ISSN 0001-4346, Mathematical Notes, 2015, Vol. 97, No. 4, pp. 531–555. © Pleiades Publishing, Ltd., 2015.
Original Russian Text © S. B. Gashkov, 2015, published in Matematicheskie Zametki, 2015, Vol. 97, No. 4, pp. 529–555.
Arithmetic Complexity of Certain Linear Transformations
S. B. Gashkov
*
Lomonosov Moscow State University, Moscow, Russia
Received September 1, 2012; in final form, September 5, 2014
Abstract—We obtain order-sharp quadratic and slightly higher estimates of the computa-
tional complexity of certain linear transformations (binomial, Stirling, Lah, Gauss, Serpi ´ nski,
Sylvester) in the basis {x + y}∪{ax : |a|≤ C} consisting of the operations of addition and in-
ner multiplication by a bounded constant as well as upper bounds O(n log n) for the compu-
tational complexity in the basis {ax + by : a, b ∈ R} consisting of all linear functions. Lower
bounds of the form Ω(n log n) are obtained for the basis consisting of all monotone linear
functions {ax + by : a, b > 0}. For the finite arithmetic basis B
+,*,/
= {x ± y, xy, 1/x, 1} and
for the bases {x ± y}∪{nx : n ∈ N}∪{x/n : n ∈ N} and B
+,*
= {x + y, xy, −1}, estimates
O(n log n log log n) are obtained in certain cases. In the case of a field of characteristic p, com-
putations in the basis {x + y mod p} are also considered.
DOI: 10.1134/S0001434615030256
Keywords: linear transformation (binomial, Stirling, Lah, Gauss, Serpi ´ nski, Sylvester), arith-
metic computational complexity of linear transformations, finite arithmetic basis, field of
characteristic p,
1. INTRODUCTION
In the present paper, we consider widely used finite-dimensional linear transformations over fields
of real and complex numbers (and over finite fields) and estimated their computational complexity
using various collections of elementary arithmetic operations in these fields. These sets of elementary
operations are called bases (there is usually no mistake of identifying them with the notion of basis in
a linear space). By computational complexity we mean the number of performed operations (in what
follows, we shall present the formal definition).
In combinatorics [1], [2], we have the binomial transformation R
n
→ R
n
and its inverse defined,
respectively, by the formulas
y
k
=
k
m=0
k
m
x
m
, x
k
=
k
m=0
(−1)
k−m
k
m
y
m
, k =0,...,n − 1,
where the
(
k
m
)
= k!/(m!(k − m)!) are the binomial coefficients. The matrix of this transformation is
triangular; its nonzero elements constitute a Pascal triangle of order n. Using Pascal’s identity, we
can easily compute all the numbers generating the Pascal triangle by using (n − 1)(n − 2)/2 operations
of addition
1
. This algorithm is optimal in the case of the additive basis B
+
= {x + y, 1} and it is also
optimal in a wider basis
2
.
The computational model used in the case of the basis B
+
is known as addition chains [3]. By an
addition chain of length l we mean a sequence of natural numbers a
0
,...,a
l
, in which a
0
=1 and
each number a
i
is equal to the sum of two preceding numbers a
j
, a
k
, j<i, k<i (not necessarily j = k;
in the case j = k, the corresponding step of the chain is called a doubling step). An addition chain can
*
E-mail: sbgashkov@gmail.com
1
In view of its symmetry, the actual number of additions is almost twice less.
2
Because the number of operations in a computation cannot be less than the number of numbers to be computed.
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