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 ==