Re: Calculating Sums Of Roots For Trans. Functions
- To: mathgroup at smc.vnet.net
- Subject: [mg5536] Re: [mg5498] Calculating Sums Of Roots For Trans. Functions
- From: Allan Hayes <hay at haystack.demon.co.uk>
- Date: Thu, 19 Dec 1996 01:02:33 -0500
- Sender: owner-wri-mathgroup at wolfram.com
frankst at deakin.edu.au (Frank Stagnitti)
[mg5498] Calculating Sums Of Roots For Trans. Functions
writes
>>>>>>>>>>>
How do i calculate the first n roots of tan(z) = -z, and then sum
them in one command.
I think i need the right combinations of Findroot, Table and N/Sum
commands.
Findroot and Table give me a non-unique and non-ordered set of roots e.g.
RootTable = Table[{FindRoot[Tan[z]==-z, {z,i},
MaxIterations ->50],i},{i,0,EndRange,Pi}]
// TableForm
In short, i'd like a procedure/command that essentially sums the first n
(unique and ordered) roots
of tan(z) = - z with no user intervention
Any ideas??
<<<<<<<<<<<<<<
Frank,
From
Plot[{-x, Tan[x]}, {x,0,20}]
and a little experimentation we get the sum of the first fifty
positive roots as follows
Sum[ z/.FindRoot[Tan[z] == -z, {z,Pi/2 + n Pi + .02/(n+1)},
MaxIterations ->50
],
{n,0,49}
]
3928.67
The erratic behaviour shown by your code:
RootTable = Table[{FindRoot[Tan[z]==-z, {z,i},
MaxIterations ->50],i},{i,0,60,Pi}]// TableForm
z -> 0. 0
z -> 2.02876 Pi
z -> 2.02876 2 Pi
z -> 2.02876 3 Pi
z -> 2.02876 4 Pi
z -> 7.97867 5 Pi
z -> 2.02876 6 Pi
z -> 2.02876 7 Pi
z -> 2.02876 8 Pi
z -> 14.2074 9 Pi
z -> 7.97867 10 Pi
z -> 4.91318 11 Pi
z -> 2.02876 12 Pi
z -> 20.4692 13 Pi
z -> 2.02876 14 Pi
z -> 20.4692 15 Pi
z -> 2.02876 16 Pi
z -> 26.7409 17 Pi
z -> 14.2074 18 Pi
z -> 23.6043 19 Pi
is explained by
Plot[{-x, Tan[x]}, {x,0,20}]
Try drawing a tangent to the curve at the points with these x
coordinates, where it meets the x-axis is the first step in the root
finding process.
We can track what happens starting at 12Pi as follows:
FindRoot[Tan[Print[z];z] == -z, {z,12Pi}]
z
37.6991
18.8496
9.42478
4.71239
7.06858
4.37906
3.6763
1.86049
1.97323
2.02324
2.02871
2.02876
2.02876
{z -> 2.02876}
Allan Hayes
hay at haystack.demon.co.uk
http://www,haystack.demon.co.uk