Using decimals in recurrence tables

Questions
,
1

I’m trying to have the recurrence table find roots of a function, but I can only use these numbers in the equation:

RecurrenceTable[{ a[n + 1] == a[n] + 1, a[1] == 0, p[n + 1] == b[n], p[1] == 1, r[n] == p[n]^2*(p[n]^2*(231*p[n]^2 - 315) + 105) - 5, d[n] == p[n]*(p[n]^2*(1386*p[n]^2 - 1260) + 210), b[n] == (d[n]*p[n] - r[n])/d[n]}, {a, p, r, d, b}, {n, 2}, DependentVariables -> {a, p, r, d, b}]

and it gives exact fraction answers.
There are three problems though.

  1. when I try to have p[1]==0.5 or any decimal, it wants to abort
  2. when using decimals in p,d,or b, it stalls
  3. it runs out of memory when n>4

All I want is to have p[1] at a decimal and I also want to have decimal approximation. I would use N but it needs n and Scientific form doesn’t seem to work. How can I get itterations approximate to 10^-10 or even make it to n=20 ?

2 
In[1]:= f[{a_, p_, r_, d_, b_}] := {a+1, b, p^2*(p^2*(231*p^2-315)+105)-5, p*(p^2*(1386*p^2-1260)+210), (d*p-r)/d};
NestList[f, {0., 1., 2., 3., 1.(*<--Need correct initial values for r,d and b *)}, 10]

Out[2]={{0., 1., 2., 3., 1.},
{1., 1., 16., 336., 0.33333},
{2., 0.33333, 16., 336., 0.95238},
{3., 0.95238, 3.09465, 29.0370, 0.28571},
{4., 0.28571, 3.46257, 197.531, 0.84580},
{5., 0.84580, 1.59796, 33.2511, 0.26818},
{6., 0.26818, -6.52096, 15.1710, 0.79774},
{7., 0.79774, 1.00840, 33.9378, 0.69801},
{8., 0.69801, -6.21544, -24.3524, 0.76803},
{9., 0.76803, -1.90085, -52.2697, 0.44278},
{10., 0.44278, -5.25580, -39.1541, 0.73166}}

I am concerned about your not including \[n\] after all your variables in your example code.
I attempted to define a function f that does what I think your RecurrenceTable was doing.
I was not able to convince myself I knew what the initial values for r, d and b should be.
But I was able to use decimal points and carry out the iterations using NestList.
Will this do what you need, after you correct any of my errors or misunderstandings?
This can rapidly do as many iterations as you like, just as long as your initial values all have decimal points.

3

Thanks, that helped a lot. r, d, and b were actually the functions I was trying to use but I can maipulate the original functions to help me in my itterations using NestList.