$3\mathrm{.}2\mathrm{.}2$ Implementation and Assessment: Recursion Relations

A program implementing recurrence relations is most easily written using subscripts. If you need to polish up on your skills with subscripts, you may want to study our program Bessel. py in Listing. $3\mathrm{.}1$ before writing your own.

Write a program that uses both upward and downward recursion to calculate $j_l(x)$ for the first $25l$ values for $x=0\mathrm{.}1,1,$ and $10.$

Tune your program so that at least one method gives "good" values $($meaning a relative error ${10}^{-10}.$ See Table $3_{1}1$ for some sample values.

Show the convergence and stability of your results.

Compare the upward and downward recursion methods, printing out $L_l^{(up)},j_l^{(down)},$ and the relative difference $|)/(|j_l^{(up)}|+|j_l^{(down)}||.$

The errors in computation depend on $x,$ and for certain values of $x,$ both up and down recursions give similar answers. Explain the reason for this.

Listing $3.$I

Bessel.py determines spherical Bessel functions by downward recursion $($you should modify this to also work by upward recursion$).$

Bessel.py

from visual import.

from visual.graph import

$x_{\text{m}\text{a}\text{x}}=40.$

$x_{\text{m}\text{i}\text{n}}=0\mathrm{.}25$

step $=0\mathrm{.}1,$ # Global class variables

order $=10$; start $=50,$# Plot j$\_$order

graph $1=$ gdisplay $($width $=500,$ height $=500,$ title $=\text{'}$ Sperical Bessel,$\frac{\frac{?}{?}}{?}$

$L=1($red$),{10}^{\text{'}},$ xtitle $={?}^{\text{'}}x\text{'},$ ytitle $=\text{'}(x{)}^{\text{'}},1$

xmin$=$xmin,xmax$=$xmax,ymin$=-0\mathrm{.}2,$ymax$=0\mathrm{.}5)$

funct$1=$ gcurve color $=$ color, red