Download or view systemSolverTest.frink in plain text format
// Demonstration and test of solving systems of equations.
use systemSolver2.frink
symbolicMode[true]
// Ellipse equations.
ellipse = new System[[x^2/a^2 + y^2/b^2 === 1,
e === sqrt[(a^2-b^2)/a^2],
area===pi a b],
["pi"]]
println["Ellipse:"]
println[join["\n",ellipse.solveAll[]]]
println[]
println["System of equations with 2 unknowns:"]
simult = new System[[x*y===8,
x+y===4]]
println[join["\n",simult.solveAll[]]]
println[]
println["System of equations with 4 unknowns:"]
simult2 = new System[[3a - 2b + c === 3,
4a + b - 3c === 10,
a + 6b - 2c === 5,
d === 4c - 10 + 2b]]
println[join["\n",simult2.solveAll[]]]
println[]
// Rump equation
println["Rump equation:"]
rump = new System[[z === ((333 + 3/4) - x^2) y^6 + x^2 (11x^2 y^2 - 121 y^4 - 2) + (5 + 1/2) y^8 + x/(2y), x===77617, y===33096]]
println[join["\n",rump.solveAll[]]]
println[]
// Cohen, p. 11
/*cc = new System[[d0+d1+d2+d3+d4 === 1,
d1+2d2+3d3+4d4 === 2(1-m),
3d0-d2+3d4 === 2g20 + g11,
theta d0 + phi d1 - phi d3 - theta d4 === m,
2 theta d0 + phi d1 + phi d3 + 2 theta d4 === 2 g10],
["g20", "g11", "theta", "phi", "g10", "m"], true]
println[join["\n",cc.solveAll[true]]]
println[] */
// Expansion of the universe
// TODO: Don't have solvers for exponential functions yet.
println["Expansion of the universe:"]
u = new System[[D0 hubbleconstant^2 e^(hubbleconstant t) === G m / d^2, d === D0, D0 === 4.3 ly], ["D0", "hubbleconstant", "ly", "G"]]
println[join["\n",u.solveFor["t"]]]
println[]
// Circle
println["Circle:"]
circle = new System[[d === 2 r, c === pi d, a === pi r^2, e===f, g===h], ["pi"]]
println[join["\n",circle.solveAll[]]]
println[]
// See http://answers.yahoo.com/question/index?qid=20091120001614AAInec3
// TODO:
// eliminate pAprime and pAB, as those are what we want to solve for.
println["Bayes equation:"]
bayes = new System[[pAB === pBA pA / pB,
pA === 1 - pAprime,
pAB === ( pBA pA ) / ( pBA pA + pBAprime pAprime )],
[]]
println[join["\n", bayes.solveAll[]]]
println[]
// Cylinder. See http://ask.metafilter.com/59183/Calculate-length-of-a-pipe-from-the-volume-it-holds
// Basically, we want the solution for "L" given "d" and "v".
println["Cylinder:"]
cyl = new System[[d === 2 r, c === pi d, a === pi r ^2, v === a L], ["pi"]]
println[join["\n", cyl.solveAll[]]]
sols = cyl.solveFor["L"]
println["Solutions for L:\n" + join["\n", sols]]
// Plug in values and solve.
args = [["d", 3/4 inch], ["v", 27 gallons]]
nsols = cyl.solveForValues["L", args]
println["Solutions for L (fully specified, symbolic):\n" + (nsols->"ft")]
println["Solutions for L (evaluated):\n" + (eval[nsols]->"ft")]
println[]
// Not fully specified case.
args = [["d", 3/4 inch]]
nsols = cyl.solveForValues["L", args, false]
println["Solutions for L (not fully specified):\n" + join["\n",nsols]]
println[]
// Phi
println["The golden ratio Phi:"]
phi = new System[[x+1===1/x]]
sols = phi.solveAll[]
println[join["\n", sols]]
println[]
// Test of parsing user-generated systems of equations. This will generally
// fail with eval because of security constraints on creating objects.
// Thus we're using unsafeEval. Hopefully in the future this can be done in
// such a way that we can solve user-submitted systems of equations. Or we
// may just have to do that in an applet or in the Frink UI. Burn your CPU
// cycles, not mine!
println["User-submitted system:"]
s = unsafeEval["new System[[density === 3.5 oz / (170 yards),v === density], []]"]
println[join["\n", s.solveFor["v"]]]
println[]
// Shweeb collision
println["Shweeb collision"]
phi = new System[[v === a t,
d === 1/2 a t^2,
E === 1/2 m v^2,
E === m gravity h],
["gravity"]]
println[join["\n", phi.solveAll[]]]
args = [["d", 1.2 m], ["v", 30 km/hr], ["gravity", gee]]
println["Acceleration: " + (eval[phi.solveForValues["a", args]]->"gee")]
println["Height: " + (eval[phi.solveForValues["h", args]]->"ft")]
println["Collision time:" + eval[phi.solveForValues["t", args]]]
println[]
// Physical system // Suppose the stone is thrown at an angle of 35.0° below
// the horizontal from the same building (h = 50.0 m) as in the example above.
// If it strikes the ground 60.8 m away, find the following. (Hint: For part
// (a), use the equation for the x-displacement to eliminate v0t from the
// equation for the y-displacement.)
// (a) the time of flight
// (b) the initial speed
// (c) the speed and angle of the velocity vector with respect to the
// horizontal
println["Ballistics problem:"]
phys = new System[[vx === v0 cos[35 degrees],
vy0 === v0 sin[35 degrees],
h === h0 - 1/2 g t^2 - vy0 t,
x === vx t],
["g", "degrees"]]
println[join["\n", phys.solveAll[]]]
args = [["x", 60.8 m], ["h0", 50 m], ["g", gravity], ["h", 0 m]]
println[join["\n", eval[phys.solveForValues["v0", args]]]]
println[]
// Jupiter gravity
println["Jupiter gravity"]
jup = new System[[ E === m g h,
E === 1/2 m v^2],
["g"]]
println[join["\n", jup.solveAll[]]]
println[]
// Lorentz equation
println["Lorentz equation:"]
lorentz = new System[[ gamma === 1/sqrt[1-v^2/c^2],
dprime === d / gamma ],
["c"]]
println[join["\n", lorentz.solveAll[]]]
// Solve for when a distance of 5 km is compressed into 1 m
args=[["d", 5 km], ["dprime", 1 m]]
sols = lorentz.solveForValues["v", args]
println[join["\n", sols]]
println["Evaluated: " + eval[sols]]
println[]
// Photon travel from quantum electrodynamic calculations
println["Photon equations:"]
light = new System[[ f === c / lambda,
omega === 2 pi f,
f === 1/T,
d === v t,
v === c,
phase === omega t],
["c", "pi"]]
println[join["\n", light.solveAll[false]]]
println[]
// Deep fryer. How much oil do I need to buy to fill a cylinder to the
// specified height?
println["Deep fryer:"]
fryer = new System[[ a === pi r^2,
v === a h,
d === 2 r ],
["pi"]]
println[join["\n", fryer.solveAll[]]]
args = [["d", 10.5 in], ["h", 7.5 in]]
println[eval[fryer.solveForValues["v",args]] -> "gallons"]
println[]
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Alan Eliasen was born 19965 days, 21 hours, 41 minutes ago.