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LambertW function - Stevetuc - 05-17-2020 08:48 AM
This cas program uses fsolve to calc Lambert fn Code: `#cas` Graph from wikipedia [attachment=8478] If x <0 lmb(x) returns both principal and negative branch solutions : lmb(-0.5/e) returns [−2.67834699002,−0.231960952987] lmb(-1/e) returns −0.999999972234 (negative and principal branch converge at -1 when x=-1/e) lmb(1) returns 0.56714329041 lmb(2) returns 0.852605502014 RE: LambertW function - Stevetuc - 11-07-2020 08:41 AM
Simplified the code and change to fsolve: Code: `#cas` 8.92180498562ᴇ−2+1.62562367443*i Code: `#cas` Edit: add initial guess of 0 to avoid terminal screen in cas. This forces going direct to iterative solver rather than first trying and failing with bisection solver. Edit: it doesn't fail with Bisectional solver. The cas terminal screen is just for information: Quote:Solving by bisection with change of variable x=tan(t) and t=-1.57..1.57. Try fsolve(equation,x=guess) for iterative solver or fsolve(equation,x=xmin..xmax) for bisection. RE: LambertW function - Albert Chan - 11-07-2020 11:57 AM
(11-07-2020 08:41 AM)Stevetuc Wrote: Edit: add initial guess of 0 to avoid terminal screen in cas. You might want to mention guess of 0 will iterate for W0(x), i.e. principle branch. (*) Also, guess 0 is same as guess x, but wasted 1 Newton iteration. We might as well use guess = x Newton: w - (w*exp(w) - x) / (w*exp(w) + exp(w)) With guess 0, first iteration of w = 0 - (0 - x) / (0 + 1) = x (*) Assumed W0 is not complex (x ≥ -1/e), see comment below. Comment: For fsolve, some randomization of guess is going on. With complex ON, if we fsolve again and again, we got different solutions. XCas> fsolve(w*e^w = -1.78, w=0) 0.0892180498562+1.62562367443*i -1.4781113814-7.66344321151*i -3.68225172433+70.6337502365*i -2.07259091944+13.9900896316*i -2.9207293675-32.8981741284*i ... Without randomization, fsolve should not even converge. (w will not flip to complex) Maybe this is the reason guess randomization kick in ... RE: LambertW function - Stevetuc - 11-07-2020 01:50 PM
(11-07-2020 11:57 AM)Albert Chan Wrote:(11-07-2020 08:41 AM)Stevetuc Wrote: Edit: add initial guess of 0 to avoid terminal screen in cas. Thanks for pointing that out. I want to retain the secondary branch result when 0 >x ≥ -1/e so I've removed the initial guess on my prime. |