[ Pobierz całość w formacie PDF ]
exception of algebraic number computation, practical
ond order ODE's are developed in two of the references
and desirable.
([4] [8]). Providing a complete complement of tools
ranging from closed form to series solutions to numer-
ical solutions, should be one of the goals of algebraic
5 Notes on Tools and Timings
manipulation systems in the next few years. (Since the
As far as the tools in MACSYMA or similar programs rst draft of this paper was written in 1980, in fact,
for handling series, it appears that systems should pro- some progress has been made on closed form solutions:
vide appropriate access to the data structures on power See, for example, [2], [9])
5
In summary: these iterative tools should be more [7] J. D. Lipson. Newton's method: A great algebraic
widely used than they are their applicability can and algorithm, Proc. 1976 ACM Symp. on Symbolic
should be extended to help solve iterative algebraic and and Algebraic Computation, R.D. Jenks, ed., 260-
transcendental equations, and di erential equations. 270.
While we have illustrated only the simplest forms, it
[8] Y-M Shen. SOODESIS, Term Project for CS 292s,
is straightforward to extend these examples to compu-
Winter, 1980 (University of California, Berkeley)
tations over systems of equations, and even to exploit
20 pp.
parallel processors.
[9] M. F. Singer. Liouvillian solutions of n-th order
homogeneous linear di erential equations, Amer.
7 Acknowledgments
J. Math. 103, (1981) 661-681.
This work was supported in part by the follow-
ing grants: National Science Foundation under grant
Appendix
number CCR-8812843 Army Research O ce, grant
DAAG29-85-K-0070, through the Center for Pure
This is a listing of the \ tweek" program.
and Applied Mathematics, University of California at
Berkeley the Defense Advanced Research Projects
Agency (DoD) ARPA order #4871, monitored by
tweek(h,n) changes the truncation upper degree
Space & Naval Warfare Systems Command under con-
of a taylor series h to degree n.
tract N00039-84-C-0089, through the Electronics Re-
e. g. if f: taylor(1+x,x,0, 10)
search Laboratory, University of California at Berkeley
f: tweek(f,20) then
the IBM Corporation a matching grant from the State
f i s same as taylor(1+x,x,0,20)
of California MICRO program.
(defun $tweek (h n)
(cond
References
((atom h)) do nothing
((memq ' trunc (car h))
[1] M. Anderson. An implementation of the Newton
patch Macsyma' s taylor series
polygon procedure, Term Project for CS 292s,
representation in 2 places.
Winter, 1980 (University of California, Berkeley),
(rplaca (car (cadaar(cdddr(cdar h)))) n)
1980.
(or (numberp (cadr h))
[2] J. Della Dora, C. Dicrescenzo, E. Tournier. An al-
don' t change a constant powerseries
gorithm to obtain formal solutions of a linear ho- (rplaca (car (cadddr h)) n))
mogeneous di erential equation at an irregular sin-
)) h)
gular point, Proc. EUROCAM ' 82, Lecture Notes
in Computer Science, 144, Springer Verlag, 1982.
(setq taylor_simplifier nil)
[3] W. G. Dubuque. Series Solution of Functional
Equations:Taylor-solve, MACSYMA Newsletter 3
no. 3 July, 1986, 18-19.
[4] R. Fateman. Some comments on series solutions,
Proc. 1977 Macsyma Users' Conf., NASA CP-
2012 (July 1977) 43-52.
[5] K. O. Geddes. Convergence Behaviour of the New-
ton iteration for First Order Di erential Equa-
tions, Symbolic and Algebraic Computation, Lec-
ture Notes in Computer Science 72, Springer Ver-
lag, 1979, 189-199.
[6] H. T. Kung and J. F. Traub. All algebraic func-
tions can be computed fast, J. ACM, 25, 2 (April
1978), 245-260. See also: R.P. Brent and J. F.
Traub. On the Complexity of Composition and
Generalized Composition of Power Series, SIAM.
J. on Comp, 9, 1 (Feb. 1980), 54-66.
6
[ Pobierz całość w formacie PDF ]
zanotowane.pldoc.pisz.plpdf.pisz.plcs-sysunia.htw.pl
|
