Thu Feb 25 19:44:05 MST 1999 aquarius% maple |\^/| Maple V Release 5.1 (WMI Campus Wide License) ._|\| |/|_. Copyright (c) 1981-1998 by Waterloo Maple Inc. All rights \ MAPLE / reserved. Maple and Maple V are registered trademarks of <____ ____> Waterloo Maple Inc. | Type ? for help. # ----------[ M a p l e ]---------- #interface(echo = 3); # ---------- Initialization ---------- > readlib(showtime): > _EnvAllSolutions:= true: > on; # ---------- Equations ---------- # Manipulate an equation using a natural syntax: # (x = 2)/2 + (1 = 1) => x/2 + 1 = 2 O1 := (x = 2)/2 + (1 = 1); 1/2 x + 1 = 2 time = 0.03, bytes = 43574 # Solve various nonlinear equations---this cubic polynomial has all real roots O2 := solve(3*x^3 - 18*x^2 + 33*x - 19 = 0, x); bytes used=1001888, alloc=786288, time=0.50 2 1 1/6 %2 + ------------------- + 2, - 1/12 %2 - ------------------- + 2 1/2 1/3 1/2 1/3 (36 + 12 I 3 ) (36 + 12 I 3 ) 1/2 / 2 \ + 1/2 I 3 |1/6 %2 - -------------------|, - 1/12 %2 | 1/2 1/3| \ (36 + 12 I 3 ) / 1 1/2 / 2 \ - ------------------- + 2 - 1/2 I 3 |1/6 %2 - -------------------| 1/2 1/3 | 1/2 1/3| (36 + 12 I 3 ) \ (36 + 12 I 3 ) / 1 %1 := ------------------- 1/2 1/3 (36 + 12 I 3 ) 1/2 1/3 %2 := (36 + 12 I 3 ) time = 0.64, bytes = 1167982 O3 := simplify(evalc({%})); bytes used=2005364, alloc=1507052, time=1.43 1/2 1/2 {- 1/3 3 cos(1/18 Pi) + 2 + sin(1/18 Pi), 2/3 3 cos(1/18 Pi) + 2, 1/2 - 1/3 3 cos(1/18 Pi) + 2 - sin(1/18 Pi)} time = 0.79, bytes = 695358 # Some simple seeming problems can have messy answers: # x = { [sqrt(5) - 1]/4 +/- 5^(1/4) sqrt(sqrt(5) + 1)/[2 sqrt(2)] i, # - [sqrt(5) + 1]/4 +/- 5^(1/4) sqrt(sqrt(5) - 1)/[2 sqrt(2)] i} O4 := eqn:= x^4 + x^3 + x^2 + x + 1 = 0; 4 3 2 x + x + x + x + 1 = 0 time = 0.02, bytes = 16070 O5 := solve(eqn, x); 1/2 1/2 1/2 1/2 1/4 5 - 1/4 + 1/4 I 2 (5 + 5 ) , 1/2 1/2 1/2 1/2 - 1/4 5 - 1/4 + 1/4 I 2 (5 - 5 ) , 1/2 1/2 1/2 1/2 - 1/4 5 - 1/4 - 1/4 I 2 (5 - 5 ) , 1/2 1/2 1/2 1/2 1/4 5 - 1/4 - 1/4 I 2 (5 + 5 ) time = 0.15, bytes = 148662 # Check one of the answers O6 := subs(x = %[1], eqn); 1/2 4 1/2 3 (1/4 5 - 1/4 + 1/4 %1) + (1/4 5 - 1/4 + 1/4 %1) 1/2 2 1/2 + (1/4 5 - 1/4 + 1/4 %1) + 1/4 5 + 3/4 + 1/4 %1 = 0 1/2 1/2 1/2 %1 := I 2 (5 + 5 ) time = 0.01, bytes = 8514 O7 := simplify(%); 0 = 0 time = 0.14, bytes = 169230 O8 := eqn:= 'eqn': time = 0.01, bytes = 3950 # x = {2^(1/3) +- sqrt(3), +- sqrt(3) - 1/2^(2/3) +- i sqrt(3)/2^(2/3)} # [Mohamed Omar Rayes] O9 := solve(x^6 - 9*x^4 - 4*x^3 + 27*x^2 - 36*x - 23 = 0, x); 6 4 3 2 RootOf(_Z - 9 _Z - 4 _Z + 27 _Z - 36 _Z - 23) time = 0.18, bytes = 180038 # x = {1, e^(+- 2 pi i/7), e^(+- 4 pi i/7), e^(+- 6 pi i/7)} O10 := solve(x^7 - 1 = 0, x); bytes used=3005620, alloc=1834672, time=2.61 1, cos(2/7 Pi) + I sin(2/7 Pi), -cos(3/7 Pi) + I sin(3/7 Pi), -cos(1/7 Pi) + I sin(1/7 Pi), -cos(1/7 Pi) - I sin(1/7 Pi), -cos(3/7 Pi) - I sin(3/7 Pi), cos(2/7 Pi) - I sin(2/7 Pi) time = 0.86, bytes = 610534 # x = 1 +- sqrt(+-sqrt(+-4 sqrt(3) - 3) - 3)/sqrt(2) [Richard Liska] O11 := solve(x^8 - 8*x^7 + 34*x^6 - 92*x^5 + 175*x^4 - 236*x^3 + 226*x^2 - 140*x + 46 O11 := = 0, x); bytes used=4006540, alloc=2031244, time=3.85 1/2 1/2 1/2 1/2 1/2 1/2 1 + 1/2 I (6 - 2 (-3 + 4 3 ) ) , 1 - 1/2 I (6 - 2 (-3 + 4 3 ) ) , 1/2 1/2 1/2 1 + 1/2 I (6 + 2 (-3 + 4 3 ) ) , 1/2 1/2 1/2 1/2 1/2 1/2 1 - 1/2 I (6 + 2 (-3 + 4 3 ) ) , 1 + 1/2 (-6 + 2 (-3 - 4 3 ) ) , 1/2 1/2 1/2 1/2 1/2 1/2 1 - 1/2 (-6 + 2 (-3 - 4 3 ) ) , 1 + 1/2 (-6 - 2 (-3 - 4 3 ) ) , 1/2 1/2 1/2 1 - 1/2 (-6 - 2 (-3 - 4 3 ) ) time = 1.50, bytes = 1263942 # The following equations have an infinite number of solutions (let n be an # arbitrary integer): # x = {log(sqrt(z) - 1), log(sqrt(z) + 1) + i pi} [+ n 2 pi i, + n 2 pi i] O12 := exp(2*x) + 2*exp(x) + 1 = z; exp(2 x) + 2 exp(x) + 1 = z time = 0.01, bytes = 16590 O13 := solve(%, x); 1/2 1/2 ln(-1 + z ), ln(-1 - z ) time = 0.21, bytes = 148182 # x = (1 +- sqrt(9 - 8 n pi i))/2. Real solutions correspond to n = 0 => # x = {-1, 2} O14 := solve(exp(2 - x^2) = exp(-x), x); -1, 2 time = 0.11, bytes = 75162 # x = -W[n](-1) [e.g., -W[0](-1) = 0.31813 - 1.33724 i] where W[n](x) is the # nth branch of Lambert's W function O15 := solve(exp(x) = x, x); -LambertW(_NN1, -1) time = 0.10, bytes = 68078 # x = {-1, 1} O16 := solve(x^x = x, x); bytes used=5006768, alloc=2293340, time=5.32 1 time = 0.44, bytes = 121126 # This equation is already factored and so *should* be easy to solve: # x = {-1, 2*{+-arcsinh(1) i + n pi}, 3*{pi/6 + n pi/3}} O17 := (x + 1) * (sin(x)^2 + 1)^2 * cos(3*x)^3 = 0; 2 2 3 (x + 1) (sin(x) + 1) cos(3 x) = 0 time = 0.02, bytes = 11286 O18 := solve(%, x); -1, I arcsinh(1) - 2 I arcsinh(1) _B1~ + 2 Pi _Z1~ + Pi _B1~, -I arcsinh(1) + 2 I arcsinh(1) _B1~ + 2 Pi _Z1~ + Pi _B1~, 1/2 Pi + 2 Pi _Z~, - 1/2 Pi + 2 Pi _Z~, 1/6 Pi + 2 Pi _Z~, 5/6 Pi + 2 Pi _Z~, - 1/6 Pi + 2 Pi _Z~, - 5/6 Pi + 2 Pi _Z~ time = 0.91, bytes = 656754 # x = pi/4 [+ n pi] O19 := solve(sin(x) = cos(x), x); 1/4 Pi + Pi _Z~ time = 0.06, bytes = 41654 O20 := solve(tan(x) = 1, x); 1/4 Pi + Pi _Z~ time = 0.07, bytes = 50390 # x = {pi/6, 5 pi/6} [ + n 2 pi, + n 2 pi ] O21 := solve(sin(x) = 1/2, x); 1/6 Pi + 2/3 Pi _B1~ + 2 Pi _Z1~ time = 0.07, bytes = 41110 # x = 0, 0 [+ n pi, + n 2 pi] O22 := solve(sin(x) = tan(x), x); Pi _Z~ time = 0.11, bytes = 89158 # x = {0, 0, 0} O23 := solve(arcsin(x) = arctan(x), x); bytes used=6006936, alloc=2358864, time=6.98 0 time = 0.57, bytes = 181594 # x = sqrt[(sqrt(5) - 1)/2] O24 := solve(arccos(x) = arctan(x), x); 1/2 -1/2 + 1/2 5 cos(arctan(2 ----------------) + 2 Pi _Z~), 1/2 1/2 (-2 + 2 5 ) 1/2 -1/2 + 1/2 5 -cos(-arctan(2 ----------------) + 2 Pi _Z~), 1/2 1/2 (-2 + 2 5 ) 1/2 1/2 1/2 cos(arctan(-1/2 - 1/2 5 , 1/2 (-2 - 2 5 ) ) + 2 Pi _Z~), 1/2 1/2 1/2 cos(arctan(-1/2 - 1/2 5 , - 1/2 (-2 - 2 5 ) ) + 2 Pi _Z~) time = 0.91, bytes = 658574 O25 := _EnvAllSolutions:= false: time = 0.01, bytes = 3982 O26 := solve(arccos(x) = arctan(x), x); 1/2 -1/2 + 1/2 5 cos(arctan(2 ----------------) + 2 Pi _Z~), 1/2 1/2 (-2 + 2 5 ) 1/2 -1/2 + 1/2 5 -cos(-arctan(2 ----------------) + 2 Pi _Z~), 1/2 1/2 (-2 + 2 5 ) 1/2 1/2 1/2 cos(arctan(-1/2 - 1/2 5 , 1/2 (-2 - 2 5 ) ) + 2 Pi _Z~), 1/2 1/2 1/2 cos(arctan(-1/2 - 1/2 5 , - 1/2 (-2 - 2 5 ) ) + 2 Pi _Z~) time = 0.04, bytes = 40318 O27 := _EnvAllSolutions:= true: time = 0.01, bytes = 3994 # x = 2 O28 := solve((x - 2)/x^(1/3) = 0, x); 2 time = 0.04, bytes = 24654 # This equation has no solutions O29 := solve(sqrt(x^2 + 1) = x - 2, x); time = 0.08, bytes = 51306 O30 := _SolutionsMayBeLost; _SolutionsMayBeLost time = 0.00, bytes = 3738 # x = 1 O31 := solve(x + sqrt(x) = 2, x); bytes used=7007104, alloc=2489912, time=8.69 1 time = 0.45, bytes = 79022 # x = 1/16 O32 := solve(2*sqrt(x) + 3*x^(1/4) - 2 = 0, x); 1/16 time = 0.14, bytes = 82698 # x = {sqrt[(sqrt(5) - 1)/2], -i sqrt[(sqrt(5) + 1)/2]} O33 := solve(x = 1/sqrt(1 + x^2), x); 2 4 2 4 RootOf(_Z + _Z - 1, -1.272019650 I), RootOf(_Z + _Z - 1, .7861513778) time = 0.35, bytes = 278926 O34 := solve(x^4 + x^2 - 1 = 0, x); 1/2 1/2 1/2 1/2 1/2 1/2 1/2 (-2 + 2 5 ) , - 1/2 (-2 + 2 5 ) , 1/2 (-2 - 2 5 ) , 1/2 1/2 - 1/2 (-2 - 2 5 ) time = 0.10, bytes = 74822 # This problem is from a computational biology talk => 1 - log_2[m (m - 1)] O35 := solve(binomial(m, 2)*2^k = 1, k); 1 ln(--------------) + 2 I Pi _Z~ binomial(m, 2) ------------------------------- ln(2) time = 0.44, bytes = 276498 # x = log(c/a) / log(b/d) for a, b, c, d != 0 and b, d != 1 [Bill Pletsch] O36 := solve(a*b^x = c*d^x, x); ln(a/c) + 2 I Pi _Z~ -------------------- ln(d/b) time = 0.39, bytes = 209234 # x = {1, e^4} O37 := solve(sqrt(log(x)) = log(sqrt(x)), x); bytes used=8007312, alloc=2620960, time=10.51 2 1, exp(2) time = 0.69, bytes = 232118 O38 := simplify([%]); [1, exp(4)] time = 0.02, bytes = 23754 # Recursive use of inverses, including multiple branches of rational # fractional powers [Richard Liska] # => x = +-(b + sin(1 + cos(1/e^2)))^(3/2) O39 := solve(log(arccos(arcsin(x^(2/3) - b) - 1)) + 2 = 0, x); bytes used=9007504, alloc=2686484, time=12.49 3/2 3/2 (b + sin(1 + cos(exp(-2)))) , -(b + sin(1 + cos(exp(-2)))) time = 1.79, bytes = 915374 # x = {-0.784966, -0.016291, 0.802557} From Metha Kamminga-van Hulsen, # ``Hoisting the Sails and Casting Off with Maple'', _Computer Algebra # Nederland Nieuwsbrief_, Number 13, December 1994, ISSN 1380-1260, 27--40. O40 := eqn:= 5*x + exp((x - 5)/2) = 8*x^3; 3 5 x + exp(1/2 x - 5/2) = 8 x time = 0.03, bytes = 14310 O41 := solve(eqn, x); time = 0.13, bytes = 75550 O42 := fsolve(eqn, x, (-1)..(-0.5)); -.7849661465 time = 0.19, bytes = 205434 O43 := fsolve(eqn, x, (-0.5)..(0.5)); -.01629073773 time = 0.12, bytes = 121338 O44 := fsolve(eqn, x, (0.5)..1); .8025567019 time = 0.13, bytes = 117430 O45 := eqn:= 'eqn': time = 0.01, bytes = 3302 # x = {-1, 3} O46 := solve(abs(x - 1) = 2, x); 3, -1 time = 0.07, bytes = 46870 # x = {-1, -7} O47 := solve(abs(2*x + 5) = abs(x - 2), x); -1, -7 time = 0.20, bytes = 97898 # x = +-3/2 O48 := solve(1 - abs(x) = max(-x - 2, x - 2), x); 3/2, -3/2 time = 0.14, bytes = 76274 # x = {-1, 3} O49 := solve(max(2 - x^2, x) = max(-x, x^3/9), x); bytes used=10008144, alloc=2752008, time=14.11 -1, 3 time = 1.04, bytes = 424978 # x = {+-3, -3 [1 + sqrt(3) sin t + cos t]} = {+-3, -1.554894} # where t = (arctan[sqrt(5)/2] - pi)/3. The third answer is the root of # x^3 + 9 x^2 - 18 = 0 in the interval (-2, -1). O50 := solve(max(2 - x^2, x) = x^3/9, x); 1/2 1/3 1 3, -3, - 1/2 (-18 + 9 I 5 ) - 9/2 ------------------- - 3 1/2 1/3 (-18 + 9 I 5 ) 1/2 / 1/2 1/3 9 \ - 1/2 I 3 |(-18 + 9 I 5 ) - -------------------| | 1/2 1/3| \ (-18 + 9 I 5 ) / time = 0.37, bytes = 246474 O51 := simplify(evalc({%})); 1/2 {-3, 3, -3 cos(- 1/3 arctan(1/2 5 ) + 1/3 Pi) - 3 1/2 1/2 + 3 3 sin(- 1/3 arctan(1/2 5 ) + 1/3 Pi)} time = 0.32, bytes = 266014 O52 := evalf(%); {3., -1.554894089, -3.} time = 0.01, bytes = 7046 # z = 2 + 3 i O53 := eqn:= (1 + I)*z + (2 - I)*conjugate(z) = -3*I; _ (1 + I) z + (2 - I) z = -3 I time = 0.02, bytes = 9706 O54 := solve(eqn, z); bytes used=11008376, alloc=2752008, time=15.89 __ __ RootOf(_Z + I _Z + 2 _Z - I _Z + 3 I) time = 0.67, bytes = 270038 O55 := assume(x, real): time = 0.01, bytes = 4882 O56 := assume(y, real): time = 0.01, bytes = 4866 O57 := simplify(subs(z = x + I*y, eqn)); 3 x~ - I y~ - 2 y~ = -3 I time = 0.11, bytes = 83078 O58 := solve(%, {x, y}); {x~ = 1/3 I y~ + 2/3 y~ - I, y~ = y~} time = 0.04, bytes = 24058 O59 := x:= 'x': time = 0.01, bytes = 3362 O60 := y:= 'y': time = 0.01, bytes = 3450 O61 := eqn:= 'eqn': time = 0.01, bytes = 3434 # => {f^(-1)(1), f^(-1)(-2)} assuming f is invertible O62 := solve(f(x)^2 + f(x) - 2 = 0, x); RootOf(f(_Z) + 2), RootOf(f(_Z) - 1) time = 0.14, bytes = 81934 O63 := eqns:= 'eqns': time = 0.00, bytes = 3330 O64 := vars:= 'vars': time = 0.01, bytes = 3374 # Solve a 3 x 3 system of linear equations O65 := eqn1:= x + y + z = 6; x + y + z = 6 time = 0.01, bytes = 7038 O66 := eqn2:= 2*x + y + 2*z = 10; 2 x + y + 2 z = 10 time = 0.01, bytes = 3694 O67 := eqn3:= x + 3*y + z = 10; x + 3 y + z = 10 time = 0.01, bytes = 3730 # Note that the solution is parametric: x = 4 - z, y = 2 O68 := solve({eqn1, eqn2, eqn3}, {x, y, z}); {z = z, x = 4 - z, y = 2} time = 0.08, bytes = 43770 # A linear system arising from the computation of a truncated power series # solution to a differential equation. There are 189 equations to be solved # for 49 unknowns. 42 of the equations are repeats of other equations; many # others are trivial. Solving this system directly by Gaussian elimination # is *not* a good idea. Solving the easy equations first is probably a better # method. The solution is actually rather simple. [Stanly Steinberg] # => k1 = ... = k22 = k24 = k25 = k27 = ... = k30 = k32 = k33 = k35 = ... # = k38 = k40 = k41 = k44 = ... = k49 = 0, k23 = k31 = k39, # k34 = b/a k26, k42 = c/a k26, {k23, k26, k43} are arbitrary O69 := eqns:= { O69 := -b*k8/a+c*k8/a = 0, -b*k11/a+c*k11/a = 0, -b*k10/a+c*k10/a+k2 = 0, O69 := -k3-b*k9/a+c*k9/a = 0, -b*k14/a+c*k14/a = 0, -b*k15/a+c*k15/a = 0, O69 := -b*k18/a+c*k18/a-k2 = 0, -b*k17/a+c*k17/a = 0, -b*k16/a+c*k16/a+k4 = 0, O69 := -b*k13/a+c*k13/a-b*k21/a+c*k21/a+b*k5/a-c*k5/a = 0, b*k44/a-c*k44/a = 0, O69 := -b*k45/a+c*k45/a = 0, -b*k20/a+c*k20/a = 0, -b*k44/a+c*k44/a = 0, O69 := b*k46/a-c*k46/a = 0, b^2*k47/a^2-2*b*c*k47/a^2+c^2*k47/a^2 = 0, k3 = 0, O69 := -k4 = 0, -b*k12/a+c*k12/a-a*k6/b+c*k6/b = 0, O69 := -b*k19/a+c*k19/a+a*k7/c-b*k7/c = 0, b*k45/a-c*k45/a = 0, O69 := -b*k46/a+c*k46/a = 0, -k48+c*k48/a+c*k48/b-c^2*k48/(a*b) = 0, O69 := -k49+b*k49/a+b*k49/c-b^2*k49/(a*c) = 0, a*k1/b-c*k1/b = 0, O69 := a*k4/b-c*k4/b = 0, a*k3/b-c*k3/b+k9 = 0, -k10+a*k2/b-c*k2/b = 0, O69 := a*k7/b-c*k7/b = 0, -k9 = 0, k11 = 0, b*k12/a-c*k12/a+a*k6/b-c*k6/b = 0, O69 := a*k15/b-c*k15/b = 0, k10+a*k18/b-c*k18/b = 0, -k11+a*k17/b-c*k17/b = 0, O69 := a*k16/b-c*k16/b = 0, -a*k13/b+c*k13/b+a*k21/b-c*k21/b+a*k5/b-c*k5/b = 0, O69 := -a*k44/b+c*k44/b = 0, a*k45/b-c*k45/b = 0, O69 := a*k14/c-b*k14/c+a*k20/b-c*k20/b = 0, a*k44/b-c*k44/b = 0, O69 := -a*k46/b+c*k46/b = 0, -k47+c*k47/a+c*k47/b-c^2*k47/(a*b) = 0, O69 := a*k19/b-c*k19/b = 0, -a*k45/b+c*k45/b = 0, a*k46/b-c*k46/b = 0, O69 := a^2*k48/b^2-2*a*c*k48/b^2+c^2*k48/b^2 = 0, O69 := -k49+a*k49/b+a*k49/c-a^2*k49/(b*c) = 0, k16 = 0, -k17 = 0, O69 := -a*k1/c+b*k1/c = 0, -k16-a*k4/c+b*k4/c = 0, -a*k3/c+b*k3/c = 0, O69 := k18-a*k2/c+b*k2/c = 0, b*k19/a-c*k19/a-a*k7/c+b*k7/c = 0, O69 := -a*k6/c+b*k6/c = 0, -a*k8/c+b*k8/c = 0, -a*k11/c+b*k11/c+k17 = 0, O69 := -a*k10/c+b*k10/c-k18 = 0, -a*k9/c+b*k9/c = 0, O69 := -a*k14/c+b*k14/c-a*k20/b+c*k20/b = 0, O69 := -a*k13/c+b*k13/c+a*k21/c-b*k21/c-a*k5/c+b*k5/c = 0, a*k44/c-b*k44/c = 0, O69 := -a*k45/c+b*k45/c = 0, -a*k44/c+b*k44/c = 0, a*k46/c-b*k46/c = 0, O69 := -k47+b*k47/a+b*k47/c-b^2*k47/(a*c) = 0, -a*k12/c+b*k12/c = 0, O69 := a*k45/c-b*k45/c = 0, -a*k46/c+b*k46/c = 0, O69 := -k48+a*k48/b+a*k48/c-a^2*k48/(b*c) = 0, O69 := a^2*k49/c^2-2*a*b*k49/c^2+b^2*k49/c^2 = 0, k8 = 0, k11 = 0, -k15 = 0, O69 := k10-k18 = 0, -k17 = 0, k9 = 0, -k16 = 0, -k29 = 0, k14-k32 = 0, O69 := -k21+k23-k31 = 0, -k24-k30 = 0, -k35 = 0, k44 = 0, -k45 = 0, k36 = 0, O69 := k13-k23+k39 = 0, -k20+k38 = 0, k25+k37 = 0, b*k26/a-c*k26/a-k34+k42 = 0, bytes used=12008552, alloc=2817532, time=17.60 O69 := -2*k44 = 0, k45 = 0, k46 = 0, b*k47/a-c*k47/a = 0, k41 = 0, k44 = 0, O69 := -k46 = 0, -b*k47/a+c*k47/a = 0, k12+k24 = 0, -k19-k25 = 0, O69 := -a*k27/b+c*k27/b-k33 = 0, k45 = 0, -k46 = 0, -a*k48/b+c*k48/b = 0, O69 := a*k28/c-b*k28/c+k40 = 0, -k45 = 0, k46 = 0, a*k48/b-c*k48/b = 0, O69 := a*k49/c-b*k49/c = 0, -a*k49/c+b*k49/c = 0, -k1 = 0, -k4 = 0, -k3 = 0, O69 := k15 = 0, k18-k2 = 0, k17 = 0, k16 = 0, k22 = 0, k25-k7 = 0, O69 := k24+k30 = 0, k21+k23-k31 = 0, k28 = 0, -k44 = 0, k45 = 0, -k30-k6 = 0, O69 := k20+k32 = 0, k27+b*k33/a-c*k33/a = 0, k44 = 0, -k46 = 0, O69 := -b*k47/a+c*k47/a = 0, -k36 = 0, k31-k39-k5 = 0, -k32-k38 = 0, O69 := k19-k37 = 0, k26-a*k34/b+c*k34/b-k42 = 0, k44 = 0, -2*k45 = 0, k46 = 0, O69 := a*k48/b-c*k48/b = 0, a*k35/c-b*k35/c-k41 = 0, -k44 = 0, k46 = 0, O69 := b*k47/a-c*k47/a = 0, -a*k49/c+b*k49/c = 0, -k40 = 0, k45 = 0, -k46 = 0, O69 := -a*k48/b+c*k48/b = 0, a*k49/c-b*k49/c = 0, k1 = 0, k4 = 0, k3 = 0, O69 := -k8 = 0, -k11 = 0, -k10+k2 = 0, -k9 = 0, k37+k7 = 0, -k14-k38 = 0, O69 := -k22 = 0, -k25-k37 = 0, -k24+k6 = 0, -k13-k23+k39 = 0, O69 := -k28+b*k40/a-c*k40/a = 0, k44 = 0, -k45 = 0, -k27 = 0, -k44 = 0, O69 := k46 = 0, b*k47/a-c*k47/a = 0, k29 = 0, k32+k38 = 0, k31-k39+k5 = 0, O69 := -k12+k30 = 0, k35-a*k41/b+c*k41/b = 0, -k44 = 0, k45 = 0, O69 := -k26+k34+a*k42/c-b*k42/c = 0, k44 = 0, k45 = 0, -2*k46 = 0, O69 := -b*k47/a+c*k47/a = 0, -a*k48/b+c*k48/b = 0, a*k49/c-b*k49/c = 0, k33 = 0, O69 := -k45 = 0, k46 = 0, a*k48/b-c*k48/b = 0, -a*k49/c+b*k49/c = 0 O69 := }: time = 1.78, bytes = 1061518 O70 := vars:= {k1, k2, k3, k4, k5, k6, k7, k8, k9, k10, k11, k12, k13, k14, k15, k16, O70 := k17, k18, k19, k20, k21, k22, k23, k24, k25, k26, k27, k28, k29, k30, O70 := k31, k32, k33, k34, k35, k36, k37, k38, k39, k40, k41, k42, k43, k44, O70 := k45, k46, k47, k48, k49}: time = 0.02, bytes = 15766 O71 := solve(eqns, vars); bytes used=13012080, alloc=2948580, time=19.20 bytes used=14012548, alloc=3210676, time=19.95 bytes used=15013440, alloc=3210676, time=21.79 a k42 b k42 {k26 = -----, k34 = -----, k10 = 0, k23 = k39, k39 = k39, k42 = k42, k43 = k43, c c k3 = 0, k9 = 0, k8 = 0, k16 = 0, k11 = 0, k33 = 0, k29 = 0, k1 = 0, k4 = 0, k28 = 0, k22 = 0, k45 = 0, k46 = 0, k41 = 0, k15 = 0, k17 = 0, k44 = 0, k36 = 0, k49 = 0, k48 = 0, k7 = 0, k19 = 0, k6 = 0, k12 = 0, k47 = 0, k20 = 0, k35 = 0, k27 = 0, k40 = 0, k14 = 0, k32 = 0, k30 = 0, k24 = 0, k37 = 0, k25 = 0, k38 = 0, k31 = k39, k5 = 0, k2 = 0, k18 = 0, k21 = 0, k13 = 0} time = 3.47, bytes = 2473686 # Solve a 3 x 3 system of nonlinear equations O72 := eqn1:= x^2*y + 3*y*z - 4 = 0; 2 x y + 3 y z - 4 = 0 time = 0.02, bytes = 6754 O73 := eqn2:= -3*x^2*z + 2*y^2 + 1 = 0; 2 2 -3 x z + 2 y + 1 = 0 time = 0.01, bytes = 3870 O74 := eqn3:= 2*y*z^2 - z^2 - 1 = 0; 2 2 2 y z - z - 1 = 0 time = 0.01, bytes = 3670 # Solving this by hand would be a nightmare O75 := solve({eqn1, eqn2, eqn3}, {x, y, z}); {x = 1, y = 1, z = 1}, {y = 1, z = 1, x = -1}, { 2 2 x = RootOf(_Z - 3 RootOf(3 _Z - 2 _Z + 1) + 2), 2 2 z = RootOf(3 _Z - 2 _Z + 1), y = 1 - 3 RootOf(3 _Z - 2 _Z + 1)}, { 2 3 4 y = 21/2 %1 + 12 %1 + 3/2 - 9 %1 + 6 %1, z = %1, 2 4 2 3 x = RootOf(3 _Z + 12 %1 - 30 %1 - 12 %1 + 7 %1)} 5 4 3 2 %1 := RootOf(6 _Z - 6 _Z - 9 _Z - 7 _Z - 3 _Z - 1) time = 1.01, bytes = 858742 O76 := evalf([%]); bytes used=16013612, alloc=3210676, time=23.31 [{z = 1., x = 1., y = 1.}, {z = 1., x = -1., y = 1.}, { -9 z = .3333333333 - .4714045208 I, y = .1 10 + 1.414213562 I, x = -.6050003336 + 1.168770894 I}, {y = 1.069034854 - 1.481223285 I, z = -.4626596394 - .3178876919 I, x = -1.800995670 - .7577068583 I}] time = 0.80, bytes = 568862 O77 := eqn1:= 'eqn1': time = 0.01, bytes = 3418 O78 := eqn2:= 'eqn2': time = 0.00, bytes = 3446 O79 := eqn3:= 'eqn3': time = 0.01, bytes = 3302 # ---------- Quit ---------- O80 := quit bytes used=16595352, alloc=3210676, time=23.77 real 25.29 user 23.83 sys 1.41