|\^/| Maple 7 (SUN SPARC SOLARIS) ._|\| |/|_. Copyright (c) 2001 by Waterloo Maple Inc. \ MAPLE / All rights reserved. Maple is a registered trademark of <____ ____> Waterloo Maple Inc. | Type ? for help. > # S -> EP | ED | G # EP -> EP+F | F+F # F -> ED | G # ED -> ED G | GG # G -> ES | C | P # C -> m | e | a (a in Sigma) # ES -> G* # P -> (S) # grammar for all valid regular expressions > > with(Groebner); [MulMatrix, SetBasis, fglm, gbasis, gsolve, hilbertdim, hilbertpoly, hilbertseries, inter_reduce, is_finite, is_solvable, leadcoeff, leadmon, leadterm, normalf, pretend_gbasis, reduce, spoly, termorder, testorder, univpoly] > > gp := [S-(EP+ED+G), > EP-(EP*F*x+F^2*x), > F-(ED+G), > ED-(ED*G+G^2), > G-(ES+C+P), > C-(x+x+k*x), > ES-(G*x), > P-S*x^2]; 2 2 gp := [S - EP - ED - G, EP - EP F x - F x, F - ED - G, ED - ED G - G , 2 G - ES - C - P, C - 2 x - k x, ES - G x, P - S x ] > > tt := gbasis(gp,plex(EP,F,ED,G,C,ES,P,S)); bytes used=1000084, alloc=917336, time=0.24 3 2 2 2 2 2 tt := [x S + k x S + 3 S x + x S + k x + 2 x + k x S + 3 x S - S, 2 3 2 2 P - S x , ES x - ES + S x + 2 x + k x , C - 2 x - k x, 2 2 2 2 G x - G + S x + 2 x + k x, 2 ED x - 3 ED x + ED k x + ED - ED k x + S x 3 3 2 2 2 2 - x S - S x - S x k - 2 x - 3 k x - k x , 2 S x + 2 F x + 2 x + k x + F k x + x S - F, 2 -S x - 3 x S - k x S + 2 EP x - 2 x - k x + EP k x - EP + S] > # the equation that S satisfies > uu := collect(tt[1],S); 3 2 2 2 2 uu := (x + x ) S + (k x + 3 x + k x + 3 x - 1) S + 2 x + k x > > dd := discrim(uu,S); 4 2 4 4 3 2 3 3 2 2 2 2 dd := 2 x k + k x + x + 10 x + 2 k x + 8 k x + 4 k x + 3 x + k x - 2 k x - 6 x + 1 > > ee := collect(dd,x); 2 4 2 3 2 2 ee := (2 k + k + 1) x + (10 + 2 k + 8 k) x + (4 k + 3 + k ) x + (-2 k - 6) x + 1 > > gg := solve(uu=0,S); bytes used=2046676, alloc=1441528, time=0.47 2 2 4 2 4 4 3 gg := 1/2 (-k x - 3 x - k x - 3 x + 1 + (2 x k + k x + x + 10 x 2 3 3 2 2 2 2 1/2 / + 2 k x + 8 k x + 4 k x + 3 x + k x - 2 k x - 6 x + 1) ) / ( / 3 2 2 2 4 2 4 4 3 x + x ), 1/2 (-k x - 3 x - k x - 3 x + 1 - (2 x k + k x + x + 10 x 2 3 3 2 2 2 2 1/2 / + 2 k x + 8 k x + 4 k x + 3 x + k x - 2 k x - 6 x + 1) ) / ( / 3 2 x + x ) > > vv := RootOf(uu=0,S); 3 2 2 2 2 vv := RootOf((x + x ) _Z + (k x + 3 x + k x + 3 x - 1) _Z + k x + 2 x) > > q1 := subs(k=1,uu); 3 2 2 2 q1 := (x + x ) S + (4 x + 4 x - 1) S + 3 x > q2 := subs(k=2,uu); 3 2 2 2 q2 := (x + x ) S + (5 x + 5 x - 1) S + 4 x > # generating function for k = 1 > taylor(RootOf(q1=0,S),x,10); 2 3 4 5 6 7 8 3 x + 12 x + 60 x + 297 x + 1509 x + 7800 x + 40962 x + 218052 x + 9 10 1174494 x + O(x ) > # generating function for k = 2 > taylor(RootOf(q2=0,S),x,10); 2 3 4 5 6 7 8 4 x + 20 x + 120 x + 716 x + 4356 x + 26880 x + 168068 x + 1063156 x + 9 10 6795128 x + O(x ) > > p1 := subs(k=1,ee); 4 3 2 p1 := 4 x + 20 x + 8 x - 8 x + 1 > p2 := subs(k=2,ee); 4 3 2 p2 := 9 x + 34 x + 15 x - 10 x + 1 > > Digits := 30; Digits := 30 > > r1 := fsolve(p1,x); bytes used=3048952, alloc=1638100, time=0.79 r1 := -4.44615693626056659979810826544, -1.04690077816302674987504961064, .162462502623911455184518370706, .330595211799681894488639505382 > > evalf(1/r1[3]); 6.15526650057167481379677563497 > > r2 := fsolve(p2,x); r2 := -3.12768932253813751568803129059, -1.03675165824086454303490103416, .137551275773557552757270978292, .249111927227666728187883568685 > > evalf(1/r2[3]); 7.27001617670373522777805502430 > > hh := series(RootOf(dd,x,index=1),k=infinity,10); bytes used=4116972, alloc=2031244, time=1.09 11159 - 5507/4 - ----- %1 %1 -15 - 19/2 %1 305/2 + 643/8 %1 16 hh := 1/k + ---- + ------------- + ---------------- + ------------------- 2 3 4 5 k k k k 808867 403043 15224925 7605149 590648743 295241127 ------ %1 + ------ - -------- %1 - ------- --------- %1 + --------- 128 32 256 64 1024 256 + ------------------ + ----------------------- + ------------------------ 6 7 8 k k k 11731051743 5865024223 - ----------- %1 - ---------- 2048 512 1 + ----------------------------- + O(---) 9 10 k k 2 %1 := RootOf(_Z + 8 _Z + 12) > > subs(RootOf(_Z^2+8*_Z+12)=-6,hh); 6 42 1 1 1620515 1 30464477 1 1/k - ---- + ---- - 1319/4 ---- + 22463/8 ---- - ------- ---- + -------- ---- 2 3 4 5 64 6 128 7 k k k k k k 1181463975 1 23463106783 1 1 - ---------- ---- + ----------- ---- + O(---) 512 8 1024 9 10 k k k > > taylor(1/%,k=infinity,10); 1 1 177923 1 3207309 1 k + 6 - 6/k + 167/4 ---- - 2615/8 ---- + ------ ---- - ------- ---- 2 3 64 4 128 5 k k k k 120566167 1 2337716671 1 1 + --------- ---- - ---------- ---- + O(----) 512 6 1024 7 8 k k k > quit bytes used=4902972, alloc=2096768, time=1.32