5
- 1.390350657*p3 + 0.4641393136*p3*p2 - 0.6266605268*p2 - 1.400171891*p4
- 2.090683800*p4*p3*p2 + 4.089263882*p4*p3 + 1.129827638*p4*p2
+ 1.881614464*p5*p4*p3*p2 + 0.4716169661*p5*p3*p2 - 0.8625849122*p5*p4*p2
- 1.398871056*p5*p2 + 0.9599693844*p5 + 0.714025397E-01*p5*p4
+ 0.1073802376*p5*p3 - 0.9259664538*p5*p4*p3 - 0.2067814278;
- 1.196136754*p3 - 0.9249804195*p4 + 0.3188761009*p4*p3 - 1.045301323*p3*p1
- 0.306661782E-01*p1 + 0.5987012929*p4*p3*p1 - 0.4448182692*p4*p1
- 0.3908068031*p5*p4*p3*p1 - 1.212939725*p5 + 2.586129779*p5*p4
- 0.1180169224*p5*p3 - 1.051519507*p5*p4*p3 + 2.134979375*p5*p3*p1
- 1.337061849*p5*p4*p1 - 0.2961272671*p5*p1 + 0.7316111016;
2.272943163*p2 - 0.4131564265*p4 - 1.920446680*p4*p2 + 0.80509234E-2*p1
+ 1.342851102*p4*p1 - 2.979502184*p2*p1 + 3.391571834*p4*p2*p1
- 0.5975693742*p5*p4*p2 + 0.3002794716*p5*p2 - 0.7893445350*p5
+ 1.276948001*p5*p4 - 4.601376311*p5*p4*p1 + 2.356804322*p5*p1
+ 3.498840190*p5*p4*p2*p1 - 1.355375015*p5*p2*p1 - 1.231070236;
- 2.206336116*p3 + 2.318673689*p3*p2 - 1.267478048*p2 + 1.110654516*p3*p1
+ 1.533592098*p1 - 1.872504375*p2*p1 + 0.3299103675*p3*p2*p1
- 3.400750472*p5*p3*p2 + 2.093674516*p5*p2 - 1.772874182*p5
+ 2.993821915*p5*p3 - 1.356762392*p5*p3*p1 + 0.637534233E-01*p5*p1
+ 0.5870371377*p5*p2*p1 + 1.018269743*p5*p3*p2*p1 + 1.400431557;
- 2.522718869*p3 + 0.8323646978*p3*p2 - 1.375039881*p2 - 0.3055443755*p4
+ 0.6760632172*p4*p3*p2 - 0.4262974456*p4*p3 + 1.268255245*p3*p1
+ 0.5352674901*p4*p2 - 1.024495558*p1+1.818275404*p4*p3*p1
- 1.354832512*p4*p1 - 1.595112039*p2*p1 + 2.237956242*p4*p2*p1
+ 3.370102170*p3*p2*p1 - 3.465040669*p4*p3*p2*p1 + 2.132631128;
TITLE : totally mixed Nash equilibria for 5 players with two pure strategies.
ROOT COUNTS :
total degree : 1024
5-homogeneous Bezout number : 44
with partition : {p3 }{p2 }{p4 }{p5 }{p1 }
generalized Bezout number : 44
based on the set structure :
{p3 }{p2 }{p4 }{p5 }
{p3 }{p4 }{p5 }{p1 }
{p2 }{p4 }{p5 }{p1 }
{p3 }{p2 }{p5 }{p1 }
{p3 }{p2 }{p4 }{p1 }
mixed volume : 44
REFERENCES :
Andrew McLennan: "The maximal generic number of pure Nash equilibria",
Journal of Economic Theory, Volume 72, pages 408-410, 1997.
Richard D. McKelvey and Andrew McLennan: "The maximal number of regular
totally mixed Nash equilibria",
Journal of Economic Theory, Volume 72, pages 411-425, 1997.
David M. Kreps and Robert Wilson: "Sequential Equibria",
Econometrica, Volume 50, Number 4, pages 863-894,1982.
DESCRIPTION :
The index set I = { 1,2,..,n } identifies people invited to attend a party.
The economical relevance is for instance a group of fishermen who have to
exploit a certain site. The mathematical problem is then to decide the
probabilities of attendance or frequency of participation, the variables
are probabilities p1,p2,..,pn, all in [0,1].
At an equilibrium state, the payoff remains unchanged when you vary your
frequency of participation. Formally, for the i-th participant we have
\sum_{S^i \subseteq I^i} p(S^i) u_i(S^i \cup \{ i \})
= \sum_{S^i \subseteq I^i} p(S^i) u_i(S^i),
for I^i = I \setminus \{ i \},
p(S^i) = \prod_{j \in S^i} p_j \prod_{j \not\in S^i} (1-p_j),
with p(S^i) expressing the probability that the group S^i attends
and u_i(S^i) a fixed constant meaning the utility for S^i to attend.
The equilibrium system then becomes:
\sum_{S^i \subseteq I^i} p(S^i) v_i(S^i) = 0, for i=1,2,..,n,
where the constants v_i(S^i) are the differences of the utilities.
The n-homogeneous Bezout number provides a generically exact root count.
It equals #derangements of I, which is #permutations without a fixed point.
Asymptotically, this number is n!/e, illustrating the exploding complexity.
The above instance is generated by a naive Maple program and has 10 real
and 34 conjugated complex solutions.
# MapleV5 program to generate the equations for totally mixed Nash equilibria,
# for games with n players each having two pure strategies.
# The utility constants are chosen as random real numbers.
# Type "eqs(5)" to generate the 5-dimensional game.
prdp := proc ( n,i ) # returns product of p_j, j=1,2,..,n, j/= i
local res,j;
res := 1;
for j from 1 to n do
if j <> i
then res := res*p.j;
fi;
od;
RETURN(res);
end;
die := convert(rand(-10^14..10^14)/10^14,float);
recp := proc ( n,i,k,acc ) # generates all monomials for i-th equation
local res;
res := 0;
if k <> i
then res := die()*p.k*acc + die()*(1-p.k)*acc;
if k < n
then res := res + recp(n,i,k+1,p.k*acc)
+ recp(n,i,k+1,(1-p.k)*acc);
fi;
else if k < n
then res := recp(n,i,k+1,acc);
fi;
fi;
RETURN(res);
end;
nash := proc ( n,i ) # generates i-th equation
local res,acc;
acc := 1;
res := recp(n,i,1,acc);
RETURN(res);
end:
eqs := proc ( n ) # prints the equations
local i;
for i from 1 to n do
print(equation.i);
lprint(expand(nash(n,i)));
od;
end;
TIMING (black-box PHC, on Pentium II PC running Linux) :
---------------------------------------------------------------------
| TIMING INFORMATION SUMMARY |
---------------------------------------------------------------------
| root counts | start system | continuation | total time |
---------------------------------------------------------------------
| 0h 1m23s370ms | 0h 0m 0s460ms | 0h 0m30s840ms | 0h 1m55s930ms |
---------------------------------------------------------------------
THE SOLUTIONS :
44 5
===========================================================================
solution 1 : start residual : 1.071E-14
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
p3 : 9.68676942518830E-01 7.49765877380210E-01
p2 : 1.04714394118568E-01 -1.30758750730968E+00
p4 : 6.97073318730708E-01 2.67202351190649E-01
p5 : 5.81281518681907E-01 -8.93829248107594E-01
p1 : -1.23402195986078E+00 2.77271973715537E+00
== err : 8.139E-15 = rco : 2.485E-02 = res : 1.071E-14 = complex regular ==
solution 2 : start residual : 3.622E-15
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
p3 : 2.57755419497223E-01 3.15932696721481E-01
p2 : 7.02702750892653E-01 3.70317529765048E-01
p4 : 1.49189204499473E+00 -8.23201409816983E-01
p5 : 2.08493308415588E-01 8.59102443844596E-01
p1 : 8.44293748511489E-01 2.60196654708793E-01
== err : 4.388E-15 = rco : 5.701E-02 = res : 3.622E-15 = complex regular ==
solution 3 : start residual : 8.986E-16
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
p3 : 3.57329480736462E-02 0.00000000000000E+00
p2 : 2.87587470893108E+00 -4.27642353614751E-50
p4 : 8.45449288811566E-01 2.67276471009220E-50
p5 : -1.02963635659300E-01 0.00000000000000E+00
p1 : -6.30080437281213E-01 0.00000000000000E+00
== err : 7.535E-16 = rco : 5.957E-02 = res : 8.986E-16 = real regular ==
solution 4 : start residual : 1.769E-15
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
p3 : -1.16767884751289E-02 -4.06732769987895E-01
p2 : 5.95697594347170E-01 -4.39499087900653E-01
p4 : 3.97344768374143E-01 8.34323995673460E-01
p5 : 1.02174500227171E+00 -6.10111344112390E-01
p1 : 9.62421920712938E-01 4.93513847847347E-01
== err : 1.140E-15 = rco : 6.355E-02 = res : 1.769E-15 = complex regular ==
solution 5 : start residual : 6.994E-15
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
p3 : 6.10794596879131E-01 1.31726531251068E-02
p2 : 8.02593743364926E-01 -1.60065180711071E-01
p4 : 7.08909309217533E-01 4.44292111043910E-02
p5 : -2.07346832731395E+00 -3.72917578985534E+00
p1 : -2.89479806764954E-02 -8.42378661874933E-01
== err : 2.044E-14 = rco : 5.103E-03 = res : 6.994E-15 = complex regular ==
solution 6 : start residual : 4.632E-14
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
p3 : 1.31332676290301E+00 -2.27111357311342E+00
p2 : 7.97274972431726E-01 7.19095370581277E-01
p4 : 1.25315786065640E+00 -9.51549363527369E-01
p5 : 3.51070605933374E-01 1.37699955195814E+00
p1 : -1.86595939857726E+00 -5.34560615960333E+00
== err : 9.126E-15 = rco : 1.490E-02 = res : 4.632E-14 = complex regular ==
solution 7 : start residual : 3.743E-15
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
p3 : 6.33535795526992E-01 -1.66209061667017E-02
p2 : 7.63814591143358E-01 -5.33943530680257E-01
p4 : -3.41034998496300E-02 -4.22262280736296E-01
p5 : 6.59493092552209E-01 -1.81691253226117E+00
p1 : 1.63674938158538E+00 2.65331142527986E-01
== err : 2.287E-15 = rco : 3.466E-02 = res : 3.743E-15 = complex regular ==
solution 8 : start residual : 4.413E-14
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
p3 : 9.46970935591909E-01 -1.33210823092709E-02
p2 : 1.25383913644346E+00 3.63841070504421E-01
p4 : 6.47383507492214E+00 3.27211325558079E-01
p5 : -1.61756213582246E+00 2.24047214015871E+00
p1 : 1.06884459654476E+00 1.54999033378930E-01
== err : 4.953E-14 = rco : 1.732E-03 = res : 4.413E-14 = complex regular ==
solution 9 : start residual : 1.857E-14
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
p3 : 1.47977621808815E+00 1.31826332571720E-02
p2 : 8.71090072145853E-01 -5.89981849604203E-02
p4 : -3.74728737487147E-02 -1.11425043359847E+00
p5 : -1.41590767270165E+00 7.33115265908787E+00
p1 : 4.51662612989230E-01 -1.69228911106828E-02
== err : 3.100E-14 = rco : 7.827E-04 = res : 1.857E-14 = complex regular ==
solution 10 : start residual : 2.200E-15
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
p3 : 2.57755419497223E-01 -3.15932696721481E-01
p2 : 7.02702750892653E-01 -3.70317529765048E-01
p4 : 1.49189204499473E+00 8.23201409816982E-01
p5 : 2.08493308415588E-01 -8.59102443844596E-01
p1 : 8.44293748511489E-01 -2.60196654708793E-01
== err : 1.879E-15 = rco : 5.701E-02 = res : 2.200E-15 = complex regular ==
solution 11 : start residual : 7.216E-16
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
p3 : 4.00971058834554E-01 -3.50324616081204E-46
p2 : -3.52939902609422E-01 -5.60519385729927E-45
p4 : 2.50160999928267E+00 -1.12103877145985E-44
p5 : 3.44434239536088E-01 -2.80259692864963E-45
p1 : -1.51317146896049E-01 -3.50324616081204E-45
== err : 4.687E-16 = rco : 1.369E-02 = res : 7.216E-16 = real regular ==
solution 12 : start residual : 6.953E-15
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
p3 : 1.06843518262231E+00 -2.68110661752041E-01
p2 : 9.89898588133995E-01 -3.25763420505250E-02
p4 : 3.82499737799405E-01 -3.70830934851723E-01
p5 : 4.26845766818034E-03 4.15463298753307E+00
p1 : 4.09176884859392E-01 3.59512644981166E-02
== err : 5.116E-15 = rco : 8.830E-03 = res : 6.953E-15 = complex regular ==
solution 13 : start residual : 9.055E-15
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
p3 : 4.34109040384804E-01 -3.26265223399926E-55
p2 : 6.85427832703092E-01 -1.54975981114965E-54
p4 : 6.90223091850030E-01 -6.52530446799852E-55
p5 : -1.19527057122906E+01 -1.04404871487976E-52
p1 : 6.51874120434089E-01 2.52855548134943E-54
== err : 2.344E-14 = rco : 4.132E-04 = res : 9.055E-15 = real regular ==
solution 14 : start residual : 1.471E-15
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
p3 : 4.66801367842685E-01 -3.82352947550051E-01
p2 : 5.10894379375690E-01 -1.22244306901515E+00
p4 : 1.50113404559223E+00 -3.62017617064430E-02
p5 : 7.09161314103614E-01 -1.24304720303601E-01
p1 : 2.26947298985926E-01 2.27459079306298E-01
== err : 3.623E-15 = rco : 6.837E-02 = res : 1.471E-15 = complex regular ==
solution 15 : start residual : 4.136E-15
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
p3 : 8.40567741887775E-01 -2.58229080736474E-01
p2 : -2.31988029925942E-01 2.03251478278364E+00
p4 : 6.20732282923283E-01 4.71106366815826E-01
p5 : -1.16989346993189E+00 1.39277431099195E-01
p1 : 4.33584619856260E-01 -4.66083201809455E-01
== err : 3.641E-15 = rco : 1.097E-02 = res : 4.136E-15 = complex regular ==
solution 16 : start residual : 3.336E-14
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
p3 : 1.31332676290300E+00 2.27111357311342E+00
p2 : 7.97274972431726E-01 -7.19095370581278E-01
p4 : 1.25315786065640E+00 9.51549363527369E-01
p5 : 3.51070605933374E-01 -1.37699955195813E+00
p1 : -1.86595939857726E+00 5.34560615960333E+00
== err : 1.743E-14 = rco : 1.490E-02 = res : 3.336E-14 = complex regular ==
solution 17 : start residual : 8.410E-15
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
p3 : 6.09650038805020E-01 -4.00754370029421E-02
p2 : 7.50226822740757E-01 2.45293190300973E-01
p4 : 5.98780997016144E-01 4.21752680483449E-02
p5 : 6.18242380174657E-02 2.65681040749625E+00
p1 : -1.33986571663983E+00 -1.64857220529506E+00
== err : 3.051E-14 = rco : 4.657E-03 = res : 8.410E-15 = complex regular ==
solution 18 : start residual : 4.419E-14
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
p3 : 9.46970935591910E-01 1.33210823092707E-02
p2 : 1.25383913644346E+00 -3.63841070504420E-01
p4 : 6.47383507492209E+00 -3.27211325558041E-01
p5 : -1.61756213582246E+00 -2.24047214015870E+00
p1 : 1.06884459654476E+00 -1.54999033378930E-01
== err : 8.042E-14 = rco : 1.732E-03 = res : 4.419E-14 = complex regular ==
solution 19 : start residual : 9.548E-15
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
p3 : 1.47977621808815E+00 -1.31826332571710E-02
p2 : 8.71090072145853E-01 5.89981849604198E-02
p4 : -3.74728737487211E-02 1.11425043359846E+00
p5 : -1.41590767270163E+00 -7.33115265908791E+00
p1 : 4.51662612989230E-01 1.69228911106829E-02
== err : 9.592E-14 = rco : 7.827E-04 = res : 9.548E-15 = complex regular ==
solution 20 : start residual : 5.933E-15
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
p3 : 6.33535795526992E-01 1.66209061667019E-02
p2 : 7.63814591143358E-01 5.33943530680257E-01
p4 : -3.41034998496306E-02 4.22262280736296E-01
p5 : 6.59493092552209E-01 1.81691253226117E+00
p1 : 1.63674938158538E+00 -2.65331142527987E-01
== err : 2.341E-15 = rco : 3.466E-02 = res : 5.933E-15 = complex regular ==
solution 21 : start residual : 2.692E-15
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
p3 : -1.16767884751289E-02 4.06732769987894E-01
p2 : 5.95697594347170E-01 4.39499087900653E-01
p4 : 3.97344768374143E-01 -8.34323995673460E-01
p5 : 1.02174500227171E+00 6.10111344112390E-01
p1 : 9.62421920712938E-01 -4.93513847847347E-01
== err : 1.660E-15 = rco : 6.355E-02 = res : 2.692E-15 = complex regular ==
solution 22 : start residual : 1.285E-14
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
p3 : 6.09650038805020E-01 4.00754370029420E-02
p2 : 7.50226822740757E-01 -2.45293190300974E-01
p4 : 5.98780997016144E-01 -4.21752680483452E-02
p5 : 6.18242380174638E-02 -2.65681040749625E+00
p1 : -1.33986571663983E+00 1.64857220529507E+00
== err : 2.087E-14 = rco : 4.657E-03 = res : 1.285E-14 = complex regular ==
solution 23 : start residual : 9.867E-15
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
p3 : -3.54778843444794E+01 0.00000000000000E+00
p2 : -1.56203574382778E-01 6.15988741779061E-52
p4 : 4.30396175401929E-01 -3.44536075910322E-52
p5 : 8.72620618786704E-01 7.51715074713430E-52
p1 : 1.63515599482588E+00 -1.24241797070692E-51
== err : 2.317E-13 = rco : 7.944E-06 = res : 9.867E-15 = real regular ==
solution 24 : start residual : 4.940E-15
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
p3 : 1.06843518262231E+00 2.68110661752041E-01
p2 : 9.89898588133995E-01 3.25763420505247E-02
p4 : 3.82499737799406E-01 3.70830934851722E-01
p5 : 4.26845766818048E-03 -4.15463298753307E+00
p1 : 4.09176884859392E-01 -3.59512644981165E-02
== err : 2.092E-14 = rco : 8.830E-03 = res : 4.940E-15 = complex regular ==
solution 25 : start residual : 1.443E-15
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
p3 : 9.92199026704117E-01 -1.52763237395737E-01
p2 : -2.70901090367897E-02 -3.70705520657547E-01
p4 : 2.23393020455874E-01 2.48411048789681E-01
p5 : 9.68729217843368E-01 -1.04788472565744E+00
p1 : 6.11433157416092E-01 6.71993522398345E-01
== err : 2.099E-15 = rco : 1.790E-01 = res : 1.443E-15 = complex regular ==
solution 26 : start residual : 2.581E-15
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
p3 : -7.96706202625038E-02 -1.07458749527053E+00
p2 : 2.87644569690750E-01 -7.27859867425151E-01
p4 : 6.23793863466399E-01 3.14440872068831E-01
p5 : 9.05512263133153E-01 -8.72603398077507E-02
p1 : 9.28716375929711E-01 5.58957406906969E-01
== err : 3.150E-15 = rco : 4.274E-02 = res : 2.581E-15 = complex regular ==
solution 27 : start residual : 1.443E-15
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
p3 : -2.83637650932959E-01 -3.34095588761524E-52
p2 : -8.45626244412587E-01 -5.34552942018439E-51
p4 : -1.66873948722793E-01 -1.33638235504610E-51
p5 : -7.59912132722694E-01 -2.67276471009220E-51
p1 : -2.78632762462768E+00 -1.06910588403688E-50
== err : 9.108E-16 = rco : 1.224E-01 = res : 1.443E-15 = real regular ==
solution 28 : start residual : 2.436E-15
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
p3 : 5.97493432045101E-01 4.96010338649300E-04
p2 : 9.18846386955377E-01 -2.62527260184234E-03
p4 : 6.22749473165076E-01 8.37753386866848E-04
p5 : -5.08994994383198E+00 -8.03294003235348E-02
p1 : -1.58430154461839E-01 -3.58785624654741E-03
== err : 2.168E-12 = rco : 6.022E-05 = res : 2.436E-15 = complex regular ==
solution 29 : start residual : 5.107E-15
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
p3 : 8.40567741887775E-01 2.58229080736473E-01
p2 : -2.31988029925943E-01 -2.03251478278364E+00
p4 : 6.20732282923283E-01 -4.71106366815827E-01
p5 : -1.16989346993189E+00 -1.39277431099195E-01
p1 : 4.33584619856261E-01 4.66083201809456E-01
== err : 4.544E-15 = rco : 1.097E-02 = res : 5.107E-15 = complex regular ==
solution 30 : start residual : 3.941E-15
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
p3 : 8.93945676490794E-01 2.42219301852105E-51
p2 : 1.22601023541330E+00 -9.10410479375154E-51
p4 : 9.57777655807107E-01 4.84438603704210E-51
p5 : 4.35954144115817E+00 -1.01565058983503E-49
p1 : 1.96467716282676E-01 4.00914706513829E-51
== err : 6.682E-15 = rco : 4.297E-03 = res : 3.941E-15 = real regular ==
solution 31 : start residual : 9.548E-15
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
p3 : 9.68676942518830E-01 -7.49765877380211E-01
p2 : 1.04714394118568E-01 1.30758750730968E+00
p4 : 6.97073318730709E-01 -2.67202351190649E-01
p5 : 5.81281518681907E-01 8.93829248107594E-01
p1 : -1.23402195986078E+00 -2.77271973715538E+00
== err : 4.919E-15 = rco : 2.485E-02 = res : 9.548E-15 = complex regular ==
solution 32 : start residual : 1.832E-15
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
p3 : 9.92199026704117E-01 1.52763237395737E-01
p2 : -2.70901090367893E-02 3.70705520657547E-01
p4 : 2.23393020455874E-01 -2.48411048789681E-01
p5 : 9.68729217843368E-01 1.04788472565744E+00
p1 : 6.11433157416092E-01 -6.71993522398345E-01
== err : 2.300E-15 = rco : 1.790E-01 = res : 1.832E-15 = complex regular ==
solution 33 : start residual : 1.846E-15
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
p3 : -7.96706202625038E-02 1.07458749527053E+00
p2 : 2.87644569690750E-01 7.27859867425152E-01
p4 : 6.23793863466399E-01 -3.14440872068831E-01
p5 : 9.05512263133153E-01 8.72603398077506E-02
p1 : 9.28716375929711E-01 -5.58957406906969E-01
== err : 1.702E-15 = rco : 4.274E-02 = res : 1.846E-15 = complex regular ==
solution 34 : start residual : 1.471E-15
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
p3 : 5.10675517957555E-01 1.27694760920207E-01
p2 : 1.20526570732874E-01 -4.20852035909150E-01
p4 : 7.75521062491410E-01 -1.02961136852815E+00
p5 : 2.26511709877061E-01 1.61809746926616E-01
p1 : -9.04507405335885E-02 1.52167498721838E-01
== err : 8.289E-16 = rco : 4.646E-02 = res : 1.471E-15 = complex regular ==
solution 35 : start residual : 4.030E-14
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
p3 : 1.76206741762025E+00 -8.05746616986770E-45
p2 : 8.99927978188588E+00 6.72623262875912E-44
p4 : -1.17204756102038E+01 2.24207754291971E-44
p5 : 8.20035156532604E-01 -6.70543210467930E-47
p1 : 4.01365108470145E-01 5.47382212626882E-46
== err : 3.547E-14 = rco : 4.445E-04 = res : 4.030E-14 = real regular ==
solution 36 : start residual : 1.853E-15
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
p3 : -3.14362278374122E-01 8.12179996012148E-01
p2 : 9.80333484151592E-01 -1.17604233495929E-01
p4 : -1.20268804726358E-01 -1.62812118901921E-01
p5 : 7.61514757179786E-01 -4.89094876227610E-01
p1 : 3.24439471325064E-01 1.75667972880234E-02
== err : 1.292E-15 = rco : 3.390E-02 = res : 1.853E-15 = complex regular ==
solution 37 : start residual : 4.885E-15
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
p3 : 8.92754127053145E-01 4.99593623331137E-55
p2 : 2.81808533765359E+00 1.10930175955975E-53
p4 : -1.07875744289497E+00 -2.28385656379948E-54
p5 : -6.63214251306746E-01 5.30180988024880E-54
p1 : 7.21969559469404E-01 -8.15663058499816E-55
== err : 4.514E-15 = rco : 1.553E-02 = res : 4.885E-15 = real regular ==
solution 38 : start residual : 2.581E-15
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
p3 : 4.66801367842685E-01 3.82352947550051E-01
p2 : 5.10894379375690E-01 1.22244306901515E+00
p4 : 1.50113404559223E+00 3.62017617064433E-02
p5 : 7.09161314103614E-01 1.24304720303601E-01
p1 : 2.26947298985926E-01 -2.27459079306298E-01
== err : 2.707E-15 = rco : 6.837E-02 = res : 2.581E-15 = complex regular ==
solution 39 : start residual : 2.661E-15
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
p3 : 5.97493432045101E-01 -4.96010338638549E-04
p2 : 9.18846386955378E-01 2.62527260178542E-03
p4 : 6.22749473165075E-01 -8.37753386848682E-04
p5 : -5.08994994383198E+00 8.03294003217919E-02
p1 : -1.58430154461837E-01 3.58785624646954E-03
== err : 6.086E-13 = rco : 6.022E-05 = res : 2.661E-15 = complex regular ==
solution 40 : start residual : 2.005E-15
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
p3 : -3.14362278374122E-01 -8.12179996012148E-01
p2 : 9.80333484151592E-01 1.17604233495929E-01
p4 : -1.20268804726358E-01 1.62812118901921E-01
p5 : 7.61514757179786E-01 4.89094876227610E-01
p1 : 3.24439471325064E-01 -1.75667972880234E-02
== err : 1.670E-15 = rco : 3.390E-02 = res : 2.005E-15 = complex regular ==
solution 41 : start residual : 5.964E-15
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
p3 : -1.88220294036595E+00 -8.55284707229503E-50
p2 : 7.43456930682186E-01 -5.34552942018439E-51
p4 : 1.07883623286192E+00 1.26725012078931E-50
p5 : -1.77058600109542E+00 -1.28292706084425E-49
p1 : 6.23360447205667E-01 -1.06910588403688E-50
== err : 3.310E-15 = rco : 3.612E-03 = res : 5.964E-15 = real regular ==
solution 42 : start residual : 8.951E-16
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
p3 : 5.10675517957555E-01 -1.27694760920207E-01
p2 : 1.20526570732873E-01 4.20852035909150E-01
p4 : 7.75521062491410E-01 1.02961136852815E+00
p5 : 2.26511709877061E-01 -1.61809746926616E-01
p1 : -9.04507405335886E-02 -1.52167498721838E-01
== err : 1.260E-15 = rco : 4.646E-02 = res : 8.951E-16 = complex regular ==
solution 43 : start residual : 2.776E-15
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
p3 : 7.84506213243864E-01 7.52650542361962E-47
p2 : 1.62937900015137E+00 5.20013101995538E-47
p4 : 3.88785458492806E-01 -1.75162308040602E-46
p5 : -2.70099733910887E+00 -9.63392694223312E-46
p1 : 4.25357850091020E-01 1.08792214759593E-46
== err : 7.775E-15 = rco : 1.694E-02 = res : 2.776E-15 = real regular ==
solution 44 : start residual : 8.549E-15
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
p3 : 6.10794596879131E-01 -1.31726531251069E-02
p2 : 8.02593743364927E-01 1.60065180711071E-01
p4 : 7.08909309217533E-01 -4.44292111043912E-02
p5 : -2.07346832731397E+00 3.72917578985533E+00
p1 : -2.89479806764972E-02 8.42378661874932E-01
== err : 2.568E-14 = rco : 5.103E-03 = res : 8.549E-15 = complex regular ==
===========================================================================