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Re: Problems with FindRoot and recursive functions

• To: mathgroup at smc.vnet.net
• Subject: [mg89694] Re: [mg89641] Problems with FindRoot and recursive functions
• From: "Daniel Kuang Chen Liu" <dkliu1 at gmail.com>
• Date: Wed, 18 Jun 2008 04:24:52 -0400 (EDT)
• References: <200806161039.GAA09440@smc.vnet.net>

I have weird function.
It doesn't work the way you think it does.

x1[0.5] = 0.5 + x1[-0.5] = 0.5 + {-0.5,1} = {0,1.5}

It only takes a scalar has input, and is only called recursively with scalar
input t-1. The function works when you plot it. Additionally, my own roots
finding function based on the secant method works fine ...

NSecant[f_, int_, k_: 0, plevel_: defplevel] :=
  Module[{old = int[[1]], est, new = int[[2]]}, i = k;
   i++;
   est = old - (new - old)/(f[new] - f[old]) f[old];
   If[Abs[N[f[est], 100]] < plevel,
     {est, N[Abs[est - new]], N[f[est], 5], i, plevel},
    NSecant[f, {new, est}, i, plevel]
    ]
   ];

In[183]:= NSecant[x1[#][[1]] &, {0.4, 0.6}, 0, 10^-10]

Out[183]= {0.5, 0.1, 0., 1, 1/10000000000}

So I really can't see why FindRoot should have any problem with it.

Daniel

On Tue, Jun 17, 2008 at 1:39 AM, Murray Eisenberg <murray at math.umass.edu>
wrote:

> You will see what is wrong if you just evaluate your function x1 a couple
> of times, beginning with the initial guess you used in the FindRoot.
>
>   x1[t_] := If[t < 0, {t, 1}, 0.5 + x1[t - 1]]
>
>   x1[0.5]
> {0., 1.5}
>  (* so far, as expected *)
>
>   x1[%]
> If[{0., 1.5} < 0, {{0., 1.5}, 1}, 0.5+ x1[{0., 1.5} - 1]]
>  (* NOT as expected! *)
>
>
> Daniel Kuang Chen Liu wrote:
>
>> Hello,
>>
>> I have a recursive function of the form
>>
>>  x1[t_] := If[t < 0, {t, 1}, 0.5 + x1[t - 1]]
>>>>
>>>
>> which returns a list of length 2, and the first element has a root at
>> t=0.5
>>
>>> In[3]:= x1[0.5]
>>>> Out[3]= {0., 1.5}
>>>>
>>>
>> I want to use FindRoot to determine t0 such that x1[t0][[1]] == 0, so I
>> use
>> the following code
>>
>>  FindRoot[x1[t][[1]] == 0, {t, 0.5}]
>>>>
>>>
>> to which Mathematica complains
>>
>> During evaluation of In[6]:= FindRoot::nlnum: The function value
>> {False} is not a list of numbers with dimensions {1} at {t} = {0.5}.
>>
>> It would much appreciated if someone could tell me what is wrong with the
>> code.
>>
>> Daniel Liu
>>
>>
>>
> --
> Murray Eisenberg                     murray at math.umass.edu
> Mathematics & Statistics Dept.
> Lederle Graduate Research Tower      phone 413 549-1020 (H)
> University of Massachusetts                413 545-2859 (W)
> 710 North Pleasant Street            fax   413 545-1801
> Amherst, MA 01003-9305
>

• References:
  • Problems with FindRoot and recursive functions
    • From: "Daniel Kuang Chen Liu" <dkliu1@student.monash.edu.au>