By Christoph KOUTSCHAN Doron ZEILBERGER
If you want to see a whole computer-generated paper, done completely automatically
(both the Pekeris part and the Accad part) reproducing, ab initio sections 1-4
of Pekeris' seminal paper, including Table III, using the largest ω as 11 (as in that paper)
The above output file only reproduced the values in Pekeris' paper, for Z from 1 to 10.
If you want an extended table for Z from 1 to 100 (in addition to doing all the rest)
(Note that the dependence on Z, not surprisingly, is almost linear)
If you want to see a whole computer-generated paper, done completely automatically
(both the Pekeris part and the Accad part) reproducing, ab initio sections 1-4
of Pekeris' seminal paper, including Table III, but with the largest ω equal to 16,
getting a higher accuracy
If you want to see a whole computer-generated paper, done completely automatically
(both the Pekeris part and the Accad part) reproducing, ab initio sections 1-4
of Pekeris' seminal paper, including Table III, but with the largest ω equal to 21,
getting even a higher accuracy
If you want to see a whole computer-generated paper, done completely automatically
(both the Pekeris part and the Accad part) reproducing, ab initio
the energy calculation in C. L. Pekeris' follow-up paper (1 1S and 2 3S States of Helium,
Phys. Rev. v.115(5), 1216-1221), for the ortho state (sec. 3) but not just for Helium (Z=2) but
for all Z from 1 to 10,
If you want to see the above but with Z from 1 to 62
.pdf
.ps
.tex
Appeared in the Mathematical Intelligencer Volume 33, Issue 2 (2011), 52-57
Written: May 26, 2010.
It is amazing what the eminent applied mathematician,
Chaim Leib Pekeris, in collaboration with
machine-language-programming-whiz Yigal Accad, could squeeze out of the
"primitive" (by today's standards) computer WEIZAC.
It is also amazing how Pekeris had the patience to derive, purely by hand, a monstrous (33-term!)
(see Eq. (22) of Pekeris' 1958 paper)
partial linear recurrence equation (with (fairly complicated) polynomial coefficients) that today can be derived
by Maple or Mathematica in a fraction of a second (after a couple of hours of programming).
Note: This will form the subject of Doron Zeilbeger's
17th Chaim Leib Pekeris' Memorial Lecture
to be delivered on June 30, 2010, 4:00-5:00pm, local time.
Mathematica Package and Matlab code:
See
Christoph Koutschan Mathematica/Matlab webpage .
Maple Package:
Important: This article is accompanied by the Maple
package
Sample Input and Output for the Maple package PEKERIS:
the input
gives the output.
the input
gives the output.
the input
gives the output.
the input
gives the output.
the input
gives the output.
the input
gives the output.
Doron Zeilberger's List of Papers