```
9
x1      + x3      + x5        + 2*x7                  - 1;
x1*x2   + x3*x4   + 2*x5*x6   + 2*x7*x8   + 2*x7*x9   - 2/3;
x1*x2^2 + x3*x4^2 + 2*x5*x6^2 + 2*x7*x8^2 + 2*x7*x9^2 - 2/5;
x1*x2^3 + x3*x4^3 + 2*x5*x6^3 + 2*x7*x8^3 + 2*x7*x9^3 - 2/7;
x1*x2^4 + x3*x4^4 + 2*x5*x6^4 + 2*x7*x8^4 + 2*x7*x9^4 - 2/9;
x5*x6^2 + 2*x7*x8*x9     - 1/9;
x5*x6^4 + 2*x7*x8^2*x9^2 - 1/25;
x5*x6^3 + x7*x8*x9^2 + x7*x8^2*x9 - 1/15;
x5*x6^4 + x7*x8*x9^3 + x7*x8^3*x9 - 1/21;

TITLE : optimal multi-dimensional quadrature formulas

ROOT COUNTS :

total degree : 36,000

4-homogeneous Bezout number : 22,740
with partition : {{x1 x3 }{x5 x7 }{x2 x4 }{x6 x8 x9 }}

Set structure analysis :
the generalized Bezout number : 7,090
The set structure :
{x1 x3 x5 x7 }
{x1 x3 x5 x7 }{x2 x4 x6 x8 x9 }
{x1 x3 x5 x7 }{x2 x4 x6 x8 x9 }{x2 x4 x6 x8 x9 }
{x1 x3 x5 x7 }{x2 x4 x6 x8 x9 }{x2 x4 x6 x8 x9 }{x2 x4 x6 x8 x9 }
{x1 x3 x5 x7 }{x2 x4 x6 x8 x9 }{x2 x4 x6 x8 x9 }{x2 x4 x6 x8 x9 }{x2 x4 x6 x8 x9
}
{x5 x7 }{x6 x8 }{x6 x9 }
{x5 x7 }{x6 x8 }{x6 x8 }{x6 x9 }{x6 x9 }
{x5 x7 }{x6 x8 }{x6 x8 x9 }{x6 x9 }
{x5 x7 }{x6 x8 }{x6 x8 x9 }{x6 x8 x9 }{x6 x9 }

mixed volume : 136

REFERENCES :

The system has been given by Rabinowitz (1977), and appears in the
derivation of optimal multi-dimensional quadrature formulas.

Ramon E. Moore:
`Methods and applications of interval analysis',
chapter 6, page 64, SIAM Philadelphia, 1979.

Sandie T. Jones:
`Locating safe starting regions for iterative methods:
a heuristic algorithm', in Interval Mathematics 1980, pages 377-386,
Academic Press 1980, Editor Karl L.E. Nickel,
Proceedings of an International Symposium on Interval Mathematics,
May 27-31, 1980.

THE SOLUTIONS :

16 9
===========================================================
solution 1 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 :  3.90052939160734E-01  -1.36498786345297E-65
x3 :  1.84758425074899E-02  -7.41841230137484E-66
x5 :  8.30951780264823E-02   2.96736492054994E-66
x7 :  2.54188020152647E-01   9.19883125370481E-66
x2 :  2.04098888634611E-01  -8.60535826959482E-66
x4 :  1.25714721355817E+00   1.37685732313517E-64
x6 :  7.95395389473177E-01  -6.52820282520986E-66
x8 :  1.69433845109034E-01  -5.34125685698989E-66
x9 :  6.79629376104577E-01   1.78041895232996E-66
== err :  7.687E-16 = rco :  3.747E-04 = res :  5.551E-17 ==
solution 2 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 :  2.72881639647540E-01   5.53793027399353E-03
x3 :  1.03967523027174E-01   5.53811822782991E-02
x5 :  4.17809813948082E-01  -5.13485523015876E-02
x7 :  1.02670511688602E-01  -4.78528012535252E-03
x2 :  7.24743971081928E-01  -3.83952929482356E-03
x4 : -1.87149444062415E-02   8.79806031776077E-02
x6 :  2.26077500792937E-01   2.71122017311246E-02
x8 :  4.84886923426993E-01   3.68985361557209E-03
x9 :  8.97538072447382E-01   9.53733463716465E-03
== err :  1.460E-15 = rco :  1.207E-04 = res :  1.121E-16 ==
solution 3 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 :  1.84758425074895E-02   1.12667136827130E-66
x3 :  3.90052939160733E-01   1.40949833726122E-66
x5 :  8.30951780264824E-02  -1.27967612198716E-66
x7 :  2.54188020152648E-01  -6.12019014863425E-67
x2 :  1.25714721355818E+00  -5.22998067246926E-66
x4 :  2.04098888634610E-01   6.12019014863425E-67
x6 :  7.95395389473177E-01   1.84533005996699E-66
x8 :  6.79629376104577E-01   3.57011092003664E-67
x9 :  1.69433845109033E-01   2.35534590568651E-66
== err :  5.274E-15 = rco :  3.736E-04 = res :  4.510E-17 ==
solution 4 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 :  7.16134247098115E-02   2.29228940112483E-65
x3 :  4.54090352551545E-01   2.03264497057671E-65
x5 :  4.27846154667776E-02  -1.75074530312446E-65
x7 :  2.15755803635933E-01  -1.28709453428854E-65
x2 :  9.69318641299284E-01  -7.89319068866283E-65
x4 :  2.39009856217790E-01   5.76781556431894E-66
x6 :  8.82786687592638E-01   6.52820282520986E-65
x8 :  2.57431190464460E-01   4.64021689450996E-65
x9 :  7.00084168001749E-01   1.40949833726122E-65
== err :  2.381E-15 = rco :  3.996E-04 = res :  9.714E-17 ==
solution 5 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 :  3.90052939160734E-01  -1.43705844756772E-74
x3 :  1.84758425074899E-02   1.83501309458647E-73
x5 :  8.30951780264823E-02  -1.24913541980886E-73
x7 :  2.54188020152647E-01  -2.21085915010418E-74
x2 :  2.04098888634611E-01  -6.41149153530212E-74
x4 :  1.25714721355817E+00   3.89111210418335E-72
x6 :  7.95395389473177E-01   2.52037943111876E-73
x8 :  6.79629376104577E-01  -7.18529223783858E-74
x9 :  1.69433845109034E-01   1.74657872858230E-73
== err :  6.353E-16 = rco :  3.747E-04 = res :  8.327E-17 ==
solution 6 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 :  7.16134247098114E-02  -2.36561929061147E-73
x3 :  4.54090352551546E-01  -1.12532730740303E-72
x5 :  4.27846154667778E-02  -1.28229830706042E-73
x7 :  2.15755803635932E-01   7.45059533585108E-73
x2 :  9.69318641299285E-01   2.17548540370251E-72
x4 :  2.39009856217790E-01  -5.92510252227920E-73
x6 :  8.82786687592637E-01   6.36727435230003E-73
x8 :  7.00084168001750E-01  -3.00124129626642E-73
x9 :  2.57431190464460E-01  -5.04075886223753E-73
== err :  6.408E-15 = rco :  4.043E-04 = res :  2.776E-17 ==
solution 7 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 :  1.84758425074899E-02   5.98220767982867E-64
x3 :  3.90052939160734E-01   1.03501688428782E-63
x5 :  8.30951780264823E-02  -1.08249472301662E-63
x7 :  2.54188020152647E-01  -2.56380329135515E-64
x2 :  1.25714721355817E+00  -1.14250671116982E-61
x4 :  2.04098888634611E-01   7.54304162803794E-64
x6 :  7.95395389473177E-01   1.80415787169436E-63
x8 :  1.69433845109034E-01   1.42433516186397E-63
x9 :  6.79629376104577E-01  -1.51929083932157E-64
== err :  4.344E-16 = rco :  3.736E-04 = res :  3.123E-17 ==
solution 8 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 :  1.03967523027174E-01   5.53811822782987E-02
x3 :  2.72881639647540E-01   5.53793027399352E-03
x5 :  4.17809813948082E-01  -5.13485523015872E-02
x7 :  1.02670511688602E-01  -4.78528012535250E-03
x2 : -1.87149444062420E-02   8.79806031776072E-02
x4 :  7.24743971081928E-01  -3.83952929482357E-03
x6 :  2.26077500792937E-01   2.71122017311245E-02
x8 :  8.97538072447382E-01   9.53733463716463E-03
x9 :  4.84886923426993E-01   3.68985361557208E-03
== err :  2.413E-15 = rco :  1.187E-04 = res :  7.076E-17 ==
solution 9 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 :  1.03967523027174E-01   5.53811822783006E-02
x3 :  2.72881639647540E-01   5.53793027399373E-03
x5 :  4.17809813948083E-01  -5.13485523015890E-02
x7 :  1.02670511688602E-01  -4.78528012535264E-03
x2 : -1.87149444062410E-02   8.79806031776100E-02
x4 :  7.24743971081928E-01  -3.83952929482369E-03
x6 :  2.26077500792936E-01   2.71122017311253E-02
x8 :  4.84886923426993E-01   3.68985361557219E-03
x9 :  8.97538072447382E-01   9.53733463716489E-03
== err :  3.466E-15 = rco :  1.157E-04 = res :  6.612E-17 ==
solution 10 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 :  1.03967523027174E-01  -5.53811822783013E-02
x3 :  2.72881639647541E-01  -5.53793027399384E-03
x5 :  4.17809813948082E-01   5.13485523015897E-02
x7 :  1.02670511688602E-01   4.78528012535270E-03
x2 : -1.87149444062408E-02  -8.79806031776110E-02
x4 :  7.24743971081928E-01   3.83952929482377E-03
x6 :  2.26077500792937E-01  -2.71122017311257E-02
x8 :  4.84886923426993E-01  -3.68985361557224E-03
x9 :  8.97538072447383E-01  -9.53733463716503E-03
== err :  2.403E-15 = rco :  1.157E-04 = res :  8.327E-17 ==
solution 11 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 :  2.72881639647540E-01  -5.53793027399364E-03
x3 :  1.03967523027175E-01  -5.53811822782998E-02
x5 :  4.17809813948082E-01   5.13485523015882E-02
x7 :  1.02670511688602E-01   4.78528012535260E-03
x2 :  7.24743971081928E-01   3.83952929482363E-03
x4 : -1.87149444062414E-02  -8.79806031776085E-02
x6 :  2.26077500792937E-01  -2.71122017311251E-02
x8 :  4.84886923426993E-01  -3.68985361557216E-03
x9 :  8.97538072447383E-01  -9.53733463716483E-03
== err :  1.870E-15 = rco :  1.207E-04 = res :  8.331E-17 ==
solution 12 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 :  4.54090352551546E-01   4.00165506168856E-73
x3 :  7.16134247098124E-02   1.49232992632032E-73
x5 :  4.27846154667769E-02  -1.64709006682761E-73
x7 :  2.15755803635932E-01  -1.93450175634116E-73
x2 :  2.39009856217790E-01   2.35456499486095E-73
x4 :  9.69318641299282E-01  -8.26861322138963E-73
x6 :  8.82786687592640E-01   6.19040562029170E-74
x8 :  7.00084168001750E-01   1.05015809629948E-73
x9 :  2.57431190464462E-01   3.71424337217502E-73
== err :  2.142E-15 = rco :  4.043E-04 = res :  4.684E-17 ==
solution 13 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 :  1.03967523027175E-01  -5.53811822782980E-02
x3 :  2.72881639647540E-01  -5.53793027399343E-03
x5 :  4.17809813948082E-01   5.13485523015865E-02
x7 :  1.02670511688602E-01   4.78528012535245E-03
x2 : -1.87149444062422E-02  -8.79806031776059E-02
x4 :  7.24743971081928E-01   3.83952929482351E-03
x6 :  2.26077500792937E-01  -2.71122017311242E-02
x8 :  8.97538072447383E-01  -9.53733463716455E-03
x9 :  4.84886923426993E-01  -3.68985361557204E-03
== err :  3.581E-15 = rco :  1.187E-04 = res :  4.221E-17 ==
solution 14 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 :  4.54090352551546E-01   4.84957635911444E-60
x3 :  7.16134247098117E-02   1.03068690539575E-60
x5 :  4.27846154667777E-02   2.62533457034767E-61
x7 :  2.15755803635932E-01  -3.06987906993316E-60
x2 :  2.39009856217790E-01   2.59434103722551E-60
x4 :  9.69318641299284E-01  -9.10115984387192E-60
x6 :  8.82786687592637E-01  -7.95379140201627E-60
x8 :  2.57431190464460E-01   3.44210532556694E-60
x9 :  7.00084168001750E-01   1.66392732722498E-60
== err :  6.002E-15 = rco :  3.996E-04 = res :  5.551E-17 ==
solution 15 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 :  2.72881639647540E-01  -5.53793027399407E-03
x3 :  1.03967523027173E-01  -5.53811822783031E-02
x5 :  4.17809813948083E-01   5.13485523015915E-02
x7 :  1.02670511688602E-01   4.78528012535287E-03
x2 :  7.24743971081928E-01   3.83952929482394E-03
x4 : -1.87149444062401E-02  -8.79806031776138E-02
x6 :  2.26077500792936E-01  -2.71122017311266E-02
x8 :  8.97538072447382E-01  -9.53733463716537E-03
x9 :  4.84886923426993E-01  -3.68985361557238E-03
== err :  4.592E-15 = rco :  1.229E-04 = res :  5.565E-17 ==
solution 16 :
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 :  2.72881639647541E-01   5.53793027399405E-03
x3 :  1.03967523027173E-01   5.53811822783033E-02
x5 :  4.17809813948083E-01  -5.13485523015916E-02
x7 :  1.02670511688602E-01  -4.78528012535285E-03
x2 :  7.24743971081928E-01  -3.83952929482392E-03
x4 : -1.87149444062397E-02   8.79806031776140E-02
x6 :  2.26077500792936E-01   2.71122017311266E-02
x8 :  8.97538072447382E-01   9.53733463716533E-03
x9 :  4.84886923426993E-01   3.68985361557236E-03
== err :  8.864E-15 = rco :  1.229E-04 = res :  5.606E-17 ==
```