Math. 55
Computing x
n
March 20, 1999 5:18 pm
Prof. W. Kahan Page 1
This note shows how a nontrivial but still simple algorithm
npow
is proved correct.
Given x and n , how quickly can x
n
be computed? We assume that n is an integer and that
x is a real number or a matrix or a polynomial for which multiplication takes so much longer
than everything else in our algorithm that only the number of such multiplications matters. In
the algorithm that follows, “
Matrix
” or “
Polynomial
” can be substituted for “
Real
” ,
whereas a formula like x
n
= exp(n·ln(x)) would not be practical for polynomials x .
As it happens, no simple algorithm is known that computes x
n
for every integer n
≥
0 in the
least possible number of multiplications ( and no divisions ) for that n . However, here is an
algorithm that is optimal in this regard for |n| < 15 and otherwise not much worse than optimal:
Real Function npow(Real x, Integer n) :
... returns x
n
.
Real z ;
Begin
If ( n < 0 ) Return reciprocal( npow(x, -n) ) and Exit ;
If ( n = 0 ) Return Real(1) and Exit ;
While ( n is even ) Do { x := x
2
; n = n/2 } ;
z := x ;
Do { n := floor(n/2) ;
If ( n = 0 ) Return z and Exit ;
x := x
2
;
If ( n is odd ) z := x·z ; } forever :
End npow .
Let’s follow the progress of
npow(X, 15)
as each variable changes its value:
n = 15 , x = X ,
( Arguments are copied, “passed by value”.)
z = X ,
n = 7 , x = X
2
, z = X
3
,
2 Real multiplications
n = 3 , x = X
4
, z = X
7
,
2 more Real multiplications
n = 1 , x = X
8
, z = X
15
,
2 more Real multiplications
n = 0 , return X
15
in 6 Real multiplications.
But there is a faster way: X
15
= [(X
2
)
2
·X]
2
·[(X
2
)
2
·X] in 5 Real multiplications. Anyway,
the algorithm’s number of multiplications must grow like
Ω
(log
2
(|n|)) as n —›
∞
since k
multiplications are insufficient to compute x
n
if n > 2
k
; can you see why?
How can
npow
be “ proved ” correct? At best we can try to prove that this algorithm fulfils
some expectation expressed as a set of specifications. Here a question arises reminiscent of the
question raised by the Roman satirist Juvenal (
ca
. 60 - 140 AD.):
“
Sed quis custodiet ipsos Custodes?
” ( “But who is to watch the Watchers?” )
Many a program has been proved correctly to implement incorrect specifications. This is no
excuse for shirking such proofs; rather it is a warning that, first, our specifications also have to
be proved correct instead of mere wishful thinking perhaps inconsistent, and, secondly, good
reasons exist to test programs even if tests can prove no more than that something is wrong.
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