Tue Feb 23 17:02:19 MST 1999 aquarius% macsyma Starting Macsyma math engine with no window system... /opt/local/macsyma_422/macsyma.422 local This is Macsyma 422.0 for Sparc (Solaris 2.x) computers. Copyright (c) 1982 - 1998 Macsyma Inc. All rights reserved. Portions copyright (c) 1982 Massachusetts Institute of Technology. All rights reserved. Type "DESCRIBE(TRADE_SECRET);" to see important legal notices. Type "HELP();" for more information. /aquarius/data2/opt/local/macsyma_422/system/init.lsp being loaded. /aquarius/home/wester/macsyma-init.lsp being loaded. (c1) (c2) /* ----------[ M a c s y m a ]---------- */ /* ---------- Initialization ---------- */ symbol_display_case: lower_case$ Time= 0 msecs (c3) showtime: all$ Time= 0 msecs (c4) prederror: false$ Time= 0 msecs (c5) /* ---------- Equations ---------- */ /* Manipulate an equation using a natural syntax: (x = 2)/2 + (1 = 1) => x/2 + 1 = 2 */ (x = 2)/2 + (1 = 1); Time= 0 msecs x (d5) - + 1 = 2 2 (c6) /* Solve various nonlinear equations---this cubic polynomial has all real roots */ solve(3*x^3 - 18*x^2 + 33*x - 19 = 0, x); Time= 40 msecs sqrt(3) %i 1 ---------- - - 2 2 %i 1 1/3 sqrt(3) %i 1 (d6) [x = -------------------- + (--------- + -) (- ---------- - -) + 2, %i 1 1/3 6 sqrt(3) 6 2 2 3 (--------- + -) 6 sqrt(3) 6 sqrt(3) %i 1 - ---------- - - %i 1 1/3 sqrt(3) %i 1 2 2 x = (--------- + -) (---------- - -) + -------------------- + 2, 6 sqrt(3) 6 2 2 %i 1 1/3 3 (--------- + -) 6 sqrt(3) 6 %i 1 1/3 1 x = (--------- + -) + -------------------- + 2] 6 sqrt(3) 6 %i 1 1/3 3 (--------- + -) 6 sqrt(3) 6 (c7) ratsimp(rectform(%)); Time= 920 msecs %pi %pi sqrt(3) sin(---) - cos(---) + 2 sqrt(3) 18 18 (d7) [x = ---------------------------------------, sqrt(3) %pi %pi %pi sqrt(3) sin(---) + cos(---) - 2 sqrt(3) 2 cos(---) + 2 sqrt(3) 18 18 18 x = - ---------------------------------------, x = ----------------------] sqrt(3) sqrt(3) (c8) /* 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} */ eqn: x^4 + x^3 + x^2 + x + 1 = 0; Time= 0 msecs 4 3 2 (d8) x + x + x + x + 1 = 0 (c9) solve(eqn, x); Time= 300 msecs 1/4 5 sqrt(sqrt(5) - 1) %i sqrt(5) 1 (d9) [x = - ------------------------- - ------- - -, 2 sqrt(2) 4 4 1/4 5 sqrt(sqrt(5) - 1) %i sqrt(5) 1 x = ------------------------- - ------- - -, 2 sqrt(2) 4 4 1/4 5 sqrt(sqrt(5) + 1) %i sqrt(5) 1 x = - ------------------------- + ------- - -, 2 sqrt(2) 4 4 1/4 5 sqrt(sqrt(5) + 1) %i sqrt(5) 1 x = ------------------------- + ------- - -] 2 sqrt(2) 4 4 (c10) /* Check one of the answers */ ev(eqn, %[1]); Time= 0 msecs 1/4 5 sqrt(sqrt(5) - 1) %i sqrt(5) 1 4 (d10) (- ------------------------- - ------- - -) 2 sqrt(2) 4 4 1/4 5 sqrt(sqrt(5) - 1) %i sqrt(5) 1 3 + (- ------------------------- - ------- - -) 2 sqrt(2) 4 4 1/4 1/4 5 sqrt(sqrt(5) - 1) %i sqrt(5) 1 2 5 sqrt(sqrt(5) - 1) %i + (- ------------------------- - ------- - -) - ------------------------- 2 sqrt(2) 4 4 2 sqrt(2) sqrt(5) 3 - ------- + - = 0 4 4 (c11) radcan(%); Time= 1050 msecs (d11) 0 = 0 (c12) remvalue(eqn)$ Time= 0 msecs (c13) /* x = {2^(1/3) +- sqrt(3), +- sqrt(3) - 1/2^(2/3) +- i sqrt(3)/2^(2/3)} [Mohamed Omar Rayes] */ solve(x^6 - 9*x^4 - 4*x^3 + 27*x^2 - 36*x - 23 = 0, x); /aquarius/data2/opt/local/macsyma_422/library1/combin.so being loaded. Time= 440 msecs 6 4 3 2 (d13) [0 = x - 9 x - 4 x + 27 x - 36 x - 23] (c14) /* x = {1, e^(+- 2 pi i/7), e^(+- 4 pi i/7), e^(+- 6 pi i/7)} */ solve(x^7 - 1 = 0, x); Time= 680 msecs 2 %i %pi 4 %i %pi 6 %i %pi %i %pi -------- -------- -------- ------ 7 7 7 7 (d14) [x = %e , x = %e , x = %e , x = - %e , 3 %i %pi 5 %i %pi -------- -------- 7 7 x = - %e , x = - %e , x = 1] (c15) /* x = 1 +- sqrt(+-sqrt(+-4 sqrt(3) - 3) - 3)/sqrt(2) [Richard Liska] */ solve(x^8 - 8*x^7 + 34*x^6 - 92*x^5 + 175*x^4 - 236*x^3 + 226*x^2 - 140*x + 46 = 0, x); Time= 1660 msecs sqrt(- sqrt(4 sqrt(3) + 3) %i - 3) - sqrt(2) (d15) [x = - --------------------------------------------, sqrt(2) sqrt(- sqrt(4 sqrt(3) + 3) %i - 3) + sqrt(2) x = --------------------------------------------, sqrt(2) sqrt(sqrt(4 sqrt(3) + 3) %i - 3) - sqrt(2) x = - ------------------------------------------, sqrt(2) sqrt(sqrt(4 sqrt(3) + 3) %i - 3) + sqrt(2) x = ------------------------------------------, sqrt(2) sqrt(sqrt(4 sqrt(3) - 3) + 3) %i - sqrt(2) x = - ------------------------------------------, sqrt(2) sqrt(sqrt(4 sqrt(3) - 3) + 3) %i + sqrt(2) x = ------------------------------------------, sqrt(2) sqrt(3 - sqrt(4 sqrt(3) - 3)) %i - sqrt(2) x = - ------------------------------------------, sqrt(2) sqrt(3 - sqrt(4 sqrt(3) - 3)) %i + sqrt(2) x = ------------------------------------------] sqrt(2) (c16) /* 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] */ %e^(2*x) + 2*%e^x + 1 = z; Time= 0 msecs 2 x x (d16) %e + 2 %e + 1 = z (c17) solve(%, x); Time= 140 msecs (d17) [x = log(sqrt(z) + 1) + 2 %i %pi %n2 + %i %pi, x = log(sqrt(z) - 1) + 2 %i %pi %n1] (c18) /* x = (1 +- sqrt(9 - 8 n pi i))/2. Real solutions correspond to n = 0 => x = {-1, 2} */ solve(exp(2 - x^2) = exp(-x), x); Time= 40 msecs sqrt(9 - 8 %i %pi %n3) - 1 sqrt(9 - 8 %i %pi %n3) + 1 (d18) [x = - --------------------------, x = --------------------------] 2 2 (c19) /* 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 */ solve(exp(x) = x, x); /aquarius/data2/opt/local/macsyma_422/library1/lambertw.so being loaded. Time= 150 msecs (d19) [x = - lambert_w(- 1)] (c20) /* x = {-1, 1} */ solve(x^x = x, x); Time= 700 msecs x (d20) [x = x ] (c21) /* 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}} */ (x + 1) * (sin(x)^2 + 1)^2 * cos(3*x)^3 = 0; Time= 10 msecs 2 2 3 (d21) (x + 1) (sin (x) + 1) cos (3 x) = 0 (c22) solve(%, x); Time= 890 msecs 12 %pi %n11 + 5 %pi 12 %pi %n10 - 5 %pi 12 %pi %n9 + %pi (d22) [x = -------------------, x = -------------------, x = ----------------, 6 6 6 12 %pi %n8 - %pi 4 %pi %n7 - %pi 4 %pi %n6 + %pi x = ----------------, x = ---------------, x = ---------------, 6 2 2 %n5 %n4 x = %pi %n5 - %i asinh(1) (- 1) , x = %i asinh(1) (- 1) + %pi %n4, x = - 1] (c23) multiplicities; Time= 0 msecs (d23) [3, 3, 3, 3, 3, 3, 2, 2, 1] (c24) /* x = pi/4 [+ n pi] */ solve(sin(x) = cos(x), x); Time= 110 msecs 8 %pi %n13 - 3 %pi 8 %pi %n12 + %pi (d24) [x = ------------------, x = ----------------] 4 4 (c25) solve(tan(x) = 1, x); Time= 20 msecs 4 %pi %n14 + %pi (d25) [x = ----------------] 4 (c26) /* x = {pi/6, 5 pi/6} [ + n 2 pi, + n 2 pi ] */ solve(sin(x) = 1/2, x); Time= 80 msecs %n15 %pi (- 1) + 6 %pi %n15 (d26) [x = --------------------------] 6 (c27) /* x = {0, 0} [+ n pi, + n 2 pi] */ solve(sin(x) = tan(x), x); Time= 30 msecs (d27) [x = %pi %n17, x = 2 %pi %n16] (c28) multiplicities; Time= 0 msecs (d28) [1, 1] (c29) /* x = {0, 0, 0} */ solve(asin(x) = atan(x), x); Time= 140 msecs (d29) [x = 0] (c30) multiplicities; Time= 0 msecs (d30) [4] (c31) /* x = sqrt[(sqrt(5) - 1)/2] */ solve(acos(x) = atan(x), x); Time= 1410 msecs sqrt(sqrt(5) - 1) (d31) [x = -----------------] sqrt(2) (c32) /* x = 2 */ solve((x - 2)/x^(1/3) = 0, x); Time= 10 msecs (d32) [x = 2] (c33) /* This equation has no solutions */ solve(sqrt(x^2 + 1) = x - 2, x); Time= 60 msecs (d33) [] (c34) /* x = 1 */ solve(x + sqrt(x) = 2, x); Time= 90 msecs (d34) [x = 1] (c35) /* x = 1/16 */ solve(2*sqrt(x) + 3*x^(1/4) - 2 = 0, x); Time= 10 msecs 1/4 3 x - 2 (d35) [sqrt(x) = - ----------] 2 (c36) /* x = {sqrt[(sqrt(5) - 1)/2], -i sqrt[(sqrt(5) + 1)/2]} */ solve(x = 1/sqrt(1 + x^2), x); Time= 580 msecs sqrt(sqrt(5) - 1) sqrt(sqrt(5) + 1) %i (d36) [x = -----------------, x = - --------------------] sqrt(2) sqrt(2) (c37) /* This problem is from a computational biology talk => 1 - log_2[m (m - 1)] */ solve(binomial(m, 2)*2^k = 1, k); /aquarius/data2/opt/local/macsyma_422/library1/binoml.so being loaded. Time= 200 msecs 1 log(------) + 2 %i %pi %n18 + log(2) 2 m - m (d37) [k = ------------------------------------] log(2) (c38) /* x = log(c/a) / log(b/d) for a, b, c, d != 0 and b, d != 1 [Bill Pletsch] */ solve(a*b^x = c*d^x, x); Time= 40 msecs log(c) - log(a) (d38) [x = - ---------------] log(d) - log(b) (c39) logcontract(%); Time= 10 msecs a log(-) c (d39) [x = ------] d log(-) b (c40) /* x = {1, e^4} */ errcatch(solve(sqrt(log(x)) = log(sqrt(x)), x)); Division by 0 Time= 20 msecs (d40) [] (c41) /* Recursive use of inverses, including multiple branches of rational fractional powers [Richard Liska] => x = +-(b + sin(1 + cos(1/e^2)))^(3/2) */ assume(sin(cos(1/%e^2) + 1) + b > 0)$ Time= 1600 msecs (c42) solve(log(acos(asin(x^(2/3) - b) - 1)) + 2 = 0, x); Time= 3980 msecs - 2 3/2 (d42) [x = (b + sin(cos(%e ) + 1)) ] (c43) forget(sin(cos(1/%e^2) + 1) + b > 0)$ Time= 10 msecs (c44) /* 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. */ eqn: 5*x + exp((x - 5)/2) = 8*x^3; Time= 10 msecs x - 5 ----- 2 3 (d44) 5 x + %e = 8 x (c45) solve(eqn, x); Time= 3890 msecs x - 5 ----- x - 5 2 sqrt(3) %i 1 sqrt(54 %e - 125) %e 1/3 (d45) [x = (- ---------- - -) (---------------------- + -------) 2 2 48 sqrt(6) 16 sqrt(3) %i 1 5 (---------- - -) 2 2 + ----------------------------------------, x - 5 ----- x - 5 2 sqrt(54 %e - 125) %e 1/3 24 (---------------------- + -------) 48 sqrt(6) 16 x - 5 ----- x - 5 2 sqrt(3) %i 1 sqrt(54 %e - 125) %e 1/3 x = (---------- - -) (---------------------- + -------) 2 2 48 sqrt(6) 16 sqrt(3) %i 1 5 (- ---------- - -) 2 2 + ----------------------------------------, x - 5 ----- x - 5 2 sqrt(54 %e - 125) %e 1/3 24 (---------------------- + -------) 48 sqrt(6) 16 x - 5 ----- x - 5 2 sqrt(54 %e - 125) %e 1/3 x = (---------------------- + -------) 48 sqrt(6) 16 5 + ----------------------------------------] x - 5 ----- x - 5 2 sqrt(54 %e - 125) %e 1/3 24 (---------------------- + -------) 48 sqrt(6) 16 (c46) root_by_bisection(eqn, x, -1, -0.5); /aquarius/data2/opt/local/macsyma_422/share/bisect.so being loaded. Time= 2120 msecs (d46) - 0.7849662 (c47) root_by_bisection(eqn, x, -0.5, 0.5); Time= 2070 msecs (d47) - 0.01629074 (c48) root_by_bisection(eqn, x, 0.5, 1); Time= 10 msecs (d48) 0.8025567 (c49) remvalue(eqn)$ Time= 0 msecs (c50) /* x = {-1, 3} */ solve(abs(x - 1) = 2, x); Time= 20 msecs (d50) [x = 3, x = - 1] (c51) /* x = {-1, -7} */ solve(abs(2*x + 5) = abs(x - 2), x); Time= 40 msecs (d51) [x = - 7, x = - 1] (c52) /* x = +-3/2 */ solve(1 - abs(x) = max(-x - 2, x - 2), x); Time= 30 msecs (d52) [abs(x) = 1 - max(x - 2, - x - 2)] (c53) /* x = {-1, 3} */ solve(max(2 - x^2, x) = max(-x, x^3/9), x); Time= 370 msecs 3 x 2 (d53) [max(--, - x) = max(2 - x , x)] 9 (c54) /* 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). */ solve(max(2 - x^2, x) = x^3/9, x); Time= 190 msecs 1/3 1/3 1/3 2 (sqrt(3) 9 %i - 9 ) max (2 - x , x) (d54) [x = ------------------------------------------, 2 1/3 1/3 1/3 2 (sqrt(3) 9 %i + 9 ) max (2 - x , x) 1/3 1/3 2 x = - ------------------------------------------, x = 9 max (2 - x , x)] 2 (c55) /* z = 2 + 3 i */ declare(z, complex)$ Time= 10 msecs (c56) eqn: (1 + %i)*z + (2 - %i)*conjugate(z) = -3*%i; Time= 10 msecs (d56) (2 - %i) z*^ + (%i + 1) z = - 3 %i (c57) solve(eqn, z); Time= 20 msecs (%i - 2) z*^ - 3 %i (d57) [z = -------------------] %i + 1 (c58) declare([x, y], real)$ Time= 0 msecs (c59) subst(z = x + %i*y, eqn); Time= 0 msecs (d59) (2 - %i) (%i y + x)*^ + (%i + 1) (%i y + x) = - 3 %i (c60) ratsimp(ev(%, conjugate)); Time= 10 msecs (d60) (- %i - 2) y + 3 x = - 3 %i (c61) solve(%, [x, y]); Time= 20 msecs 2 %r1 + %i (%r1 - 3) (d61) [[x = --------------------, y = %r1]] 3 (c62) remove([x, y], real, z, complex)$ Time= 0 msecs (c63) remvalue(eqn)$ Time= 0 msecs (c64) /* => {f^(-1)(1), f^(-1)(-2)} assuming f is invertible */ solve(f(x)^2 + f(x) - 2 = 0, x); Time= 20 msecs (d64) [f(x) = 1, f(x) = - 2] (c65) remvalue(eqns, vars)$ Time= 0 msecs (c66) /* Solve a 3 x 3 system of linear equations */ eqn1: x + y + z = 6; Time= 0 msecs (d66) z + y + x = 6 (c67) eqn2: 2*x + y + 2*z = 10; Time= 0 msecs (d67) 2 z + y + 2 x = 10 (c68) eqn3: x + 3*y + z = 10; Time= 0 msecs (d68) z + 3 y + x = 10 (c69) /* Note that the solution is parametric: x = 4 - z, y = 2 */ solve([eqn1, eqn2, eqn3], [x, y, z]); Dependent equations eliminated: (3) Time= 20 msecs (d69) [[x = 4 - %r2, y = 2, z = %r2]] (c70) /* 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 */ eqns: [ -b*k8/a+c*k8/a = 0, -b*k11/a+c*k11/a = 0, -b*k10/a+c*k10/a+k2 = 0, -k3-b*k9/a+c*k9/a = 0, -b*k14/a+c*k14/a = 0, -b*k15/a+c*k15/a = 0, -b*k18/a+c*k18/a-k2 = 0, -b*k17/a+c*k17/a = 0, -b*k16/a+c*k16/a+k4 = 0, -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, -b*k45/a+c*k45/a = 0, -b*k20/a+c*k20/a = 0, -b*k44/a+c*k44/a = 0, 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, -k4 = 0, -b*k12/a+c*k12/a-a*k6/b+c*k6/b = 0, -b*k19/a+c*k19/a+a*k7/c-b*k7/c = 0, b*k45/a-c*k45/a = 0, -b*k46/a+c*k46/a = 0, -k48+c*k48/a+c*k48/b-c^2*k48/(a*b) = 0, -k49+b*k49/a+b*k49/c-b^2*k49/(a*c) = 0, a*k1/b-c*k1/b = 0, a*k4/b-c*k4/b = 0, a*k3/b-c*k3/b+k9 = 0, -k10+a*k2/b-c*k2/b = 0, a*k7/b-c*k7/b = 0, -k9 = 0, k11 = 0, b*k12/a-c*k12/a+a*k6/b-c*k6/b = 0, a*k15/b-c*k15/b = 0, k10+a*k18/b-c*k18/b = 0, -k11+a*k17/b-c*k17/b = 0, 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, -a*k44/b+c*k44/b = 0, a*k45/b-c*k45/b = 0, a*k14/c-b*k14/c+a*k20/b-c*k20/b = 0, a*k44/b-c*k44/b = 0, -a*k46/b+c*k46/b = 0, -k47+c*k47/a+c*k47/b-c^2*k47/(a*b) = 0, a*k19/b-c*k19/b = 0, -a*k45/b+c*k45/b = 0, a*k46/b-c*k46/b = 0, a^2*k48/b^2-2*a*c*k48/b^2+c^2*k48/b^2 = 0, -k49+a*k49/b+a*k49/c-a^2*k49/(b*c) = 0, k16 = 0, -k17 = 0, -a*k1/c+b*k1/c = 0, -k16-a*k4/c+b*k4/c = 0, -a*k3/c+b*k3/c = 0, k18-a*k2/c+b*k2/c = 0, b*k19/a-c*k19/a-a*k7/c+b*k7/c = 0, -a*k6/c+b*k6/c = 0, -a*k8/c+b*k8/c = 0, -a*k11/c+b*k11/c+k17 = 0, -a*k10/c+b*k10/c-k18 = 0, -a*k9/c+b*k9/c = 0, -a*k14/c+b*k14/c-a*k20/b+c*k20/b = 0, -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, -a*k45/c+b*k45/c = 0, -a*k44/c+b*k44/c = 0, a*k46/c-b*k46/c = 0, -k47+b*k47/a+b*k47/c-b^2*k47/(a*c) = 0, -a*k12/c+b*k12/c = 0, a*k45/c-b*k45/c = 0, -a*k46/c+b*k46/c = 0, -k48+a*k48/b+a*k48/c-a^2*k48/(b*c) = 0, a^2*k49/c^2-2*a*b*k49/c^2+b^2*k49/c^2 = 0, k8 = 0, k11 = 0, -k15 = 0, k10-k18 = 0, -k17 = 0, k9 = 0, -k16 = 0, -k29 = 0, k14-k32 = 0, -k21+k23-k31 = 0, -k24-k30 = 0, -k35 = 0, k44 = 0, -k45 = 0, k36 = 0, k13-k23+k39 = 0, -k20+k38 = 0, k25+k37 = 0, b*k26/a-c*k26/a-k34+k42 = 0, -2*k44 = 0, k45 = 0, k46 = 0, b*k47/a-c*k47/a = 0, k41 = 0, k44 = 0, -k46 = 0, -b*k47/a+c*k47/a = 0, k12+k24 = 0, -k19-k25 = 0, -a*k27/b+c*k27/b-k33 = 0, k45 = 0, -k46 = 0, -a*k48/b+c*k48/b = 0, a*k28/c-b*k28/c+k40 = 0, -k45 = 0, k46 = 0, a*k48/b-c*k48/b = 0, a*k49/c-b*k49/c = 0, -a*k49/c+b*k49/c = 0, -k1 = 0, -k4 = 0, -k3 = 0, k15 = 0, k18-k2 = 0, k17 = 0, k16 = 0, k22 = 0, k25-k7 = 0, k24+k30 = 0, k21+k23-k31 = 0, k28 = 0, -k44 = 0, k45 = 0, -k30-k6 = 0, k20+k32 = 0, k27+b*k33/a-c*k33/a = 0, k44 = 0, -k46 = 0, -b*k47/a+c*k47/a = 0, -k36 = 0, k31-k39-k5 = 0, -k32-k38 = 0, k19-k37 = 0, k26-a*k34/b+c*k34/b-k42 = 0, k44 = 0, -2*k45 = 0, k46 = 0, a*k48/b-c*k48/b = 0, a*k35/c-b*k35/c-k41 = 0, -k44 = 0, k46 = 0, b*k47/a-c*k47/a = 0, -a*k49/c+b*k49/c = 0, -k40 = 0, k45 = 0, -k46 = 0, -a*k48/b+c*k48/b = 0, a*k49/c-b*k49/c = 0, k1 = 0, k4 = 0, k3 = 0, -k8 = 0, -k11 = 0, -k10+k2 = 0, -k9 = 0, k37+k7 = 0, -k14-k38 = 0, -k22 = 0, -k25-k37 = 0, -k24+k6 = 0, -k13-k23+k39 = 0, -k28+b*k40/a-c*k40/a = 0, k44 = 0, -k45 = 0, -k27 = 0, -k44 = 0, k46 = 0, b*k47/a-c*k47/a = 0, k29 = 0, k32+k38 = 0, k31-k39+k5 = 0, -k12+k30 = 0, k35-a*k41/b+c*k41/b = 0, -k44 = 0, k45 = 0, -k26+k34+a*k42/c-b*k42/c = 0, k44 = 0, k45 = 0, -2*k46 = 0, -b*k47/a+c*k47/a = 0, -a*k48/b+c*k48/b = 0, a*k49/c-b*k49/c = 0, k33 = 0, -k45 = 0, k46 = 0, a*k48/b-c*k48/b = 0, -a*k49/c+b*k49/c = 0 ]$ Time= 210 msecs (c71) vars: [k1, k2, k3, k4, k5, k6, k7, k8, k9, k10, k11, k12, k13, k14, k15, k16, k17, k18, k19, k20, k21, k22, k23, k24, k25, k26, k27, k28, k29, k30, k31, k32, k33, k34, k35, k36, k37, k38, k39, k40, k41, k42, k43, k44, k45, k46, k47, k48, k49]$ Time= 0 msecs (c72) solve(eqns, vars); Dependent equations eliminated: (53 114 153 26 113 152 12 21 39 45 180 64 69 177 86 93 103 107 125 138 147 166 27 4 60 78 157 11 14 38 41 179 165 63 65 176 85 92 97 124 129 137 142 2 155 74 119 15 169 22 42 139 46 148 66 70 187 94 98 104 130 108 143 36 9 52 79 118 80 8 35 58 77 117 1 57 73 6 33 75 25 51 151 128 102 106 164 132 141 175 121 83 101 120 90 172 81 89 116 100 10 62 37 133 109 47 149 188 140 23 71 105 55 29 20 43 144 95 131 182 16 99 67 40 61 5 184 72 48 145 24 150 111 189 3 59 54 28 34 19 56 32 136) Time= 49350 msecs (d72) [[k1 = 0, k2 = 0, k3 = 0, k4 = 0, k5 = 0, k6 = 0, k7 = 0, k8 = 0, k9 = 0, k10 = 0, k11 = 0, k12 = 0, k13 = 0, k14 = 0, k15 = 0, k16 = 0, k17 = 0, k18 = 0, k19 = 0, k20 = 0, k21 = 0, k22 = 0, k23 = %r4, k24 = 0, k25 = 0, %r3 a k26 = -----, k27 = 0, k28 = 0, k29 = 0, k30 = 0, k31 = %r4, k32 = 0, k33 = 0, c %r3 b k34 = -----, k35 = 0, k36 = 0, k37 = 0, k38 = 0, k39 = %r4, k40 = 0, k41 = 0, c k42 = %r3, k43 = %r5, k44 = 0, k45 = 0, k46 = 0, k47 = 0, k48 = 0, k49 = 0]] (c73) /* Solve a 3 x 3 system of nonlinear equations */ eqn1: x^2*y + 3*y*z - 4 = 0; Time= 10 msecs 2 (d73) 3 y z + x y - 4 = 0 (c74) eqn2: -3*x^2*z + 2*y^2 + 1 = 0; Time= 0 msecs 2 2 (d74) - 3 x z + 2 y + 1 = 0 (c75) eqn3: 2*y*z^2 - z^2 - 1 = 0; Time= 0 msecs 2 2 (d75) 2 y z - z - 1 = 0 (c76) /* Solving this by hand would be a nightmare */ solve([eqn1, eqn2, eqn3], [x, y, z]); /aquarius/data2/opt/local/macsyma_422/library1/algsys.so being loaded. /aquarius/data2/opt/local/macsyma_422/library1/triangsy.so being loaded. Time= 3970 msecs sqrt(2) %i + 1 sqrt(- 2 %i - sqrt(2)) (d76) [[y = sqrt(2) %i, z = ----------------, x = ----------------------], 2 sqrt(2) %i - 1 1/4 2 sqrt(2) %i + 1 sqrt(- 2 %i - sqrt(2)) [y = sqrt(2) %i, z = ----------------, x = - ----------------------], 2 sqrt(2) %i - 1 1/4 2 sqrt(2) %i - 1 sqrt(2 %i - sqrt(2)) [y = - sqrt(2) %i, z = ----------------, x = --------------------], 2 sqrt(2) %i + 1 1/4 2 sqrt(2) %i - 1 sqrt(2 %i - sqrt(2)) [y = - sqrt(2) %i, z = ----------------, x = - --------------------], 2 sqrt(2) %i + 1 1/4 2 [y = 1, z = 1, x = 1], [y = 1, z = 1, x = - 1], 4 3 2 5 3 2 2 y - y + y + 4 y [y = root_of(4 y - 3 y + 19 y + 18 y - 18), z = --------------------, 12 y - 6 5 4 3 2 2 y - y + y + 4 y - 16 y + 8 sqrt(--------------------------------) 2 y - 2 y x = --------------------------------------], sqrt(2) 4 3 2 5 3 2 2 y - y + y + 4 y [y = root_of(4 y - 3 y + 19 y + 18 y - 18), z = --------------------, 12 y - 6 5 4 3 2 2 y - y + y + 4 y - 16 y + 8 sqrt(--------------------------------) 2 y - 2 y x = - --------------------------------------]] sqrt(2) (c77) rectform(sfloat(%[1..5])); Time= 30 msecs (d77) [[y = 1.414214 %i, z = 0.3333333 - 0.4714045 %i, x = 0.6050004 - 1.168771 %i], [y = 1.414214 %i, z = 0.3333333 - 0.4714045 %i, x = 1.168771 %i - 0.6050004], [y = - 1.414214 %i, z = 0.4714045 %i + 0.3333333, x = 1.168771 %i + 0.6050004], [y = - 1.414214 %i, z = 0.4714045 %i + 0.3333333, x = - 1.168771 %i - 0.6050004], [y = 1.0, z = 1.0, x = 1.0]] (c78) remvalue(eqn1, eqn2, eqn3)$ Time= 0 msecs (c79) /* ---------- Quit ---------- */ quit(); Bye. real 83.94 user 80.21 sys 1.18