7
x1 + x1*x2 + x2*x3 + x3*x4 + x4*x5 + x5*x6 - u7;
x2 + x1*x3 + x2*x4 + x3*x5 + x4*x6 - 2*u7;
x3 + x1*x4 + x2*x5 + x3*x6 - 3*u7;
x4 + x1*x5 + x2*x6 - 4*u7;
x5 + x1*x6 - 5*u7;
x6 - 6*u7;
x1 + x2 + x3 + x4 + x5 + x6 + 1;
TITLE : reduced 7-dimensional economics problem
ROOT COUNTS :
total degree : 32
2-homogeneous Bezout number : 64
with partition : {x1 x2 x3 x4 x5 }{x6 u7 }
generalized Bezout number : 32
based on the set structure :
{x1 x3 x5 u7 }{x2 x4 x6 }
{x1 x2 x5 x6 u7 }{x3 x4 }
{x1 x2 x3 u7 }{x4 x5 x6 }
{x1 x2 x4 u7 }{x5 x6 }
{x1 x5 u7 }{x6 }
{x6 u7 }
{x1 x2 x3 x4 x5 x6 }
mixed volume : 32
REFERENCE :
This is the reduced economics problem (u7 = 1/x7).
Alexander Morgan:
`Solving polynomial systems using continuation for engineering
and scientific problems', Prentice-Hall, Englewood Cliffs, New Jersey, 1987,
(p 148).
THE SOLUTIONS :
32 7
===========================================================
solution 1 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : -6.81648480864347E-01 -3.40321148008031E-62
x2 : 9.58129145190158E-01 5.83407682299482E-62
x3 : 6.31940496052593E-01 -3.88938454866321E-62
x4 : -4.34421392403970E-01 4.86173068582902E-62
x5 : -8.87912123263308E-01 -3.88938454866321E-62
x6 : -5.86087644711127E-01 -9.72346137165803E-63
u7 : -9.76812741185211E-02 -3.03858167864314E-63
== err : 6.304E-16 = rco : 2.962E-02 = res : 2.220E-16 ==
solution 2 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 5.93369639826469E-03 -1.28636083334685E-01
x2 : 5.95495251474475E-01 -2.08845973813888E+00
x3 : -6.99468204461880E-01 4.29895386284063E-01
x4 : -1.32701859744045E+00 6.35115643463731E-01
x5 : 1.49441734602828E-01 5.41035850350458E-01
x6 : 2.75616119426760E-01 6.11048941375316E-01
u7 : 4.59360199044601E-02 1.01841490229219E-01
== err : 2.366E-15 = rco : 2.512E-02 = res : 2.118E-16 ==
solution 3 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 6.93515873660876E-01 1.60919707978251E-54
x2 : 7.22177850183674E-01 -7.83036536159823E-54
x3 : -1.65871370380082E+00 2.61012178719941E-54
x4 : 1.16901800599298E-01 0.00000000000000E+00
x5 : -1.07196055981947E-01 -3.26265223399926E-55
x6 : -7.66685764661078E-01 2.61012178719941E-54
u7 : -1.27780960776846E-01 6.52530446799852E-55
== err : 2.177E-15 = rco : 2.812E-02 = res : 2.220E-16 ==
solution 4 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : -1.66666666666667E-01 3.98272977783113E-59
x2 : -1.66666666666667E-01 0.00000000000000E+00
x3 : -1.66666666666667E-01 -9.95682444457783E-60
x4 : -1.66666666666667E-01 9.95682444457783E-60
x5 : -1.66666666666667E-01 -4.48057100006002E-59
x6 : -1.66666666666667E-01 0.00000000000000E+00
u7 : -2.77777777777778E-02 -3.38667498114907E-61
== err : 3.132E-16 = rco : 3.765E-02 = res : 2.220E-16 ==
solution 5 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 5.93369639826462E-03 3.71479583210633E+00
x2 : -6.29608516467496E+00 -2.89485362873526E-01
x3 : 6.85454641806507E-01 -6.41622249799802E+00
x4 : 4.94031447364827E+00 2.36147158475215E-01
x5 : 2.97245117069881E-01 2.53379026615599E+00
x6 : -6.32862764247967E-01 2.20974604134012E-01
u7 : -1.05477127374661E-01 3.68291006890019E-02
== err : 5.334E-15 = rco : 1.183E-02 = res : 1.067E-14 ==
solution 6 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : -2.51557210700575E-01 -1.79307987438582E+00
x2 : -5.18994747263476E-01 1.65067431386248E+00
x3 : 2.54014532482806E+00 -1.49593314929199E+00
x4 : -1.73247661935806E+00 -1.65911587822018E+00
x5 : -1.53306651761352E+00 2.14239297415820E+00
x6 : 4.95949770107570E-01 1.15506161387731E+00
u7 : 8.26582950179284E-02 1.92510268979552E-01
== err : 3.738E-15 = rco : 2.485E-02 = res : 9.930E-16 ==
solution 7 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 3.97308366771297E-01 -1.92171595772051E+00
x2 : -2.38300160166170E+00 -1.65159813713614E+00
x3 : -1.17635241772214E+00 2.17748014148655E+00
x4 : 5.01732358554671E-01 1.22966143182879E+00
x5 : 1.19054568318791E+00 6.79107470333445E-01
x6 : 4.69767610869959E-01 -5.12934948792143E-01
u7 : 7.82946018116599E-02 -8.54891581320238E-02
== err : 4.453E-15 = rco : 1.532E-02 = res : 8.882E-16 ==
solution 8 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 5.93369639826422E-03 -3.71479583210633E+00
x2 : -6.29608516467496E+00 2.89485362873526E-01
x3 : 6.85454641806506E-01 6.41622249799802E+00
x4 : 4.94031447364827E+00 -2.36147158475213E-01
x5 : 2.97245117069882E-01 -2.53379026615599E+00
x6 : -6.32862764247966E-01 -2.20974604134012E-01
u7 : -1.05477127374661E-01 -3.68291006890020E-02
== err : 4.397E-15 = rco : 1.183E-02 = res : 3.878E-15 ==
solution 9 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 5.69908729836229E-01 -1.79307987438582E+00
x2 : -1.27629537390668E+00 1.77720268392520E-01
x3 : 2.62145169634245E+00 8.47407879790141E-01
x4 : 2.99663982501566E-01 -4.45879059994048E+00
x5 : -4.31844602385446E+00 2.65619747762992E+00
x6 : 1.10371698908089E+00 2.57054484851373E+00
u7 : 1.83952831513482E-01 4.28424141418954E-01
== err : 4.440E-15 = rco : 2.857E-02 = res : 1.831E-15 ==
solution 10 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 4.36024966562036E-01 -1.92171595772051E+00
x2 : -8.16560321165552E-01 -1.72600044478268E+00
x3 : -3.45095823467105E+00 -8.67158668830608E-01
x4 : -5.62843410349119E-01 4.13034050901962E+00
x5 : 2.42330915003328E+00 1.44479097161200E+00
x6 : 9.71027849590407E-01 -1.06025640929782E+00
u7 : 1.61837974931735E-01 -1.76709401549637E-01
== err : 5.610E-15 = rco : 1.976E-02 = res : 2.402E-15 ==
solution 11 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : -4.24157573765507E-01 -1.92171595772051E+00
x2 : -1.48391565543637E+00 -7.29739304826770E-02
x3 : -5.60052371456210E-01 1.17922334308334E+00
x4 : 5.48670634671490E-01 9.30133142907219E-01
x5 : 7.08367213492226E-01 1.15818184498564E-01
x6 : 2.11087752494370E-01 -2.30484782285939E-01
u7 : 3.51812920823950E-02 -3.84141303809898E-02
== err : 2.990E-15 = rco : 1.721E-02 = res : 8.006E-16 ==
solution 12 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 1.25749090709884E+00 1.92171595772051E+00
x2 : -1.00903560786738E+00 3.30462465143614E+00
x3 : -3.99271237385452E+00 1.22680685072913E+00
x4 : -3.03453290093010E+00 -3.65908289972281E+00
x5 : 3.61780513583610E+00 -5.15362410077954E+00
x6 : 2.16098483971706E+00 2.35955954061657E+00
u7 : 3.60164139952843E-01 3.93259923436095E-01
== err : 7.855E-15 = rco : 1.195E-02 = res : 5.403E-15 ==
solution 13 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 8.27399636935069E-01 1.28636083334685E-01
x2 : 4.97146350108965E-02 2.19412989932238E+00
x3 : 1.55565100551911E-01 1.21486906126276E+00
x4 : -2.47476413001306E+00 -2.09079838123884E+00
x5 : -1.71288232676718E-01 -8.69709294847598E-02
x6 : 6.13372990191896E-01 -1.35986573319623E+00
u7 : 1.02228831698649E-01 -2.26644288866038E-01
== err : 5.124E-15 = rco : 2.241E-02 = res : 1.057E-15 ==
solution 14 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 3.97308366771297E-01 1.92171595772051E+00
x2 : -2.38300160166170E+00 1.65159813713614E+00
x3 : -1.17635241772214E+00 -2.17748014148655E+00
x4 : 5.01732358554671E-01 -1.22966143182879E+00
x5 : 1.19054568318791E+00 -6.79107470333445E-01
x6 : 4.69767610869959E-01 5.12934948792143E-01
u7 : 7.82946018116599E-02 8.54891581320238E-02
== err : 4.453E-15 = rco : 1.532E-02 = res : 8.882E-16 ==
solution 15 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 1.25749090709884E+00 -1.92171595772051E+00
x2 : -1.00903560786738E+00 -3.30462465143614E+00
x3 : -3.99271237385452E+00 -1.22680685072913E+00
x4 : -3.03453290093010E+00 3.65908289972281E+00
x5 : 3.61780513583610E+00 5.15362410077954E+00
x6 : 2.16098483971706E+00 -2.35955954061657E+00
u7 : 3.60164139952843E-01 -3.93259923436095E-01
== err : 1.282E-14 = rco : 1.195E-02 = res : 7.324E-15 ==
solution 16 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 1.00000000000000E+00 0.00000000000000E+00
x2 : 1.00000000000000E+00 0.00000000000000E+00
x3 : 1.00000000000000E+00 5.93472984109987E-67
x4 : 1.00000000000000E+00 -9.08755506918418E-67
x5 : 1.00000000000000E+00 -5.93472984109987E-67
x6 : -6.00000000000000E+00 -4.74778387287990E-66
u7 : -1.00000000000000E+00 -1.18694596821997E-66
== err : 2.940E-50 = rco : 4.762E-02 = res : 5.657E-66 ==
solution 17 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : -4.24157573765507E-01 1.92171595772051E+00
x2 : -1.48391565543637E+00 7.29739304826769E-02
x3 : -5.60052371456210E-01 -1.17922334308334E+00
x4 : 5.48670634671490E-01 -9.30133142907219E-01
x5 : 7.08367213492226E-01 -1.15818184498564E-01
x6 : 2.11087752494370E-01 2.30484782285939E-01
u7 : 3.51812920823950E-02 3.84141303809898E-02
== err : 2.809E-15 = rco : 1.721E-02 = res : 4.475E-16 ==
solution 18 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 2.63424603497105E-01 -1.79307987438582E+00
x2 : -1.42230137654078E+00 7.27270787149925E-01
x3 : 6.48059125615115E-01 7.41546225335957E-01
x4 : -1.21625875444223E-01 -4.44285120925777E-01
x5 : -5.08590494417704E-01 4.40081292362660E-01
x6 : 1.41034017290485E-01 3.28466690463057E-01
u7 : 2.35056695484141E-02 5.47444484105095E-02
== err : 2.618E-15 = rco : 2.056E-02 = res : 3.554E-16 ==
solution 19 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 2.63424603497105E-01 1.79307987438582E+00
x2 : -1.42230137654078E+00 -7.27270787149925E-01
x3 : 6.48059125615115E-01 -7.41546225335957E-01
x4 : -1.21625875444223E-01 4.44285120925777E-01
x5 : -5.08590494417704E-01 -4.40081292362661E-01
x6 : 1.41034017290485E-01 -3.28466690463057E-01
u7 : 2.35056695484141E-02 -5.47444484105095E-02
== err : 2.558E-15 = rco : 2.056E-02 = res : 2.483E-16 ==
solution 20 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 8.27399636935069E-01 3.71479583210633E+00
x2 : -6.84186578113854E+00 2.76209288924990E+00
x3 : -4.12071064151692E+00 -8.69959556204953E+00
x4 : 8.72511873116428E+00 -3.50394294906873E+00
x5 : 1.81846960499532E+00 5.23487937380794E+00
x6 : -1.40841155043922E+00 4.91770415954087E-01
u7 : -2.34735258406536E-01 8.19617359923479E-02
== err : 1.048E-14 = rco : 3.851E-03 = res : 1.421E-14 ==
solution 21 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 6.54799273870137E-01 2.91703841149741E-62
x2 : -8.54232602712374E-01 -2.91703841149741E-62
x3 : 1.57319825015525E-01 1.97800470374835E-62
x4 : -5.20756185121201E-01 -1.59209508504746E-62
x5 : -6.62201048157115E-02 1.21543267145725E-63
x6 : -3.70910206236376E-01 -1.09388940431153E-62
u7 : -6.18183677060627E-02 -1.79360058965448E-63
== err : 4.779E-16 = rco : 2.506E-02 = res : 1.110E-16 ==
solution 22 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 8.27399636935069E-01 -1.28636083334685E-01
x2 : 4.97146350108963E-02 -2.19412989932238E+00
x3 : 1.55565100551911E-01 -1.21486906126276E+00
x4 : -2.47476413001305E+00 2.09079838123884E+00
x5 : -1.71288232676718E-01 8.69709294847598E-02
x6 : 6.13372990191896E-01 1.35986573319623E+00
u7 : 1.02228831698649E-01 2.26644288866038E-01
== err : 4.845E-15 = rco : 2.241E-02 = res : 1.116E-15 ==
solution 23 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : -2.51557210700575E-01 1.79307987438582E+00
x2 : -5.18994747263476E-01 -1.65067431386248E+00
x3 : 2.54014532482806E+00 1.49593314929199E+00
x4 : -1.73247661935806E+00 1.65911587822018E+00
x5 : -1.53306651761352E+00 -2.14239297415820E+00
x6 : 4.95949770107570E-01 -1.15506161387731E+00
u7 : 8.26582950179284E-02 -1.92510268979552E-01
== err : 3.232E-15 = rco : 2.485E-02 = res : 1.351E-15 ==
solution 24 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 4.36024966562036E-01 1.92171595772051E+00
x2 : -8.16560321165552E-01 1.72600044478268E+00
x3 : -3.45095823467105E+00 8.67158668830609E-01
x4 : -5.62843410349120E-01 -4.13034050901962E+00
x5 : 2.42330915003328E+00 -1.44479097161200E+00
x6 : 9.71027849590408E-01 1.06025640929782E+00
u7 : 1.61837974931735E-01 1.76709401549637E-01
== err : 6.794E-15 = rco : 1.976E-02 = res : 3.202E-15 ==
solution 25 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 5.93369639826470E-03 1.28636083334685E-01
x2 : 5.95495251474475E-01 2.08845973813888E+00
x3 : -6.99468204461880E-01 -4.29895386284063E-01
x4 : -1.32701859744045E+00 -6.35115643463731E-01
x5 : 1.49441734602828E-01 -5.41035850350458E-01
x6 : 2.75616119426760E-01 -6.11048941375316E-01
u7 : 4.59360199044601E-02 -1.01841490229219E-01
== err : 2.370E-15 = rco : 2.512E-02 = res : 2.178E-16 ==
solution 26 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 8.27399636935069E-01 -3.71479583210633E+00
x2 : -6.84186578113854E+00 -2.76209288924990E+00
x3 : -4.12071064151691E+00 8.69959556204954E+00
x4 : 8.72511873116428E+00 3.50394294906872E+00
x5 : 1.81846960499532E+00 -5.23487937380794E+00
x6 : -1.40841155043922E+00 -4.91770415954087E-01
u7 : -2.34735258406536E-01 -8.19617359923478E-02
== err : 1.532E-14 = rco : 3.851E-03 = res : 2.383E-14 ==
solution 27 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 1.08489054403391E+00 1.79307987438582E+00
x2 : -1.75656198282473E+00 7.45683258320038E-01
x3 : -2.96247415748368E-01 -2.32634270785216E+00
x4 : 1.04373156113984E+00 8.97470024637929E-01
x5 : -1.38967844523799E+00 -3.78900562083039E-01
x6 : 3.13865738637350E-01 -7.30989887408594E-01
u7 : 5.23109564395583E-02 -1.21831647901432E-01
== err : 4.986E-15 = rco : 1.450E-02 = res : 1.299E-15 ==
solution 28 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 1.08489054403391E+00 -1.79307987438582E+00
x2 : -1.75656198282473E+00 -7.45683258320039E-01
x3 : -2.96247415748368E-01 2.32634270785216E+00
x4 : 1.04373156113984E+00 -8.97470024637929E-01
x5 : -1.38967844523800E+00 3.78900562083039E-01
x6 : 3.13865738637350E-01 7.30989887408594E-01
u7 : 5.23109564395583E-02 1.21831647901432E-01
== err : 4.894E-15 = rco : 1.450E-02 = res : 1.986E-15 ==
solution 29 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 1.39817459672457E-01 -2.62619866977784E-63
x2 : -1.52476811214796E-01 -4.70980160189686E-63
x3 : 2.16348572029908E+00 -3.03858167864314E-63
x4 : -9.41947644616723E-01 5.01365976976117E-63
x5 : -9.04563389105642E-01 9.11574503592941E-64
x6 : -1.30431533503437E+00 4.86173068582902E-63
u7 : -2.17385889172396E-01 8.35609961626862E-64
== err : 4.735E-15 = rco : 4.114E-02 = res : 1.943E-16 ==
solution 30 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 1.51498181419768E+00 2.22552369041245E-67
x2 : 7.41222573661469E-01 5.93472984109987E-67
x3 : -1.07823548534975E+00 7.78933291644359E-67
x4 : -1.63478816493804E+00 0.00000000000000E+00
x5 : 1.16304871296224E+00 -3.70920615068742E-67
x6 : -1.70622945053360E+00 -1.11276184520623E-66
u7 : -2.84371575088934E-01 -1.85460307534371E-67
== err : 5.014E-15 = rco : 2.480E-02 = res : 4.441E-16 ==
solution 31 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 1.78534059463196E-01 9.11574503592941E-64
x2 : 1.40399529688145E+00 -4.55787251796470E-63
x3 : -4.24145311642535E-01 -1.01602574879630E-63
x4 : 2.30307117019040E+00 -7.29259602874353E-63
x5 : -1.76538489062995E+00 4.25401435010039E-63
x6 : -2.69607032426256E+00 5.92523427335412E-63
u7 : -4.49345054043761E-01 1.06350358752510E-63
== err : 2.862E-15 = rco : 3.735E-02 = res : 7.772E-16 ==
solution 32 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 5.69908729836229E-01 1.79307987438582E+00
x2 : -1.27629537390668E+00 -1.77720268392520E-01
x3 : 2.62145169634245E+00 -8.47407879790140E-01
x4 : 2.99663982501565E-01 4.45879059994048E+00
x5 : -4.31844602385446E+00 -2.65619747762992E+00
x6 : 1.10371698908089E+00 -2.57054484851373E+00
u7 : 1.83952831513482E-01 -4.28424141418954E-01
== err : 5.346E-15 = rco : 2.857E-02 = res : 2.220E-15 ==