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    Subsections


    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


    \begin{displaymath}
n-8\frac{ln(n)}{ln(2)}=0
\end{displaymath}

    Range: [1,1000]

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

    Sun Blade 0.06s

    Box3

    Origin: [9]
    Let

    \begin{displaymath}
t_i =0.1 i
\end{displaymath}

    and

    \begin{displaymath}
f_i = e^{-t_ix_1}-e^{-t_ix_2}-x_3(e^{-t_i}- e^{-10 t_i})
\end{displaymath}

    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

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

    and

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

    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

    \begin{displaymath}
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}}
\end{displaymath}

    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 $v$:

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

    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:

    \begin{displaymath}
x_1e^{-x_2(t_i-x_3)^2/2}-y_i
\end{displaymath}

    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

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

    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

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

    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]

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

    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]

    \begin{displaymath}
e^x-2x-1=0
\end{displaymath}

    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

    \begin{displaymath}
2.77675 x^3 (1 + cos(x) cosh(x)) -( sin(x) cosh(x) - sinh(x) cos(x))
\end{displaymath}

    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

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

    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

    \begin{displaymath}
\sin(x)-x/100
\end{displaymath}

    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

    \begin{displaymath}
(3x^3-5x+2)sin(x)^2+(x^3+5x)sin(x)-2x^2-x-2
\end{displaymath}

    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]

    \begin{displaymath}
exp(x)+arctan(x)=0
\end{displaymath}

    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:

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

    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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