8
x1^2 + x2^2 - 1;
x3^2 + x4^2 - 1;
x5^2 + x6^2 - 1;
x7^2 + x8^2 - 1;
0.004731*x1*x3 - 0.3578*x2*x3 - 0.1238*x1 - 0.001637*x2 - 0.9338*x4 + x7
- 0.3571;
0.2238*x1*x3 + 0.7623*x2*x3 + 0.2638*x1 - 0.07745*x2 -0.6734*x4 -0.6022;
x6*x8 + 0.3578*x1 + 0.004731*x2;
-0.7623*x1 + 0.2238*x2 + 0.3461;
TITLE : hand position and orientation of PUMA robot
ROOT COUNTS :
total degree : 128
3-homogeneous Bezout number : 16
with partition {{x1 x2 }{x3 x4 x7 x8 }{x5 x6 }}
mixed volume : 16
REFERENCES :
A. Morgan and V. Shapiro.
`Box-Bisection for solving second-degree systems and the problem
of clustering', ACM Transactions on Mathematical Software,
Vol. 13, No. 2, pp. 152-167, 1987.
NOTE :
The system can be easily reduced to a lower dimensional one,
by eliminating x1 or x2 by the last equation.
It is not a hard problem to solve, but a very good test example
for algorithms that verify whether the mixed volume is zero or not.
Every fine mixed subdivision will have only one mixed cell.
More precisely, there are 6336 different configurations, which all
yield a mixed cell with mixed volume 16.
THE SOLUTIONS :
16 8
===========================================================
solution 1 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 1.64431665854327E-01 8.56002160304148E-55
x2 : -9.86388476850967E-01 2.61012178719941E-54
x3 : 2.39616017204621E-01 5.94496235550677E-54
x4 : -9.70867737799024E-01 1.46725258920162E-54
x5 : 9.97635398227387E-01 2.66040587238087E-52
x6 : 6.87285399501796E-02 -3.86173644000454E-51
x7 : -6.15508407155080E-01 0.00000000000000E+00
x8 : -7.88130319630844E-01 -4.42836640644059E-50
== err : 5.847E-16 = rco : 1.742E-01 = res : 1.631E-16 ==
solution 2 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 1.64431665854327E-01 8.56002160304148E-55
x2 : -9.86388476850967E-01 2.61012178719941E-54
x3 : 2.39616017204621E-01 5.94496235550677E-54
x4 : -9.70867737799024E-01 1.46725258920162E-54
x5 : -9.97635398227387E-01 -8.57311908487996E-50
x6 : -6.87285399501796E-02 2.56585412168851E-49
x7 : -6.15508407155080E-01 0.00000000000000E+00
x8 : 7.88130319630844E-01 -2.10522022959584E-49
== err : 5.847E-16 = rco : 2.040E-01 = res : 1.631E-16 ==
solution 3 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 1.64431665854327E-01 8.56002160304148E-55
x2 : -9.86388476850967E-01 2.61012178719941E-54
x3 : 2.39616017204621E-01 5.94496235550677E-54
x4 : -9.70867737799024E-01 1.46725258920162E-54
x5 : -9.97635398227387E-01 -2.66040587238087E-52
x6 : 6.87285399501796E-02 -3.86173644000454E-51
x7 : -6.15508407155080E-01 0.00000000000000E+00
x8 : -7.88130319630844E-01 -4.42836640644059E-50
== err : 5.847E-16 = rco : 2.040E-01 = res : 1.631E-16 ==
solution 4 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 1.64431665854327E-01 8.56002160304148E-55
x2 : -9.86388476850967E-01 2.61012178719941E-54
x3 : 2.39616017204621E-01 5.94496235550677E-54
x4 : -9.70867737799024E-01 1.46725258920162E-54
x5 : 9.97635398227387E-01 8.57311908487996E-50
x6 : -6.87285399501796E-02 2.56585412168851E-49
x7 : -6.15508407155080E-01 0.00000000000000E+00
x8 : 7.88130319630844E-01 -2.10522022959584E-49
== err : 5.847E-16 = rco : 1.742E-01 = res : 1.631E-16 ==
solution 5 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 1.64431665854327E-01 0.00000000000000E+00
x2 : -9.86388476850967E-01 2.51345585423244E-88
x3 : -9.54728434485888E-01 4.11804607157442E-83
x4 : 2.97478766274377E-01 0.00000000000000E+00
x5 : 9.11154796098708E-01 7.46761833343337E-60
x6 : -4.12064239586890E-01 3.48488855560224E-59
x7 : 9.91322415106624E-01 3.89318277576152E-64
x8 : 1.31452916693283E-01 3.69491532123005E-61
== err : 5.116E-15 = rco : 9.651E-03 = res : 1.110E-16 ==
solution 6 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 1.64431665854327E-01 0.00000000000000E+00
x2 : -9.86388476850967E-01 2.51345585423244E-88
x3 : -9.54728434485888E-01 4.11804607157442E-83
x4 : 2.97478766274377E-01 0.00000000000000E+00
x5 : -9.11154796098708E-01 -7.46761833343337E-60
x6 : 4.12064239586890E-01 -3.48488855560224E-59
x7 : 9.91322415106624E-01 3.89318277576152E-64
x8 : -1.31452916693283E-01 -3.69491532123005E-61
== err : 5.116E-15 = rco : 9.729E-03 = res : 1.110E-16 ==
solution 7 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 1.64431665854327E-01 0.00000000000000E+00
x2 : -9.86388476850967E-01 2.51345585423244E-88
x3 : -9.54728434485888E-01 4.11804607157442E-83
x4 : 2.97478766274377E-01 0.00000000000000E+00
x5 : -9.11154796098708E-01 -7.46761833343337E-60
x6 : -4.12064239586890E-01 3.48488855560224E-59
x7 : 9.91322415106624E-01 3.89318277576152E-64
x8 : 1.31452916693283E-01 3.69491532123005E-61
== err : 5.116E-15 = rco : 9.729E-03 = res : 1.110E-16 ==
solution 8 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 1.64431665854327E-01 0.00000000000000E+00
x2 : -9.86388476850967E-01 2.51345585423244E-88
x3 : -9.54728434485888E-01 4.11804607157442E-83
x4 : 2.97478766274377E-01 0.00000000000000E+00
x5 : 9.11154796098708E-01 7.46761833343337E-60
x6 : 4.12064239586890E-01 -3.48488855560224E-59
x7 : 9.91322415106624E-01 3.89318277576152E-64
x8 : -1.31452916693283E-01 -3.69491532123005E-61
== err : 5.116E-15 = rco : 9.651E-03 = res : 1.110E-16 ==
solution 9 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 6.71554261818887E-01 2.08809742975953E-53
x2 : 7.40955378840649E-01 2.50571691571143E-52
x3 : -2.39611659195710E-01 2.77934733962389E-52
x4 : -9.70868813371549E-01 0.00000000000000E+00
x5 : -9.57917101877785E-01 -1.04404871487976E-53
x6 : -2.87044989383310E-01 4.17619485951906E-53
x7 : -5.27909026365623E-01 8.35238971903811E-53
x8 : 8.49300924220444E-01 8.35238971903811E-53
== err : 5.564E-16 = rco : 1.733E-01 = res : 1.110E-16 ==
solution 10 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 6.71554261818887E-01 2.08809742975953E-53
x2 : 7.40955378840649E-01 2.50571691571143E-52
x3 : -2.39611659195710E-01 2.50571691571143E-52
x4 : -9.70868813371549E-01 -8.35238971903811E-53
x5 : 9.57917101877785E-01 6.26429228927858E-53
x6 : 2.87044989383310E-01 -2.08809742975953E-52
x7 : -5.27909026365623E-01 0.00000000000000E+00
x8 : -8.49300924220444E-01 -8.35238971903811E-53
== err : 5.564E-16 = rco : 1.667E-01 = res : 1.110E-16 ==
solution 11 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 6.71554261818887E-01 2.08809742975953E-53
x2 : 7.40955378840649E-01 2.50571691571143E-52
x3 : -2.39611659195710E-01 2.77934733962389E-52
x4 : -9.70868813371549E-01 0.00000000000000E+00
x5 : 9.57917101877785E-01 1.04404871487976E-53
x6 : -2.87044989383310E-01 4.17619485951906E-53
x7 : -5.27909026365623E-01 8.35238971903811E-53
x8 : 8.49300924220444E-01 8.35238971903811E-53
== err : 5.564E-16 = rco : 1.667E-01 = res : 1.110E-16 ==
solution 12 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 6.71554261818887E-01 2.08809742975953E-53
x2 : 7.40955378840649E-01 2.50571691571143E-52
x3 : -2.39611659195710E-01 2.50571691571143E-52
x4 : -9.70868813371549E-01 -8.35238971903811E-53
x5 : -9.57917101877785E-01 -6.26429228927858E-53
x6 : 2.87044989383310E-01 -2.08809742975953E-52
x7 : -5.27909026365623E-01 0.00000000000000E+00
x8 : -8.49300924220444E-01 -8.35238971903811E-53
== err : 5.564E-16 = rco : 1.733E-01 = res : 1.110E-16 ==
solution 13 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 6.71554261818887E-01 2.10843958864610E-81
x2 : 7.40955378840649E-01 -6.32531876593832E-81
x3 : 9.54729769791417E-01 -1.76673203694632E-80
x4 : 2.97474480710579E-01 3.37350334183377E-80
x5 : -1.28778236227690E-01 2.59425224673324E-49
x6 : 9.91673416944351E-01 -5.54759771060745E-71
x7 : 9.69311807770115E-01 2.26391976970668E-72
x8 : -2.45834536461890E-01 -1.37524217994990E-71
== err : 5.075E-14 = rco : 3.169E-03 = res : 1.110E-16 ==
solution 14 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 6.71554261818887E-01 2.10843958864610E-81
x2 : 7.40955378840649E-01 -6.32531876593832E-81
x3 : 9.54729769791417E-01 -1.76673203694632E-80
x4 : 2.97474480710579E-01 3.37350334183377E-80
x5 : 1.28778236227690E-01 -2.59425224673324E-49
x6 : -9.91673416944351E-01 5.54759771060745E-71
x7 : 9.69311807770115E-01 2.26391976970668E-72
x8 : 2.45834536461890E-01 1.37524217994990E-71
== err : 5.075E-14 = rco : 3.187E-03 = res : 1.110E-16 ==
solution 15 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 6.71554261818887E-01 2.10843958864610E-81
x2 : 7.40955378840649E-01 -6.32531876593832E-81
x3 : 9.54729769791417E-01 -1.76673203694632E-80
x4 : 2.97474480710579E-01 3.37350334183377E-80
x5 : 1.28778236227690E-01 -2.59425224673324E-49
x6 : 9.91673416944351E-01 -5.54759771060745E-71
x7 : 9.69311807770115E-01 2.26391976970668E-72
x8 : -2.45834536461890E-01 -1.37524217994990E-71
== err : 5.075E-14 = rco : 3.187E-03 = res : 1.110E-16 ==
solution 16 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
x1 : 6.71554261818887E-01 2.10843958864610E-81
x2 : 7.40955378840649E-01 -6.32531876593832E-81
x3 : 9.54729769791417E-01 -1.76673203694632E-80
x4 : 2.97474480710579E-01 3.37350334183377E-80
x5 : -1.28778236227690E-01 2.59425224673324E-49
x6 : -9.91673416944351E-01 5.54759771060745E-71
x7 : 9.69311807770115E-01 2.26391976970668E-72
x8 : 2.45834536461890E-01 1.37524217994990E-71
== err : 5.075E-14 = rco : 3.169E-03 = res : 1.110E-16 ==