P-adic Reconstruction
Paul S. Wang*
Kent State University
Department of Mathematical Sciences
Kent
Ohio U.S.A.
44242
of Rational Numbers
M.J.T. Guy
University of Cambridge
Computer Laboratory
Corn Exchange Street
Cambridge, England
CB2 3QG
J.H. Davenport
Emmanuel College
Cambridge
England
CB2 3AP
I. Introduction
clearly satisfies
In a recent paper, Wang [1981] introduces a p-adic
algorithm for the construction of partial fraction
decompositions. This differs from the usual p-adic
algorithms for factorisation or the computation of
greatest common divisors ([Wang, 1978], [Wang, 1980],
[Moore & Norman, 1981]) in that the p-adic image is
used to reconstruct rational numbers, rather than
integers.
In order to generate these rational numbers, Wang
[1981] gives an algorithm RATCONVERT (p. 216) to
deduce the rational u/v such that uv -I = c (mod m),
with luJ, IvI < J(m/2). In this paper we prove that
RATCONVERT always reconstructs such a number, if it
exists. As Wang points out, the representation, if it
exists, is unique. Such a number need not exist: e.g.
modulo 9 we have that 0/I = 0/2 = O, I/I = 2/2 = I,
2/I = 2,
I/2 = 5,
-I/I = -2/2 = 8,
-2/I = 7,
-I/2 = 4, leaving
3 and
6
without
any
such
representation.
v3/v2 5. c (mod m).
We have to prove that any solution to a/b = c
satisfying the size constraint la~, Ibl < ./(m/2) will
be generated by this algorithm.
3. Continued
Fractions.
The algorithm can be viewed another way: as the
calculation of approximations to c/m, since
-Vl/V 2 = c/m - v3/(mv2).
The numbers q computed in step [3] are, in fact, the
successive
terms
in
the
continued
fraction
approximation to c/m (see [Hardy & Wright, 1979]
Theorem 161 or Knuth [1981] pp. 339-343):
I
C
m
ql +
1
q2 +
I
2. The Algorithm
q3
Wang's
algorithm
slightly) :
is
(we
adapt
the
+
"'"
"
notation
After
one
iteration
of step
[33, we have u I : O,
u 2 : I and -Ul/U 2 is the first approximation
(0) to
[I] u:= (I, O, m), v:: (0, I, c)
c/m, and
second
[2] While J(m/2) ~ v 3 do
approximation
Algorithm
[3]
{q::
RATCONVERT(c ,m)
Lu3/v3],
[4] If ~v21 ~ J(m/2)
[5] Return
r::
u-qv,
u::
v, v::
v I = I, v 2 = q1' and
-Vl/V 2 is the
(I/qi) to c/re. The definition
of v I, v 2
in terms of the previous values of Ul, u2, Vl, v 2 and
r}
q is
then error()
then
(after
convention)
(v3,v 2)
allowing
just
Pn+1 = qPn+Pn-1
for
the
for computing
our
different
standard
sign
iteration
a convergent
in terms
[Wang ensures that v 2 is positive, but we do not
need this complication]
of its two immediate predecessors. Hence we have that
Throughout
hold:
convergents
the
the iteration, the following equations
ulm + u2c : u 3,
vlm + v2c = v 3.
If the algorithm terminates with a solution, then
it
sequence
of
-Vl/V 2
is
the
sequence
of
to c/m.
Conversely, suppose that a/b = c (mod m), with
lal, Ibl < J(m/2). Then a+dm = bc for some integer d,
so that
d
_
C
_
b
a
_
=
_ -
m
bm.
Hence d/b is an approximation
lal, Ib~ < J(m/2), the error is
I al
I--I_<
Ibml
J(m/2)
1
......
We can now a p p e a l
184 (p. 153): I f
is
is
Running
<--2bY(m/2) -
Hardy & W r i g h t
IP
I--xl
Iq
then p/q
(d, b, a)
1
_<
bm
to
to c/m, and, since
I
I
2b 2 .
[1979]
Theorem
Time
1
a convergent [to
indeed generated
and
2q 2
Czechoslovak
x ] . Hence t h e t r i p l e
by t h e a l g o r i t h m , and
The
users
of
with
Option
Academy
Astronomical
acquainted
a
Kalina
of S c i e n c e s
Institute
6, B u d e c s k a ,
so the algorithm is guaranteed to find a rational in
the appropriate range which corresponds to c mod m,
if such a value exists.
Lisp/360
under
Sharing
Zden~k
<--- -
REDUCE
interactively
120
23 P r a g u e
algebraic
many
systems
advantages
of
are
using
4. References
diaglog
versions
We
Hardy & Wright,1979
Hardy, G.H. & Wright,E.M., An Introduction to the
Theory of Numbers (5th. ed.). Clarendon Press,
Oxford, 1979.
Knuth,198 la
Knuth,D.E., The Art of Computer Programming, Vol.
II, Semi-numerical Algorithms. Second Edition,
Addison-Wesley, 1981.
Moore & Norman,1981
Moore,P.M.A.
&
Norman,A.C., Implementing
a
Polynomial
Factorization
and
GCD Package.
Proceedings of the 1 9 8 1 ACM Symposium on
Symbolic and Algebraic Computation, ACM Inc.,
New York, 1981, pp. 109-116.
Wang, 1978
Wang,P.S., An Improved Multivariable Polynomial
Factorising Algorithm. Math. Comp. 32(1978) pp.
1215-1231. Zbl. 388.10035.
REDUCE
Time
Wang, 1981
Wang,P.S., A p-adic Algorithm for Univariate
Partial Fractions. Proceedings of the 1981 ACM
Symposium
on
Symbolic
and
Algebraic
Computation, ACM Inc., New York, 1981, pp. 212217.
* R e s e a r c h s u p p o r t e d in p a r t by g r a n t s f r o m
the N a t i o n a l S c i e n c e F o u n d a t i o n
( N S F - M C S 8 0 - 0 2 4 1 4 ) and the D e p a r t m e n t of
E n e r g y (DE A S O 2 E R 7 6 0 2 0 7 5 - A 0 1 0 ) .
in use
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made
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COSYMS
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Option
centre.
dialog
these
of Professor
Lisp/360
with
Wang, 1980
Wang,P.S., The EEZ-GCD Algorithm. SIGSAM Bulletin
14(1980) 2 pp. 50-60. Zbl. 445.68026.
have
of
Analyser
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