Sat Jan 31 10:56:28 MST 1998 aquarius% reduce REDUCE 3.6, 15-Jul-95, patched to 15 Apr 96 ... 1: % ----------[ R e d u c e ]---------- % ---------- Initialization ---------- on multiplicities; on time; Time: 0 ms % ---------- Equations ---------- % Manipulate an equation using a natural syntax: % (x = 2)/2 + (1 = 1) => x/2 + 1 = 2 (x = 2)/2 + (1 = 1); x + 2 -------=2 2 Time: 10 ms % Solve various nonlinear equations---this cubic polynomial has all real roots on fullroots; Time: 0 ms solve(3*x^3 - 18*x^2 + 33*x - 19 = 0, x); pi 2*(cos(----) + sqrt(3)) 18 {x=-------------------------, sqrt(3) pi pi - cos(----) + sqrt(3)*sin(----) + 2*sqrt(3) 18 18 x=----------------------------------------------, sqrt(3) pi pi - cos(----) - sqrt(3)*sin(----) + 2*sqrt(3) 18 18 x=----------------------------------------------} sqrt(3) Time: 370 ms % 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; 4 3 2 eqn := x + x + x + x + 1=0 Time: 0 ms solve(ws, x); 2*sqrt( - sqrt(5) - 5) + sqrt(10) - sqrt(2) {x=---------------------------------------------, 4*sqrt(2) - 2*sqrt( - sqrt(5) - 5) + sqrt(10) - sqrt(2) x=------------------------------------------------, 4*sqrt(2) 2*sqrt(sqrt(5) - 5) - sqrt(10) - sqrt(2) x=------------------------------------------, 4*sqrt(2) - 2*sqrt(sqrt(5) - 5) - sqrt(10) - sqrt(2) x=---------------------------------------------} 4*sqrt(2) Time: 80 ms % Check one of the answers sub(first(ws), part(eqn, 1)); 0 Time: 20 ms off fullroots; Time: 0 ms clear eqn; Time: 0 ms % 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); 6 4 3 2 {x=root_of(x_ - 9*x_ - 4*x_ + 27*x_ - 36*x_ - 23,x_,tag_1)} Time: 50 ms % x = {1, e^(+- 2 pi i/7), e^(+- 4 pi i/7), e^(+- 6 pi i/7)} solve(x^7 - 1 = 0, x); 6 5 4 3 2 {x=root_of(x_ + x_ + x_ + x_ + x_ + x_ + 1,x_,tag_2), x=1} Time: 20 ms % 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); sqrt( - sqrt( - 4*sqrt(3) - 3) - 3)*sqrt(2) + 2 {x=-------------------------------------------------, 2 - sqrt( - sqrt( - 4*sqrt(3) - 3) - 3)*sqrt(2) + 2 x=----------------------------------------------------, 2 sqrt( - sqrt(4*sqrt(3) - 3) - 3)*sqrt(2) + 2 x=----------------------------------------------, 2 - sqrt( - sqrt(4*sqrt(3) - 3) - 3)*sqrt(2) + 2 x=-------------------------------------------------, 2 sqrt(sqrt( - 4*sqrt(3) - 3) - 3)*sqrt(2) + 2 x=----------------------------------------------, 2 - sqrt(sqrt( - 4*sqrt(3) - 3) - 3)*sqrt(2) + 2 x=-------------------------------------------------, 2 sqrt(sqrt(4*sqrt(3) - 3) - 3)*sqrt(2) + 2 x=-------------------------------------------, 2 - sqrt(sqrt(4*sqrt(3) - 3) - 3)*sqrt(2) + 2 x=----------------------------------------------} 2 Time: 300 ms % 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; 2*x x e + 2*e + 1=z Time: 0 ms solve(ws, x); {x=2*arbint(2)*i*pi + log( - sqrt(z) - 1), x=2*arbint(1)*i*pi + log(sqrt(z) - 1)} Time: 40 ms % 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); 2 x x 2 { - e + e *e =0} Time: 1470 ms plus GC time: 80 ms % 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); {x= - lambert_w(-1)} Time: 190 ms plus GC time: 50 ms % x = {-1, 1} solve(x^x = x, x); x_ {x=root_of( - x_ + x_,x_,tag_6)} Time: 120 ms % 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}} load_package(assist)$ Time: 80 ms (x + 1) * (sin(x)^2 + 1)^2 * cos(3*x)^3 = 0; 3 4 3 4 3 2 cos(3*x) *sin(x) *x + cos(3*x) *sin(x) + 2*cos(3*x) *sin(x) *x 3 2 3 3 + 2*cos(3*x) *sin(x) + cos(3*x) *x + cos(3*x) =0 Time: 10 ms solve(ws, x); pi*(4*arbint(12) + 1) {x=-----------------------, 6 pi*(4*arbint(12) + 1) x=-----------------------, 6 pi*(4*arbint(12) + 1) x=-----------------------, 6 pi*(4*arbint(12) - 1) x=-----------------------, 6 pi*(4*arbint(12) - 1) x=-----------------------, 6 pi*(4*arbint(12) - 1) x=-----------------------, 6 x=2*arbint(11)*pi + asinh(1)*i + pi, x=2*arbint(11)*pi + asinh(1)*i + pi, x=2*arbint(11)*pi - asinh(1)*i, x=2*arbint(11)*pi - asinh(1)*i, x=2*arbint(10)*pi + asinh(1)*i, x=2*arbint(10)*pi + asinh(1)*i, x=2*arbint(10)*pi - asinh(1)*i + pi, x=2*arbint(10)*pi - asinh(1)*i + pi, x=-1} Time: 70 ms frequency(ws); 4*arbint(12)*pi + pi {{x=----------------------,3}, 6 4*arbint(12)*pi - pi {x=----------------------,3}, 6 {x=2*arbint(11)*pi + asinh(1)*i + pi,2}, {x=2*arbint(11)*pi - asinh(1)*i,2}, {x=2*arbint(10)*pi + asinh(1)*i,2}, {x=2*arbint(10)*pi - asinh(1)*i + pi,2}, {x=-1,1}} Time: 290 ms % x = pi/4 [+ n pi] solve(sin(x) = cos(x), x); {x=2*(arbint(13)*pi + atan(sqrt(2) - 1)), x=2*(arbint(13)*pi - atan(sqrt(2) + 1))} Time: 80 ms solve(tan(x) = 1, x); pi*(4*arbint(14) + 1) {x=-----------------------} 4 Time: 10 ms % x = {pi/6, 5 pi/6} [ + n 2 pi, + n 2 pi ] solve(sin(x) = 1/2, x); pi*(12*arbint(15) + 5) pi*(12*arbint(15) + 1) {x=------------------------,x=------------------------} 6 6 Time: 20 ms % x = {0, 0} [+ n pi, + n 2 pi] solve(sin(x) = tan(x), x); {x=pi*(2*arbint(16) + 1), x=pi*(2*arbint(16) - 1), x=2*arbint(16)*pi} Time: 80 ms % x = {0, 0, 0} solve(asin(x) = atan(x), x); {x=one_of(0,0,0)} Time: 80 ms plus GC time: 50 ms % x = sqrt[(sqrt(5) - 1)/2] solve(acos(x) = atan(x), x); sqrt(sqrt(5) - 1) sqrt(sqrt(5) - 1) {x=one_of(-------------------,-------------------)} sqrt(2) sqrt(2) Time: 390 ms % x = 2 solve((x - 2)/x^(1/3) = 0, x); {x=2} Time: 0 ms % This equation has no solutions solve(sqrt(x^2 + 1) = x - 2, x); {} Time: 490 ms plus GC time: 60 ms % x = 1 solve(x + sqrt(x) = 2, x); {x=1} Time: 10 ms % x = 1/16 solve(2*sqrt(x) + 3*x^(1/4) - 2 = 0, x); 1 {x=----} 16 Time: 120 ms % x = {sqrt[(sqrt(5) - 1)/2], -i sqrt[(sqrt(5) + 1)/2]} solve(x = 1/sqrt(1 + x^2), x); sqrt(sqrt(5) - 1) {x=-------------------} sqrt(2) Time: 190 ms % This problem is from a computational biology talk => 1 - log_2[m (m - 1)] load_package(specfn)$ Time: 1280 ms plus GC time: 60 ms solve(Binomial(m, 2)*2^k = 1, k); 1 2*arbint(17)*i*pi + log(---------------) binomial(m,2) {k=------------------------------------------} log(2) Time: 30 ms % 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); x_ x_ {x=root_of(b *a - d *c,x_,tag_11)} Time: 430 ms plus GC time: 80 ms % x = {1, e^4} solve(sqrt(log(x)) = log(sqrt(x)), x); {x=root_of(sqrt(log(x_)) - log(sqrt(x_)),x_,tag_15)} Time: 13550 ms plus GC time: 1550 ms % Recursive use of inverses, including multiple branches of rational % fractional powers [Richard Liska] % => x = +-(b + sin(1 + cos(1/e^2)))^(3/2) solve(log(acos(asin(x^(2/3) - b) - 1)) + 2 = 0, x); 1 1 {x=sqrt(sin(cos(----) + 1) + b)*(sin(cos(----) + 1) + b), 2 2 e e 1 1 x= - sqrt(sin(cos(----) + 1) + b)*(sin(cos(----) + 1) + b)} 2 2 e e Time: 360 ms plus GC time: 80 ms % 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; x/2 2 e + 5*sqrt(e)*e *x 3 eqn := -----------------------=8*x 2 sqrt(e)*e Time: 10 ms solve(eqn, x); x_/2 2 3 2 {x=root_of( - e + 8*sqrt(e)*e *x_ - 5*sqrt(e)*e *x_,x_,tag_17)} Time: 2080 ms plus GC time: 250 ms clear eqn; Time: 10 ms % x = {-1, 3} solve(abs(x - 1) = 2, x); {x=3,x=-1} Time: 10 ms % x = {-1, -7} solve(abs(2*x + 5) = abs(x - 2), x); {x=root_of(abs(2*x_ + 5) - abs(x_ - 2),x_,tag_19)} Time: 50 ms plus GC time: 80 ms % x = +-3/2 solve(1 - abs(x) = max(-x - 2, x - 2), x); {x=root_of(abs(x_) + max( - x_ - 2,x_ - 2) - 1,x_,tag_21)} Time: 40 ms % x = {-1, 3} solve(max(2 - x^2, x) = max(-x, x^3/9), x); 3 x_ 2 {x=root_of(max( - x_,-----) - max( - x_ + 2,x_),x_,tag_23)} 9 Time: 240 ms % 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); 2 3 {x=root_of( - 9*max( - x_ + 2,x_) + x_ ,x_,tag_25)} Time: 240 ms plus GC time: 80 ms % z = 2 + 3 i eqn:= (1 + i)*z + (2 - i)*conj(z) = -3*i; eqn := - 2*impart(z)*i - impart(z) - repart(z)*i + 2*repart(z) + i*z + z= - 3*i Time: 0 ms solve(eqn, z); {z=root_of( - 2*impart(z_)*i - impart(z_) - repart(z_)*i + 2*repart(z_) + i*z_ + 3*i + z_,z_,tag_27)} Time: 40 ms sub(z = x + i*y, eqn); - 2*impart(x)*i - impart(x) + impart(y)*i - 2*impart(y) - repart(x)*i + 2*repart(x) - 2*repart(y)*i - repart(y) + i*x + i*y + x - y= - 3*i Time: 20 ms (1 + i)*(x + i*y) + (2 - i)*(x - i*y) = -3*i; - i*y + 3*x - 2*y= - 3*i Time: 0 ms solve(ws, {x, y}); arbcomplex(20)*i + 2*arbcomplex(20) - 3*i {{x=-------------------------------------------,y=arbcomplex(20)}} 3 Time: 10 ms clear eqn; Time: 0 ms % => {f^(-1)(1), f^(-1)(-2)} assuming f is invertible operator f; Time: 0 ms solve(f(x)^2 + f(x) - 2 = 0, x); {f(x) + 2=0,f(x) - 1=0} Time: 20 ms clear eqns, vars; Time: 0 ms % Solve a 3 x 3 system of linear equations eqn1:= x + y + z - 6; eqn1 := x + y + z - 6 Time: 0 ms eqn2:= 2*x + y + 2*z - 10; eqn2 := 2*x + y + 2*z - 10 Time: 0 ms eqn3:= x + 3*y + z - 10; eqn3 := x + 3*y + z - 10 Time: 0 ms % Note that the solution is parametric: x = 4 - z, y = 2 solve({eqn1, eqn2, eqn3}, {x, y, z}); {{x= - arbcomplex(21) + 4,y=2,z=arbcomplex(21)}} Time: 10 ms % 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: 280 ms 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: 20 ms solve(eqns, vars); {{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=arbcomplex(22), k24=0, k25=0, arbcomplex(23)*a k26=------------------, c k27=0, k28=0, k29=0, k30=0, k31=arbcomplex(22), k32=0, k33=0, arbcomplex(23)*b k34=------------------, c k35=0, k36=0, k37=0, k38=0, k39=arbcomplex(22), k40=0, k41=0, k42=arbcomplex(23), k43=arbcomplex(24), k44=0, k45=0, k46=0, k47=0, k48=0, k49=0}} Time: 35130 ms plus GC time: 33880 ms % Solve a 3 x 3 system of nonlinear equations eqn1:= x^2*y + 3*y*z - 4; 2 eqn1 := x *y + 3*y*z - 4 Time: 0 ms eqn2:= -3*x^2*z + 2*y^2 + 1; 2 2 eqn2 := - 3*x *z + 2*y + 1 Time: 10 ms eqn3:= 2*y*z^2 - z^2 - 1; 2 2 eqn3 := 2*y*z - z - 1 Time: 0 ms % Solving this by hand would be a nightmare solve({eqn1, eqn2, eqn3}, {x, y, z}); sqrt(2)*i + 1 {{z=---------------, 3 y= - sqrt(2)*i, x=sqrt(sqrt(2)*i - 1)}, sqrt(2)*i + 1 {z=---------------, 3 y= - sqrt(2)*i, x= - sqrt(sqrt(2)*i - 1)}, - sqrt(2)*i + 1 {z=------------------, 3 y=sqrt(2)*i, x=sqrt( - sqrt(2)*i - 1)}, - sqrt(2)*i + 1 {z=------------------, 3 y=sqrt(2)*i, x= - sqrt( - sqrt(2)*i - 1)}, 5 4 3 2 {z=root_of(6*z_ - 6*z_ - 9*z_ - 7*z_ - 3*z_ - 1,z_,tag_28), 4 3 2 3*( - 6*z + 8*z + 7*z + 4*z + 1) y=-------------------------------------, 2 3 2 sqrt(z)*sqrt( - 12*z + 12*z + 30*z - 7) x=-------------------------------------------}, sqrt(3) 5 4 3 2 {z=root_of(6*z_ - 6*z_ - 9*z_ - 7*z_ - 3*z_ - 1,z_,tag_28), 4 3 2 3*( - 6*z + 8*z + 7*z + 4*z + 1) y=-------------------------------------, 2 3 2 - sqrt(z)*sqrt( - 12*z + 12*z + 30*z - 7) x=----------------------------------------------}, sqrt(3) {z=1,y=1,x=1}, {z=1,y=1,x=-1}} Time: 3240 ms plus GC time: 450 ms on rounded; Time: 0 ms ws; {{z=0.471404520791*i + 0.333333333333, y= - 1.41421356237*i, 0.5 x=(1.41421356237*i - 1) }, {z=0.471404520791*i + 0.333333333333, y= - 1.41421356237*i, 0.5 x= - (1.41421356237*i - 1) }, {z= - 0.471404520791*i + 0.333333333333, y=1.41421356237*i, 0.5 x=( - 1.41421356237*i - 1) }, {z= - 0.471404520791*i + 0.333333333333, y=1.41421356237*i, 0.5 x= - ( - 1.41421356237*i - 1) }, {z=one_of({ - 0.0701123791218 + 0.501151860893*i, - 0.0701123791218 - 0.501151860893*i, - 0.462659639448 + 0.317887691935*i, - 0.462659639448 - 0.317887691935*i,2.06554403714},tag_28), 4 3 2 y= - 9.0*z + 12.0*z + 10.5*z + 6.0*z + 1.5, 0.5 3 2 0.5 x=0.57735026919*z *( - 12*z + 12*z + 30*z - 7) }, {z=one_of({ - 0.0701123791218 + 0.501151860893*i, - 0.0701123791218 - 0.501151860893*i, - 0.462659639448 + 0.317887691935*i, - 0.462659639448 - 0.317887691935*i,2.06554403714},tag_28), 4 3 2 y= - 9.0*z + 12.0*z + 10.5*z + 6.0*z + 1.5, 0.5 3 2 0.5 x= - 0.57735026919*z *( - 12*z + 12*z + 30*z - 7) }, {z=1,y=1,x=1}, {z=1,y=1,x=-1}} Time: 290 ms plus GC time: 90 ms off rounded; Time: 0 ms clear eqn1, eqn2, eqn3; Time: 0 ms % *** The numerics package causes conflicts with solve, so do numerical % solutions last. *** % 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; x/2 2 e + 5*sqrt(e)*e *x 3 eqn := -----------------------=8*x 2 sqrt(e)*e Time: 10 ms load_package(numeric)$ *** .. redefined Time: 60 ms num_solve(eqn, x = -0.75); {x= - 0.784966146486} Time: 30 ms num_solve(eqn, x = 0); {x= - 0.0162907377299} Time: 30 ms num_solve(eqn, x = 0.75); {x=0.802556701916} Time: 30 ms clear eqn; Time: 10 ms % x = {-1, -7} num_solve(abs(2*x + 5) = abs(x - 2), x); {x= - 7.0} Time: 30 ms % x = +-3/2 num_solve(1 - abs(x) = max(-x - 2, x - 2), x); ***** 74.0 invalid as kernel ***** error during function evaluation (e.g. singularity) Cont? (Y or N) ?y Time: 20 ms % x = {-1, 3} num_solve(max(2 - x^2, x) = max(-x, x^3/9), x); ***** 61.0 invalid as kernel ***** error during function evaluation (e.g. singularity) Cont? (Y or N) ?y Time: 20 ms % 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). num_solve(max(2 - x^2, x) = x^3/9, x); ***** 58.0 invalid as kernel ***** error during function evaluation (e.g. singularity) Cont? (Y or N) ?y Time: 10 ms % ---------- Quit ---------- quit; Quitting real 136.85 user 99.19 sys 3.68