[CSC 335] Derivatives of Hermite Polynomails

[Andrew J. Pounds]

First -- please forgive me for not getting back to you all sooner about 
this.   I got home from campus on Wednesday with a fever that blossomed 
into something that kept me bedridden for two days.

Anyway, in looking at problems in the text that deal with the 
derivatives of Hermite polynomials, you have to remember that the 
polynomial can be constructed as shown in the equation just above Table 
3.16.  In that equation the divided differences are the Q values that 
you calculate in Algorithm 3.3.

As way of an example, looking at problem 1a on page 139, I get the 
following Q values

  Q00 =  17.56492
  Q11 =  3.116256
  Q22 =  0.05948
  Q33 =  -0.002022222

Which translates into

$H(x) =  Q_{00} + Q_{11} (x-8.3) + Q_{22} (x-8.3)^2 + Q_{33} (x-8.3)^2 
(x-8.6)$

this can be simplified (approximately) to...

$H(x) = -0.00202222 x^3 + 0.11044 x^2 + 1.70088 x - 3.00435$

which can easily have it's derivative taken wrt $x$.

My recommendation is to use your program to generate the Q values for 
the Hermite polynomial, then enter the polynomial into Sage or 
Mathematica in its "pre-simplified" canonical form --  and let the CAS 
simplify the expression and provide you with the derivative.



-- 
Andrew J. Pounds, Ph.D.  (pounds_aj at mercer.edu)
Professor of Chemistry and Computer Science
Mercer University,  Macon, GA 31207   (478) 301-5627
http://faculty.mercer.edu/pounds_aj

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