recursive definition

[Shigeharu T. <sh...@ie...>]

shige 02/23 2009
----------------

The demo script demo/approximate.dem is the sample for the 
'filledcurve' and the pseudofile '+'. This treats the Taylor 
approximation of sin(x) and it defines them explicitly as

 approx_1(x) = x - x**3/6
 approx_2(x) = x - x**3/6 + x**5/120 
 approx_3(x) = x - x**3/6 + x**5/120 - x**7/5040

For the "Approximation demo", there is another way to define it 
by the recursive definition:

 set title "Taylor polynomial approximation of sin(x)"
 set zeroaxis
 # f(x,n) = (-1)^{n-1}*x^{2*n-1}/(2*n-1)!
 f(x,n) = (n>1)? (-f(x,n-1)*x*x/((2.0*n-1.0)*(2.0*n-2.0))) : x
 # sum_of_f(x,n) = f(x,1) + f(x,2) + ... + f(x,n)
 sum_of_f(x,n) = (n>0)? sum_of_f(x,n-1) + f(x,n) : 0
 plot [-8:8][-2:2] for [n=1:5] sum_of_f(x,n) \
  title sprintf("degree %d",2*n-1), sin(x)

+========================================================+
 Shigeharu TAKENO     NIigata Institute of Technology
                       kashiwazaki,Niigata 945-1195 JAPAN
 sh...@ie...   TEL(&FAX): +81-257-22-8161
+========================================================+