Sat Jan 31 19:01:18 MET 1998 anne % axiom Axiom Computer Algebra System (Release 2.1) Digital Unix on DEC Alpha (AXIOM Sockets) The AXIOM server number is undefined. ----------------------------------------------------------------------------- Issue )copyright to view copyright notices. Issue )summary for a summary of useful system commands. Issue )quit to leave AXIOM and return to shell. ----------------------------------------------------------------------------- initial (1) -> -- ----------[ A x i o m ]---------- -- ---------- Initialization ---------- )set messages autoload off )set messages time on )set quit unprotected -- ---------- Equations ---------- -- Manipulate an equation using a natural syntax: -- (x = 2)/2 + (1 = 1) => x/2 + 1 = 2 (x = 2)/2 + (1 = 1) x + 2 (1) -----= 2 2 Type: Equation Fraction Polynomial Integer Time: 0.55 (IN) + 0.08 (EV) + 0.20 (OT) + 0.08 (GC) = 0.92 sec -- Solve various nonlinear equations---this cubic polynomial has all real roots radicalSolve(3*x**3 - 18*x**2 + 33*x - 19 = 0, x) (2) +----------+2 +----------+ | +---+ | +---+ +---+ |\|- 3 + 1 +---+ |\|- 3 + 1 (- 3\|- 3 + 3) |---------- + (6\|- 3 + 6) |---------- - 2 3| +---+ 3| +---+ \| 6\|- 3 \| 6\|- 3 [x= --------------------------------------------------------------, +----------+ | +---+ +---+ |\|- 3 + 1 (3\|- 3 + 3) |---------- 3| +---+ \| 6\|- 3 +----------+2 +----------+ | +---+ | +---+ +---+ |\|- 3 + 1 +---+ |\|- 3 + 1 (- 3\|- 3 - 3) |---------- + (6\|- 3 - 6) |---------- + 2 3| +---+ 3| +---+ \| 6\|- 3 \| 6\|- 3 x= --------------------------------------------------------------, +----------+ | +---+ +---+ |\|- 3 + 1 (3\|- 3 - 3) |---------- 3| +---+ \| 6\|- 3 +----------+2 +----------+ | +---+ | +---+ |\|- 3 + 1 |\|- 3 + 1 3 |---------- + 6 |---------- + 1 3| +---+ 3| +---+ \| 6\|- 3 \| 6\|- 3 x= ------------------------------------] +----------+ | +---+ |\|- 3 + 1 3 |---------- 3| +---+ \| 6\|- 3 Type: List Equation Expression Integer Time: 0.10 (IN) + 0.55 (EV) + 0.15 (OT) + 0.08 (GC) = 0.88 sec map(e +-> lhs(e) = simplify(complexForm(rhs(e))), %) (3) +-+ %pi %pi +-+ +-+ %pi %pi +-+ - \|3 sin(---) - cos(---) + 2\|3 \|3 sin(---) - cos(---) + 2\|3 18 18 18 18 [x= ---------------------------------, x= -------------------------------, +-+ +-+ \|3 \|3 %pi +-+ 2cos(---) + 2\|3 18 x= -----------------] +-+ \|3 Type: List Equation Expression Integer Time: 0.25 (IN) + 1.62 (EV) + 0.12 (OT) + 0.02 (GC) = 2.0 sec -- 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 (4) x + x + x + x + 1= 0 Type: Equation Polynomial Integer Time: 0.02 (IN) + 0.02 (OT) = 0.03 sec radicalSolve(eqn, x) (5) [ x = - 2 * ROOT +-------------+2 +-------------+ | +---+ | +---+ | | 5 | | 5 |45 |- - - 25 |45 |- - - 25 | \| 3 | \| 3 (- 4 3|------------- - 10 3|------------- - 40) \| 2 \| 2 * +-------------------------------------------+ | +-------------+2 +-------------+ | | +---+ | +---+ | | | 5 | | 5 | |45 |- - - 25 |45 |- - - 25 | | \| 3 | \| 3 |4 3|------------- - 5 3|------------- + 40 | \| 2 \| 2 |------------------------------------------- | +-------------+ | | +---+ | | | 5 | |45 |- - - 25 | | \| 3 | 12 3|------------- \| \| 2 + +-------------+ | +---+ | | 5 |45 |- - - 25 | \| 3 - 15 3|------------- \| 2 / +-------------+ | +---+ | | 5 |45 |- - - 25 | \| 3 12 3|------------- \| 2 * +-------------------------------------------+ | +-------------+2 +-------------+ | | +---+ | +---+ | | | 5 | | 5 | |45 |- - - 25 |45 |- - - 25 | | \| 3 | \| 3 |4 3|------------- - 5 3|------------- + 40 | \| 2 \| 2 |------------------------------------------- | +-------------+ | | +---+ | | | 5 | |45 |- - - 25 | | \| 3 | 12 3|------------- \| \| 2 + +-------------------------------------------+ | +-------------+2 +-------------+ | | +---+ | +---+ | | | 5 | | 5 | |45 |- - - 25 |45 |- - - 25 | | \| 3 | \| 3 |4 3|------------- - 5 3|------------- + 40 | \| 2 \| 2 2 |------------------------------------------- - 1 | +-------------+ | | +---+ | | | 5 | |45 |- - - 25 | | \| 3 | 12 3|------------- \| \| 2 / 4 , x = 2 * ROOT +-------------+2 +-------------+ | +---+ | +---+ | | 5 | | 5 |45 |- - - 25 |45 |- - - 25 | \| 3 | \| 3 (- 4 3|------------- - 10 3|------------- - 40) \| 2 \| 2 * +-------------------------------------------+ | +-------------+2 +-------------+ | | +---+ | +---+ | | | 5 | | 5 | |45 |- - - 25 |45 |- - - 25 | | \| 3 | \| 3 |4 3|------------- - 5 3|------------- + 40 | \| 2 \| 2 |------------------------------------------- | +-------------+ | | +---+ | | | 5 | |45 |- - - 25 | | \| 3 | 12 3|------------- \| \| 2 + +-------------+ | +---+ | | 5 |45 |- - - 25 | \| 3 - 15 3|------------- \| 2 / +-------------+ | +---+ | | 5 |45 |- - - 25 | \| 3 12 3|------------- \| 2 * +-------------------------------------------+ | +-------------+2 +-------------+ | | +---+ | +---+ | | | 5 | | 5 | |45 |- - - 25 |45 |- - - 25 | | \| 3 | \| 3 |4 3|------------- - 5 3|------------- + 40 | \| 2 \| 2 |------------------------------------------- | +-------------+ | | +---+ | | | 5 | |45 |- - - 25 | | \| 3 | 12 3|------------- \| \| 2 + +-------------------------------------------+ | +-------------+2 +-------------+ | | +---+ | +---+ | | | 5 | | 5 | |45 |- - - 25 |45 |- - - 25 | | \| 3 | \| 3 |4 3|------------- - 5 3|------------- + 40 | \| 2 \| 2 2 |------------------------------------------- - 1 | +-------------+ | | +---+ | | | 5 | |45 |- - - 25 | | \| 3 | 12 3|------------- \| \| 2 / 4 , x = - 2 * ROOT +-------------+2 +-------------+ | +---+ | +---+ | | 5 | | 5 |45 |- - - 25 |45 |- - - 25 | \| 3 | \| 3 (- 4 3|------------- - 10 3|------------- - 40) \| 2 \| 2 * +-------------------------------------------+ | +-------------+2 +-------------+ | | +---+ | +---+ | | | 5 | | 5 | |45 |- - - 25 |45 |- - - 25 | | \| 3 | \| 3 |4 3|------------- - 5 3|------------- + 40 | \| 2 \| 2 |------------------------------------------- | +-------------+ | | +---+ | | | 5 | |45 |- - - 25 | | \| 3 | 12 3|------------- \| \| 2 + +-------------+ | +---+ | | 5 |45 |- - - 25 | \| 3 15 3|------------- \| 2 / +-------------+ | +---+ | | 5 |45 |- - - 25 | \| 3 12 3|------------- \| 2 * +-------------------------------------------+ | +-------------+2 +-------------+ | | +---+ | +---+ | | | 5 | | 5 | |45 |- - - 25 |45 |- - - 25 | | \| 3 | \| 3 |4 3|------------- - 5 3|------------- + 40 | \| 2 \| 2 |------------------------------------------- | +-------------+ | | +---+ | | | 5 | |45 |- - - 25 | | \| 3 | 12 3|------------- \| \| 2 + +-------------------------------------------+ | +-------------+2 +-------------+ | | +---+ | +---+ | | | 5 | | 5 | |45 |- - - 25 |45 |- - - 25 | | \| 3 | \| 3 |4 3|------------- - 5 3|------------- + 40 | \| 2 \| 2 - 2 |------------------------------------------- - 1 | +-------------+ | | +---+ | | | 5 | |45 |- - - 25 | | \| 3 | 12 3|------------- \| \| 2 / 4 , x = 2 * ROOT +-------------+2 +-------------+ | +---+ | +---+ | | 5 | | 5 |45 |- - - 25 |45 |- - - 25 | \| 3 | \| 3 (- 4 3|------------- - 10 3|------------- - 40) \| 2 \| 2 * +-------------------------------------------+ | +-------------+2 +-------------+ | | +---+ | +---+ | | | 5 | | 5 | |45 |- - - 25 |45 |- - - 25 | | \| 3 | \| 3 |4 3|------------- - 5 3|------------- + 40 | \| 2 \| 2 |------------------------------------------- | +-------------+ | | +---+ | | | 5 | |45 |- - - 25 | | \| 3 | 12 3|------------- \| \| 2 + +-------------+ | +---+ | | 5 |45 |- - - 25 | \| 3 15 3|------------- \| 2 / +-------------+ | +---+ | | 5 |45 |- - - 25 | \| 3 12 3|------------- \| 2 * +-------------------------------------------+ | +-------------+2 +-------------+ | | +---+ | +---+ | | | 5 | | 5 | |45 |- - - 25 |45 |- - - 25 | | \| 3 | \| 3 |4 3|------------- - 5 3|------------- + 40 | \| 2 \| 2 |------------------------------------------- | +-------------+ | | +---+ | | | 5 | |45 |- - - 25 | | \| 3 | 12 3|------------- \| \| 2 + +-------------------------------------------+ | +-------------+2 +-------------+ | | +---+ | +---+ | | | 5 | | 5 | |45 |- - - 25 |45 |- - - 25 | | \| 3 | \| 3 |4 3|------------- - 5 3|------------- + 40 | \| 2 \| 2 - 2 |------------------------------------------- - 1 | +-------------+ | | +---+ | | | 5 | |45 |- - - 25 | | \| 3 | 12 3|------------- \| \| 2 / 4 ] Type: List Equation Expression Integer Time: 0.05 (EV) + 0.52 (OT) = 0.57 sec -- Check one of the answers eval(eqn, %.1) (6) 0= 0 Type: Equation Expression Integer Time: 0.47 (IN) + 0.58 (EV) + 0.07 (OT) = 1.12 sec )clear properties eqn -- 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 (7) [x - 9x - 4x + 27x - 36x - 23= 0] Type: List Equation Fraction Polynomial Integer Time: 0.23 (IN) + 0.08 (EV) + 0.13 (OT) = 0.45 sec -- 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 (8) [x= 1,x + x + x + x + x + x + 1= 0] Type: List Equation Fraction Polynomial Integer Time: 0.05 (EV) = 0.05 sec -- 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) 8 7 6 5 4 3 2 (9) [x - 8x + 34x - 92x + 175x - 236x + 226x - 140x + 46= 0] Type: List Equation Fraction Polynomial Integer Time: 0.05 (IN) + 0.02 (EV) + 0.05 (OT) = 0.12 sec -- 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 2x x (10) %e + 2%e + 1= z Type: Equation Expression Integer Time: 0.20 (IN) + 0.07 (EV) + 0.05 (OT) = 0.32 sec solve(%, x) +-+ +-+ (11) [x= log(\|z - 1),x= log(- \|z - 1)] Type: List Equation Expression Integer Time: 0.32 (IN) + 0.85 (EV) + 0.12 (OT) + 0.38 (GC) = 1.67 sec -- 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) (12) [] Type: List Equation Expression Integer Time: 0.17 (EV) + 0.02 (OT) = 0.18 sec -- 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) (13) [] Type: List Equation Expression Integer Time: 0.07 (IN) + 0.07 (EV) + 0.02 (OT) = 0.15 sec -- x = {-1, 1} solve(x**x = x, x) (14) [] Type: List Equation Expression Integer Time: 0.03 (IN) + 0.08 (EV) + 0.02 (OT) = 0.13 sec -- 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 3 4 3 2 3 (15) (x + 1)cos(3x) sin(x) + (2x + 2)cos(3x) sin(x) + (x + 1)cos(3x) = 0 Type: Equation Expression Integer Time: 0.05 (IN) + 0.03 (EV) + 0.03 (OT) = 0.12 sec solve(%, x) (16) [] Type: List Equation Expression Integer Time: 2.88 (EV) + 0.02 (GC) = 2.90 sec -- x = pi/4 [+ n pi] solve(sin(x) = cos(x), x) %pi (17) [x= ---] 4 Type: List Equation Expression Integer Time: 0.02 (IN) + 0.15 (EV) = 0.17 sec solve(tan(x) = 1, x) %pi (18) [x= ---] 4 Type: List Equation Expression Integer Time: 0.03 (EV) = 0.03 sec -- x = {pi/6, 5 pi/6} [ + n 2 pi, + n 2 pi ] solve(sin(x) = 1/2, x) 1 (19) [x= asin(-)] 2 Type: List Equation Expression Integer Time: 0.08 (IN) + 0.02 (EV) + 0.02 (OT) = 0.12 sec map(e +-> lhs(e) = normalize(rhs(e)), %) %pi (20) [x= ---] 6 Type: List Equation Expression Integer Time: 0.05 (IN) + 0.02 (EV) = 0.07 sec -- x = {0, 0} [+ n pi, + n 2 pi] solve(sin(x) = tan(x), x) (21) [x= 0] Type: List Equation Expression Integer Time: 0.15 (EV) = 0.15 sec -- x = {0, 0, 0} solve(asin(x) = atan(x), x) (22) [x= 0] Type: List Equation Expression Integer Time: 0.68 (EV) + 0.02 (OT) = 0.70 sec -- x = sqrt[(sqrt(5) - 1)/2] solve(acos(x) = atan(x), x) (23) +-----------+ +-----------+ +---------+ +---------+ | +-+ | +-+ | +-+ | +-+ \|- 2\|5 - 2 \|- 2\|5 - 2 \|2\|5 - 2 \|2\|5 - 2 [x= - --------------,x= --------------,x= - ------------,x= ------------] 2 2 2 2 Type: List Equation Expression Integer Time: 0.02 (IN) + 1.08 (EV) + 0.03 (OT) + 0.38 (GC) = 1.52 sec -- x = 2 solve((x - 2)/x**(1/3) = 0, x) (24) [x= 2] Type: List Equation Expression Integer Time: 0.10 (IN) + 0.03 (EV) + 0.03 (OT) = 0.17 sec -- This equation has no solutions solve(sqrt(x**2 + 1) = x - 2, x) 3 (25) [x= -] 4 Type: List Equation Expression Integer Time: 0.05 (IN) + 0.05 (EV) = 0.10 sec -- x = 1 solve(x + sqrt(x) = 2, x) (26) [x= 4,x= 1] Type: List Equation Expression Integer Time: 0.03 (IN) + 0.03 (EV) = 0.07 sec -- x = 1/16 solve(2*sqrt(x) + 3*x**(1/4) - 2 = 0, x) +---+ +---+ 1 3\|- 7 - 31 - 3\|- 7 - 31 (27) [x= 16,x= --,x= ------------,x= --------------] 16 32 32 Type: List Equation Expression Integer Time: 0.03 (IN) + 0.18 (EV) + 0.03 (OT) = 0.25 sec -- x = {sqrt[(sqrt(5) - 1)/2], -i sqrt[(sqrt(5) + 1)/2]} solve(x = 1/sqrt(1 + x**2), x) (28) +---------+ +---------+ +-----------+ +-----------+ | +-+ | +-+ | +-+ | +-+ \|2\|5 - 2 \|2\|5 - 2 \|- 2\|5 - 2 \|- 2\|5 - 2 [x= ------------,x= - ------------,x= --------------,x= - --------------] 2 2 2 2 Type: List Equation Expression Integer Time: 0.07 (IN) + 0.17 (EV) + 0.03 (OT) = 0.27 sec -- This problem is from a computational biology talk => 1 - log_2[m (m - 1)] solve(binomial(m, 2)*2**k = 1, k) 2 log(------) 2 m - m (29) [k= -----------] log(2) Type: List Equation Expression Integer Time: 0.13 (IN) + 0.12 (EV) = 0.25 sec -- 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) (30) [] Type: List Equation Expression Integer Time: 0.15 (IN) + 0.18 (EV) + 0.02 (OT) = 0.35 sec -- x = {1, e^4} solve(sqrt(log(x)) = log(sqrt(x)), x) (31) [x= 0,x= 1] Type: List Equation Expression Integer Time: 0.02 (IN) + 0.10 (EV) + 0.03 (OT) = 0.15 sec -- 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) (32) +---------------------+ 1 | 1 [x= (- sin(cos(---) + 1) - b) |sin(cos(---) + 1) + b , 2 | 2 %e \| %e +---------------------+ 1 | 1 x= (sin(cos(---) + 1) + b) |sin(cos(---) + 1) + b ] 2 | 2 %e \| %e Type: List Equation Expression Integer Time: 0.05 (IN) + 0.25 (EV) + 0.03 (OT) + 0.02 (GC) = 0.35 sec -- 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 - 5 ----- 2 3 (33) %e + 5x= 8x Type: Equation Expression Integer Time: 0.50 (IN) + 0.02 (EV) + 0.02 (OT) = 0.53 sec solve(eqn, x) (34) [] Type: List Equation Expression Integer Time: 0.08 (EV) = 0.08 sec --root_by_bisection(eqn, x, -1, -0.5) --root_by_bisection(eqn, x, -0.5, 0.5) --root_by_bisection(eqn, x, 0.5, 1) )clear properties eqn -- x = {-1, 3} solve(abs(x - 1) = 2, x) (35) [] Type: List Equation Expression Integer Time: 0.03 (IN) + 0.03 (EV) = 0.07 sec -- x = {-1, -7} solve(abs(2*x + 5) = abs(x - 2), x) (36) [] Type: List Equation Expression Integer Time: 0.02 (IN) + 0.05 (EV) + 0.02 (OT) = 0.08 sec -- x = +-3/2 solve(1 - abs(x) = max(-x - 2, x - 2), x) (37) [] Type: List Equation Expression Integer Time: 0.03 (IN) + 0.03 (EV) + 0.02 (OT) = 0.08 sec -- x = {-1, 3} solve(max(2 - x**2, x) = max(-x, x**3/9), x) (38) [x= 3,x= 0,x= - 3] Type: List Equation Fraction Polynomial Integer Time: 0.58 (IN) + 0.02 (EV) + 0.12 (OT) = 0.72 sec -- 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) (39) [x= 3,x= 0,x= - 3] Type: List Equation Fraction Polynomial Integer Time: 0.08 (IN) + 0.03 (OT) = 0.12 sec -- z = 2 + 3 i z : Complex Expression Integer Type: Void Time: 0 sec (1 + %i)*z + (2 - %i)*conjugate(z) = -3*%i (41) 3z= - 3%i Type: Equation Complex Expression Integer Time: 0.47 (IN) + 0.02 (EV) + 0.05 (OT) = 0.53 sec )clear properties z (1 + %i)*(x + %i*y) + (2 - %i)*conjugate(x + %i*y) = -3*%i There are 4 exposed and 1 unexposed library operations named conjugate having 1 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op conjugate to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named conjugate with argument type(s) Polynomial Complex Integer (1 + %i)*(x + %i*y) + (2 - %i)*(x - %i*y) = -3*%i (42) (- 2 - %i)y + 3x= - 3%i Type: Equation Polynomial Complex Integer Time: 0.07 (IN) + 0.03 (OT) = 0.10 sec solve(%, [x, y]) There are 18 exposed and 3 unexposed library operations named solve having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op solve to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named solve with argument type(s) Equation Polynomial Complex Integer List OrderedVariableList [x,y] -- => {f^(-1)(1), f^(-1)(-2)} assuming f is invertible f:= operator('f); Type: BasicOperator Time: 0.05 (IN) = 0.05 sec solve(f(x)**2 + f(x) - 2 = 0, x) (44) [] Type: List Equation Expression Integer Time: 0.32 (IN) + 0.02 (EV) + 0.07 (OT) = 0.40 sec )clear properties f -- Solve a 3 x 3 system of linear equations eqn1:= x + y + z = 6 (45) z + y + x= 6 Type: Equation Polynomial Integer Time: 0.08 (IN) = 0.08 sec eqn2:= 2*x + y + 2*z = 10 (46) 2z + y + 2x= 10 Type: Equation Polynomial Integer Time: 0.02 (IN) + 0.02 (OT) = 0.03 sec eqn3:= x + 3*y + z = 10 (47) z + 3y + x= 10 Type: Equation Polynomial Integer Time: 0.02 (OT) = 0.02 sec -- Note that the solution is parametric: x = 4 - z, y = 2 solve([eqn1, eqn2, eqn3], [x, y, z]) (48) [[x= - %DW + 4,y= 2,z= %DW]] Type: List List Equation Fraction Polynomial Integer Time: 0.68 (IN) + 0.05 (EV) + 0.08 (OT) = 0.82 sec -- 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 _ ]; Type: List Equation Fraction Polynomial Integer Time: 6.35 (IN) + 0.72 (EV) + 1.33 (OT) + 0.87 (GC) = 9.26 sec 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]; Type: List OrderedVariableList [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: 0.05 (IN) = 0.05 sec solve(eqns, vars) (51) [ [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, %DY a k20= 0, k21= 0, k22= 0, k23= %DX, k24= 0, k25= 0, k26= -----, k27= 0, c %DY b k28= 0, k29= 0, k30= 0, k31= %DX, k32= 0, k33= 0, k34= -----, k35= 0, c k36= 0, k37= 0, k38= 0, k39= %DX, k40= 0, k41= 0, k42= %DY, k43= %DZ, k44= 0, k45= 0, k46= 0, k47= 0, k48= 0, k49= 0] ] Type: List List Equation Fraction Polynomial Integer Time: 0.69 (IN) + 1.42 (EV) + 0.15 (OT) = 2.25 sec )clear properties eqns vars -- Solve a 3 x 3 system of nonlinear equations eqn1:= x**2*y + 3*y*z - 4 = 0 2 (52) 3y z + x y - 4= 0 Type: Equation Polynomial Integer Time: 0.03 (OT) = 0.03 sec eqn2:= -3*x**2*z + 2*y**2 + 1 = 0 2 2 (53) - 3x z + 2y + 1= 0 Type: Equation Polynomial Integer Time: 0.02 (IN) + 0.02 (OT) = 0.03 sec eqn3:= 2*y*z**2 - z**2 - 1 = 0 2 (54) (2y - 1)z - 1= 0 Type: Equation Polynomial Integer Time: 0.02 (IN) + 0.02 (OT) = 0.03 sec -- Solving this by hand would be a nightmare solve([eqn1, eqn2, eqn3], [x, y, z]) (55) [[x= 1,y= 1,z= 1], [x= - 1,y= 1,z= 1], 2 2 [- 3z + x + 2= 0,y= - 3z + 1,3z - 2z + 1= 0], 4 3 2 4 3 2 2 - 18z + 24z + 21z + 12z + 3 [12z - 12z - 30z + 7z + 3x = 0, y= ------------------------------, 2 5 4 3 2 6z - 6z - 9z - 7z - 3z - 1= 0] ] Type: List List Equation Fraction Polynomial Integer Time: 0.02 (IN) + 1.18 (EV) + 0.03 (OT) + 0.03 (GC) = 1.27 sec )clear properties eqn1 eqn2 eqn3 -- ---------- Quit ---------- )quit real 118.9 user 38.5 sys 0.9