11
z0*z1*z2*z3*z4*z5*z6*z7*z8 - 1
+( 8.09814453125000E-01 - 8.36517333984375E-01*i)*zz1
+( 9.01000976562500E-01 + 9.64630126953125E-01*i)*zz2;
z0 + z1 + z2 + z3 + z4 + z5 + z6 + z7 + z8
+( 3.57177734375000E-01 - 6.44622802734375E-01*i)*zz1
+( 6.20239257812500E-01 + 2.03399658203125E-01*i)*zz2;
z0*z1 + z0*z8 + z1*z2 + z2*z3 + z3*z4 + z4*z5
+ z5*z6 + z6*z7 + z7*z8
+( 9.49401855468750E-01 + 2.92816162109375E-01*i)*zz1
+(-5.12145996093750E-01 - 5.68145751953125E-01*i)*zz2;
z0*z1*z2 + z0*z1*z8 + z0*z7*z8 + z1*z2*z3
+ z2*z3*z4 + z3*z4*z5 + z4*z5*z6 + z5*z6*z7 + z6*z7*z8
+( 4.01977539062500E-01 + 4.66918945312500E-03*i)*zz1
+(-9.40429687500000E-01 - 9.71527099609375E-01*i)*zz2;
z0*z1*z2*z3 + z0*z1*z2*z8 + z0*z1*z7*z8 + z0*z6*z7*z8
+ z1*z2*z3*z4 + z2*z3*z4*z5 + z3*z4*z5*z6 + z4*z5*z6*z7
+ z5*z6*z7*z8
+( 5.74279785156250E-01 + 7.25311279296875E-01*i)*zz1
+(-3.05236816406250E-01 - 2.72369384765625E-01*i)*zz2;
z0*z1*z2*z3*z4 + z0*z1*z2*z3*z8 + z0*z1*z2*z7*z8
+ z0*z1*z6*z7*z8 + z0*z5*z6*z7*z8 + z1*z2*z3*z4*z5
+ z2*z3*z4*z5*z6 + z3*z4*z5*z6*z7 + z4*z5*z6*z7*z8
+(-4.24316406250000E-01 + 9.39117431640625E-01*i)*zz1
+( 2.80761718750000E-03 + 5.13702392578125E-01*i)*zz2;
z0*z1*z2*z3*z4*z5 + z0*z1*z2*z3*z4*z8 + z0*z1*z2*z3*z7*z8
+ z0*z1*z2*z6*z7*z8 + z0*z1*z5*z6*z7*z8 + z0*z4*z5*z6*z7*z8
+ z1*z2*z3*z4*z5*z6 + z2*z3*z4*z5*z6*z7 + z3*z4*z5*z6*z7*z8
+(-2.34436035156250E-01 - 6.19537353515625E-01*i)*zz1
+( 8.43078613281250E-01 + 6.21063232421875E-01*i)*zz2;
z0*z1*z2*z3*z4*z5*z6 + z0*z1*z2*z3*z4*z5*z8
+ z0*z1*z2*z3*z4*z7*z8 + z0*z1*z2*z3*z6*z7*z8
+ z0*z1*z2*z5*z6*z7*z8 + z0*z1*z4*z5*z6*z7*z8
+ z0*z3*z4*z5*z6*z7*z8 + z1*z2*z3*z4*z5*z6*z7
+ z2*z3*z4*z5*z6*z7*z8
+(-2.46704101562500E-01 - 9.66278076171875E-01*i)*zz1
+(-6.75048828125000E-01 - 4.65911865234375E-01*i)*zz2;
z0*z1*z2*z3*z4*z5*z6*z7 + z0*z1*z2*z3*z4*z5*z6*z8
+ z0*z1*z2*z3*z4*z5*z7*z8 + z0*z1*z2*z3*z4*z6*z7*z8
+ z0*z1*z2*z3*z5*z6*z7*z8 + z0*z1*z2*z4*z5*z6*z7*z8
+ z0*z1*z3*z4*z5*z6*z7*z8 + z0*z2*z3*z4*z5*z6*z7*z8
+ z1*z2*z3*z4*z5*z6*z7*z8
+( 3.98254394531250E-01 - 8.66729736328125E-01*i)*zz1
+( 8.07800292968750E-01 + 9.87152099609375E-01*i)*zz2;
+( 2.93814126877713E-01 + 1.92096677720035E-01*i)*z0
+(-1.33379543011714E-01 + 1.78361504432095E-01*i)*z1
+(-1.94140987554216E-01 + 1.94664546378996E-01*i)*z2
+(-2.95401059232047E-01 - 5.14158221596368E-02*i)*z3
+(-1.86538211186990E-01 - 2.59379181252348E-01*i)*z4
+( 3.83069103439169E-01 - 1.28725110967683E-01*i)*z5
+(-2.43984412828121E-01 + 1.12755116662622E-01*i)*z6
+( 2.63192775134840E-02 + 4.33791685409140E-02*i)*z7
+( 1.94127276329902E-01 + 2.82778182872871E-02*i)*z8
+( 3.47351074218750E-01 + 8.58734130859375E-01*i)*zz1
+(-5.36071777343750E-01 - 2.99102783203125E-01*i)*zz2
-4.13655969736451E-01;
+( 4.00055227354677E-02 - 1.78212168127364E-02*i)*z0
+(-5.67824735956847E-02 - 1.50308926563653E-01*i)*z1
+(-1.20729956099808E-01 + 1.76605995309073E-01*i)*z2
+(-1.82210059338039E-01 + 1.38012495099721E-01*i)*z3
+(-4.15664590028213E-03 - 2.46700553101298E-02*i)*z4
+(-2.91758849749721E-01 + 1.90546837675407E-01*i)*z5
+( 3.37332357998628E-01 - 7.81789304912989E-02*i)*z6
+(-1.10719165850518E-01 - 2.98797965629384E-01*i)*z7
+( 1.38748795590589E-02 - 2.66821946699176E-01*i)*z8
+( 3.39898284836732E-01 - 2.05749730862605E-01*i)*zz1
+( 6.05957031250000E-01 - 3.19671630859375E-01*i)*zz2
+(-1.98377408987400E-01 + 2.15444748723150E-01*i);
TITLE : embedded cyclic 9-roots system
ROOT COUNTS :
total degree : 362,880 (= 9!)
Mixed volume : 20,376
NOTE : the original last equation is moved up to the front.
The zz1 and zz2 are two "slack variables", used to "square" the
system which became overdetermined after adding two random hyperplanes.
For zz1 = 0 and zz2 = 0 we find 18 solutions. These 18 solutions are
generic points on the two dimensional solution components.
With these 18 generic points we then can figure out the decomposition
into irreducible components either by interpolation, or by application
of monodromy group actions.
We found six two dimensional cubic surfaces.
REFERENCES :
J.C. Faug\`ere:
"A new efficient algorithm for computing Gr\"obner bases (F_4)",
Journal of Pure and Applied Algebra vol. 139, Nos 1-3, pages 61-88, 1999.
Proceedings of MEGA'98, 22-27 June 1998, Saint-Malo, France.
T.Y. Li and X. Li:
"Finding mixed cells in the mixed volume computation",
Found. Comput. Math. 1(2): 161--181, 2001.
A.J. Sommese and J. Verschelde:
"Numerical homotopies to compute generic points on positive
dimensional algebraic sets",
Journal of Complexity 16(3):572--602, 2000.
A.J. Sommese, J. Verschelde and C.W. Wampler:
"Numerical decomposition of the solution sets of polynomial
systems into irreducible components",
SIAM J.\ Numer.\ Anal. 38(6):2022--2046, 2001.
A.J. Sommese, J. Verschelde and C.W. Wampler:
"Numerical irreducible decomposition
using projections from points on the components",
to appear in Contemporary Mathematics.
A.J. Sommese, J. Verschelde and C.W. Wampler:
"Using monodromy to decompose solution sets of polynomial
systems into irreducible components",
In {\em Application of Algebraic Geometry to Coding Theory,
Physics and Computation}, edited by
C. Ciliberto, F. Hirzebruch, R. Miranda, and M. Teicher.
Proceedings of a NATO Conference, February 25 - March 1, 2001, Eilat,
Israel. Pages 297--315, Kluwer Academic Publishers.
A.J. Sommese, J. Verschelde and C.W. Wampler:
"Symmetric functions applied to decomposing solution sets
of polynomial systems",
Submitted for Publication, 2001.
TIMING (on 800 Mhz Pentium III Linux machine) :
mixed volume calculation : 13m 4s 540ms
polyhedral continuation : 4h 4m 29s 730ms
solving target system : 4h 54m 53s 550ms
The mixed volume computation was done with the fabulous program
of TY Li (http://www.math.msu.edu/~li)
and Xing Li (http://www.math.msu.edu/~xing).
In about 2.5 minutes the irreducible decomposition is found :
Found 6 irreducible components of dimension 2 :
Component 1 has degree 3 and spans space of dimension 3;
contains 3 generic points 1 10 15.
Component 2 has degree 3 and spans space of dimension 3;
contains 3 generic points 2 6 18.
Component 3 has degree 3 and spans space of dimension 3;
contains 3 generic points 3 4 9.
Component 4 has degree 3 and spans space of dimension 3;
contains 3 generic points 5 13 14.
Component 5 has degree 3 and spans space of dimension 3;
contains 3 generic points 7 8 16.
Component 6 has degree 3(=2+1) and spans space of dimension 4;
contains 3 generic points 11 12 17.
The sum of the degrees is 18.
TIMING INFORMATION for Numerical Irreducible Decomposition
The elapsed time in seconds was 157.530000000 = 0h 2m37s530ms
User time in seconds was 157.460000000 = 0h 2m37s460ms
System CPU time in seconds was 0.070000000 = 0h 0m 0s 70ms
The labels of the generic points refer to the solution list below.
This calculation is done with 32 decimal places.
THE SOLUTIONS with zz1 = 0 = zz2 :
18 11
===========================================================
solution 1 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
z0 : 1.94777894754863E-01 3.03980869930257E-01
z1 : -1.75205569745961E+00 7.38019746148818E-01
z2 : -1.14504250637622E-01 -1.45242797165602E+00
z3 : 1.65866208246664E-01 -3.20673039918491E-01
z4 : 1.51517169738923E+00 1.14831486977088E+00
z5 : -1.20058739530241E+00 8.25377575721490E-01
z6 : -3.60644103001527E-01 1.66921699882347E-02
z7 : 2.36884000070388E-01 -1.88633461591970E+00
z8 : 1.31509164594003E+00 6.27050395934529E-01
zz1 : 3.03056941271789E-16 7.72065916513053E-17
zz2 : 1.77123029769856E-16 3.52712221967583E-17
== err : 3.240E-14 = rco : 1.149E-03 = res : 8.065E-15 ==
solution 2 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
z0 : 6.96134794996519E-01 -3.81765630552550E-01
z1 : 4.19870265222308E-01 1.05890553195944E+00
z2 : -1.03657206997467E+00 -3.84873632169384E-01
z3 : -6.78686131848552E-01 -4.11987601648983E-01
z4 : 7.07103958273594E-01 -8.93071081955948E-01
z5 : 1.84975692281861E-01 1.09013456153618E+00
z6 : -1.74486631479666E-02 7.93753232201533E-01
z7 : -1.12697422349590E+00 -1.65834450003490E-01
z8 : 8.51596377692809E-01 -7.05260929366793E-01
zz1 : -6.98104655000615E-16 3.60615461130046E-17
zz2 : -2.43066814426445E-16 -7.24956832500021E-17
== err : 1.256E-14 = rco : 6.213E-03 = res : 3.856E-15 ==
solution 3 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
z0 : -2.06112760332844E-01 -9.61078642544502E-02
z1 : 4.51764436091336E-01 2.63534164408104E+00
z2 : -1.59051675824592E+00 -4.18105097364720E-01
z3 : 1.86288232114242E-01 -1.30444954365151E-01
z4 : -2.50815502947090E+00 -9.26431343859074E-01
z5 : 1.15734801489257E+00 -1.16837536910348E+00
z6 : 1.98245282186015E-02 2.26552818619601E-01
z7 : 2.05639059337957E+00 -1.70891030022197E+00
z8 : 4.33168743353350E-01 1.58648046646820E+00
zz1 : -9.65192080773141E-17 -2.59890032017615E-16
zz2 : -6.61393537829865E-17 8.29064848659207E-17
== err : 1.186E-13 = rco : 1.476E-04 = res : 8.695E-15 ==
solution 4 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
z0 : 4.47843464214542E-01 6.26516284437912E-01
z1 : 1.17677553670748E+00 -1.20139074685270E+00
z2 : -2.78602804862677E-01 7.20114913529667E-01
z3 : -7.66500750315139E-01 7.45856747096647E-02
z4 : 4.52047138292262E-01 1.61981288276709E+00
z5 : -4.84336406329388E-01 -6.01334563341511E-01
z6 : 3.18657286100597E-01 -7.01101959147576E-01
z7 : -1.62882267499974E+00 -4.18422135914392E-01
z8 : 7.62939211192065E-01 -1.18780350188156E-01
zz1 : -3.94901288356343E-18 3.82617038167079E-16
zz2 : 4.64375384070048E-17 1.72104852511891E-16
== err : 2.655E-14 = rco : 4.480E-03 = res : 7.085E-15 ==
solution 5 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
z0 : -1.92184149839757E-01 2.12806797245316E-01
z1 : 1.40446168485247E+00 1.84336840303556E+00
z2 : -1.49963421888538E+00 1.25431835283928E-01
z3 : -8.82040175925694E-02 -2.72839754588603E-01
z4 : -2.29863470798858E+00 2.94615296206350E-01
z5 : 6.41189953643502E-01 -1.36143724758114E+00
z6 : 2.80388167432326E-01 6.00329573432866E-02
z7 : 8.94173023136108E-01 -2.13798369924191E+00
z8 : 8.58444265241879E-01 1.23600541229721E+00
zz1 : -2.69694578969180E-16 4.06319303028357E-16
zz2 : -1.10178254985569E-16 2.48439847053124E-16
== err : 7.264E-14 = rco : 2.525E-04 = res : 7.750E-15 ==
solution 6 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
z0 : -1.85038777085167E-01 2.68531654821430E-01
z1 : -2.12831064369090E+00 -1.73279877544696E-01
z2 : 5.09374417678794E-01 -1.34265735168901E+00
z3 : 3.25074623338216E-01 2.59824542302457E-02
z4 : 9.14090545927086E-01 1.92981102335348E+00
z5 : -1.41746258398002E+00 2.30197490096766E-01
z6 : -1.40035846253049E-01 -2.94514109051676E-01
z7 : 1.21422009776381E+00 -1.75653114580878E+00
z8 : 9.08088166301223E-01 1.11245986159225E+00
zz1 : 2.01523006191032E-16 -2.58269088124487E-16
zz2 : 1.19376980179133E-16 -2.19780485301289E-16
== err : 3.098E-14 = rco : 6.045E-04 = res : 6.224E-15 ==
solution 7 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
z0 : 1.93243706560192E-01 -9.70807989598383E-01
z1 : 1.16145738574929E+00 -8.00421070840451E-01
z2 : -2.83464525987396E-01 6.57725566680197E-01
z3 : -9.37366234469194E-01 3.18050035796601E-01
z4 : -1.27391367394682E+00 -6.05641066051723E-01
z5 : 7.11339312457264E-01 -8.33753027633003E-02
z6 : 7.44122527909003E-01 6.52757953801783E-01
z7 : 1.12456288197528E-01 1.40606213689217E+00
z8 : -4.27874786469869E-01 -5.74350263916897E-01
zz1 : 4.83657441345604E-16 6.07616449753450E-16
zz2 : 4.54097585502138E-17 6.72479715158070E-17
== err : 1.278E-14 = rco : 7.001E-03 = res : 5.623E-15 ==
solution 8 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
z0 : 5.45943598887387E-01 3.83857867828646E-02
z1 : -7.21829459608911E-01 1.22131869639108E+00
z2 : -7.31461724390237E-01 -1.06007529474556E+00
z3 : -2.39728732945480E-01 -4.91993919061411E-01
z4 : 1.41860774699602E+00 1.44633010257721E-02
z5 : -5.52321272978810E-01 1.16350208259069E+00
z6 : -3.06214865941907E-01 4.53608132278547E-01
z7 : -6.96778287387110E-01 -1.23578199741685E+00
z8 : 1.28378299736905E+00 -1.03426787845138E-01
zz1 : 1.19195718174989E-16 3.85094396130255E-16
zz2 : 1.39962710386250E-16 1.55333050142353E-17
== err : 2.465E-14 = rco : 2.454E-03 = res : 4.112E-15 ==
solution 9 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
z0 : 7.96177224979220E-01 -7.49191407260689E-02
z1 : -1.90587283271681E+00 -6.16360362675215E-01
z2 : 1.87251624620590E-01 -5.95539090395260E-01
z3 : -3.33206733391133E-01 7.26969273109638E-01
z4 : 1.48672014832093E+00 -1.34235410817776E+00
z5 : 4.22126168918677E-01 4.59934209018970E-01
z6 : -4.62970491588087E-01 -6.52050132383568E-01
z7 : 4.19152684395879E-01 1.95871447085297E+00
z8 : -6.09377793539268E-01 1.35604881376291E-01
zz1 : -2.46956292680576E-16 -8.88834886960506E-17
zz2 : -2.34695128092660E-17 -2.00326040102526E-17
== err : 1.592E-14 = rco : 4.032E-03 = res : 9.338E-15 ==
solution 10 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
z0 : 5.52774027869717E-01 -7.73376118820404E-01
z1 : 1.14284224731358E+00 2.59523071516570E-01
z2 : -8.51974745252514E-01 2.82581007736281E-01
z3 : -9.46150379513541E-01 -9.20282912772205E-02
z4 : -3.46667550855275E-01 -1.11949195444994E+00
z5 : 6.70709703952884E-01 5.96541268903312E-01
z6 : 3.93376351643823E-01 8.65404410097625E-01
z7 : -7.96174696458305E-01 8.59968882933374E-01
z8 : 1.81265041299630E-01 -8.79122276639593E-01
zz1 : -1.10051371462021E-16 6.41271852090500E-16
zz2 : -1.00541921886301E-16 2.03947746557514E-17
== err : 1.184E-14 = rco : 1.209E-02 = res : 3.969E-15 ==
solution 11 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
z0 : 4.05687846341250E-02 -2.84124455231732E-01
z1 : -7.45503479959813E-01 2.30261221569105E+00
z2 : -1.19441060242073E+00 -8.03634866104804E-01
z3 : 2.25774603750032E-01 1.77195825709678E-01
z4 : -1.62136893387291E+00 -1.79693106010042E+00
z5 : 1.29317351062403E+00 -6.32572491193425E-01
z6 : -2.66343388384157E-01 1.06928629522054E-01
z7 : 2.36687241383272E+00 -5.05681155590625E-01
z8 : -9.87629082033027E-02 1.43620735729823E+00
zz1 : -6.05741985763463E-16 -5.19033432124912E-16
zz2 : -9.28090149268392E-17 -8.48307517897459E-17
== err : 8.557E-14 = rco : 2.100E-04 = res : 7.083E-15 ==
solution 12 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
z0 : 9.24298094925836E-01 2.32885928443073E-01
z1 : -1.23303583935353E+00 -1.77027805764844E+00
z2 : 4.79922155660824E-01 -7.84514060547976E-02
z3 : -6.63834177678543E-01 6.84022666653798E-01
z4 : 2.14962368936249E+00 -1.82701331832603E-01
z5 : -1.72020167224349E-01 4.54850481668662E-01
z6 : -2.60463917247292E-01 -9.16908595096871E-01
z7 : -9.16587850008959E-01 1.95297938948105E+00
z8 : -3.07901988436475E-01 -3.76399075613864E-01
zz1 : -4.65135445905829E-16 -2.08218303580685E-17
zz2 : -1.22274873369727E-16 3.18610865935046E-17
== err : 1.596E-14 = rco : 4.280E-03 = res : 5.612E-15 ==
solution 13 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
z0 : 7.84155191193138E-01 5.12740994760677E-01
z1 : -3.14127291818504E-04 -1.99996564889732E+00
z2 : 2.91994218795347E-01 4.46711535309179E-01
z3 : -8.36124322621019E-01 4.22727818702362E-01
z4 : 1.73217812228722E+00 9.99710782233925E-01
z5 : -5.32860647138971E-01 2.95186435803730E-02
z6 : 5.19691314278816E-02 -9.35468813463040E-01
z7 : -1.73186399499540E+00 1.00025486666340E+00
z8 : 2.40866428343624E-01 -4.76230178889551E-01
zz1 : 1.31314186516337E-17 3.85451463706670E-16
zz2 : -1.21446648329924E-16 6.84575408465178E-17
== err : 1.408E-14 = rco : 6.668E-03 = res : 6.135E-15 ==
solution 14 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
z0 : 4.45936887507539E-01 -2.70058512548602E-01
z1 : -1.68148041747863E+00 9.74187780414907E-01
z2 : -4.74227938397982E-01 -8.65672644823417E-01
z3 : 1.09090886215586E-02 5.21221929340393E-01
z4 : -2.93115715636793E-03 -1.94329864771001E+00
z5 : 9.86808470977334E-01 2.21428805747345E-02
z6 : -4.56845976129097E-01 -2.51163416791791E-01
z7 : 1.68441157463500E+00 9.69110867295107E-01
z8 : -5.12580532579352E-01 8.43529764248682E-01
zz1 : 2.45289674316459E-16 8.62179578576891E-17
zz2 : -8.79542777985422E-19 1.04268115608854E-16
== err : 1.772E-14 = rco : 4.613E-03 = res : 4.120E-15 ==
solution 15 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
z0 : -4.34438371978545E-01 -5.21695823767205E-01
z1 : -6.21405113054966E-01 -1.69512822071102E+00
z2 : 8.15853637264480E-01 -1.17235338698019E-03
z3 : -2.34582650441377E-01 6.37082578395776E-01
z4 : -1.15732154528017E+00 1.38571672430265E+00
z5 : -4.08942106447577E-01 -7.05963798947484E-01
z6 : 6.69021022419921E-01 -1.15386754628571E-01
z7 : 1.77872665833514E+00 3.09411496408367E-01
z8 : -4.06911530816902E-01 7.07136152334464E-01
zz1 : -1.20220844837185E-16 2.48995482476794E-16
zz2 : 6.54593217365234E-17 2.08406667546686E-16
== err : 1.370E-14 = rco : 6.067E-03 = res : 5.094E-15 ==
solution 16 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
z0 : -4.26073754801543E-01 -5.86688698418328E-02
z1 : -1.67024648934138E+00 -1.11848302859626E+00
z2 : 8.64300891751977E-01 -7.68669589241359E-01
z3 : 1.62228145706422E-01 3.98325130464874E-01
z4 : -1.33511471795426E-01 2.00571740464954E+00
z5 : -1.09783783727555E+00 -3.64171734150078E-01
z6 : 2.63845609095121E-01 -3.39656260623042E-01
z7 : 1.80375796113680E+00 -8.87234376053278E-01
z8 : 2.33536945523579E-01 1.13284132339143E+00
zz1 : -4.79484852526130E-16 -2.82254292151564E-16
zz2 : -2.54175648815729E-16 8.74064936487334E-18
== err : 3.067E-14 = rco : 1.779E-03 = res : 6.147E-15 ==
solution 17 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
z0 : 7.30410493009578E-02 5.06727806246054E-01
z1 : 1.70120645939535E+00 2.85256376510535E-01
z2 : -9.67379491728109E-01 5.88556997929289E-01
z3 : -4.75359677663520E-01 -1.90108498909326E-01
z4 : -1.09764249834730E+00 1.33065982266328E+00
z5 : -2.60155659178145E-02 -1.13205371390126E+00
z6 : 4.02318628362563E-01 -3.16619307336728E-01
z7 : -6.03563961048051E-01 -1.61591619917382E+00
z8 : 9.93395057645923E-01 5.43496715971976E-01
zz1 : -1.06313604887849E-16 -2.24208133955173E-16
zz2 : -1.31000694300005E-16 1.10220314581333E-16
== err : 3.015E-14 = rco : 2.769E-03 = res : 3.465E-15 ==
solution 18 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
z0 : -1.97982467265316E-01 -8.77857096926233E-01
z1 : 4.77821815267592E-01 -1.58321105746037E+00
z2 : 3.76572293670222E-01 5.56511666551676E-01
z3 : -6.61255313197918E-01 6.10386394618801E-01
z4 : -1.61001190294690E+00 3.77799698226055E-01
z5 : 2.93667093901055E-01 -6.04377005955624E-01
z6 : 8.59237780463233E-01 2.67470702307432E-01
z7 : 1.13219008767931E+00 1.20541135923431E+00
z8 : -6.70239387571277E-01 4.78653394039482E-02
zz1 : 3.55740528279679E-16 -4.47108512460978E-16
zz2 : 1.99268167352691E-16 -6.20718779211461E-17
== err : 1.248E-14 = rco : 5.931E-03 = res : 6.578E-15 ==