Re: FixedPoint vs. FixedPointList

*To*: mathgroup at smc.vnet.net*Subject*: [mg24488] Re: [mg24384] FixedPoint vs. FixedPointList*From*: "John D. Hendrickson" <jdh at hend.net>*Date*: Wed, 19 Jul 2000 01:22:00 -0400 (EDT)*Organization*: home*References*: <8l0qd5$dvj@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

(Mathematica book page 243) FixedPoint simply calls your function unil the result no longer changes. You can easily write a function which does the same. The difference being, your function would keep track of the iterations. In[467]:= myFixedPoint[f_, x_] := Module[{n = 0, y = x + 1, y1 = x}, While[y != y1, y = y1; ++n; y1 = f[y]]; {y1, n} ] (* test - same as one in Mathematica Book *) In[472]:= myFixedPoint[Log, 1. + \[ImaginaryI]] Out[472]= {0.318132\[InvisibleSpace] + 1.33724 \[ImaginaryI], 101} In[473]:= Log[%[[1]]] Out[473]= 0.318132\[InvisibleSpace] + 1.33724 \[ImaginaryI] so we have { solution, iterations } as the output BobHanlon at aol.com wrote in message <8l0qd5$dvj at smc.vnet.net>... > >In a message dated 7/12/2000 11:43:32 PM, linsuain+ at andrew.cmu.edu writes: > >>Hi all. I need to know the fixed point of a function (to a certain >>accuracy) starting from a certain value of the arguement, say: >> >> x = FixedPoint[ f, x, SameTest -> ( #2-#1 < somesmallnumber &) ] >> >> I do not care about the intermediate results, but I would like to know >>how many iterations it takes for the process to converge. FixedPointList >>will do, for example: >> >> {n,x}={Length[#],Last[#]}& @ FixedPointList[....] >> >> However I believe this would cause problems. The reason is that x is >>really a very long list (FixedPoint works for lists too), and it could >>take many iterations for this to settle. Mathematica, I believe, stores >>only a maximum of two values at any given moment when executing >>FixedPoint, but it stores all intermediate values for FixedPointList, >>and most certainly would run out of memory. >> >> Could any think of how to trick Mathematica to count the iteration >>without trying to store all the results? >> > >poly[x_] := x^5 - x^4 + 2 x^3 - x^2 + 3; > >n = 1; > >FixedPoint[(n++; # - poly[#]/poly'[#]) &, 2.] > >-0.8574417862415682 > >n > >23 > >Length[FixedPointList[(n++; # - poly[#]/poly'[#]) &, 2.]] > >23 > > >Bob Hanlon >