Re: FindRoot and Bose-Einstein distribution

*To*: mathgroup at smc.vnet.net*Subject*: [mg82887] Re: FindRoot and Bose-Einstein distribution*From*: danl at wolfram.com*Date*: Fri, 2 Nov 2007 03:32:11 -0500 (EST)*References*: <fg6ql3$dmc$1@smc.vnet.net>

On Oct 30, 3:40 am, P_ter <peter_van_summe... at yahoo.co.uk> wrote: > Hello, > I have two functions and two values: > Clear[p, q, k, mm, mn] > n[p_, q_] := Sum[k/(Exp[p + k q ] - 1), {k, 1, 10000}] > m[p_, q_] := Sum[1/(Exp[p + k q ] - 1), {k, 1, Infinity}] > mm = 0.501 > mn = 41.0959 > The first two equations are the Bose-Einstein distribution. Given mm and mn, find p and q. > A first estimation is: p=5.086,q= 0.01226 > I know that n[p_,q_] is stable until k = 50000 for p=5.086 and q= 0.01226. My check is: n[5.086,0.01226]= 41.1969 > m[p_,q_] is also stable until 100000 and m[5.086,0.01226]= 0.502762 It is not clear to me what is meant by "stable until k=...". Do you actually intend to do infinite summations for both m and n, and are checking to see how far finite truncations can be computed wo some precision? > Everything seems ok. > So, I tried: > FindRoot[{m[p, q] - mm, n[p, q] - mn}, {{p, 5}, {q, 0.1}}] > The answer from FindRoot is that the Sum does not converge. > What did I do wrong, what went wrong, why? > with friendly greetings, > P_ter There is not much hope to working with the infinite summation since Sum cannot compute a closed form for the result. Hence anything it does will rely on some sort of approximation via numeric summing of some initial summands, and extrapolation for the tail. This might or might not give reliable results. Since it is fairly easy to bound the error of finite summations "by hand", given that {p,q} are in the whereabouts of {5,.01}, it is probably more reliable just to truncate the sum at some sufficiently high, but finite, bound. 10000 will do just fine (check it). n[p_Real, q_Real] := Sum[k/(Exp[p + k*q] - 1), {k, 1, 10000}] m[p_Real, q_Real] := Sum[1/(Exp[p + k*q] - 1), {k, 1, 10000}] mm = 0.501; mn = 41.0959; In[13]:= rt = FindRoot[{m[p, q] - mm, n[p, q] - mn}, {{p, 5}, {q, 0.1}}] Out[13]= {p -> 5.09055, q -> 0.0122471} In[14]:= m[p, q] /. rt Out[14]= 0.501 In[15]:= n[p, q] /. rt Out[15]= 41.0959 I'll sum more terms for n[] just to show that the result really does not change (that is, the truncation approximation was fine). In[26]:= Sum[1/(Exp[p + k q] - 1) /. rt, {k, 1, 10^5}] // InputForm Out[26]//InputForm= 0.501000000000001 This is the same as I get if opnly summing to 10^4. Daniel Lichtblau Wolfram Research