License expires in 12 days |\^/| Maple 12 (X86 64 LINUX) ._|\| |/|_. Copyright (c) Maplesoft, a division of Waterloo Maple Inc. 2008 \ MAPLE / All rights reserved. Maple is a trademark of <____ ____> Waterloo Maple Inc. | Type ? for help. > restart; > > kernelopts(printbytes=false): > > with(LinearAlgebra): > > Digits:=14: > > read`polysolver.mpl`: > > read`multiplerefiner.mpl`: > > > # cbms1, (0,0,0) is 11-fold and index is 5. > > lsys:=[x^3-y*z,y^3-x*z,z^3-x*y]; 3 3 3 lsys := [x - y z, y - x z, z - x y] > > mroot:=convert(RandomVector(3,generator=-1..1.)*1/100.,list); mroot := [-0.00746026367412987862, 0.00811583874151238470, 0.00629447372786357852] > > mroot:=Refineroot(lsys,[x,y,z],10.^(-2), mroot, 3); `k=`, 2 `the dimension of the matrix`, 4, 3 `check the dimension d0`, 3 `the dimension of the matrix`, 4, 12 `check the dimension d1`, 3 `the system is involutive after t prolongation`, 3, 1 `k=`, 3 `the dimension of the matrix`, 10, 3 `check the dimension d0`, 7 `the dimension of the matrix`, 10, 12 `check the dimension d1`, 5 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 10, 30 `check the dimension d1`, 5 `the system is involutive after t prolongation`, 5, 2 `k=`, 4 `the dimension of the matrix`, 20, 3 `check the dimension d0`, 17 `the dimension of the matrix`, 20, 12 `check the dimension d1`, 8 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 20, 30 `check the dimension d1`, 8 `the system is involutive after t prolongation`, 8, 2 `the order of the primary ideal and dimension`, 4, 8 `k=`, 5 `the dimension of the matrix`, 35, 12 `check the dimension d0`, 23 `the dimension of the matrix`, 35, 30 `check the dimension d1`, 10 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 35, 60 `check the dimension d1`, 10 `the system is involutive after t prolongation`, 10, 2 `the order of the primary ideal and dimension`, 5, 10 `k=`, 6 `the dimension of the matrix`, 56, 30 `check the dimension d0`, 26 `the dimension of the matrix`, 56, 60 `check the dimension d1`, 11 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 56, 105 `check the dimension d1`, 11 `the system is involutive after t prolongation`, 11, 2 `the order of the primary ideal and dimension`, 6, 11 `k=`, 7 `the dimension of the matrix`, 84, 60 `check the dimension d0`, 27 `the dimension of the matrix`, 84, 105 `check the dimension d1`, 11 `the system is involutive after t prolongation`, 11, 1 `the order of the primary ideal and dimension`, 7, 11 `the index and multiplicity are `, 6, 11 [.277578618e-7, -.235492783e-7, -.360849027e-7] `k=`, 2 `the dimension of the matrix`, 4, 3 `check the dimension d0`, 4 `the dimension of the matrix`, 4, 12 `check the dimension d1`, 4 `the system is involutive after t prolongation`, 4, 1 `k=`, 3 `the dimension of the matrix`, 10, 3 `check the dimension d0`, 7 `the dimension of the matrix`, 10, 12 `check the dimension d1`, 7 `the system is involutive after t prolongation`, 7, 1 `k=`, 4 `the dimension of the matrix`, 20, 3 `check the dimension d0`, 17 `the dimension of the matrix`, 20, 12 `check the dimension d1`, 10 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 20, 30 `check the dimension d1`, 10 `the system is involutive after t prolongation`, 10, 2 `the order of the primary ideal and dimension`, 4, 10 `k=`, 5 `the dimension of the matrix`, 35, 12 `check the dimension d0`, 23 `the dimension of the matrix`, 35, 30 `check the dimension d1`, 11 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 35, 60 `check the dimension d1`, 11 `the system is involutive after t prolongation`, 11, 2 `the order of the primary ideal and dimension`, 5, 11 `k=`, 6 `the dimension of the matrix`, 56, 30 `check the dimension d0`, 26 `the dimension of the matrix`, 56, 60 `check the dimension d1`, 11 `the system is involutive after t prolongation`, 11, 1 `the order of the primary ideal and dimension`, 6, 11 `the index and multiplicity are `, 5, 11 [-.91818e-16, .590909e-15, .1318182e-14] `k=`, 2 `the dimension of the matrix`, 4, 3 `check the dimension d0`, 4 `the dimension of the matrix`, 4, 12 `check the dimension d1`, 4 `the system is involutive after t prolongation`, 4, 1 `k=`, 3 `the dimension of the matrix`, 10, 3 `check the dimension d0`, 7 `the dimension of the matrix`, 10, 12 `check the dimension d1`, 7 `the system is involutive after t prolongation`, 7, 1 `k=`, 4 `the dimension of the matrix`, 20, 3 `check the dimension d0`, 17 `the dimension of the matrix`, 20, 12 `check the dimension d1`, 10 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 20, 30 `check the dimension d1`, 10 `the system is involutive after t prolongation`, 10, 2 `the order of the primary ideal and dimension`, 4, 10 `k=`, 5 `the dimension of the matrix`, 35, 12 `check the dimension d0`, 23 `the dimension of the matrix`, 35, 30 `check the dimension d1`, 11 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 35, 60 `check the dimension d1`, 11 `the system is involutive after t prolongation`, 11, 2 `the order of the primary ideal and dimension`, 5, 11 `k=`, 6 `the dimension of the matrix`, 56, 30 `check the dimension d0`, 26 `the dimension of the matrix`, 56, 60 `check the dimension d1`, 11 `the system is involutive after t prolongation`, 11, 1 `the order of the primary ideal and dimension`, 6, 11 `the index and multiplicity are `, 5, 11 [.24454563636364e-15, .590909e-15, -.11363634545455e-14] `the number of iteration is`, 2 -15 -15 -14 mroot := [0.24454563636364 10 , 0.590909 10 , -0.11363634545455 10 ] > # cbms2, (0,0,0) is 8-fold and index is 4. > > lsys:=[x^3-3*x^2*y+3*x*y^2-y^3-z^2,z^3-3*z^2*x+3*z*x^2-x^3-y^2,y^3-3*y^2*z+3*y*z^2-z^3-x^2]; 3 2 2 3 2 3 2 2 3 2 lsys := [x - 3 x y + 3 x y - y - z , z - 3 z x + 3 z x - x - y , 3 2 2 3 2 y - 3 y z + 3 y z - z - x ] > > mroot:=convert(RandomVector(3,generator=-1..1.)*1/100., list); mroot := [-0.00804919190001180881, 0.00264718492450819040, 0.00826751712278038856] > > mroot:=Refineroot(lsys,[x,y,z],10^(-2), mroot, 3); `k=`, 2 `the dimension of the matrix`, 4, 3 `check the dimension d0`, 2 `the dimension of the matrix`, 4, 12 `check the dimension d1`, 2 `the system is involutive after t prolongation`, 2, 1 `k=`, 3 `the dimension of the matrix`, 10, 3 `check the dimension d0`, 7 `the dimension of the matrix`, 10, 12 `check the dimension d1`, 4 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 10, 30 `check the dimension d1`, 4 `the system is involutive after t prolongation`, 4, 2 `k=`, 4 `the dimension of the matrix`, 20, 3 `check the dimension d0`, 17 `the dimension of the matrix`, 20, 12 `check the dimension d1`, 8 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 20, 30 `check the dimension d1`, 7 ` the system is not involutive yet, prolong the system`, 2 `the dimension of the matrix`, 20, 60 `check the dimension d1`, 7 `the system is involutive after t prolongation`, 7, 3 `the order of the primary ideal and dimension`, 4, 7 `k=`, 5 `the dimension of the matrix`, 35, 12 `check the dimension d0`, 23 `the dimension of the matrix`, 35, 30 `check the dimension d1`, 8 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 35, 60 `check the dimension d1`, 8 `the system is involutive after t prolongation`, 8, 2 `the order of the primary ideal and dimension`, 5, 8 `k=`, 6 `the dimension of the matrix`, 56, 30 `check the dimension d0`, 26 `the dimension of the matrix`, 56, 60 `check the dimension d1`, 8 `the system is involutive after t prolongation`, 8, 1 `the order of the primary ideal and dimension`, 6, 8 `the index and multiplicity are `, 5, 8 [.22691179076e-5, -.23409113657e-5, -.44070227744e-5] `k=`, 2 `the dimension of the matrix`, 4, 3 `check the dimension d0`, 4 `the dimension of the matrix`, 4, 12 `check the dimension d1`, 4 `the system is involutive after t prolongation`, 4, 1 `k=`, 3 `the dimension of the matrix`, 10, 3 `check the dimension d0`, 7 `the dimension of the matrix`, 10, 12 `check the dimension d1`, 7 `the system is involutive after t prolongation`, 7, 1 `k=`, 4 `the dimension of the matrix`, 20, 3 `check the dimension d0`, 17 `the dimension of the matrix`, 20, 12 `check the dimension d1`, 8 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 20, 30 `check the dimension d1`, 8 `the system is involutive after t prolongation`, 8, 2 `the order of the primary ideal and dimension`, 4, 8 `k=`, 5 `the dimension of the matrix`, 35, 12 `check the dimension d0`, 23 `the dimension of the matrix`, 35, 30 `check the dimension d1`, 8 `the system is involutive after t prolongation`, 8, 1 `the order of the primary ideal and dimension`, 5, 8 `the index and multiplicity are `, 4, 8 [.176e-16, -.758e-16, -.3570e-15] `k=`, 2 `the dimension of the matrix`, 4, 3 `check the dimension d0`, 4 `the dimension of the matrix`, 4, 12 `check the dimension d1`, 4 `the system is involutive after t prolongation`, 4, 1 `k=`, 3 `the dimension of the matrix`, 10, 3 `check the dimension d0`, 7 `the dimension of the matrix`, 10, 12 `check the dimension d1`, 7 `the system is involutive after t prolongation`, 7, 1 `k=`, 4 `the dimension of the matrix`, 20, 3 `check the dimension d0`, 17 `the dimension of the matrix`, 20, 12 `check the dimension d1`, 8 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 20, 30 `check the dimension d1`, 8 `the system is involutive after t prolongation`, 8, 2 `the order of the primary ideal and dimension`, 4, 8 `k=`, 5 `the dimension of the matrix`, 35, 12 `check the dimension d0`, 23 `the dimension of the matrix`, 35, 30 `check the dimension d1`, 8 `the system is involutive after t prolongation`, 8, 1 `the order of the primary ideal and dimension`, 5, 8 `the index and multiplicity are `, 4, 8 [.12676000000000e-14, .30416863120875e-15, .51692085360970e-15] `the number of iteration is`, 2 mroot := -14 -15 -15 [0.12676000000000 10 , 0.30416863120875 10 , 0.51692085360970 10 ] > # mth191, 15 regular roots and 3 roots of multiplicity 4, (0,0,1), (0,1,0) and (1,0,0) are 4-fold zeros and indices are 3. > # > > lsys:=[x^3+y^2+z^2-1, x^2+y^3+z^2-1, x^2+y^2+z^3-1]; 3 2 2 2 3 2 2 2 3 lsys := [x + y + z - 1, x + y + z - 1, x + y + z - 1] > > mroot:=RandomVector(3,generator=-1..1.)*1/100.; [0.00915013670868595162 ] [ ] mroot := [0.000937630384099676930] [ ] [-0.00443003562265903254] > > mroot:=convert([mroot[1]+1,mroot[2],mroot[3]], list); mroot := [1.0091501367087, 0.000937630384099676930, -0.00443003562265903254] > > mroot:=Refineroot(lsys,[x,y,z],10.^(-2),mroot,3); `k=`, 2 `the dimension of the matrix`, 4, 3 `check the dimension d0`, 3 `the dimension of the matrix`, 4, 12 `check the dimension d1`, 1 `the system is involutive after t prolongation`, 1, 1 `k=`, 3 `the dimension of the matrix`, 10, 3 `check the dimension d0`, 7 `the dimension of the matrix`, 10, 12 `check the dimension d1`, 4 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 10, 30 `check the dimension d1`, 3 ` the system is not involutive yet, prolong the system`, 2 `the dimension of the matrix`, 10, 60 `check the dimension d1`, 3 `the system is involutive after t prolongation`, 3, 3 `k=`, 4 `the dimension of the matrix`, 20, 3 `check the dimension d0`, 17 `the dimension of the matrix`, 20, 12 `check the dimension d1`, 8 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 20, 30 `check the dimension d1`, 4 ` the system is not involutive yet, prolong the system`, 2 `the dimension of the matrix`, 20, 60 `check the dimension d1`, 4 `the system is involutive after t prolongation`, 4, 3 `the order of the primary ideal and dimension`, 4, 4 `k=`, 5 `the dimension of the matrix`, 35, 12 `check the dimension d0`, 23 `the dimension of the matrix`, 35, 30 `check the dimension d1`, 6 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 35, 60 `check the dimension d1`, 4 `the system is involutive after t prolongation`, 4, 2 `the order of the primary ideal and dimension`, 5, 4 `the index should be less or equal to multiplicity,`, 4, 4 [1.0000000970606, .12211049988e-6, .1691489935e-6] `k=`, 2 `the dimension of the matrix`, 4, 3 `check the dimension d0`, 3 `the dimension of the matrix`, 4, 12 `check the dimension d1`, 3 `the system is involutive after t prolongation`, 3, 1 `k=`, 3 `the dimension of the matrix`, 10, 3 `check the dimension d0`, 7 `the dimension of the matrix`, 10, 12 `check the dimension d1`, 4 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 10, 30 `check the dimension d1`, 4 `the system is involutive after t prolongation`, 4, 2 `k=`, 4 `the dimension of the matrix`, 20, 3 `check the dimension d0`, 17 `the dimension of the matrix`, 20, 12 `check the dimension d1`, 8 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 20, 30 `check the dimension d1`, 4 `the system is involutive after t prolongation`, 4, 2 `the order of the primary ideal and dimension`, 4, 4 `the index and multiplicity are `, 3, 4 [1.0000000000000, -.603918e-14, -.2185155e-13] `k=`, 2 `the dimension of the matrix`, 4, 3 `check the dimension d0`, 3 `the dimension of the matrix`, 4, 12 `check the dimension d1`, 3 `the system is involutive after t prolongation`, 3, 1 `k=`, 3 `the dimension of the matrix`, 10, 3 `check the dimension d0`, 7 `the dimension of the matrix`, 10, 12 `check the dimension d1`, 4 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 10, 30 `check the dimension d1`, 4 `the system is involutive after t prolongation`, 4, 2 `k=`, 4 `the dimension of the matrix`, 20, 3 `check the dimension d0`, 17 `the dimension of the matrix`, 20, 12 `check the dimension d1`, 8 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 20, 30 `check the dimension d1`, 4 `the system is involutive after t prolongation`, 4, 2 `the order of the primary ideal and dimension`, 4, 4 `the index and multiplicity are `, 3, 4 [1.0000000000000, .2511495089078e-15, -.202317693222e-15] `the number of iteration is`, 2 -15 -15 mroot := [1.0000000000000, 0.2511495089078 10 , -0.202317693222 10 ] > # LVZ, IMA08 Leykin, Verschelde and Zhao, (0,0,-1) is 18-fold zero and index is 7. > > lsys:=[2*x1+2*x1^2+2*x2+2*x2^2+x3^2-1, (x1+x2-x3-1)^3-x1^3, (2*x1^3+2*x2^2+10*x3+5*x3^2+5)^3-1000*x1^5]; 2 2 2 3 3 lsys := [2 x1 + 2 x1 + 2 x2 + 2 x2 + x3 - 1, (x1 + x2 - x3 - 1) - x1 , 3 2 2 3 5 (2 x1 + 2 x2 + 10 x3 + 5 x3 + 5) - 1000 x1 ] > > mroot:=RandomVector(3,generator=-1..1.)*1/10000.; [0.0000941185563521231396 ] [ ] mroot := [-0.0000684773836644903434] [ ] [0.0000929777070398553075 ] > > mroot:=[mroot[1],mroot[2],mroot[3]-1]; mroot := [0.0000941185563521231396, -0.0000684773836644903434, -0.99990702229296] > > mroot:=Refineroot(lsys,[x1,x2,x3],10^(-3), mroot, 3); `k=`, 2 `the dimension of the matrix`, 4, 3 `check the dimension d0`, 3 `the dimension of the matrix`, 4, 12 `check the dimension d1`, 3 `the system is involutive after t prolongation`, 3, 1 `k=`, 3 `the dimension of the matrix`, 10, 3 `check the dimension d0`, 9 `the dimension of the matrix`, 10, 12 `check the dimension d1`, 6 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 10, 30 `check the dimension d1`, 6 `the system is involutive after t prolongation`, 6, 2 `k=`, 4 `the dimension of the matrix`, 20, 6 `check the dimension d0`, 15 `the dimension of the matrix`, 20, 18 `check the dimension d1`, 9 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 20, 40 `check the dimension d1`, 9 `the system is involutive after t prolongation`, 9, 2 `the order of the primary ideal and dimension`, 4, 9 `k=`, 5 `the dimension of the matrix`, 35, 15 `check the dimension d0`, 20 `the dimension of the matrix`, 35, 34 `check the dimension d1`, 12 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 35, 65 `check the dimension d1`, 12 `the system is involutive after t prolongation`, 12, 2 `the order of the primary ideal and dimension`, 5, 12 `k=`, 6 `the dimension of the matrix`, 56, 31 `check the dimension d0`, 26 `the dimension of the matrix`, 56, 59 `check the dimension d1`, 15 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 56, 101 `check the dimension d1`, 15 `the system is involutive after t prolongation`, 15, 2 `the order of the primary ideal and dimension`, 6, 15 `k=`, 7 `the dimension of the matrix`, 84, 56 `check the dimension d0`, 32 `the dimension of the matrix`, 84, 95 `check the dimension d1`, 17 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 84, 150 `check the dimension d1`, 17 `the system is involutive after t prolongation`, 17, 2 `the order of the primary ideal and dimension`, 7, 17 `k=`, 8 `the dimension of the matrix`, 120, 92 `check the dimension d0`, 38 `the dimension of the matrix`, 120, 144 `check the dimension d1`, 18 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 120, 214 `check the dimension d1`, 18 `the system is involutive after t prolongation`, 18, 2 `the order of the primary ideal and dimension`, 8, 18 `k=`, 9 `the dimension of the matrix`, 165, 141 `check the dimension d0`, 44 `the dimension of the matrix`, 165, 208 `check the dimension d1`, 20 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 165, 295 `check the dimension d1`, 18 `the system is involutive after t prolongation`, 18, 2 `the order of the primary ideal and dimension`, 9, 18 `the index and multiplicity are `, 8, 18 [-.6584694655e-8, .10416197999e-7, -1.0000000077680] `k=`, 2 `the dimension of the matrix`, 4, 3 `check the dimension d0`, 3 `the dimension of the matrix`, 4, 12 `check the dimension d1`, 3 `the system is involutive after t prolongation`, 3, 1 `k=`, 3 `the dimension of the matrix`, 10, 3 `check the dimension d0`, 9 `the dimension of the matrix`, 10, 12 `check the dimension d1`, 6 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 10, 30 `check the dimension d1`, 6 `the system is involutive after t prolongation`, 6, 2 `k=`, 4 `the dimension of the matrix`, 20, 6 `check the dimension d0`, 15 `the dimension of the matrix`, 20, 18 `check the dimension d1`, 9 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 20, 40 `check the dimension d1`, 9 `the system is involutive after t prolongation`, 9, 2 `the order of the primary ideal and dimension`, 4, 9 `k=`, 5 `the dimension of the matrix`, 35, 15 `check the dimension d0`, 20 `the dimension of the matrix`, 35, 34 `check the dimension d1`, 12 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 35, 65 `check the dimension d1`, 12 `the system is involutive after t prolongation`, 12, 2 `the order of the primary ideal and dimension`, 5, 12 `k=`, 6 `the dimension of the matrix`, 56, 31 `check the dimension d0`, 26 `the dimension of the matrix`, 56, 59 `check the dimension d1`, 15 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 56, 101 `check the dimension d1`, 15 `the system is involutive after t prolongation`, 15, 2 `the order of the primary ideal and dimension`, 6, 15 `k=`, 7 `the dimension of the matrix`, 84, 56 `check the dimension d0`, 32 `the dimension of the matrix`, 84, 95 `check the dimension d1`, 17 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 84, 150 `check the dimension d1`, 17 `the system is involutive after t prolongation`, 17, 2 `the order of the primary ideal and dimension`, 7, 17 `k=`, 8 `the dimension of the matrix`, 120, 92 `check the dimension d0`, 38 `the dimension of the matrix`, 120, 144 `check the dimension d1`, 18 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 120, 214 `check the dimension d1`, 18 `the system is involutive after t prolongation`, 18, 2 `the order of the primary ideal and dimension`, 8, 18 `k=`, 9 `the dimension of the matrix`, 165, 141 `check the dimension d0`, 44 `the dimension of the matrix`, 165, 208 `check the dimension d1`, 20 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 165, 295 `check the dimension d1`, 18 `the system is involutive after t prolongation`, 18, 2 `the order of the primary ideal and dimension`, 9, 18 `the index and multiplicity are `, 8, 18 [.35458333e-14, .5511000e-14, -.99999999999999] `k=`, 2 `the dimension of the matrix`, 4, 3 `check the dimension d0`, 3 `the dimension of the matrix`, 4, 12 `check the dimension d1`, 3 `the system is involutive after t prolongation`, 3, 1 `k=`, 3 `the dimension of the matrix`, 10, 3 `check the dimension d0`, 9 `the dimension of the matrix`, 10, 12 `check the dimension d1`, 6 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 10, 30 `check the dimension d1`, 6 `the system is involutive after t prolongation`, 6, 2 `k=`, 4 `the dimension of the matrix`, 20, 6 `check the dimension d0`, 15 `the dimension of the matrix`, 20, 18 `check the dimension d1`, 9 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 20, 40 `check the dimension d1`, 9 `the system is involutive after t prolongation`, 9, 2 `the order of the primary ideal and dimension`, 4, 9 `k=`, 5 `the dimension of the matrix`, 35, 15 `check the dimension d0`, 21 `the dimension of the matrix`, 35, 34 `check the dimension d1`, 12 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 35, 65 `check the dimension d1`, 12 `the system is involutive after t prolongation`, 12, 2 `the order of the primary ideal and dimension`, 5, 12 `k=`, 6 `the dimension of the matrix`, 56, 31 `check the dimension d0`, 26 `the dimension of the matrix`, 56, 59 `check the dimension d1`, 15 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 56, 101 `check the dimension d1`, 15 `the system is involutive after t prolongation`, 15, 2 `the order of the primary ideal and dimension`, 6, 15 `k=`, 7 `the dimension of the matrix`, 84, 56 `check the dimension d0`, 32 `the dimension of the matrix`, 84, 95 `check the dimension d1`, 17 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 84, 150 `check the dimension d1`, 17 `the system is involutive after t prolongation`, 17, 2 `the order of the primary ideal and dimension`, 7, 17 `k=`, 8 `the dimension of the matrix`, 120, 92 `check the dimension d0`, 38 `the dimension of the matrix`, 120, 144 `check the dimension d1`, 18 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 120, 214 `check the dimension d1`, 18 `the system is involutive after t prolongation`, 18, 2 `the order of the primary ideal and dimension`, 8, 18 `k=`, 9 `the dimension of the matrix`, 165, 141 `check the dimension d0`, 44 `the dimension of the matrix`, 165, 208 `check the dimension d1`, 20 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 165, 295 `check the dimension d1`, 18 `the system is involutive after t prolongation`, 18, 2 `the order of the primary ideal and dimension`, 9, 18 `the index and multiplicity are `, 8, 18 [-.20097222555556e-14, .5110000000000e-15, -1.0000000000000] `the number of iteration is`, 2 -14 -15 mroot := [-0.20097222555556 10 , 0.5110000000000 10 , -1.0000000000000] > # KSS, (1,1,1,1,1) is 16-fold zero and index is 5. > > lsys:=[x1^2+x1+x2+x3+x4+x5-2*x1-4,x2^2+x1+x2+x3+x4+x5-2*x2-4,x3^2+x1+x2+x3+x4+x5-2*x3-4,x4^2+x1+x2+x3+x4+x5-2*x4-4,x5^2+x1+x2+x3+x4+x5-2*x5-4]; 2 2 lsys := [x1 - x1 + x2 + x3 + x4 + x5 - 4, x2 + x1 - x2 + x3 + x4 + x5 - 4, 2 2 x3 + x1 + x2 - x3 + x4 + x5 - 4, x4 + x1 + x2 + x3 - x4 + x5 - 4, 2 x5 + x1 + x2 + x3 + x4 - x5 - 4] > > mroot:=[1,1,1,1,1]+convert(RandomVector(5,generator=-1..1.)*1/1000,list); mroot := [0.99984352256525, 0.99928377267725, 1.0006005609378, 0.99997075129745, 1.0009143338965] > > mroot:=Refineroot(lsys,[x1,x2,x3,x4,x5], 0.05, mroot, 3); `k=`, 2 `the dimension of the matrix`, 6, 5 `check the dimension d0`, 5 `the dimension of the matrix`, 6, 30 `check the dimension d1`, 5 `the system is involutive after t prolongation`, 5, 1 `k=`, 3 `the dimension of the matrix`, 21, 5 `check the dimension d0`, 16 `the dimension of the matrix`, 21, 30 `check the dimension d1`, 11 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 21, 105 `check the dimension d1`, 11 `the system is involutive after t prolongation`, 11, 2 `k=`, 4 `the dimension of the matrix`, 56, 30 `check the dimension d0`, 26 `the dimension of the matrix`, 56, 105 `check the dimension d1`, 15 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 56, 280 `check the dimension d1`, 15 `the system is involutive after t prolongation`, 15, 2 `the order of the primary ideal and dimension`, 4, 15 `k=`, 5 `the dimension of the matrix`, 126, 105 `check the dimension d0`, 31 `the dimension of the matrix`, 126, 280 `check the dimension d1`, 16 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 126, 630 `check the dimension d1`, 16 `the system is involutive after t prolongation`, 16, 2 `the order of the primary ideal and dimension`, 5, 16 `k=`, 6 `the dimension of the matrix`, 252, 280 `check the dimension d0`, 32 `the dimension of the matrix`, 252, 630 `check the dimension d1`, 16 `the system is involutive after t prolongation`, 16, 1 `the order of the primary ideal and dimension`, 6, 16 `the index and multiplicity are `, 5, 16 [1.0000000474565, .99999999740814, 1.0000000331521, 1.0000000512105, 1.00000000\ 39172] `k=`, 2 `the dimension of the matrix`, 6, 5 `check the dimension d0`, 5 `the dimension of the matrix`, 6, 30 `check the dimension d1`, 5 `the system is involutive after t prolongation`, 5, 1 `k=`, 3 `the dimension of the matrix`, 21, 5 `check the dimension d0`, 16 `the dimension of the matrix`, 21, 30 `check the dimension d1`, 11 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 21, 105 `check the dimension d1`, 11 `the system is involutive after t prolongation`, 11, 2 `k=`, 4 `the dimension of the matrix`, 56, 30 `check the dimension d0`, 26 `the dimension of the matrix`, 56, 105 `check the dimension d1`, 15 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 56, 280 `check the dimension d1`, 15 `the system is involutive after t prolongation`, 15, 2 `the order of the primary ideal and dimension`, 4, 15 `k=`, 5 `the dimension of the matrix`, 126, 105 `check the dimension d0`, 31 `the dimension of the matrix`, 126, 280 `check the dimension d1`, 16 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 126, 630 `check the dimension d1`, 16 `the system is involutive after t prolongation`, 16, 2 `the order of the primary ideal and dimension`, 5, 16 `k=`, 6 `the dimension of the matrix`, 252, 280 `check the dimension d0`, 32 `the dimension of the matrix`, 252, 630 `check the dimension d1`, 16 `the system is involutive after t prolongation`, 16, 1 `the order of the primary ideal and dimension`, 6, 16 `the index and multiplicity are `, 5, 16 [1.0000000000000, .99999999999997, 1.0000000000000, 1.0000000000000, 1.00000000\ 00000] `k=`, 2 `the dimension of the matrix`, 6, 5 `check the dimension d0`, 5 `the dimension of the matrix`, 6, 30 `check the dimension d1`, 5 `the system is involutive after t prolongation`, 5, 1 `k=`, 3 `the dimension of the matrix`, 21, 5 `check the dimension d0`, 16 `the dimension of the matrix`, 21, 30 `check the dimension d1`, 11 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 21, 105 `check the dimension d1`, 11 `the system is involutive after t prolongation`, 11, 2 `k=`, 4 `the dimension of the matrix`, 56, 30 `check the dimension d0`, 26 `the dimension of the matrix`, 56, 105 `check the dimension d1`, 15 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 56, 280 `check the dimension d1`, 15 `the system is involutive after t prolongation`, 15, 2 `the order of the primary ideal and dimension`, 4, 15 `k=`, 5 `the dimension of the matrix`, 126, 105 `check the dimension d0`, 31 `the dimension of the matrix`, 126, 280 `check the dimension d1`, 16 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 126, 630 `check the dimension d1`, 16 `the system is involutive after t prolongation`, 16, 2 `the order of the primary ideal and dimension`, 5, 16 `k=`, 6 `the dimension of the matrix`, 252, 280 `check the dimension d0`, 32 `the dimension of the matrix`, 252, 630 `check the dimension d1`, 16 `the system is involutive after t prolongation`, 16, 1 `the order of the primary ideal and dimension`, 6, 16 `the index and multiplicity are `, 5, 16 [.99999999999999, 1.0000000000000, .99999999999999, .99999999999999, .999999999\ 99999] `the number of iteration is`, 2 mroot := [0.99999999999999, 1.0000000000000, 0.99999999999999, 0.99999999999999, 0.99999999999999] > # Caprasse, (2,-i*sqrt(3),2,i*sqrt(3)) is 4-fold zero and index is 3. > > lsys:=[-x1^3*x3+4*x1*x2^2*x3+4*x1^2*x2*x4+2*x2^3*x4+4*x1^2-10*x2^2+4*x1*x3-10*x2*x4+2,-x1*x3^3+4*x2*x3^2*x4+4*x1*x3*x4^2+2*x2*x4^3+4*x1*x3+4*x3^2-10*x2*x4-10*x4^2+2,x2^2*x3+2*x1*x2*x4-2*x1-x3,2*x2*x3*x4+x1*x4^2-x1-2*x3]; 3 2 2 3 2 2 lsys := [-x1 x3 + 4 x1 x2 x3 + 4 x1 x2 x4 + 2 x2 x4 + 4 x1 - 10 x2 3 2 2 3 + 4 x1 x3 - 10 x2 x4 + 2, -x1 x3 + 4 x2 x3 x4 + 4 x1 x3 x4 + 2 x2 x4 2 2 2 + 4 x1 x3 + 4 x3 - 10 x2 x4 - 10 x4 + 2, x2 x3 + 2 x1 x2 x4 - 2 x1 - x3, 2 2 x2 x3 x4 + x1 x4 - x1 - 2 x3] > > mroot0:=evalf([2,-I*sqrt(3),2,I*sqrt(3)]); mroot0 := [2., -1.7320508075689 I, 2., 1.7320508075689 I] > > peroot:=RandomVector(4,generator=-1..1.)*1/1000.+RandomVector(4,generator=-1..1.)*1/1000.*I; [0.000311481398313173685 + 0.000357470309715546969 I] [ ] [0.000918984852785806024 + 0.000867986495515101126 I] peroot := [ ] [0.000584414659119108799 + 0.000698258611737554271 I] [ ] [0.000831471050378134249 - 0.000928576642851620861 I] > > mroot:=[mroot0[1]+peroot[1],mroot0[2]+peroot[2],mroot0[3]+peroot[3],mroot0[4]+peroot[4]]; mroot := [2.0003114813983 + 0.00035747030971555 I, 0.00091898485278581 - 1.7311828210734 I, 2.0005844146591 + 0.00069825861173755 I, 0.00083147105037813 + 1.7311222309260 I] > > mroot:=Refineroot(evalf(lsys),[x1,x2,x3,x4],10.^(-2),mroot,3); `k=`, 2 `the dimension of the matrix`, 5, 4 `check the dimension d0`, 2 `the dimension of the matrix`, 5, 20 `check the dimension d1`, 1 `the system is involutive after t prolongation`, 1, 1 `k=`, 3 `the dimension of the matrix`, 15, 4 `check the dimension d0`, 11 `the dimension of the matrix`, 15, 20 `check the dimension d1`, 3 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 15, 60 `check the dimension d1`, 3 `the system is involutive after t prolongation`, 3, 2 `k=`, 4 `the dimension of the matrix`, 35, 4 `check the dimension d0`, 31 `the dimension of the matrix`, 35, 20 `check the dimension d1`, 15 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 35, 60 `check the dimension d1`, 4 ` the system is not involutive yet, prolong the system`, 2 `the dimension of the matrix`, 35, 140 `check the dimension d1`, 4 `the system is involutive after t prolongation`, 4, 3 `the order of the primary ideal and dimension`, 4, 4 `k=`, 5 `the dimension of the matrix`, 70, 12 `check the dimension d0`, 58 `the dimension of the matrix`, 70, 40 `check the dimension d1`, 30 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 70, 100 `check the dimension d1`, 4 `the system is involutive after t prolongation`, 4, 2 `the order of the primary ideal and dimension`, 5, 4 `the index should be less or equal to multiplicity,`, 4, 4 [2.0000000000360-.24563097e-9*I, -.22862494e-9-1.7320508075133*I, 1.99999999997\ 31+.18308477e-9*I, -.68631137e-9+1.7320508075492*I] `k=`, 2 `the dimension of the matrix`, 5, 4 `check the dimension d0`, 3 `the dimension of the matrix`, 5, 20 `check the dimension d1`, 3 `the system is involutive after t prolongation`, 3, 1 `k=`, 3 `the dimension of the matrix`, 15, 4 `check the dimension d0`, 11 `the dimension of the matrix`, 15, 20 `check the dimension d1`, 4 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 15, 60 `check the dimension d1`, 4 `the system is involutive after t prolongation`, 4, 2 `k=`, 4 `the dimension of the matrix`, 35, 4 `check the dimension d0`, 31 `the dimension of the matrix`, 35, 20 `check the dimension d1`, 15 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 35, 60 `check the dimension d1`, 4 `the system is involutive after t prolongation`, 4, 2 `the order of the primary ideal and dimension`, 4, 4 `the index and multiplicity are `, 3, 4 [2.0000000000001+.68405000e-15*I, -.24000000e-15-1.7320508075690*I, 2.000000000\ 0000+.146750000e-14*I, -.56000000e-15+1.7320508075688*I] `k=`, 2 `the dimension of the matrix`, 5, 4 `check the dimension d0`, 3 `the dimension of the matrix`, 5, 20 `check the dimension d1`, 3 `the system is involutive after t prolongation`, 3, 1 `k=`, 3 `the dimension of the matrix`, 15, 4 `check the dimension d0`, 11 `the dimension of the matrix`, 15, 20 `check the dimension d1`, 4 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 15, 60 `check the dimension d1`, 4 `the system is involutive after t prolongation`, 4, 2 `k=`, 4 `the dimension of the matrix`, 35, 4 `check the dimension d0`, 31 `the dimension of the matrix`, 35, 20 `check the dimension d1`, 15 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 35, 60 `check the dimension d1`, 4 `the system is involutive after t prolongation`, 4, 2 `the order of the primary ideal and dimension`, 4, 4 `the index and multiplicity are `, 3, 4 [1.9999999999999+.15905000000000e-15*I, .43000000000000e-15-1.7320508075689*I, 2.0000000000000-.1250000000000e-15*I, .16000000000000e-15+1.7320508075689*I] `the number of iteration is`, 2 -15 mroot := [1.9999999999999 + 0.15905000000000 10 I, -15 0.43000000000000 10 - 1.7320508075689 I, -15 2.0000000000000 - 0.1250000000000 10 I, -15 0.16000000000000 10 + 1.7320508075689 I] > > mroot-mroot0; -12 -15 -15 [-0.1 10 + 0.15905000000000 10 I, 0.43000000000000 10 + 0. I, -15 -15 0. - 0.1250000000000 10 I, 0.16000000000000 10 + 0. I] > > > # DZ1, ISSAC05 Barry Dayton and Zhonggang Zeng, (0,0,0,0) is 131-fold zero and index is 11. > > lsys:=[x1^4-x2*x3*x4,x2^4-x1*x3*x4,x3^4-x1*x2*x4,x4^4-x1*x2*x3]; 4 4 4 4 lsys := [x1 - x2 x3 x4, x2 - x1 x3 x4, x3 - x1 x2 x4, x4 - x1 x2 x3] > > mroot:=convert(RandomVector(4,generator=-1..1.)*1/100.,list); mroot := [0.00310955780355113296, -0.00215545960931663684, 0.00486264936249832403, 0.00515480261156666883] > > mroot:=Refineroot(evalf(lsys),[x1,x2,x3,x4],10.^(-2),mroot,3); `k=`, 2 `the dimension of the matrix`, 5, 4 `check the dimension d0`, 5 `the dimension of the matrix`, 5, 20 `check the dimension d1`, 5 `the system is involutive after t prolongation`, 5, 1 `k=`, 3 `the dimension of the matrix`, 15, 4 `check the dimension d0`, 15 `the dimension of the matrix`, 15, 20 `check the dimension d1`, 15 `the system is involutive after t prolongation`, 15, 1 `k=`, 4 `the dimension of the matrix`, 35, 4 `check the dimension d0`, 31 `the dimension of the matrix`, 35, 20 `check the dimension d1`, 31 `the system is involutive after t prolongation`, 31, 1 `the order of the primary ideal and dimension`, 4, 31 `k=`, 5 `the dimension of the matrix`, 70, 4 `check the dimension d0`, 66 `the dimension of the matrix`, 70, 20 `check the dimension d1`, 51 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 70, 60 `check the dimension d1`, 51 `the system is involutive after t prolongation`, 51, 2 `the order of the primary ideal and dimension`, 5, 51 `k=`, 6 `the dimension of the matrix`, 126, 20 `check the dimension d0`, 106 `the dimension of the matrix`, 126, 60 `check the dimension d1`, 71 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 126, 140 `check the dimension d1`, 70 ` the system is not involutive yet, prolong the system`, 2 `the dimension of the matrix`, 126, 280 `check the dimension d1`, 70 `the system is involutive after t prolongation`, 70, 3 `the order of the primary ideal and dimension`, 6, 70 `k=`, 7 `the dimension of the matrix`, 210, 60 `check the dimension d0`, 150 `the dimension of the matrix`, 210, 140 `check the dimension d1`, 90 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 210, 280 `check the dimension d1`, 88 ` the system is not involutive yet, prolong the system`, 2 `the dimension of the matrix`, 210, 504 `check the dimension d1`, 88 `the system is involutive after t prolongation`, 88, 3 `the order of the primary ideal and dimension`, 7, 88 `k=`, 8 `the dimension of the matrix`, 330, 140 `check the dimension d0`, 190 `the dimension of the matrix`, 330, 280 `check the dimension d1`, 106 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 330, 504 `check the dimension d1`, 106 `the system is involutive after t prolongation`, 106, 2 `the order of the primary ideal and dimension`, 8, 106 `k=`, 9 `the dimension of the matrix`, 495, 280 `check the dimension d0`, 221 `the dimension of the matrix`, 495, 504 `check the dimension d1`, 117 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 495, 840 `check the dimension d1`, 117 `the system is involutive after t prolongation`, 117, 2 `the order of the primary ideal and dimension`, 9, 117 `k=`, 10 `the dimension of the matrix`, 715, 504 `check the dimension d0`, 241 `the dimension of the matrix`, 715, 840 `check the dimension d1`, 126 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 715, 1320 `check the dimension d1`, 126 `the system is involutive after t prolongation`, 126, 2 `the order of the primary ideal and dimension`, 10, 126 `k=`, 11 `the dimension of the matrix`, 1001, 840 `check the dimension d0`, 251 `the dimension of the matrix`, 1001, 1320 `check the dimension d1`, 130 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 1001, 1980 `check the dimension d1`, 130 `the system is involutive after t prolongation`, 130, 2 `the order of the primary ideal and dimension`, 11, 130 `k=`, 12 `the dimension of the matrix`, 1365, 1320 `check the dimension d0`, 255 `the dimension of the matrix`, 1365, 1980 `check the dimension d1`, 131 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 1365, 2860 `check the dimension d1`, 131 `the system is involutive after t prolongation`, 131, 2 `the order of the primary ideal and dimension`, 12, 131 `k=`, 13 `the dimension of the matrix`, 1820, 1980 `check the dimension d0`, 256 `the dimension of the matrix`, 1820, 2860 `check the dimension d1`, 131 `the system is involutive after t prolongation`, 131, 1 `the order of the primary ideal and dimension`, 13, 131 `the index and multiplicity are `, 12, 131 [.2648919e-9, -.1337204e-9, .8515849e-9, .10249720e-8] `k=`, 2 `the dimension of the matrix`, 5, 4 `check the dimension d0`, 5 `the dimension of the matrix`, 5, 20 `check the dimension d1`, 5 `the system is involutive after t prolongation`, 5, 1 `k=`, 3 `the dimension of the matrix`, 15, 4 `check the dimension d0`, 15 `the dimension of the matrix`, 15, 20 `check the dimension d1`, 15 `the system is involutive after t prolongation`, 15, 1 `k=`, 4 `the dimension of the matrix`, 35, 4 `check the dimension d0`, 31 `the dimension of the matrix`, 35, 20 `check the dimension d1`, 31 `the system is involutive after t prolongation`, 31, 1 `the order of the primary ideal and dimension`, 4, 31 `k=`, 5 `the dimension of the matrix`, 70, 4 `check the dimension d0`, 66 `the dimension of the matrix`, 70, 20 `check the dimension d1`, 53 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 70, 60 `check the dimension d1`, 53 `the system is involutive after t prolongation`, 53, 2 `the order of the primary ideal and dimension`, 5, 53 `k=`, 6 `the dimension of the matrix`, 126, 20 `check the dimension d0`, 106 `the dimension of the matrix`, 126, 60 `check the dimension d1`, 78 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 126, 140 `check the dimension d1`, 78 `the system is involutive after t prolongation`, 78, 2 `the order of the primary ideal and dimension`, 6, 78 `k=`, 7 `the dimension of the matrix`, 210, 60 `check the dimension d0`, 150 `the dimension of the matrix`, 210, 140 `check the dimension d1`, 100 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 210, 280 `check the dimension d1`, 100 `the system is involutive after t prolongation`, 100, 2 `the order of the primary ideal and dimension`, 7, 100 `k=`, 8 `the dimension of the matrix`, 330, 140 `check the dimension d0`, 190 `the dimension of the matrix`, 330, 280 `check the dimension d1`, 116 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 330, 504 `check the dimension d1`, 116 `the system is involutive after t prolongation`, 116, 2 `the order of the primary ideal and dimension`, 8, 116 `k=`, 9 `the dimension of the matrix`, 495, 280 `check the dimension d0`, 221 `the dimension of the matrix`, 495, 504 `check the dimension d1`, 126 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 495, 840 `check the dimension d1`, 126 `the system is involutive after t prolongation`, 126, 2 `the order of the primary ideal and dimension`, 9, 126 `k=`, 10 `the dimension of the matrix`, 715, 504 `check the dimension d0`, 241 `the dimension of the matrix`, 715, 840 `check the dimension d1`, 130 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 715, 1320 `check the dimension d1`, 130 `the system is involutive after t prolongation`, 130, 2 `the order of the primary ideal and dimension`, 10, 130 `k=`, 11 `the dimension of the matrix`, 1001, 840 `check the dimension d0`, 251 `the dimension of the matrix`, 1001, 1320 `check the dimension d1`, 131 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 1001, 1980 `check the dimension d1`, 131 `the system is involutive after t prolongation`, 131, 2 `the order of the primary ideal and dimension`, 11, 131 `k=`, 12 `the dimension of the matrix`, 1365, 1320 `check the dimension d0`, 255 `the dimension of the matrix`, 1365, 1980 `check the dimension d1`, 131 `the system is involutive after t prolongation`, 131, 1 `the order of the primary ideal and dimension`, 12, 131 `the index and multiplicity are `, 11, 131 [.14427481e-15, -.17099237e-15, -.29083969e-15, .152672e-16] `k=`, 2 `the dimension of the matrix`, 5, 4 `check the dimension d0`, 5 `the dimension of the matrix`, 5, 20 `check the dimension d1`, 5 `the system is involutive after t prolongation`, 5, 1 `k=`, 3 `the dimension of the matrix`, 15, 4 `check the dimension d0`, 15 `the dimension of the matrix`, 15, 20 `check the dimension d1`, 15 `the system is involutive after t prolongation`, 15, 1 `k=`, 4 `the dimension of the matrix`, 35, 4 `check the dimension d0`, 31 `the dimension of the matrix`, 35, 20 `check the dimension d1`, 31 `the system is involutive after t prolongation`, 31, 1 `the order of the primary ideal and dimension`, 4, 31 `k=`, 5 `the dimension of the matrix`, 70, 4 `check the dimension d0`, 66 `the dimension of the matrix`, 70, 20 `check the dimension d1`, 53 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 70, 60 `check the dimension d1`, 53 `the system is involutive after t prolongation`, 53, 2 `the order of the primary ideal and dimension`, 5, 53 `k=`, 6 `the dimension of the matrix`, 126, 20 `check the dimension d0`, 106 `the dimension of the matrix`, 126, 60 `check the dimension d1`, 78 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 126, 140 `check the dimension d1`, 78 `the system is involutive after t prolongation`, 78, 2 `the order of the primary ideal and dimension`, 6, 78 `k=`, 7 `the dimension of the matrix`, 210, 60 `check the dimension d0`, 150 `the dimension of the matrix`, 210, 140 `check the dimension d1`, 100 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 210, 280 `check the dimension d1`, 100 `the system is involutive after t prolongation`, 100, 2 `the order of the primary ideal and dimension`, 7, 100 `k=`, 8 `the dimension of the matrix`, 330, 140 `check the dimension d0`, 190 `the dimension of the matrix`, 330, 280 `check the dimension d1`, 116 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 330, 504 `check the dimension d1`, 116 `the system is involutive after t prolongation`, 116, 2 `the order of the primary ideal and dimension`, 8, 116 `k=`, 9 `the dimension of the matrix`, 495, 280 `check the dimension d0`, 221 `the dimension of the matrix`, 495, 504 `check the dimension d1`, 126 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 495, 840 `check the dimension d1`, 126 `the system is involutive after t prolongation`, 126, 2 `the order of the primary ideal and dimension`, 9, 126 `k=`, 10 `the dimension of the matrix`, 715, 504 `check the dimension d0`, 241 `the dimension of the matrix`, 715, 840 `check the dimension d1`, 130 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 715, 1320 `check the dimension d1`, 130 `the system is involutive after t prolongation`, 130, 2 `the order of the primary ideal and dimension`, 10, 130 `k=`, 11 `the dimension of the matrix`, 1001, 840 `check the dimension d0`, 251 `the dimension of the matrix`, 1001, 1320 `check the dimension d1`, 131 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 1001, 1980 `check the dimension d1`, 131 `the system is involutive after t prolongation`, 131, 2 `the order of the primary ideal and dimension`, 11, 131 `k=`, 12 `the dimension of the matrix`, 1365, 1320 `check the dimension d0`, 255 `the dimension of the matrix`, 1365, 1980 `check the dimension d1`, 131 `the system is involutive after t prolongation`, 131, 1 `the order of the primary ideal and dimension`, 12, 131 `the index and multiplicity are `, 11, 131 [-.39770992282443e-15, .5801526358779e-16, .22061069167939e-15, .24427483358779\ e-15] `the number of iteration is`, 2 -15 -16 mroot := [-0.39770992282443 10 , 0.5801526358779 10 , -15 -15 0.22061069167939 10 , 0.24427483358779 10 ] > > > # DZ2, (0,0,-1) is 16-fold zero and index is 8. > > lsys:=[x^4,x^2*y+y^4,z+z^2-7*x^3-8*x^2]; 4 2 4 2 3 2 lsys := [x , x y + y , z + z - 7 x - 8 x ] > > mroot:=convert(RandomVector(3,generator=0..1.)*1/1000.,list); mroot := [ 0.0000318328463774206734, 0.000706046088019608741, 0.000171186687811561758] > > mroot:=Refineroot(evalf(lsys),[x,y,z],10.^(-3),mroot,3); `k=`, 2 `the dimension of the matrix`, 4, 3 `check the dimension d0`, 3 `the dimension of the matrix`, 4, 12 `check the dimension d1`, 3 `the system is involutive after t prolongation`, 3, 1 `k=`, 3 `the dimension of the matrix`, 10, 3 `check the dimension d0`, 9 `the dimension of the matrix`, 10, 12 `check the dimension d1`, 6 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 10, 30 `check the dimension d1`, 6 `the system is involutive after t prolongation`, 6, 2 `k=`, 4 `the dimension of the matrix`, 20, 3 `check the dimension d0`, 18 `the dimension of the matrix`, 20, 12 `check the dimension d1`, 15 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 20, 30 `check the dimension d1`, 9 ` the system is not involutive yet, prolong the system`, 2 `the dimension of the matrix`, 20, 60 `check the dimension d1`, 9 `the system is involutive after t prolongation`, 9, 3 `the order of the primary ideal and dimension`, 4, 9 `k=`, 5 `the dimension of the matrix`, 35, 6 `check the dimension d0`, 29 `the dimension of the matrix`, 35, 18 `check the dimension d1`, 20 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 35, 40 `check the dimension d1`, 11 ` the system is not involutive yet, prolong the system`, 2 `the dimension of the matrix`, 35, 75 `check the dimension d1`, 11 `the system is involutive after t prolongation`, 11, 3 `the order of the primary ideal and dimension`, 5, 11 `k=`, 6 `the dimension of the matrix`, 56, 18 `check the dimension d0`, 38 `the dimension of the matrix`, 56, 40 `check the dimension d1`, 23 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 56, 75 `check the dimension d1`, 13 ` the system is not involutive yet, prolong the system`, 2 `the dimension of the matrix`, 56, 126 `check the dimension d1`, 13 `the system is involutive after t prolongation`, 13, 3 `the order of the primary ideal and dimension`, 6, 13 `k=`, 7 `the dimension of the matrix`, 84, 40 `check the dimension d0`, 44 `the dimension of the matrix`, 84, 75 `check the dimension d1`, 25 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 84, 126 `check the dimension d1`, 15 ` the system is not involutive yet, prolong the system`, 2 `the dimension of the matrix`, 84, 196 `check the dimension d1`, 15 `the system is involutive after t prolongation`, 15, 3 `the order of the primary ideal and dimension`, 7, 15 `k=`, 8 `the dimension of the matrix`, 120, 75 `check the dimension d0`, 47 `the dimension of the matrix`, 120, 126 `check the dimension d1`, 26 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 120, 196 `check the dimension d1`, 16 ` the system is not involutive yet, prolong the system`, 2 `the dimension of the matrix`, 120, 288 `check the dimension d1`, 16 `the system is involutive after t prolongation`, 16, 3 `the order of the primary ideal and dimension`, 8, 16 `k=`, 9 `the dimension of the matrix`, 165, 126 `check the dimension d0`, 48 `the dimension of the matrix`, 165, 196 `check the dimension d1`, 26 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 165, 288 `check the dimension d1`, 16 `the system is involutive after t prolongation`, 16, 2 `the order of the primary ideal and dimension`, 9, 16 `the index and multiplicity are `, 8, 16 [.3056467364e-8, .1779227836e-7, -.3308380e-10] `k=`, 2 `the dimension of the matrix`, 4, 3 `check the dimension d0`, 3 `the dimension of the matrix`, 4, 12 `check the dimension d1`, 3 `the system is involutive after t prolongation`, 3, 1 `k=`, 3 `the dimension of the matrix`, 10, 3 `check the dimension d0`, 9 `the dimension of the matrix`, 10, 12 `check the dimension d1`, 6 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 10, 30 `check the dimension d1`, 6 `the system is involutive after t prolongation`, 6, 2 `k=`, 4 `the dimension of the matrix`, 20, 3 `check the dimension d0`, 18 `the dimension of the matrix`, 20, 12 `check the dimension d1`, 15 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 20, 30 `check the dimension d1`, 9 ` the system is not involutive yet, prolong the system`, 2 `the dimension of the matrix`, 20, 60 `check the dimension d1`, 9 `the system is involutive after t prolongation`, 9, 3 `the order of the primary ideal and dimension`, 4, 9 `k=`, 5 `the dimension of the matrix`, 35, 6 `check the dimension d0`, 29 `the dimension of the matrix`, 35, 18 `check the dimension d1`, 20 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 35, 40 `check the dimension d1`, 11 ` the system is not involutive yet, prolong the system`, 2 `the dimension of the matrix`, 35, 75 `check the dimension d1`, 11 `the system is involutive after t prolongation`, 11, 3 `the order of the primary ideal and dimension`, 5, 11 `k=`, 6 `the dimension of the matrix`, 56, 18 `check the dimension d0`, 38 `the dimension of the matrix`, 56, 40 `check the dimension d1`, 23 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 56, 75 `check the dimension d1`, 13 ` the system is not involutive yet, prolong the system`, 2 `the dimension of the matrix`, 56, 126 `check the dimension d1`, 13 `the system is involutive after t prolongation`, 13, 3 `the order of the primary ideal and dimension`, 6, 13 `k=`, 7 `the dimension of the matrix`, 84, 40 `check the dimension d0`, 44 `the dimension of the matrix`, 84, 75 `check the dimension d1`, 25 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 84, 126 `check the dimension d1`, 15 ` the system is not involutive yet, prolong the system`, 2 `the dimension of the matrix`, 84, 196 `check the dimension d1`, 15 `the system is involutive after t prolongation`, 15, 3 `the order of the primary ideal and dimension`, 7, 15 `k=`, 8 `the dimension of the matrix`, 120, 75 `check the dimension d0`, 47 `the dimension of the matrix`, 120, 126 `check the dimension d1`, 26 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 120, 196 `check the dimension d1`, 16 ` the system is not involutive yet, prolong the system`, 2 `the dimension of the matrix`, 120, 288 `check the dimension d1`, 16 `the system is involutive after t prolongation`, 16, 3 `the order of the primary ideal and dimension`, 8, 16 `k=`, 9 `the dimension of the matrix`, 165, 126 `check the dimension d0`, 48 `the dimension of the matrix`, 165, 196 `check the dimension d1`, 26 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 165, 288 `check the dimension d1`, 16 `the system is involutive after t prolongation`, 16, 2 `the order of the primary ideal and dimension`, 9, 16 `the index and multiplicity are `, 8, 16 [-.1360000e-15, -.1015000e-14, -.5862500000e-14] `k=`, 2 `the dimension of the matrix`, 4, 3 `check the dimension d0`, 3 `the dimension of the matrix`, 4, 12 `check the dimension d1`, 3 `the system is involutive after t prolongation`, 3, 1 `k=`, 3 `the dimension of the matrix`, 10, 3 `check the dimension d0`, 9 `the dimension of the matrix`, 10, 12 `check the dimension d1`, 6 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 10, 30 `check the dimension d1`, 6 `the system is involutive after t prolongation`, 6, 2 `k=`, 4 `the dimension of the matrix`, 20, 3 `check the dimension d0`, 18 `the dimension of the matrix`, 20, 12 `check the dimension d1`, 15 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 20, 30 `check the dimension d1`, 9 ` the system is not involutive yet, prolong the system`, 2 `the dimension of the matrix`, 20, 60 `check the dimension d1`, 9 `the system is involutive after t prolongation`, 9, 3 `the order of the primary ideal and dimension`, 4, 9 `k=`, 5 `the dimension of the matrix`, 35, 6 `check the dimension d0`, 29 `the dimension of the matrix`, 35, 18 `check the dimension d1`, 20 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 35, 40 `check the dimension d1`, 11 ` the system is not involutive yet, prolong the system`, 2 `the dimension of the matrix`, 35, 75 `check the dimension d1`, 11 `the system is involutive after t prolongation`, 11, 3 `the order of the primary ideal and dimension`, 5, 11 `k=`, 6 `the dimension of the matrix`, 56, 18 `check the dimension d0`, 38 `the dimension of the matrix`, 56, 40 `check the dimension d1`, 23 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 56, 75 `check the dimension d1`, 13 ` the system is not involutive yet, prolong the system`, 2 `the dimension of the matrix`, 56, 126 `check the dimension d1`, 13 `the system is involutive after t prolongation`, 13, 3 `the order of the primary ideal and dimension`, 6, 13 `k=`, 7 `the dimension of the matrix`, 84, 40 `check the dimension d0`, 44 `the dimension of the matrix`, 84, 75 `check the dimension d1`, 25 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 84, 126 `check the dimension d1`, 15 ` the system is not involutive yet, prolong the system`, 2 `the dimension of the matrix`, 84, 196 `check the dimension d1`, 15 `the system is involutive after t prolongation`, 15, 3 `the order of the primary ideal and dimension`, 7, 15 `k=`, 8 `the dimension of the matrix`, 120, 75 `check the dimension d0`, 47 `the dimension of the matrix`, 120, 126 `check the dimension d1`, 26 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 120, 196 `check the dimension d1`, 16 ` the system is not involutive yet, prolong the system`, 2 `the dimension of the matrix`, 120, 288 `check the dimension d1`, 16 `the system is involutive after t prolongation`, 16, 3 `the order of the primary ideal and dimension`, 8, 16 `k=`, 9 `the dimension of the matrix`, 165, 126 `check the dimension d0`, 48 `the dimension of the matrix`, 165, 196 `check the dimension d1`, 26 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 165, 288 `check the dimension d1`, 16 `the system is involutive after t prolongation`, 16, 2 `the order of the primary ideal and dimension`, 9, 16 `the index and multiplicity are `, 8, 16 [-.1360000e-15, -.39000000000000e-15, -.27375000000000e-14] `the number of iteration is`, 2 -15 -15 -14 mroot := [-0.1360000 10 , -0.39000000000000 10 , -0.27375000000000 10 ] > > > # DZ3, An approximate zero (1.5055,0.36528) is 5-fold and index is 5. > > lsys:=[14*x+33*y-3*sqrt(5)*(x^2+4*x*y+4*y^2+2)+sqrt(7)+x^3+6*x^2*y+12*x*y^2+8*y^3,41*x-18*y-sqrt(5)+8*x^3-12*x^2*y+6*x*y^2-y^3+3*sqrt(7)*(4*x*y-4*x^2-y^2-2)]; 1/2 2 2 1/2 3 2 lsys := [14 x + 33 y - 3 5 (x + 4 x y + 4 y + 2) + 7 + x + 6 x y 2 3 1/2 3 2 2 3 + 12 x y + 8 y , 41 x - 18 y - 5 + 8 x - 12 x y + 6 x y - y 1/2 2 2 + 3 7 (4 x y - 4 x - y - 2)] > > lsys:=evalf(lsys); 2 lsys := [14. x + 33. y - 6.7082039324994 x - 26.832815729998 x y 2 3 2 2 3 - 26.832815729998 y - 10.770656553934 + x + 6. x y + 12. x y + 8. y , 3 2 2 3 41. x - 18. y - 18.110575843888 + 8. x - 12. x y + 6. x y - 1. y 2 2 + 31.749015732774 x y - 31.749015732774 x - 7.9372539331938 y ] > > > > > > mroot0:=[1.5055,0.36528]; mroot0 := [1.5055, 0.36528] > > peroot:=RandomVector(2,generator=-1..1.)*1/100.; [-0.00907657218737692104] peroot := [ ] [-0.00446154030078220118] > > mroot:=[mroot0[1]+peroot[1],mroot0[2]+peroot[2]]; mroot := [1.4964234278126, 0.36081845969922] > > mroot:=Refineroot(lsys,[x,y],10.^(-2),mroot,3); `k=`, 2 `the dimension of the matrix`, 3, 2 `check the dimension d0`, 2 `the dimension of the matrix`, 3, 6 `check the dimension d1`, 2 `the system is involutive after t prolongation`, 2, 1 `k=`, 3 `the dimension of the matrix`, 6, 2 `check the dimension d0`, 4 `the dimension of the matrix`, 6, 6 `check the dimension d1`, 2 `the system is involutive after t prolongation`, 2, 1 the index and multiplicity are, 2, 2 [1.4959731797925, .36209790880832] `k=`, 2 `the dimension of the matrix`, 3, 2 `check the dimension d0`, 2 `the dimension of the matrix`, 3, 6 `check the dimension d1`, 2 `the system is involutive after t prolongation`, 2, 1 `k=`, 3 `the dimension of the matrix`, 6, 2 `check the dimension d0`, 4 `the dimension of the matrix`, 6, 6 `check the dimension d1`, 2 `the system is involutive after t prolongation`, 2, 1 the index and multiplicity are, 2, 2 [1.4983669804152, .36289467873373] `k=`, 2 `the dimension of the matrix`, 3, 2 `check the dimension d0`, 2 `the dimension of the matrix`, 3, 6 `check the dimension d1`, 1 `the system is involutive after t prolongation`, 1, 1 `k=`, 3 `the dimension of the matrix`, 6, 2 `check the dimension d0`, 4 `the dimension of the matrix`, 6, 6 `check the dimension d1`, 2 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 6, 12 `check the dimension d1`, 2 `the system is involutive after t prolongation`, 2, 2 `k=`, 4 `the dimension of the matrix`, 10, 2 `check the dimension d0`, 8 `the dimension of the matrix`, 10, 6 `check the dimension d1`, 4 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 10, 12 `check the dimension d1`, 3 ` the system is not involutive yet, prolong the system`, 2 `the dimension of the matrix`, 10, 20 `check the dimension d1`, 3 `the system is involutive after t prolongation`, 3, 3 `the order of the primary ideal and dimension`, 4, 3 `k=`, 5 `the dimension of the matrix`, 15, 6 `check the dimension d0`, 9 `the dimension of the matrix`, 15, 12 `check the dimension d1`, 4 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 15, 20 `check the dimension d1`, 3 `the system is involutive after t prolongation`, 3, 2 `the order of the primary ideal and dimension`, 5, 3 `the index should be less or equal to multiplicity,`, 3, 3 [1.5019409469839, .36408575853685] `the number of iteration is`, 3 mroot := [1.5019409469839, 0.36408575853685] > > expand(evalf(subs(x=mroot[1],y=mroot[2],lsys))); -6 -6 [-0.54903915 10 , -0.549002889 10 ] > # tangents, 4-fold zero and index is 4. > > lsys:=[x0**2 + x1**2 + x2**2 - 1, > > x0*x3 + x1*x4 + x2*x5, > > x3**2 + x4**2 + x5**2 - 1/4, > > x3**2 + x4**2 - 2*x2*x4 + x2**2 + x5**2 + 2*x1*x5 + x1**2 - 01/4, > > > > x3**2 + sqrt(3)*x2*x3 + 3/4*x2**2 + x4**2 - x2*x4 + 1/4*x2**2 > > + x5**2 - sqrt(3)*x0*x5 + x1*x5 > > + 3/4*x0**2 - 1/2*sqrt(3)*x0*x1 + 1/4*x1**2 - 1/4, > > > > x3**2 - 2/3*sqrt(6)*x1*x3 + 1/3*sqrt(3)*x2*x3 > > + 2/3*x1**2 - 1/9*sqrt(18)*x1*x2 + 1/12*x2**2 > > > > + x4**2 + 2/3*sqrt(6)*x0*x4 - x2*x4 + 2/3*x0**2 > > - 1/3*sqrt(6)*x0*x2 + 1/4*x2**2 > > > > + x5**2 - 1/3*sqrt(3)*x0*x5 + x1*x5 + 1/12*x0**2 > > - 1/6*sqrt(3)*x0*x1 + 1/4*x1**2 - 1/4]; 2 2 2 2 2 2 lsys := [x0 + x1 + x2 - 1, x0 x3 + x1 x4 + x2 x5, x3 + x4 + x5 - 1/4, 2 2 2 2 2 2 1/2 x3 + x4 - 2 x2 x4 + x2 + x5 + 2 x1 x5 + x1 - 1/4, x3 + 3 x2 x3 2 1/2 2 2 2 2 1/2 3 x0 3 x0 x1 x1 + x2 + x4 - x2 x4 + x5 - 3 x0 x5 + x1 x5 + ----- - ---------- + --- 4 2 4 1/2 1/2 2 1/2 2 2 2 6 x1 x3 3 x2 x3 11 x1 2 x1 x2 x2 2 - 1/4, x3 - ------------ + ---------- + ------ - ---------- + --- + x4 3 3 12 3 3 1/2 2 1/2 1/2 2 6 x0 x4 3 x0 6 x0 x2 2 3 x0 x5 + ------------ - x2 x4 + ----- - ---------- + x5 - ---------- + x1 x5 3 4 3 3 1/2 3 x0 x1 - ---------- - 1/4] 6 > > lsys:=evalf(lsys); 2 2 2 lsys := [x0 + x1 + x2 - 1., x0 x3 + x1 x4 + x2 x5, 2 2 2 x3 + x4 + x5 - 0.25000000000000, 2 2 2 2 2 2 x3 + x4 - 2. x2 x4 + x2 + x5 + 2. x1 x5 + x1 - 0.25000000000000, x3 2 2 2 + 1.7320508075689 x2 x3 + x2 + x4 - 1. x2 x4 + x5 2 - 1.7320508075689 x0 x5 + x1 x5 + 0.75000000000000 x0 2 2 - 0.86602540378445 x0 x1 + 0.25000000000000 x1 - 0.25000000000000, x3 2 - 1.6329931618555 x1 x3 + 0.57735026918963 x2 x3 + 0.91666666666667 x1 2 2 - 0.47140452079103 x1 x2 + 0.33333333333333 x2 + x4 2 + 1.6329931618555 x0 x4 - 1. x2 x4 + 0.75000000000000 x0 2 - 0.81649658092773 x0 x2 + x5 - 0.57735026918963 x0 x5 + x1 x5 - 0.28867513459482 x0 x1 - 0.25000000000000] > > > > mroot0:=[-sqrt(2)/2, sqrt(6)/6, -sqrt(3)/3, 1/4,-sqrt(3)/12, -sqrt(6)/6]; 1/2 1/2 1/2 1/2 1/2 2 6 3 3 6 mroot0 := [- ----, ----, - ----, 1/4, - ----, - ----] 2 6 3 12 6 > > evalf(subs(x0=mroot0[1],x1=mroot0[2], x2=mroot0[3], x3=mroot0[4],x4=mroot0[5], x5=mroot0[6],lsys)); -13 -14 [0., 0., 0., 0., -0.3 10 , -0.6 10 ] > > peroot:=RandomVector(6,generator=-1..1.)*1/100.; [-0.00931107838994182577] [ ] [0.00900444097676709847 ] [ ] [-0.00365801039878278951] peroot := [ ] [0.00389657245951634083 ] [ ] [0.00646915656654585298 ] [ ] [-0.00805736437528304858] > > mroot:=evalf([mroot0[1]+peroot[1],mroot0[2]+peroot[2],mroot0[3]+peroot[3],mroot0[4]+peroot[4], mroot0[5]+peroot[5], mroot0[6]+peroot[6]]); mroot := [-0.71641785957649, 0.41725273144064, -0.58100827958841, 0.25389657245952, -0.13786841073086, -0.41630565483915] > > mroot:=Refineroot(lsys,[x0,x1,x2,x3,x4,x5],10.^(-2), mroot, 3); `k=`, 2 `the dimension of the matrix`, 7, 6 `check the dimension d0`, 2 `the dimension of the matrix`, 7, 42 `check the dimension d1`, 1 `the system is involutive after t prolongation`, 1, 1 `k=`, 3 `the dimension of the matrix`, 28, 6 `check the dimension d0`, 22 `the dimension of the matrix`, 28, 42 `check the dimension d1`, 3 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 28, 168 `check the dimension d1`, 3 `the system is involutive after t prolongation`, 3, 2 `k=`, 4 `the dimension of the matrix`, 84, 42 `check the dimension d0`, 42 `the dimension of the matrix`, 84, 168 `check the dimension d1`, 4 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 84, 504 `check the dimension d1`, 4 `the system is involutive after t prolongation`, 4, 2 `the order of the primary ideal and dimension`, 4, 4 `k=`, 5 `the dimension of the matrix`, 210, 168 `check the dimension d0`, 57 `the dimension of the matrix`, 210, 504 `check the dimension d1`, 4 `the system is involutive after t prolongation`, 4, 1 `the order of the primary ideal and dimension`, 5, 4 `the index should be less or equal to multiplicity,`, 4, 4 [-.70710695744270, .40824915207037, -.57734990091059, .25000006086378, -.144337\ 25571963, -.40824889315952] `k=`, 2 `the dimension of the matrix`, 7, 6 `check the dimension d0`, 3 `the dimension of the matrix`, 7, 42 `check the dimension d1`, 3 `the system is involutive after t prolongation`, 3, 1 `k=`, 3 `the dimension of the matrix`, 28, 6 `check the dimension d0`, 22 `the dimension of the matrix`, 28, 42 `check the dimension d1`, 4 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 28, 168 `check the dimension d1`, 4 `the system is involutive after t prolongation`, 4, 2 `k=`, 4 `the dimension of the matrix`, 84, 42 `check the dimension d0`, 42 `the dimension of the matrix`, 84, 168 `check the dimension d1`, 4 `the system is involutive after t prolongation`, 4, 1 `the order of the primary ideal and dimension`, 4, 4 `the index and multiplicity are `, 3, 4 [-.70710678118683, .40824829046416, -.57735026918960, .24999999999996, -.144337\ 56729725, -.40824829046418] `k=`, 2 `the dimension of the matrix`, 7, 6 `check the dimension d0`, 3 `the dimension of the matrix`, 7, 42 `check the dimension d1`, 3 `the system is involutive after t prolongation`, 3, 1 `k=`, 3 `the dimension of the matrix`, 28, 6 `check the dimension d0`, 22 `the dimension of the matrix`, 28, 42 `check the dimension d1`, 4 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 28, 168 `check the dimension d1`, 4 `the system is involutive after t prolongation`, 4, 2 `k=`, 4 `the dimension of the matrix`, 84, 42 `check the dimension d0`, 42 `the dimension of the matrix`, 84, 168 `check the dimension d1`, 4 `the system is involutive after t prolongation`, 4, 1 `the order of the primary ideal and dimension`, 4, 4 `the index and multiplicity are `, 3, 4 [-.70710678118655, .40824829046386, -.57735026918968, .25000000000001, -.144337\ 56729744, -.40824829046386] `the number of iteration is`, 2 mroot := [-0.70710678118655, 0.40824829046386, -0.57735026918968, 0.25000000000001, -0.14433756729744, -0.40824829046386] > # D2, SNC07 Barry Dayton, (0,0,0) is an isolated 5-fold zero and index is 5. The system is not zero-dimensional. > > lsys:=[2*x^2-x-x^3+z^3,x-y-x^2+x*y+z^2,x*y^2*z-x^2*z-y^2*z+x^3*z]; lsys := 2 3 3 2 2 2 2 2 3 [2 x - x - x + z , x - y - x + x y + z , x y z - x z - y z + x z] > > mroot:=convert(RandomVector(3,generator=0..1.)*1/100.,list); mroot := [0.00765516788149002425, 0.00381558457093008393, 0.00438744359656398266] > > mroot:=Refineroot(lsys,[x,y,z],.01, mroot, 3); `k=`, 2 `the dimension of the matrix`, 4, 3 `check the dimension d0`, 2 `the dimension of the matrix`, 4, 12 `check the dimension d1`, 2 `the system is involutive after t prolongation`, 2, 1 `k=`, 3 `the dimension of the matrix`, 10, 3 `check the dimension d0`, 7 `the dimension of the matrix`, 10, 12 `check the dimension d1`, 3 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 10, 30 `check the dimension d1`, 3 `the system is involutive after t prolongation`, 3, 2 `k=`, 4 `the dimension of the matrix`, 20, 6 `check the dimension d0`, 14 `the dimension of the matrix`, 20, 18 `check the dimension d1`, 5 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 20, 40 `check the dimension d1`, 4 ` the system is not involutive yet, prolong the system`, 2 `the dimension of the matrix`, 20, 75 `check the dimension d1`, 4 `the system is involutive after t prolongation`, 4, 3 `the order of the primary ideal and dimension`, 4, 4 `k=`, 5 `the dimension of the matrix`, 35, 15 `check the dimension d0`, 20 `the dimension of the matrix`, 35, 34 `check the dimension d1`, 6 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 35, 65 `check the dimension d1`, 5 ` the system is not involutive yet, prolong the system`, 2 `the dimension of the matrix`, 35, 111 `check the dimension d1`, 5 `the system is involutive after t prolongation`, 5, 3 `the order of the primary ideal and dimension`, 5, 5 `k=`, 6 `the dimension of the matrix`, 56, 34 `check the dimension d0`, 23 `the dimension of the matrix`, 56, 65 `check the dimension d1`, 6 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 56, 111 `check the dimension d1`, 5 `the system is involutive after t prolongation`, 5, 2 `the order of the primary ideal and dimension`, 6, 5 `the index should be less or equal to multiplicity,`, 5, 5 [.12123416790e-5, -.250724724857e-4, .576308123460e-4] `k=`, 2 `the dimension of the matrix`, 4, 3 `check the dimension d0`, 2 `the dimension of the matrix`, 4, 12 `check the dimension d1`, 2 `the system is involutive after t prolongation`, 2, 1 `k=`, 3 `the dimension of the matrix`, 10, 3 `check the dimension d0`, 8 `the dimension of the matrix`, 10, 12 `check the dimension d1`, 3 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 10, 30 `check the dimension d1`, 3 `the system is involutive after t prolongation`, 3, 2 `k=`, 4 `the dimension of the matrix`, 20, 6 `check the dimension d0`, 14 `the dimension of the matrix`, 20, 18 `check the dimension d1`, 5 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 20, 40 `check the dimension d1`, 4 ` the system is not involutive yet, prolong the system`, 2 `the dimension of the matrix`, 20, 75 `check the dimension d1`, 4 `the system is involutive after t prolongation`, 4, 3 `the order of the primary ideal and dimension`, 4, 4 `k=`, 5 `the dimension of the matrix`, 35, 15 `check the dimension d0`, 20 `the dimension of the matrix`, 35, 34 `check the dimension d1`, 6 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 35, 65 `check the dimension d1`, 5 ` the system is not involutive yet, prolong the system`, 2 `the dimension of the matrix`, 35, 111 `check the dimension d1`, 5 `the system is involutive after t prolongation`, 5, 3 `the order of the primary ideal and dimension`, 5, 5 `k=`, 6 `the dimension of the matrix`, 56, 34 `check the dimension d0`, 23 `the dimension of the matrix`, 56, 65 `check the dimension d1`, 6 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 56, 111 `check the dimension d1`, 5 `the system is involutive after t prolongation`, 5, 2 `the order of the primary ideal and dimension`, 6, 5 `the index should be less or equal to multiplicity,`, 5, 5 [.27829993717e-8, -.5200642500e-8, .10380726224e-7] `k=`, 2 `the dimension of the matrix`, 4, 3 `check the dimension d0`, 2 `the dimension of the matrix`, 4, 12 `check the dimension d1`, 2 `the system is involutive after t prolongation`, 2, 1 `k=`, 3 `the dimension of the matrix`, 10, 3 `check the dimension d0`, 8 `the dimension of the matrix`, 10, 12 `check the dimension d1`, 3 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 10, 30 `check the dimension d1`, 3 `the system is involutive after t prolongation`, 3, 2 `k=`, 4 `the dimension of the matrix`, 20, 6 `check the dimension d0`, 14 `the dimension of the matrix`, 20, 18 `check the dimension d1`, 5 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 20, 40 `check the dimension d1`, 4 ` the system is not involutive yet, prolong the system`, 2 `the dimension of the matrix`, 20, 75 `check the dimension d1`, 4 `the system is involutive after t prolongation`, 4, 3 `the order of the primary ideal and dimension`, 4, 4 `k=`, 5 `the dimension of the matrix`, 35, 15 `check the dimension d0`, 20 `the dimension of the matrix`, 35, 34 `check the dimension d1`, 6 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 35, 65 `check the dimension d1`, 5 ` the system is not involutive yet, prolong the system`, 2 `the dimension of the matrix`, 35, 111 `check the dimension d1`, 5 `the system is involutive after t prolongation`, 5, 3 `the order of the primary ideal and dimension`, 5, 5 `k=`, 6 `the dimension of the matrix`, 56, 34 `check the dimension d0`, 23 `the dimension of the matrix`, 56, 65 `check the dimension d1`, 6 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 56, 111 `check the dimension d1`, 5 `the system is involutive after t prolongation`, 5, 2 `the order of the primary ideal and dimension`, 6, 5 `the index should be less or equal to multiplicity,`, 5, 5 [.2317000e-15, -.3000000e-15, .2224000e-14] `the number of iteration is`, 3 -15 -15 -14 mroot := [0.2317000 10 , -0.3000000 10 , 0.2224000 10 ] > > > # Ojika1, (1,2) is 3-fold zero and index is 3. > > lsys:=[x^2+y-3,x+0.125*y^2-1.5]; 2 2 lsys := [x + y - 3, x + 0.125 y - 1.5] > > mroot0:=[1,2]; mroot0 := [1, 2] > > peroot:=RandomVector(2,generator=-1..1.)*1/100.; [-0.00626254790891242821] peroot := [ ] [0.00590399802274126344 ] > > mroot:=[mroot0[1]+peroot[1],mroot0[2]+peroot[2]]; mroot := [0.99373745209109, 2.0059039980227] > > mroot:=Refineroot(lsys,[x,y],.01,mroot,3); `k=`, 2 `the dimension of the matrix`, 3, 2 `check the dimension d0`, 2 `the dimension of the matrix`, 3, 6 `check the dimension d1`, 2 `the system is involutive after t prolongation`, 2, 1 `k=`, 3 `the dimension of the matrix`, 6, 2 `check the dimension d0`, 4 `the dimension of the matrix`, 6, 6 `check the dimension d1`, 3 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 6, 12 `check the dimension d1`, 3 `the system is involutive after t prolongation`, 3, 2 `k=`, 4 `the dimension of the matrix`, 10, 6 `check the dimension d0`, 4 `the dimension of the matrix`, 10, 12 `check the dimension d1`, 3 `the system is involutive after t prolongation`, 3, 1 `the order of the primary ideal and dimension`, 4, 3 `the index should be less or equal to multiplicity,`, 3, 3 [.99996995456592, 2.0000988144211] `k=`, 2 `the dimension of the matrix`, 3, 2 `check the dimension d0`, 2 `the dimension of the matrix`, 3, 6 `check the dimension d1`, 2 `the system is involutive after t prolongation`, 2, 1 `k=`, 3 `the dimension of the matrix`, 6, 2 `check the dimension d0`, 4 `the dimension of the matrix`, 6, 6 `check the dimension d1`, 3 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 6, 12 `check the dimension d1`, 3 `the system is involutive after t prolongation`, 3, 2 `k=`, 4 `the dimension of the matrix`, 10, 6 `check the dimension d0`, 4 `the dimension of the matrix`, 10, 12 `check the dimension d1`, 3 `the system is involutive after t prolongation`, 3, 1 `the order of the primary ideal and dimension`, 4, 3 `the index should be less or equal to multiplicity,`, 3, 3 [.99999999916695, 2.0000000047934] `k=`, 2 `the dimension of the matrix`, 3, 2 `check the dimension d0`, 2 `the dimension of the matrix`, 3, 6 `check the dimension d1`, 2 `the system is involutive after t prolongation`, 2, 1 `k=`, 3 `the dimension of the matrix`, 6, 2 `check the dimension d0`, 4 `the dimension of the matrix`, 6, 6 `check the dimension d1`, 3 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 6, 12 `check the dimension d1`, 3 `the system is involutive after t prolongation`, 3, 2 `k=`, 4 `the dimension of the matrix`, 10, 6 `check the dimension d0`, 4 `the dimension of the matrix`, 10, 12 `check the dimension d1`, 3 `the system is involutive after t prolongation`, 3, 1 `the order of the primary ideal and dimension`, 4, 3 `the index should be less or equal to multiplicity,`, 3, 3 [1.0000000000000, 2.0000000000000] `the number of iteration is`, 3 mroot := [1.0000000000000, 2.0000000000000] > > mroot-mroot0; [0., 0.] > # Ojika2, (1,0,0), (0,1,0), (0,0,1) are 2-fold zeros and indices are 2. > > lsys:=[x^2+y+z-1,x+y^2+z-1,x+y+z^2-1]; 2 2 2 lsys := [x + y + z - 1, x + y + z - 1, x + y + z - 1] > > mroot0:=[1,0,0]; mroot0 := [1, 0, 0] > > peroot:=RandomVector(3,generator=-1..1.)*1/100.; [ 0.00292626020222529247 ] [ ] peroot := [-0.00108827598578201015 ] [ ] [-0.000204712084235378914] > > mroot:=[mroot0[1]+peroot[1],mroot0[2]+peroot[2],mroot0[3]+peroot[3]]; mroot := [1.0029262602022, -0.0010882759857820, -0.00020471208423538] > > mroot:=Refineroot(lsys,[x,y,z],10^(-2),mroot,3); `k=`, 2 `the dimension of the matrix`, 4, 3 `check the dimension d0`, 2 `the dimension of the matrix`, 4, 12 `check the dimension d1`, 2 `the system is involutive after t prolongation`, 2, 1 `k=`, 3 `the dimension of the matrix`, 10, 3 `check the dimension d0`, 7 `the dimension of the matrix`, 10, 12 `check the dimension d1`, 2 `the system is involutive after t prolongation`, 2, 1 the index and multiplicity are, 2, 2 [1.0000108703686, -.85870698990e-5, -.679493067788e-5] `k=`, 2 `the dimension of the matrix`, 4, 3 `check the dimension d0`, 2 `the dimension of the matrix`, 4, 12 `check the dimension d1`, 2 `the system is involutive after t prolongation`, 2, 1 `k=`, 3 `the dimension of the matrix`, 10, 3 `check the dimension d0`, 7 `the dimension of the matrix`, 10, 12 `check the dimension d1`, 2 `the system is involutive after t prolongation`, 2, 1 the index and multiplicity are, 2, 2 [1.0000000002822, -.2131037590e-9, -.1836179510e-9] `k=`, 2 `the dimension of the matrix`, 4, 3 `check the dimension d0`, 2 `the dimension of the matrix`, 4, 12 `check the dimension d1`, 2 `the system is involutive after t prolongation`, 2, 1 `k=`, 3 `the dimension of the matrix`, 10, 3 `check the dimension d0`, 7 `the dimension of the matrix`, 10, 12 `check the dimension d1`, 2 `the system is involutive after t prolongation`, 2, 1 the index and multiplicity are, 2, 2 [.99999999999998, .1760275418e-13, .337089777e-14] `the number of iteration is`, 3 -13 -14 mroot := [0.99999999999998, 0.1760275418 10 , 0.337089777 10 ] > > mroot-mroot0; -13 -13 -14 [-0.2 10 , 0.1760275418 10 , 0.337089777 10 ] > > > # Ojika3, (0,0,1) is 4-fold zero and index is 3, while (-2.5,2.5,1) is 2-fold zero and index is 2. > > lsys:= [x1+x2+x3-1, 0.2*x1**3+0.5*x2**2-x3+0.5*x3**2+0.5, x1+x2+0.5*x3**2-0.5]; 3 2 2 lsys := [x1 + x2 + x3 - 1, 0.2 x1 + 0.5 x2 - x3 + 0.5 x3 + 0.5, 2 x1 + x2 + 0.5 x3 - 0.5] > > mroot0:=[0,0,1]; mroot0 := [0, 0, 1] > > peroot:=RandomVector(3,generator=-1..1.)*1/100.; [-0.00447949846002843304] [ ] peroot := [0.00509373363964721760 ] [ ] [0.00418729661716145136 ] > > mroot:=convert([mroot0[1]+peroot[1],mroot0[2]+peroot[2],mroot0[3]+peroot[3]],list); mroot := [-0.0044794984600284, 0.0050937336396472, 1.0041872966172] > > mroot:=Refineroot(lsys,[x1,x2,x3],10^(-2),mroot,3); `k=`, 2 `the dimension of the matrix`, 4, 3 `check the dimension d0`, 3 `the dimension of the matrix`, 4, 12 `check the dimension d1`, 3 `the system is involutive after t prolongation`, 3, 1 `k=`, 3 `the dimension of the matrix`, 10, 6 `check the dimension d0`, 4 `the dimension of the matrix`, 10, 18 `check the dimension d1`, 4 `the system is involutive after t prolongation`, 4, 1 `k=`, 4 `the dimension of the matrix`, 20, 15 `check the dimension d0`, 6 `the dimension of the matrix`, 20, 34 `check the dimension d1`, 4 `the system is involutive after t prolongation`, 4, 1 `the order of the primary ideal and dimension`, 4, 4 `the index and multiplicity are `, 3, 4 [.181728594956e-4, -.144681600908e-4, .99999996189694] `k=`, 2 `the dimension of the matrix`, 4, 3 `check the dimension d0`, 3 `the dimension of the matrix`, 4, 12 `check the dimension d1`, 3 `the system is involutive after t prolongation`, 3, 1 `k=`, 3 `the dimension of the matrix`, 10, 6 `check the dimension d0`, 4 `the dimension of the matrix`, 10, 18 `check the dimension d1`, 4 `the system is involutive after t prolongation`, 4, 1 `k=`, 4 `the dimension of the matrix`, 20, 15 `check the dimension d0`, 6 `the dimension of the matrix`, 20, 34 `check the dimension d1`, 4 `the system is involutive after t prolongation`, 4, 1 `the order of the primary ideal and dimension`, 4, 4 `the index and multiplicity are `, 3, 4 [.276537401e-9, -.228375480e-9, .99999999995195] `k=`, 2 `the dimension of the matrix`, 4, 3 `check the dimension d0`, 3 `the dimension of the matrix`, 4, 12 `check the dimension d1`, 3 `the system is involutive after t prolongation`, 3, 1 `k=`, 3 `the dimension of the matrix`, 10, 6 `check the dimension d0`, 4 `the dimension of the matrix`, 10, 18 `check the dimension d1`, 4 `the system is involutive after t prolongation`, 4, 1 `k=`, 4 `the dimension of the matrix`, 20, 15 `check the dimension d0`, 6 `the dimension of the matrix`, 20, 34 `check the dimension d1`, 4 `the system is involutive after t prolongation`, 4, 1 `the order of the primary ideal and dimension`, 4, 4 `the index and multiplicity are `, 3, 4 [.1319397825e-13, -.500419573e-14, 1.0000000000000] `the number of iteration is`, 3 -13 -14 mroot := [0.1319397825 10 , -0.500419573 10 , 1.0000000000000] > > mroot-mroot0; -13 -14 [0.1319397825 10 , -0.500419573 10 , 0.] > > mroot0:=[-2.5,2.5,1]; mroot0 := [-2.5, 2.5, 1] > > mroot:=convert([mroot0[1]+peroot[1],mroot0[2]+peroot[2],mroot0[3]+peroot[3]],list); mroot := [-2.5044794984600, 2.5050937336396, 1.0041872966172] > > mroot:=Refineroot(lsys,[x1,x2,x3],10^(-2),mroot,3); `k=`, 2 `the dimension of the matrix`, 4, 3 `check the dimension d0`, 2 `the dimension of the matrix`, 4, 12 `check the dimension d1`, 2 `the system is involutive after t prolongation`, 2, 1 `k=`, 3 `the dimension of the matrix`, 10, 6 `check the dimension d0`, 4 `the dimension of the matrix`, 10, 18 `check the dimension d1`, 2 `the system is involutive after t prolongation`, 2, 1 the index and multiplicity are, 2, 2 [-2.5000969070826, 2.5001393955934, .99995866415600] `k=`, 2 `the dimension of the matrix`, 4, 3 `check the dimension d0`, 2 `the dimension of the matrix`, 4, 12 `check the dimension d1`, 2 `the system is involutive after t prolongation`, 2, 1 `k=`, 3 `the dimension of the matrix`, 10, 6 `check the dimension d0`, 4 `the dimension of the matrix`, 10, 18 `check the dimension d1`, 2 `the system is involutive after t prolongation`, 2, 1 the index and multiplicity are, 2, 2 [-2.5000000087111, 2.5000000112257, .99999999818120] `k=`, 2 `the dimension of the matrix`, 4, 3 `check the dimension d0`, 2 `the dimension of the matrix`, 4, 12 `check the dimension d1`, 2 `the system is involutive after t prolongation`, 2, 1 `k=`, 3 `the dimension of the matrix`, 10, 6 `check the dimension d0`, 4 `the dimension of the matrix`, 10, 18 `check the dimension d1`, 2 `the system is involutive after t prolongation`, 2, 1 the index and multiplicity are, 2, 2 [-2.5000000000000, 2.5000000000000, 1.0000000000000] `the number of iteration is`, 3 mroot := [-2.5000000000000, 2.5000000000000, 1.0000000000000] > > mroot-mroot0; [0., 0., 0.] > > > # Ojika4, six triple zeros, (0,0,1) is 3-fold and index is 3. > > lsys:=[x1+x3*x1**3+x1*x3*x2**2-x1*x3,10*x2-2*x2*x3*x1**2-x3*x2**3-x2*x3, > > -6*x3**2*x1**4-3*x1**2*x2**2*x3**2-x3**2*x1**2+28*x3*x1**2 > > -3*x3**2*x2**4+2*x3**2*x2**2+7*x3*x2**2+x3**2-11*x3+10]; 3 2 2 3 lsys := [x1 + x3 x1 + x1 x3 x2 - x1 x3, 10 x2 - 2 x2 x3 x1 - x3 x2 - x2 x3, 2 4 2 2 2 2 2 2 2 4 2 2 -6 x3 x1 - 3 x1 x2 x3 - x3 x1 + 28 x1 x3 - 3 x3 x2 + 2 x3 x2 2 2 + 7 x2 x3 + x3 - 11 x3 + 10] > > mroot0:=[0,0,1]; mroot0 := [0, 0, 1] > > peroot:=RandomVector(3,generator=-1..1.)*1/100.; [-0.00674776529610738880] [ ] peroot := [0.00310196007947681311 ] [ ] [0.00359405353707349618 ] > > mroot:=[mroot0[1]+peroot[1],mroot0[2]+peroot[2],mroot0[3]+peroot[3]]; mroot := [-0.0067477652961074, 0.0031019600794768, 1.0035940535371] > > mroot:=Refineroot(lsys,[x1,x2,x3],0.01,mroot,3); `k=`, 2 `the dimension of the matrix`, 4, 3 `check the dimension d0`, 2 `the dimension of the matrix`, 4, 12 `check the dimension d1`, 1 `the system is involutive after t prolongation`, 1, 1 `k=`, 3 `the dimension of the matrix`, 10, 3 `check the dimension d0`, 7 `the dimension of the matrix`, 10, 12 `check the dimension d1`, 2 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 10, 30 `check the dimension d1`, 2 `the system is involutive after t prolongation`, 2, 2 `k=`, 4 `the dimension of the matrix`, 20, 3 `check the dimension d0`, 17 `the dimension of the matrix`, 20, 12 `check the dimension d1`, 8 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 20, 30 `check the dimension d1`, 3 ` the system is not involutive yet, prolong the system`, 2 `the dimension of the matrix`, 20, 60 `check the dimension d1`, 3 `the system is involutive after t prolongation`, 3, 3 `the order of the primary ideal and dimension`, 4, 3 `k=`, 5 `the dimension of the matrix`, 35, 3 `check the dimension d0`, 32 `the dimension of the matrix`, 35, 12 `check the dimension d1`, 23 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 35, 30 `check the dimension d1`, 7 ` the system is not involutive yet, prolong the system`, 2 `the dimension of the matrix`, 35, 60 `check the dimension d1`, 3 `the system is involutive after t prolongation`, 3, 3 `the order of the primary ideal and dimension`, 5, 3 `the index should be less or equal to multiplicity,`, 3, 3 [.2620922856e-6, .206294313e-7, .99999964388473] `k=`, 2 `the dimension of the matrix`, 4, 3 `check the dimension d0`, 2 `the dimension of the matrix`, 4, 12 `check the dimension d1`, 2 `the system is involutive after t prolongation`, 2, 1 `k=`, 3 `the dimension of the matrix`, 10, 3 `check the dimension d0`, 7 `the dimension of the matrix`, 10, 12 `check the dimension d1`, 3 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 10, 30 `check the dimension d1`, 3 `the system is involutive after t prolongation`, 3, 2 `k=`, 4 `the dimension of the matrix`, 20, 3 `check the dimension d0`, 17 `the dimension of the matrix`, 20, 12 `check the dimension d1`, 8 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 20, 30 `check the dimension d1`, 3 `the system is involutive after t prolongation`, 3, 2 `the order of the primary ideal and dimension`, 4, 3 `the index should be less or equal to multiplicity,`, 3, 3 [-.1335251e-13, .323644e-15, .99999999999955] `k=`, 2 `the dimension of the matrix`, 4, 3 `check the dimension d0`, 2 `the dimension of the matrix`, 4, 12 `check the dimension d1`, 2 `the system is involutive after t prolongation`, 2, 1 `k=`, 3 `the dimension of the matrix`, 10, 3 `check the dimension d0`, 7 `the dimension of the matrix`, 10, 12 `check the dimension d1`, 3 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 10, 30 `check the dimension d1`, 3 `the system is involutive after t prolongation`, 3, 2 `k=`, 4 `the dimension of the matrix`, 20, 3 `check the dimension d0`, 17 `the dimension of the matrix`, 20, 12 `check the dimension d1`, 8 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 20, 30 `check the dimension d1`, 3 `the system is involutive after t prolongation`, 3, 2 `the order of the primary ideal and dimension`, 4, 3 `the index should be less or equal to multiplicity,`, 3, 3 [-.19176666667e-16, .2389188958841e-16, 1.0000000000000] `the number of iteration is`, 2 -16 -16 mroot := [-0.19176666667 10 , 0.2389188958841 10 , 1.0000000000000] > > mroot-mroot0; -16 -16 [-0.19176666667 10 , 0.2389188958841 10 , 0.] > # Cyclic9, 4-fold zero with index=4 > > lsys:=[1-x0*x1*x2*x3*x4*x5*x6*x7*x8,x0+x1+x2+x3+x4+x5+x6+x7+x8,x0*x1+x1*x2+x2*x3+x3*x4+x4*x5+x5*x6+x6*x7+x7*x8+x8*x0,x0*x1*x2+x1*x2*x3+x2*x3*x4+x3*x4*x5+x4*x5*x6+x5*x6*x7+x6*x7*x8+x7*x8*x0+x8*x0*x1,x0*x1*x2*x3+x1*x2*x3*x4+x2*x3*x4*x5+x3*x4*x5*x6+x4*x5*x6*x7+x5*x6*x7*x8+x6*x7*x8*x0+x7*x8*x0*x1+x8*x0*x1*x2,x0*x1*x2*x3*x4+x1*x2*x3*x4*x5+x2*x3*x4*x5*x6+x3*x4*x5*x6*x7+x4*x5*x6*x7*x8+x5*x6*x7*x8*x0+x6*x7*x8*x0*x1+x7*x8*x0*x1*x2+x8*x0*x1*x2*x3,x0*x1*x2*x3*x4*x5+x1*x2*x3*x4*x5*x6+x2*x3*x4*x5*x6*x7+x3*x4*x5*x6*x7*x8+x4*x5*x6*x7*x8*x0+x5*x6*x7*x8*x0*x1+x6*x7*x8*x0*x1*x2+x7*x8*x0*x1*x2*x3+x8*x0*x1*x2*x3*x4,x0*x1*x2*x3*x4*x5*x6+x1*x2*x3*x4*x5*x6*x7+x2*x3*x4*x5*x6*x7*x8+x3*x4*x5*x6*x7*x8*x0+x4*x5*x6*x7*x8*x0*x1+x5*x6*x7*x8*x0*x1*x2+x6*x7*x8*x0*x1*x2*x3+x7*x8*x0*x1*x2*x3*x4+x8*x0*x1*x2*x3*x4*x5,x0*x1*x2*x3*x4*x5*x6*x7+x1*x2*x3*x4*x5*x6*x7*x8+x2*x3*x4*x5*x6*x7*x8*x0+x3*x4*x5*x6*x7*x8*x0*x1+x4*x5*x6*x7*x8*x0*x1*x2+x5*x6*x7*x8*x0*x1*x2*x3+x6*x7*x8*x0*x1*x2*x3*x4+x7*x8*x0*x1*x2*x3*x4*x5+x8*x0*x1*x2*x3*x4*x5*x6]; lsys := [1 - x0 x1 x2 x3 x4 x5 x6 x7 x8, x0 + x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8, x0 x1 + x1 x2 + x2 x3 + x3 x4 + x4 x5 + x5 x6 + x6 x7 + x7 x8 + x8 x0, x0 x1 x2 + x1 x2 x3 + x2 x3 x4 + x3 x4 x5 + x4 x5 x6 + x5 x6 x7 + x6 x7 x8 + x7 x8 x0 + x8 x0 x1, x0 x1 x2 x3 + x1 x2 x3 x4 + x2 x3 x4 x5 + x3 x4 x5 x6 + x4 x5 x6 x7 + x5 x6 x7 x8 + x6 x7 x8 x0 + x7 x8 x0 x1 + x8 x0 x1 x2, x0 x1 x2 x3 x4 + x1 x2 x3 x4 x5 + x2 x3 x4 x5 x6 + x3 x4 x5 x6 x7 + x4 x5 x6 x7 x8 + x5 x6 x7 x8 x0 + x6 x7 x8 x0 x1 + x7 x8 x0 x1 x2 + x8 x0 x1 x2 x3, x0 x1 x2 x3 x4 x5 + x1 x2 x3 x4 x5 x6 + x2 x3 x4 x5 x6 x7 + x3 x4 x5 x6 x7 x8 + x4 x5 x6 x7 x8 x0 + x5 x6 x7 x8 x0 x1 + x6 x7 x8 x0 x1 x2 + x7 x8 x0 x1 x2 x3 + x8 x0 x1 x2 x3 x4, x0 x1 x2 x3 x4 x5 x6 + x1 x2 x3 x4 x5 x6 x7 + x2 x3 x4 x5 x6 x7 x8 + x3 x4 x5 x6 x7 x8 x0 + x4 x5 x6 x7 x8 x0 x1 + x5 x6 x7 x8 x0 x1 x2 + x6 x7 x8 x0 x1 x2 x3 + x7 x8 x0 x1 x2 x3 x4 + x8 x0 x1 x2 x3 x4 x5, x0 x1 x2 x3 x4 x5 x6 x7 + x1 x2 x3 x4 x5 x6 x7 x8 + x2 x3 x4 x5 x6 x7 x8 x0 + x3 x4 x5 x6 x7 x8 x0 x1 + x4 x5 x6 x7 x8 x0 x1 x2 + x5 x6 x7 x8 x0 x1 x2 x3 + x6 x7 x8 x0 x1 x2 x3 x4 + x7 x8 x0 x1 x2 x3 x4 x5 + x8 x0 x1 x2 x3 x4 x5 x6] > > Z9:=[z0,z1,z2,z0,-z2,-z1,z0,-z2,-z1]; Z9 := [z0, z1, z2, z0, -z2, -z1, z0, -z2, -z1] > > a1:=-1/4*(-36-16*I*15^(1/2)-4*(-163+72*I*15^(1/2))^(1/2))^(1/3)-1/4*I*3^(1/2)*(-36-16*I*15^(1/2)-4*(-163+72*I*15^(1/2))^(1/2))^(1/3); 1/2 1/2 1/2 1/3 (-36 - 16 I 15 - 4 (-163 + 72 I 15 ) ) a1 := - ------------------------------------------------ 4 1/2 1/2 1/2 1/2 (1/3) - 1/4 I 3 (-36 - 16 I 15 - 4 (-163 + 72 I 15 ) ) > > > > a0:=subs(x1=a1, 6765/2584*x1-1/2584*x1^10); / 1/3 \10 | %1 1/2 (1/3)| 1/3 |- ----- - 1/4 I 3 %1 | 6765 %1 6765 1/2 (1/3) \ 4 / a0 := - ---------- - ----- I 3 %1 - -------------------------------- 10336 10336 2584 1/2 1/2 1/2 %1 := -36 - 16 I 15 - 4 (-163 + 72 I 15 ) > > evalf(a0); -0.9396926207874 - 0.3420201433260 I > > a2:=3*a0-a1; / 1/3 \10 | %1 1/2 (1/3)| 1/3 3 |- ----- - 1/4 I 3 %1 | 17711 %1 17711 1/2 (1/3) \ 4 / a2 := - ----------- - ----- I 3 %1 - ---------------------------------- 10336 10336 2584 1/2 1/2 1/2 %1 := -36 - 16 I 15 - 4 (-163 + 72 I 15 ) > > evalf(a2); -0.358930642168 - 0.1306400699142 I > > mroot0:=subs({z0=a0,z1=a1,z2=a2},Z9): > > mroot:=evalf(mroot0); mroot := [-0.9396926207874 - 0.3420201433260 I, -2.4601472201949 - 0.89542036006370 I, -0.358930642168 - 0.1306400699142 I, -0.9396926207874 - 0.3420201433260 I, 0.358930642168 + 0.1306400699142 I, 2.4601472201949 + 0.89542036006370 I, -0.9396926207874 - 0.3420201433260 I, 0.358930642168 + 0.1306400699142 I, 2.4601472201949 + 0.89542036006370 I] > > mrootp:=RandomVector(9,generator=-1..1.)*1/1000; mrootq:=RandomVector(9,generator=-1..1.)*1/1000.*I: [ 0.0000119141033302847530 ] [ ] [ -0.000489809769081461831 ] [ ] [ 0.000502534118611305683 ] [ ] [ -0.000552376121017726044 ] [ ] mrootp := [ 0.000170535501959554684 ] [ ] [ -0.000319228546667733610 ] [ ] [ 0.000919487917032162203 ] [ ] [ -5] [-0.327189603571409460 10 ] [ ] [ -0.000762004636883246707 ] > > for i from 1 to 9 do mroot[i]:=evalf(mroot0[i])+mrootp[i]+mrootq[i]; od; mroot[1] := -0.93968070668407 - 0.34251157896806 I mroot[2] := -2.4606370299640 - 0.89473892555173 I mroot[3] := -0.35842810804939 - 0.13112505340595 I mroot[4] := -0.94024499690842 - 0.34272155531488 I mroot[5] := 0.35910117766996 + 0.12991731879986 I mroot[6] := 2.4598279916482 + 0.89551479112363 I mroot[7] := -0.93877313287037 - 0.34110156047559 I mroot[8] := 0.35892737027196 + 0.13142187641927 I mroot[9] := 2.4593852155580 + 0.89581851350901 I > > mroot; [-0.93968070668407 - 0.34251157896806 I, -2.4606370299640 - 0.89473892555173 I, -0.35842810804939 - 0.13112505340595 I, -0.94024499690842 - 0.34272155531488 I, 0.35910117766996 + 0.12991731879986 I, 2.4598279916482 + 0.89551479112363 I, -0.93877313287037 - 0.34110156047559 I, 0.35892737027196 + 0.13142187641927 I, 2.4593852155580 + 0.89581851350901 I ] > > mroot:=Refineroot(evalf(lsys),[x0,x1,x2,x3,x4,x5,x6,x7,x8],5*10.^(-2),mroot,3); `k=`, 2 `the dimension of the matrix`, 10, 9 `check the dimension d0`, 3 `the dimension of the matrix`, 10, 90 `check the dimension d1`, 3 `the system is involutive after t prolongation`, 3, 1 `k=`, 3 `the dimension of the matrix`, 55, 18 `check the dimension d0`, 37 `the dimension of the matrix`, 55, 135 `check the dimension d1`, 4 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 55, 660 `check the dimension d1`, 4 `the system is involutive after t prolongation`, 4, 2 `k=`, 4 `the dimension of the matrix`, 220, 72 `check the dimension d0`, 149 `the dimension of the matrix`, 220, 345 `check the dimension d1`, 59 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 220, 1320 `check the dimension d1`, 4 `the system is involutive after t prolongation`, 4, 2 `the order of the primary ideal and dimension`, 4, 4 `the index and multiplicity are `, 3, 4 [-.93969192570566-.34202000988499*I, -2.4601483300744-.89542031605361*I, -.3589\ 3047140597-.13063995517697*I, -.93969278219046-.34202045266137*I, .358930850653\ 39+.13064019672361*I, 2.4601471442754+.89541963853790*I, -.93969272484015-.3420\ 1994881835*I, .35893070889764+.13063998324699*I, 2.4601489468022+.8954205647029\ 6*I] `k=`, 2 `the dimension of the matrix`, 10, 9 `check the dimension d0`, 3 `the dimension of the matrix`, 10, 90 `check the dimension d1`, 3 `the system is involutive after t prolongation`, 3, 1 `k=`, 3 `the dimension of the matrix`, 55, 18 `check the dimension d0`, 37 `the dimension of the matrix`, 55, 135 `check the dimension d1`, 4 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 55, 660 `check the dimension d1`, 4 `the system is involutive after t prolongation`, 4, 2 `k=`, 4 `the dimension of the matrix`, 220, 72 `check the dimension d0`, 149 `the dimension of the matrix`, 220, 345 `check the dimension d1`, 59 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 220, 1320 `check the dimension d1`, 4 `the system is involutive after t prolongation`, 4, 2 `the order of the primary ideal and dimension`, 4, 4 `the index and multiplicity are `, 3, 4 [-.93969262078591-.34202014332618*I, -2.4601472201939-.89542036006299*I, -.3589\ 3064216282-.13064006991337*I, -.93969262078534-.34202014332523*I, .358930642162\ 90+.13064006991334*I, 2.4601472201944+.89542036006369*I, -.93969262078580-.3420\ 2014332558*I, .35893064216285+.13064006991330*I, 2.4601472201941+.8954203600631\ 1*I] `k=`, 2 `the dimension of the matrix`, 10, 9 `check the dimension d0`, 3 `the dimension of the matrix`, 10, 90 `check the dimension d1`, 3 `the system is involutive after t prolongation`, 3, 1 `k=`, 3 `the dimension of the matrix`, 55, 18 `check the dimension d0`, 37 `the dimension of the matrix`, 55, 135 `check the dimension d1`, 4 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 55, 660 `check the dimension d1`, 4 `the system is involutive after t prolongation`, 4, 2 `k=`, 4 `the dimension of the matrix`, 220, 72 `check the dimension d0`, 149 `the dimension of the matrix`, 220, 345 `check the dimension d1`, 59 ` the system is not involutive yet, prolong the system`, 1 `the dimension of the matrix`, 220, 1320 `check the dimension d1`, 4 `the system is involutive after t prolongation`, 4, 2 `the order of the primary ideal and dimension`, 4, 4 `the index and multiplicity are `, 3, 4 [-.93969262078586-.34202014332566*I, -2.4601472201950-.89542036006370*I, -.3589\ 3064216274-.13064006991329*I, -.93969262078590-.34202014332566*I, .358930642162\ 76+.13064006991330*I, 2.4601472201949+.89542036006377*I, -.93969262078592-.3420\ 2014332567*I, .35893064216275+.13064006991330*I, 2.4601472201950+.8954203600636\ 9*I] `the number of iteration is`, 2 mroot := [-0.93969262078586 - 0.34202014332566 I, -2.4601472201950 - 0.89542036006370 I, -0.35893064216274 - 0.13064006991329 I, -0.93969262078590 - 0.34202014332566 I, 0.35893064216276 + 0.13064006991330 I, 2.4601472201949 + 0.89542036006377 I, -0.93969262078592 - 0.34202014332567 I, 0.35893064216275 + 0.13064006991330 I, 2.4601472201950 + 0.89542036006369 I ] > > sol:={x0=mroot[1], x1=mroot[2],x2=mroot[3],x3=mroot[4],x4=mroot[5],x5=mroot[6],x6=mroot[7],x7=mroot[8],x8=mroot[9]}; sol := {x0 = -0.93969262078586 - 0.34202014332566 I, x1 = -2.4601472201950 - 0.89542036006370 I, x2 = -0.35893064216274 - 0.13064006991329 I, x3 = -0.93969262078590 - 0.34202014332566 I, x4 = 0.35893064216276 + 0.13064006991330 I, x5 = 2.4601472201949 + 0.89542036006377 I, x6 = -0.93969262078592 - 0.34202014332567 I, x7 = 0.35893064216275 + 0.13064006991330 I, x8 = 2.4601472201950 + 0.89542036006369 I} > > subs(sol, lsys); -13 -13 -13 -12 [0.4 10 - 0.20521208599540 10 I, 0. - 0.1 10 I, 0. - 0.2 10 I, -12 -13 -12 -13 -12 0. - 0.2 10 I, -0.2 10 - 0.1 10 I, 0.4 10 - 0.3 10 I, -13 -13 -12 -13 0.9 10 + 0.5 10 I, 0. + 0. I, -0.4 10 - 0.1 10 I] > > mroot1:=[-.93968956515135-.34202105984954*I, -2.4601512534522-.89541880437148*I, -.35893068462180-.13064033407368*I, -.93969477336376-.34201890605351*I, .35893048272467+.13063995458098*I, 2.4601468579739+.89542022286307*I, -.93969290232824-.34201949139813*I, .35893055490575+.13064003655586*I, 2.4601521132453+.89541712420394*I]; mroot1 := [-0.93968956515135 - 0.34202105984954 I, -2.4601512534522 - 0.89541880437148 I, -0.35893068462180 - 0.13064033407368 I, -0.93969477336376 - 0.34201890605351 I, 0.35893048272467 + 0.13063995458098 I, 2.4601468579739 + 0.89542022286307 I, -0.93969290232824 - 0.34201949139813 I, 0.35893055490575 + 0.13064003655586 I, 2.4601521132453 + 0.89541712420394 I ] > # > > evalf(mroot1-mroot0); -5 -6 -5 -5 [0.305563605 10 - 0.91652354 10 I, -0.40332573 10 + 0.155569222 10 I, -7 -6 -5 -5 -0.4245380 10 - 0.26415948 10 I, -0.215257636 10 + 0.123727249 10 I, -6 -6 -6 -6 -0.15944333 10 - 0.11533322 10 I, -0.3622210 10 - 0.13720063 10 I, -6 -6 -7 -7 -0.28154084 10 + 0.65192787 10 I, -0.8726225 10 - 0.3335834 10 I, -5 -5 0.48930504 10 - 0.323585976 10 I] > > mroot2:=[-.93969262078497-.34202014332892*I, -2.4601472201885-.89542036007386*I, -.35893064216345-.13064006991234*I, -.93969262078395-.34202014333222*I, .35893064216263+.13064006991250*I, 2.4601472201912+.89542036006840*I, -.93969262078580-.34202014332602*I, .35893064216319+.13064006991273*I, 2.4601472201911+.89542036007468*I]; mroot2 := [-0.93969262078497 - 0.34202014332892 I, -2.4601472201885 - 0.89542036007386 I, -0.35893064216345 - 0.13064006991234 I, -0.93969262078395 - 0.34202014333222 I, 0.35893064216263 + 0.13064006991250 I, 2.4601472201912 + 0.89542036006840 I, -0.93969262078580 - 0.34202014332602 I, 0.35893064216319 + 0.13064006991273 I, 2.4601472201911 + 0.89542036007468 I ] > > evalf(mroot2-mroot0); -11 -11 -11 -10 [0.243 10 - 0.292 10 I, 0.64 10 - 0.1016 10 I, -11 -11 -11 -11 0.455 10 + 0.186 10 I, 0.345 10 - 0.622 10 I, -11 -11 -11 -11 -0.537 10 - 0.170 10 I, -0.37 10 + 0.470 10 I, -11 -13 -11 -11 0.160 10 - 0.2 10 I, -0.481 10 - 0.147 10 I, -11 -10 -0.38 10 + 0.1098 10 I] > > evalf(mroot-mroot0); -11 -12 -12 -11 -12 [0.154 10 + 0.34 10 I, -0.1 10 + 0. I, 0.526 10 + 0.91 10 I, -11 -12 -11 -12 -13 0.150 10 + 0.34 10 I, -0.524 10 - 0.90 10 I, 0. + 0.7 10 I, -11 -12 -11 -12 0.148 10 + 0.33 10 I, -0.525 10 - 0.90 10 I, -12 -13 0.1 10 - 0.1 10 I] > # > > > > quit memory used=1543596.0MB, alloc=1711.6MB, time=30868.12