 13 (exact)
Computation time (November 2005):

Sun Blade

11.36s

Design problem

Origin: [6]
9 equations

$$\begin{eqnarray*} && 23.3037 x_2+(1-x_1 x_2)x_3 (e^{x_5 ( 0.485- 0.0052095 x_7- ... ....0202153 x_7+ 0.191267 x_9)}-1)+ 191.267\\ &&x_1 x_3-x_2 x_4\\ \end{eqnarray*}$$

 
 23.3037 x_2+(1-x_1 x_2)x_3 (e^{x_5 ( 0.485- 0.0052095 x_7- 0.0285132 x_8)}-1)- 28.5132
- 28.5132 x_1+(1-x_1 x_2)x_4 (e^{x_6 ( 0.116- 0.0052095 x_7+ 0.0233037 x_9)}-1)+ 23.3037
 101.779 x_2+(1-x_1 x_2)x_3 (e^{x_5 ( 0.752- 0.0100677 x_7- 0.1118467 x_8)}-1)- 111.8467
- 111.8467 x_1+(1-x_1 x_2)x_4 (e^{x_6 (- 0.502- 0.0100677 x_7+ 0.101779 x_9)}-1)+ 101.779
 111.461 x_2+(1-x_1 x_2)x_3 (e^{x_5 ( 0.869- 0.0229274 x_7- 0.1343884 x_8)}-1)- 134.3884
- 134.3884 x_1+(1-x_1 x_2)x_4 (e^{x_6 ( 0.166- 0.0229274 x_7+ 0.111461 x_9)}-1)+ 111.461
 191.267 x_2+(1-x_1 x_2)x_3 (e^{x_5 ( 0.982- 0.0202153 x_7-0.2114823 x_8)}-1)- 211.4823
- 211.4823 x_1+(1-x_1 x_2)x_4 (e^{x_6 (- 0.473- 0.0202153 x_7+ 0.191267 x_9)}-1)+ 191.267
x_1 x_3-x_2 x_4

Ranges: [0,10]

Solving method: HessianSolve+HullConsistencyStrong +3B
Solutions:: 1 (exact)
Computation time:

Cluster (11 PC's)	3h 49mn (April 2003)
DELL D400 (1.7GHz)	398s (May 2004)

DiGregorio

Origin:Digregorio [3]

$$\begin{eqnarray*} &&-1000+\left (10-40\,\cos(p)+q\right )^{2}+\left (-40+40\,\si... ...(-10-35\,\cos(t) \right )^{2}-1225\,\left (\sin(t)\right )^{2}=0 \end{eqnarray*}$$

Ranges: for $p,t$ [0,$2\pi$], for $q$ [-1000,1000]

Solving method: GradientSolve+3B
Solutions:: 4 (exact)
Computation time (September 2006):

DELL D400 (1.7Ghz)

0.55s

Electrical circuit

Origin:Quateroni A, Sacco, R. and Saleri, F., Numerical mathematics, Springer 2000, pp. 413

Solve in $v$:

$$v(1/R+\mu \gamma)-\mu v^2+\alpha*(e^{v/\beta}-1)-E/R =0$$

with $R=1$, $\mu =1e-3$, $\gamma = 0.4$ $\alpha = 1e-12$, $E=1.2 e-4$, $\beta = 1/40$

Ranges: [-1e8,1e8]
Solving method: HessianSolve+HullConsistencyStrong
Solutions:: 1 (exact)
Computation time:

DELL D400 (1.7GHz)

0.003s (February 2006)

Extended Powell

Origin: [8]
n=2m equations:

$$\begin{eqnarray*} &&{\rm if}~mod(i,2)=1~~10^4 x_ix_{i+1}-1\\ &&{\rm if}~mod(i,2)=0~~e^{-x_{i-1}}+e^{-x_i}-1.0001\\ \end{eqnarray*}$$

Ranges: [-100,100] Solving method: HessianSolve+HullConsistencyStrong+3B
Solutions:: 32 (n=10), 200 (n=20) (exact)
Computation time:

Evo 410C (1.2Ghz)	23.32s (n=10) (April 2003)
DELL D400 (1.7Ghz)	1.72s (n=10), 88.48s (n=20) (May 2004)

Ferraris

Origin: [6]

$$\begin{eqnarray*} &&-0.25/\pi x_2 - 0.5x_1 + 0.5 \sin(x_1~x_2)\\ && e/\pi x_2 - 2e~x_1 + (1 - 0.25/\pi)(e^{2x_1} -e)\\ \end{eqnarray*}$$

 
 -0.25/Pi*x2 - 0.5*x1  + 0.5*sin(x1*x2);
exp(1)/Pi*x2 - 2*exp(1)*x1  + (1 - 0.25/Pi)*(exp(2*x1) -exp(1));

Ranges: [-1e8,1e8]
Solving method: HessianSolve+HullConsistency+3B
Solutions:: 12 (exact)
Computation time (April 2003):

Evo 410C (1.2Ghz)

0.16s

Gaussian

Origin:COPRIN, derived from [9]
3 equations:

$$x_1e^{-x_2(t_i-x_3)^2/2}-y_i$$

with $t_i =(8-i)/2$, y1=0.0009, y2=0.0044, y3=0.0175

Ranges: [-9,9] for all unknowns

Solving method: HessianSolve+HullConsistencyStrong+3B
Solutions:: 1 (exact)
Computation time:

Evo 410C (1.2Ghz)	12.84s (April 2003)
DELL D400 (1.7GHz)	2.33s (May 2004)

Geometrica1

Origin:COPRIN, derived from a mesh problem submitted by M. Pouget and F. Cazals from the Geometrica project.
2 equations, 2 unknowns:

 
(36/25+(x+1)^2)^(1/2)+1/2*(36/25+(1+y)^2)^(1/2)+1/2*(72/25+(-x+1+y)^2)^(1/2)+
(36/25+(1-x)^2)^(1/2)+1/2*(36/25+(2-y)^2)^(1/2)+
1/2*(72/25+(x+2-y)^2)^(1/2)-4*2^(1/2)-3^(1/2)=0

signum(-1/(36/25+(1+y)^2)^(1/2))*arccos((1+y)/(36/25+(1+y)^2)^(1/2))+Pi+
2*signum(-1/(36/25+(1-x)^2)^(1/2))*arccos((1-x)/(36/25+(1-x)^2)^(1/2))+
2^(1/2)*signum(-1/(72/25+(x+2-y)^2)^(1/2))*
arccos((x+2-y)/(72/25+(x+2-y)^2)^(1/2))+
signum(-1/(36/25+(2-y)^2)^(1/2))*arccos((2-y)/(36/25+(2-y)^2)^(1/2))+
((x-1)^2+y^2+36/25)^(1/2)*
signum((366+150*y^2+150*x^2-300*x)/(72/25+(-x+1+y)^2)^(1/2)/(36/25+(1-x)^2)^(1/2))*
arccos(1/25*(61+25*x^2-50*x-25*x*y+25*y)/(72/25+(-x+1+y)^2)^(1/2)/(36/25+(1-x)^2)^(1/2))+
2*signum(-1/(36/25+(x+1)^2)^(1/2))*
arccos((x+1)/(36/25+(x+1)^2)^(1/2))+(x^2+(y-2)^2+36/25)^(1/2)*
signum((816+150*y^2-600*y+150*x^2)/(72/25+(x+2-y)^2)^(1/2)/(36/25+(2-y)^2)^(1/2))*
arccos(1/25*(136-25*x*y-100*y+25*y^2+50*x)/(72/25+(x+2-y)^2)^(1/2)/(36/25+(2-y)^2)^(1/2))+
(x^2+(1+y)^2+36/25)^(1/2)*
signum((366+300*y+150*y^2+150*x^2)/(72/25+(-x+1+y)^2)^(1/2)/(36/25+(1+y)^2)^(1/2))*
arccos(1/25*(61-25*x+50*y-25*x*y+25*y^2)/(72/25+(-x+1+y)^2)^(1/2)/(36/25+(1+y)^2)^(1/2))+
((x+1)^2+(y-1)^2+36/25)^(1/2)*
signum((300*x-300*y+516+150*y^2+150*x^2)/(72/25+(x+2-y)^2)^(1/2)/(36/25+(x+1)^2)^(1/2))*
arccos(1/25*(86+25*x^2+75*x-25*x*y-25*y)/(72/25+(x+2-y)^2)^(1/2)/(36/25+(x+1)^
2)^(1/2))
-4*2^(1/2)*arccos(1/3*3^(1/2)*2^(1/2))+2^(1/2)*
signum(-1/(72/25+(-x+1+y)^2)^(1/2))*
arccos((-x+1+y)/(72/25+(-x+1+y)^2)^(1/2))+((x-1)^2+(y-2)^2+36/25)^(1/2)*
signum((966+150*y^2-600*y+150*x^2-300*x)/(36/25+(1-x)^2)^(1/2)/(36/25+(2-y)^2)^(1/2))*
arccos((x-1)*(y-2)/(36/25+(1-x)^2)^(1/2)/(36/25+(2-y)^2)^(1/2))+((x+1)^2+(1+y)^2+36/25)^(1/2)*
signum((516+300*y+150*y^2+150*x^2+300*x)/(36/25+(x+1)^2)^(1/2)/(36/25+(1+y)^2)^(1/2))*
arccos((x+1)*(1+y)/(36/25+(x+1)^2)^(1/2)/(36/25+(1+y)^2)^(1/2))+
2*2^(1/2)*arccos(1/3*3^(1/2))-2/3*3^(1/2)*Pi =0

where signum(x)=1 if $x >=0$, -1 if $x<0$
Ranges: [-1,1] for all unknowns

Solving method: GeneralSolve
Solutions:: 0 (exact)
Computation time:

DELL D400 (1.7GHz)

1.7s (July 2004)

Helical

Origin: [9]

$$\begin{eqnarray*} &&-100\frac{1}{\pi} \arctan(\frac{x_2}{x_1})+0.5\\ &&10(\sqrt{x_1^2+x_2^2}-1)\\ \end{eqnarray*}$$

Range: x1 in [-1e8,-1e-3], x2 in [-1e8,1e8]

Solving method: GradientSolve,3B
Solutions:: 1 (exact)
Computation time (April 2003):

EVO 410C (1.2GHz)

0.04s

Helical1

Origin: COPRIN, derived from [9]

$$\begin{eqnarray*} &&10(x_3-10(\frac{1}{\pi}\arctan(\frac{x_2}{x_1})+0.5))\\ &&10(\sqrt{x_1^2+x_2^2}-1)\\ &&x_1-\sin(x_3)\\ \end{eqnarray*}$$

Range: x1 in [-1e8,-1e-3], x2,x3 in [-1e8,1e8]

Solving method: GradientSolve,3B
Solutions:: 2 (exact)
Computation time (April 2003):

EVO 410C (1.2GHz)

0.08s

Helical valley

Origin:

$$\begin{eqnarray*} &&10000x_1x_2-1\\ && e^{-x_1}+ e^{-x_2}-1.0001 \end{eqnarray*}$$

Range: x1 in [-1e8,1e8], x2 in [-1e8,1e8]

Solving method: GradientSolve,3B
Solutions:: 1 (exact)
Computation time (April 2003):

DELL D620 (1.7GHz)

0.02s

Jennrich1

Origin: COPRIN, derived from [9]
2 equations with 2 unknowns

$$\begin{eqnarray*} &&3 -(e^{x_1}+e^{x_2})\\ &&6-(e^{2x_1}+e^{2x_2})\\ \end{eqnarray*}$$

Range: [-5,5] for all unknowns

Solving method: HessianSolve,3B
Solutions:: 2 (exact)
Computation time (April 2003):

EVO 410C (1.2GHz)

0.01s

Kincox modified

Origin: COPRIN, derived from Kincox COCONUT
2 equations with 2 unknowns with a =1, b=4, l1=10, l2=6

 
-1.+6.*cos(t1)*cos(t2)-6.*sin(t1)*sin(t2)+10.*cos(t1)=0
 -4.+6.*cos(t1)*sin(t2)+6.*cos(t2)*sin(t1)+10.*sin(t1)=0

Ranges: [0,2$\pi$] for all unknowns

Solving method: GradientSolve+HullConsistencyStrong+ +3B
Solutions:: 2 (exact)
Computation time (May 2004):

DELL D400 (1.7GHz)

0.04s

Kin1 modified

Origin: COPRIN, modified from COCONUT kin1

 
sin(t2)*cos(t5)*sin(t6)-sin(t3)*cos(t5)*sin(t6)-sin(t4)*cos(t5)*sin(t6)+
cos(t2)*cos(t6)+cos(t3)*cos(t6)+cos(t4)*cos(t6)-0.4077 =0

cos(t1)*cos(t2)*sin(t5)+cos(t1)*cos(t3)*sin(t5)+cos(t1)*cos(t4)*sin(t5)
+sin(t1)*cos(t5)-1.9115 =0

sin(t2)*sin(t5)+sin(t3)*sin(t5)+sin(t4)*sin(t5)-1.9791 =0

cos(t1)*cos(t2)+cos(t1)*cos(t3)+cos(t1)*cos(t4)+cos(t1)*cos(t2)+
cos(t1)*cos(t3)+cos(t1)*cos(t2)-4.0616 =0

sin(t1)*cos(t2)+sin(t1)*cos(t3)+sin(t1)*cos(t4)+sin(t1)*cos(t2)+sin(t1)*cos(t3)+sin(t1)*cos(t2)-1.7172
=0

sin(t2)+sin(t3)+sin(t4)+sin(t2)+sin(t3)+sin(t2)-3.9701 =0

Ranges: [0,$2\pi$]

Solving method: HessianSolve+3B+HullConsistencyStrong
Solutions:: 16 (exact)
Computation time:

DELL D400 (1.7GHz)

155.18s (August 2004)

Lambert

Origin: a classical astronomy problem

We define:

r0,r;
                                     11                    10
                  .138256930224638 10  , .68876844473745 10
mu:=13.271244e10:
dnu:=Pi/4.:
A:=sqrt(r0*r*(1+cos(dnu))):
S:=(u-sin(u))/u^3:
y:=r0+r-A*sin(u)/sqrt(1-cos(u)):
x:=u*sqrt(y)/sqrt(1-cos(u)):

The equation to solve in $u$ is

$$(S~x^3+A\sqrt{y})/\sqrt{\mu}= 1e^{10}$$

Ranges: [0.1, 2$\pi$-0.1]

Solving method: GradientSolve+3B
Solutions:: 1 (exact)
Computation time:

DELL D400 (1.7GHz)

0.11s (September 2006)

Lambert1

Origin: another formulation of a classical astronomy problem

We define:

r0,r;
                                     11                    10
                  .138256930224638 10  , .68876844473745 10
mu:=13.271244e10:

We then define $S=r+r_0-2\sqrt{rr_0}\cos(x)\cos(\pi/8)$. The equation to solve in $x$ is

$$\sqrt{\frac{S^3}{\mu}}\frac{1}{\sin(x)^3}(x-\tan(x)(1- \frac{r+r_0}{S}\sin(x)^2))= 1e^{10}$$

Ranges: [0.1, $\pi$/2-0.1]

Solving method: GradientSolve+3B
Solutions:: 1 (exact)
Computation time:

DELL D400 (1.7GHz)

0.04s (September 2006)

Mixed algebraic-trigonometric

Origin:sci.math.num-analysis

$$\begin{eqnarray*} &&-x + y + z + 2 sin(y-z) = 1\\ &&-x + y - z + 2 sin(x-z) = 1\\ &&x + y - z + 2 sin(x-y) = 1\\ \end{eqnarray*}$$

Ranges: [-1e8,1e8]

Solving method: HessianSolve+HullConsistency+3B
Solutions:: 1 (exact)
Computation time:

Sun Blade	6.85s (April 2003)
DELL D400 (1.7GHz)	0.1s (May 2004)

Powell

Origin: [9]

$$\begin{eqnarray*} &&10^4 x_1x_2-1\\ &&e^{-1}+e^{-x_2}-1.0001 \end{eqnarray*}$$

Ranges: [-100,100] for all unknowns

Solving method: HessianSolve+HullConsistency+3B
Solutions:: 2 (exact)
Computation time (April 2003):

Evo 410C (1.2Ghz)

0.01s

Puma modified

Origin: COPRIN, modified from COCONUT Puma
Physical meaning: inverse kinematics of a 3R robot, the angles t1,t2,t3 are the angles of the three joints
4 equations with 4 unknowns

 
.4731e-2*cos(t1)*cos(t2)-.3578*sin(t1)*cos(t2)-.1238*cos(t1)
-.1637e-2*sin(t1)-.9338*sin(t2)+cos(t4)-.3571=0

 .2238*cos(t1)*cos(t2)+.7623*sin(t1)*cos(t2)+.2638*cos(t1)
-.7745e-1*sin(t1)-.6734*sin(t2)-.6022=0

 sin(t3)*sin(t4)+.3578*cos(t1)+.4731e-2*sin(t1)=0

 -.7623*cos(t1)+.2238*sin(t1)+.3461=0

Ranges: [0,$2\pi$] for all unknowns

Solving method: GradientSolve+HullConsistencyStrong+ +3B
Solutions:: 16 (exact)
Computation time (May 2004):

DELL D400 (1.7GHz)

0.48s

See also the algebraic puma

Reactor

Origin: [6]

$${\frac {1}{298}}\,T+{\frac {672000000}{149}}\,T{e^{- 7548.11... ...,{T}^ {-1}}}- 2020510067.0\,{e^{- 7548.119260\,{T}^{-1}}}-1 =0$$

Ranges: [100,1000]

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

Evo 410C (1.2Ghz)

0.06s

Rump univariate

Origin: mentioned by Rump[11]

$$e^x-2x-1=0$$

Ranges: [-1e8,1e8]

Solving method: HessianSolve,3B
Solutions:: 2 (exact)
Computation time (September 2006):

DELL D400 (1.7GHz)

0.006s

sci.math.num-analysis1

Origin: Article 86825 of sci.math.num-analysis

$$2.77675 x^3 (1 + cos(x) cosh(x)) -( sin(x) cosh(x) - sinh(x) cos(x))$$

Ranges: [-100,100]

Solving method: HessianSolve+HullConsistency+3B
Solutions:: 65 (49 approximate, 16 exact)
Computation time (September 2005):

DELL D400 (1.2Ghz)

0.06s

Solving method: Maple, based on intpakX
Solutions:: 65 (1 approximate, 64 exact)
Computation time (November 2005):

DELL D400 (1.2Ghz)

1069s

sci.math.num-analysis-89741

Origin: Article 89741 of sci.math.num-analysis

$$\begin{eqnarray*} &&-\cos (x)/81 + (y)^2/9 + \sin(z)/3-x\\ &&\sin(x)/3 + \cos (z)/3-y\\ &&-\cos (x)/9 + (y)/3 +\sin (z)/6-z\\ \end{eqnarray*}$$

Ranges: [-1e8,1e8] for all unknowns (may be largely improved..)

Solving method: HessianSolve+HullConsistency+3B
Solutions:: 1, exact
Computation time (April 2006):

DELL D400 (1.2Ghz)

$\approx$ 0.008s

sci.math.num-analysis90897

Origin: Article 90897 of sci.math.num-analysis

$$\begin{eqnarray*} &&x_1- 0.0000179297550\,x_2^{2}=0\\ && x_2-\frac{90}{1-x_1}=0\\ \end{eqnarray*}$$

Ranges: [-1000,0.99],[-1e8,1e8]

Solving method: HessianSolve+3B
Solutions:: 2 (exact)
Computation time (June 2006):

Sun Blade

0.13s

sci.math.num-analysis92191

Origin: Article 92191 of sci.math.num-analysis

$$\begin{eqnarray*} && 0.7\,{e^{-12\,b}}\cos(12\,a)+ 0.3\,{e^{-13\,b}}\cos(13\,a)-... ...&& 0.7\,{e^{-12\,b}}\sin(12\,a)+ 0.3\,{e^{-13\,b}}\sin(13\,a)\\ \end{eqnarray*}$$

Ranges: [0,$2\Pi$],[-10,30]

Solving method: HessianSolve or GradientSolve+3B
Solutions:: 14 (exact)
Computation time (September 2006):

DELL D400 (1.7GHz)

21.3s (GradientSolve), 68.15s (HessianSolve)

sci.math.num-analysis97384

Origin: Article 97384 of sci.math.num-analysis
Modified by COPRIN to scale the unknown

$$2.800000000\times 10^{38}\,{\frac {{ 2.7}^{- 0.2800000000\,... ...{-67}}\,{M}^{2}}{M}^{2} \right) }{{M}^{4 }}}- 0.00001900000000$$

Ranges: [1e-4,1000]

Solving method: HessianSolve+3B
Solutions:: 1 (exact)
Computation time (July 2007):

DELL D620 (1.7Ghz)

0.01s

Semi-conductor (Molenaar)

Origin: J. Molenaar, P.W. Hemker: A multigrid approach for the solution of the 2D semiconductor equations. IMPACT Comput. Sci. Eng. 2, No. 3, p. 219-243 (1990).
6 equations with a=38.683, ni=1.22e10, V=100, DDD=10.e17.

$$\begin{eqnarray*} &&eq1=x2\\ &&eq2=x3\\ &&eq3=x5-V\\ &&eq4=x6-V\\ &&eq5=e^{a... ...a*(x1-x2)}-DDD/ni\\ &&eq6=e^{a*(x6-x4)}-e^{a*(x4-x5)}+DDD/ni\\ \end{eqnarray*}$$

Ranges: [-400,400]

Solving method: HessianSolve+HullConsistency+3B
Solutions:: 1 (exact)
Computation time (April 2003):

Sun Blade

0.16s

Sinxx

Origin: COPRIN

$$\sin(x)-x/100$$

Range: [0,A], where A may be chosen arbitrary large without any change in the computation time

Solving method: GradientSolve,HullConsistency
Solutions:: 32 (exact)
Computation time (April 2003):

Sun Blade

0.051s

Sinxx1

Origin: COPRIN

$$(3x^3-5x+2)sin(x)^2+(x^3+5x)sin(x)-2x^2-x-2$$

Range: [-10,10]

Solving method: GradientSolve,HullConsistency
Solutions:: 9 (exact)
Computation time (April 2003):

Sun Blade

0.062s

Sjirk-Boon

Origin: news from Sjirk-Boon, modified by COPRIN

$$\begin{eqnarray*} &&5 + C1 cos(3 phi1) + C2 cos(3 phi2)\\ && -3 + C1 cos(phi1) ... ...phi1) + C2 sin(3 phi2)\\ && -4 + C1 sin(phi1) + C2 sin(phi2)\\ \end{eqnarray*}$$

Range: [-100,100] for $C1,C2$, $[0, 2\pi]$ for $phi1, phi2$

Solving method: GradientSolve,Simp2B, 3B
Solutions:: 8 (exact)
Computation time:

Dell D620 (1.7GHz) (May 2007)

94s

stoutemyer-eq-2007

Origin: [12]

$$exp(x)+arctan(x)=0$$

Range: [-1e8,1e8]

Solving method: GradientSolve
Solutions:: 1 (exact)
Computation time (November 2007):

DELL D620

0.01s

Trigexp1

Origin: [8]
n equations:

$$\begin{eqnarray*} &&3 x_1^3+2 x_2-5+\sin(x_1-x-2) \sin(x_1+x_2)\\ &&4 x_n-x_{n-... ...i-x_{i+1}) \sin(x_i+x_{i+1}) -4 x_i-x_{i-1} e^{x_{i-1}-x_i}-3\\ \end{eqnarray*}$$

for n=20
Ranges: [-100,100]

Solving method: HessianSolve+HullConsistencyStrong+3B
Solutions:: 1 (exact) for n=20,30, 50
Computation time:

Evo 410C (1.2Ghz)	693.88s (April 2003)
DELL D400 (1.7GHz)	0.9s (n=20), 2.3s (n=30) 12.9s (n=50) (May 2004)

Trigexp2

Origin: [8]
n=2m+1 equations:

$$\begin{eqnarray*} &&{\rm if}~mod(i,2)=1, i =1~~ 3 (x_i-x_{i+2})^3-5+2 x_{i+1}+\s... ...od(i,2)=0~~ 4 x_i-(x_{i-1}-x_{i+1}) e^{x_{i-1}-x_i-x_{i+1}}-3\\ \end{eqnarray*}$$

Ranges: [-200,200]

Solving method: HessianSolve or GradientSolve +HullConsistencyStrong+ 3B
Solutions:: 0 (n=5,7,9,11) (exact)
Computation time (May 2004):

DELL D400 (1.7Ghz)

15.31s (n=5) 58.59s (n=7) 115.3s (n=9) 327s (n=11)

This system may be simplified by introducing the new variables $u_1=x_1-x_3,u_2=x_3-x_5,\ldots,u_m=x_{n-2}-x_n$ i.e. m new variables and the corresponding new equations. We end up with a system of m+n equations in the m+n unknowns xi, ui.

Solutions:: 0 (n=21) (exact)
Computation time (April 2003):

Evo 410C (1.2Ghz)

2202s

Trigexp3

Origin: [8]
n equations with h=1/(n+1):

$$\begin{eqnarray*} &&x_1-e^{\cos(h(x_1+x_2))}\\ &&x_n-e^{\cos(h(x_{n-1}+x_n))}\\... ...\rm for }~1< i<n \\ && x_i -e^{\cos(h(x_{i-1}+x_i+x_{i+1}))}\\ \end{eqnarray*}$$

for n=20,40
Ranges: [$e^{-1},e^{1}$]

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

Evo 410C (1.2Ghz)

0.51s (n=20), 2.64s (n=40)

Trigo1

Origin: [9]
n equations:

$$n-\sum_{j=1}^{j=n}\cos(x_j)+i(1-\cos(x_i))-\sin(x_i)$$

for n=5
Ranges: [0.1,$2\pi$-0.1]

Solving method: HessianSolve+3B+HullConsistencyStrong
Solutions:: 1 (exact)
Computation time:

Evo 410C (1.2GHz)	1.09s (April 2003)
DELL D400 (1.7GHz)	0.36s (May 2004)

Troesch

Origin: [8]
n equations with h=1/(n+1),R =10:

$$\begin{eqnarray*} &&2x_1+R~h^2\sinh(R~x_1)-x_2\\ &&2x_n+R~h^2\sinh(R~x_n)-x_{n-... ...m for } 1< i<n \\ && 2 x_i+R~h^2\sinh(R~x_i)-x_{i-1}-x_{i+1}\\ \end{eqnarray*}$$

for n=10,20
Ranges: [-10,10]

Solving method: HessianSolve+HullConsistency+3B
Solutions:: 1 for n=10, 15 (exact)
Computation time (April 2003):

Evo 410C (1.2GHz)

32.83s (n=10), 407s (n=15)

Xu

Origin: [14]
2 equations

$$\begin{eqnarray*} &&2\,\sin \left( x_1 \right) + 0.8\,\cos \left( 2\,x_1 \right)... ...\left( 3\,x_2 \right) + 3.1\,\cos \left( 2\,x_2 \right) -x_2\\ \end{eqnarray*}$$

Ranges: [-20,20] for $x_1, x_2$

Solving method: GradientSolve+Simp2B+3B
Solutions:: 29 (exact)
Computation time:

DELL D620 (1.7GHz), (July 2007)

0.037s

Zufiris

Origin: [16]
4 equations

$$\begin{eqnarray*} &&0.742476763140869227613777849835\,x_3- 3.712383815704346138... ...57784299212\,\cos \left( x_1 \right) \sin \left( x_2 \right)\\ \end{eqnarray*}$$

Ranges: [$-\pi,\pi$] for $x_1, x_2$, [-1.5,1.5] for $x_3, x_4$

Solving method: HessianSolve+Simp2B+3B
Solutions:: 13 (exact)
Computation time:

DELL D620 (1.7GHz), (July 2007)

0.07s

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