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Optimum multiplication algorithms using multiples of the multiplicand
OPTIMUM MULTIPLICATION ALGORITHMS USING MULTIPLES OF THE MULTIPLICAND* V. M.
MIKHELEV
(MOSCOW) (Received
The
speed
rithms
of
multiplication
using
special
both
features
where
the
the of
relative
independent
of
the
different.
This
However,
quired.
Consequently.
of
cases
summations the
From a mathematical to
greatest
time
ments for
the
tion
of
the
does
code
the
of
the
the
is
of
factor
or the
view,
higher
realization
of
formation
which
of
than is
of
com-
can be a
is
re-
either
of
(subtractions),
or
by
algorithms is
the number of
an appropriate
de-
by re-
displacements,
construction
in
some-
may or may not
can be reduced
reducing
class
by one number,
number of
the
code,
one
and displacements
one operation the
algo-
and certain
a binary
range
summator.
for
summations
of
generally
non-binary,
determining
the
factor
Mat.
which
4,
of with
No.
5,
is
a control
this
problem
transforms
form of
transfer
a minimum as compared
l’ych.
the
multiplication,
an algorithm
Zh.
of
the
multiplication
the methods
involve
of
l
to
“optimum”
device
of
the
displace-
logical
scheme
mechanism.
Since which
point
since
basically
within
code
of
a minimum number of
interest
the
characteristic
(subtractions)
pend on the actual code value of ducing the number of summations, both at the same time.
leading
of
minimum time
in all
number of
by various
concept base. The “optimum” algorithm according to the technical possibilities
requiring
certain
is
operations
transmission
algorithms
1963)
possibilities
code.
number of
the
October
can be increased
technical
the code with any other what conditional since, puters,
26
the
recording,
the
multiplicand
any other
956-963. 243
code is factor
1964.
code
in which into
algorithm
for
reduced the the
using
the mechanism to into
the
formula-
another
number of summator, the
symbols would be
same symbols
244
V.W.
dMikhelev
for recording the factor. Naturally, the symbols used should conform to the technical possibilities of the type of machine used. All the algorithms studied below assume in particular, the presence of a certain number of discriminators or other devices in the machine for storing or obtaining the multiples of the multiplicand which can be transmitted with a plus or minus sign to the summator. In this case it is advisable + - + to use e.g. the following as symbols: 0, 1. 1. 3. 3 or similar symbols denoting the transfer to the summator of the multiplicand or trebled multiplicand with a plus or minus sign. Using these symbols, the binary cods of factor 0111101101 can be written, e.g. in one of the following forms:
fn the first case to multipiy. seven additions have to be made; in the second, three and one subtraction and in the third, one addition and two subtractions. It can be seen from the example given that even with the according to the algorithm, can be represented same symbols the factor, in different forms by different numbers of operations. As a very simple
example we will
consider
two algorithms which trans+ form a binary code to a code with symbols 0. 1 and i. The first algorithm indicates that the multiplicand is transmitted to the summator only if the value of the factor order is changed to a binary entry (transition This algorithm can be exfrom zero to unity and from unity to zero). pressed in the form of the logical functions Si = (Ui# Qi_1),
sign si = Ui,
where Si denotes the transfer of the factor to the summator on multiplying by the i-th order of the factor and sign Si is the sign of the operation. With sign Si = 0 addition and with sign Si = 1, subtraction is carried out. The meaning of the algorithm is to replace a group of units by two unities with plus and minus signs. It is easy to see that the + number of operations (additions 1 and subtractions 1) in the transformed code will be an average of (a + 1)/2, where n is the length of the factor code. Another algorithm which also uses symbols 0, l and 1 for recording a transformed code, can be expressed in the form of the logical functions si = (ai + Ui_i)Si_i’
sign si = ai+i.
Optimum multiplication
This the
algorithm optimum
was proposed
algorithm
age number of
i. e.
for
operations
a high
These method
value
factor.
orders,
denote
US
than
of
by S; the
the if,
multiplication
operation
by the
i-order
the
the with of
of
the
the
the
the
factor
is
previous
using to
factor i-th
the
general
multiples
the
of
summator
order
factor,
aver-
algorithm.
by an i-th
operation
is
case
to understand
algorithms
transmission
as a result
be seen,
In this
for
possible
class
as will
by an n-order
when multiplying
only
and,
24.5
symbols.
less
make it
a wider
Obviously, out
analogous
1% times
examples
multiplicand
can be carried out,
n.
of
Let
a j-multiple the
of
formulation
[II
by Lehman in using
q when multiplying
two simplest
of
a multiplicand. of
for
algorithms
of
of
order the
factor
in previous
has not
been
carried
i.e.
_
.vi =
where
pi
is
the
-.s.1_lsi_’
function
orders by which with tion can be effected, plicand
for
of
with the
1
-. s’lo. i-k-1
.
resolving
_.
;j,,
’ ’ ’
i-_k__l
the
operation,
s’lii
i-/(-l’
k
is
the
selected multiples of the multiplicand, irrespective of the value of orders of
and k + 1 is
code,
can be effected multiples
-Al Yi_&&
.
_
the
a certain
number of code
of
orders
by which
the multiplicand,
number of
multiplicathe multi-
multiplication using
certain
multiplicand.
The operation which has to be carried out in the i-th order of the factor in order to multiply by the i-th and subsequent orders, depends on the nearest orders of the factor and is carried out only in case of resolving
the
operation .s!,= $j (. . ., ai+l’ aj, ui_l, . . .) <.
The sign the
of
the
operation
is
also
determined
by the
nearest
orders
of
factor sign
Thus a set information
of
logical
on the
Si =
functions operations
q(. . ., cZi+l,(li,Ui_1,. . .).
is
obtained,
in previous
the orders
factor being
orders the
and the
initial
in-
formation. Logical functions y and 9 can be selected by widely different methods. Obtaining the optimum algorithm depends on the method of selection. We will now consider the subclass of algorithms of factor coding which is
246
V.M.
distinguished present, i. e.
by the
,w is
class
of
the
number of
algorithm
changing
sign”
from
condition
the
considering groups
in Ivj a factor
of
of
various
multiples
of
be termed
factor
coding
that
the
of
logical
multiplying
admissibility
units
the
will
implying
the
of
express
that
the
type
oi
f
“i-l
is
$ @i* ai_l)and~,;V~p;V~~V...VV~=”
4) = where
fact
Mikhelev
zeros
are
logical
si = (ai # ai_l)
sf_ 1
4 = $i(. . ., ai+l,
.
functions
made.
Taking
equation .
‘i-k -&
i_k_1
are
for
selected
clearly
number of
Si = (ai #
system . * .
This
“operation
Thus operations
subtraction.
not
*
with
by a maximum possible
of
following
the multiplicand.
for
the
we can
ai_l)<,
this
orders
inside
class:
?&_, . . . q’k_l. . . ij&_,,
. . .)si,
ai, ai_l,
sign si = 9 (. . ., ai+l, ai, ai_i, . . .). We will
now prove
in relation
to the
of
the
average
of
the
multiplicand.
The set are
ak+l
jc <
is
is
example
factor ation
of j<
the
are
optimum
if
of
. . . .OllOlO..
effected “with
multiples
If the selection
leads set of of the
one hand,
number of changing
three
1,
2,
of
the
algrithms
5,
of
. . For a set in the
form
from unity
multiples
jz, . . . , jl
where ak -
an operation
is
1 <
changing
dense.
changing the also follows
to unity.
At the
the sign” +-+ . ..lOlOlO...
operations
or
for
multiples
when changing
on the other, unity
number of
sign” from
multiples of 1 and 5 this + + . ..010050... and the second oper-
can be carried
to
assumptions reduction
of
multiplication and,
of
. . . , Zk - 1, jl,
“with
is dense, it is operation by the
orders
the
multiplicand,
multiplicand
multiples required
from zero
certain
character
“with an operation for multiples is not dense,
operation to
type
set
by changing the
with
the
odd.
of
the
explain,
on increasing
multiples
can be represented is
algorithm
the
operations
The fact that the algorithm not optimum, if the set of
the
of
multiples,
The subclass
f.
sign”
which
factor
of
set
I and all
-
Theorem
the
multiples
a dense
theorems
of
number of
of
termed
the set
the
from unity
possible following out
same time the
the
same set
to organize the method so that
on
by the
maximum possible
is
out
operation to
for
zero.
carried
by
multiplication
Optimum
We note that the
order
always
by
by
the
(k
dense
+ 1)
k orders.
set
of
algorithms
makes
multiples
with
almost
any
We use
the
following
247
it possible
combination
of
selection
to
factor of
multiply
orders
and
multiplicand
multiples: 1)
if
ai+k+I
= 0.
2)
if
ai+k+l
= ai+k+2
then
then
= . . .
aa . . . ap is
the
absence
carried
ples
of
out
out
the
With
real.
can
with
be
sign.
in
Since
the
optimum
“with
We will number
operations
multiples
be termed Theorem
using of
2.
used. the
For
a complete
operations n/(3 the
multiplicand.
For
a given
termined
Let of
us
+ log2
w) for
class
by the
denote
a j-order
of
orders
of
logical
by Pj factor.
of
entry
of
of
is
Is
always begins
from
1 to
to
k + 1 by
Place
the
place
0
the as
k,
following
by changing for
multi-
carried
unity
i.e.
selected
of
operation
replacing
to
j
versa.
4i+k+l,
procedure
set
the
multiplication
by cbsnging
a subclass
of
by
the
sign.
algorithms
sign”. of
the
dependence
algorithm
set
of
on the
the
type
multiples
*with
of
an operation
of
the
n values,
II is
existence
the
3.
5,
average of
multiplicand.
for
changing the
the
the
operation
in
1
sign’ number
multfplier
number
of
2k -
average
n-ordered the
multi-
. . . .
the
by one
where
the
1,
of
number
multiplicand,
multiplication
algorithms
value
vice
takes
belongs the
multiples
large
f
same the
addition
of
is
the
since
following
and takes
nature
set
case also
an optimum
for
binary
operations
(2).
the
case
The dense
necessary
is
of
algorithms set
of
order this
complete
= 0,
(or
Set,
case
the
of
When ei+k
in
the
(1)
The
the
this
involves
changing
for
factor
respective
case
constructed for
now elucidate
of
plicand will
algorithm
an operation
ei+k+k+l
following
of
the
i + k + 1-st number
the
In
In
the
case
in
1 + 1.
operation
of
the
of
OCCUrS Only
a maximum possible the
the 1,
i + k +
k.
dense,
complement
seen,
the
In
= 0;
and Sign Si = 1,
multiple
each
0 to
the
number
easily
is
sign. from
obtaining
operation
= 1 and
code
k + 1 by
selection
Si
multiple).
a suitable
multiplicand
by changing
order
of
the
replacing
this
when changing since
‘Li+k+I
a binary
of In
=
j = 2itktz+1 - ai+k+l .. . ai
where
Is
j = Oi+k.. . Oi and sign
of
multiples
operation
is
de-
function
the For
probability Pj
based
of
the
on the
logical
function
the of
j-tb order the
248
a difference
we have
operation,
Pj = $ (’ The particular teristic
solution takes
2&-
zk-i f
us consider
Let
of
equation
equation
-Pj_l
-
this
the
Pj-2
-
equation
root
to in addition X0 = 1. We have
iik (2; Consequently, Since terms of j
it
value l,‘(k
It of
+ 2);
,*+I _ Z- i
I”+
the
charac-
1
= 0.
i = r-j *
the
initial
:zjq,z -
!I=
iand
or
I z( < 1, since
solution
the
Pj = l/(k
of
independent
is
Thus,
Pj_*)-
of
the
of
theorem
I;{”
contains
I 23I # 1.
homogeneous
length
of
proved.
another
(2 /;I-i)[
equation
Ci is a constant, for Thus, the probability
the is
equation,
the
code
Since
comprises
k
higher Pj for
value a high
the of
and is
equal
k = 1 + logEm,
to then
+ log,m). is
the
j
2).
q En/(3
general
roots
the type ci(xi)j, where can be as low as desired.
of f
the
lZ/ < 1,
the
of
-
equation
1,
1) =
is
or
22”+” _ :k _ which,
*
form
. * * +z+i=O,
the
* ’
easy
to
see
that
multiplicand,
the
with
a dense
number of
but
set
incomplete
operations
of
in the
q lies
multiples range
&
consider
with of
symbols
examples
multiplier
zero
of
the coding + + 0, 1. 1, 3, orders.
to unity;
the
algorithms
algorithm 3. Table
The first
second,
vice
belonging using
to
a given
a threefold
1 shows
all
the
column
corresponds
versa,
from unity
possible to to
zero.
-
I
i
i+l
I
-
i+2
i-l
j
if-l
j
i
/
we i.e.
combinations
transition
TABLE 1
ii_?
class,
multiplicand,
i-l
-
0
0
1
0
;
0
u
0
1
0
1
1
i)
$
I)
1
0
1
1
0
1
0
i
1
0
0
I
i
1
1
0
T
I
I
u
I
from
Ootinrun mu~t~plicaiio~
249
afrrorithms
Operations in the first and second row of the first column cause DO Uncertainty. Subtraction in the third and fourth, apparently, lead to a minimum number of operations irrespective of the code value in i + 3 and subsequent orders. It should be noted that since subtraction is effected in thse orders. in the i t 3 order unity should be added. Multiplication is thus. in fact, carried out by at least three orders. In case of unity has to be effected in the (i + 3) or in the fi + 2) order, correction next order, where the operation has to be carried out, i.e. where the sign is changed. This has been taken into account when determining the operation in the second column. It is carried trebled ation is pressed
easy to see that the operation with a simple multiplicand is out in the case of equality of (i + I) and (i + 2). and with a multiplicand, in the case of inequality. The sign of the operdetermined by the (i f 2) order. Thus, the algorithm can be exin the form _ $4= (di # oi_i) s*_iSi.+ ‘t = (‘i+i # ‘$*s) si, $ = (ai+i = a*+s)si, sign si = u{+~.
This algorithm
is optimum since
the set
The average number of operations factor for this algorithm is
q2++-
1 2 o/y)
(l/Tjn
of multiples
in mUltipliC8tiOn
. s’n
[
(n
- if arc tg VT-
is dense. by one a-th order
arc tg 5v-
. 3
This formula does not take into account the operation in the (II + 1) order, if the previous operation was subtraction. This, however, is not necessary since the known methods of multiPlication by a negative number mean precisely that multiplication by an order following the sign, is not effected. The result of the formula is similar to the proof of theorem 2. In this case. however, the determination of the roots of the characteristic equation can be completed. Here
It is easy to see that q & n/4 directly also from the general
for high values of n which follows theory since this algorithm belong
to the
V.M. Mikhelev
250
subclass of algorithms “with an operation set of multiples Is complete. As an example of the algorithm multiples, an algorithm sidered (see Table 2).
using
with symbols
0,
for
changing
the sign”
and the
a dense but incomplete set of t t - + 1. 1. 5. 3, 3 and 5 may be con-
TABLE 2
i+3
I
i+2
i+l
i-f
i+3
i+.Z
i+r
i-i
I
0
0
0
0
0
0
0
1
0
1
1
0
0
1
1
1
1
0
0
0
1
0
0
1
i
i
1
0
1
1
1
1
In all the rows of the first column in Table 2, in addition to the fourth and fifth, the operation selected mahes it possible to multiply by orders of up to i + 3. In the fourth and fifth rows multiplication is carried out by orders of up to i + 2 inclusive. However, in this case, there is a change of sign in i + 3. In addition, all the rows, in which It is remembered that unity is added to the order following the last, by contain unity in this last order. which multiplication has been effected, This has been taken into account when determining the operation in the second column. It is easy to see that operation with a simple multivlicand (I or 1) is orders and with ity. The treble the equality of plicsnd, in the in the form
effected with the equality of the (i f 1) and (i + 2) treble or quintuvle multiplicand, ia the case of inequalmultivlicand is transferred to the summator in case of the (i + 2) and (i + 3) orders and the quintuple multicase of inequality. Thus, the algorithm can be expressed _ _ Si = (ai # a*_i) si-l ‘i-e’:-,* s; = (uitL=
54.2 1 ‘i,
sf =
‘I$+3 #
tai+$ =
ai+l)
“i,
Optimum
multiplication
sf=(ai+3=ai+l# sign si = (ai+ 1 = ai+ J ai+
For this ations
algorithm,
which is also
251
algorithns
ai +2) %,
V (ai+1# ai+2)‘i+ 3.
optimum, the average number of oper-
is
All the observations in this case also.
concerning
g for
the previous
algorithm
are valid
It follows from theorem 2 that the average number of orders, by which multiplication can be effected in one operation (n/q), is proportional to the logarithm of the number of multiples of the multiplicand (3 + log2 m). Thus, the number of orders has increased much more slowly than the number of multiples, which reflects increased equipment in the multiusing a trebled plying mechanism. From this point of view, multiplication multiplicand, is the most effective. However, in sequential machines a sufficiently large number of multiples can easily be obtained with only one additional summator which makes it possible to use more complicated algorithms.
REFERMCES 1.
Lehman, Y.,
IRE Trans.
Electronic
Comput.,
Sept.,
EC-6, 204-205.
1957.
MIKHELEV
(MOSCOW) (Received
The
speed
rithms
of
multiplication
using
special
both
features
where
the
the of
relative
independent
of
the
different.
This
However,
quired.
Consequently.
of
cases
summations the
From a mathematical to
greatest
time
ments for
the
tion
of
the
does
code
the
of
the
the
is
of
factor
or the
view,
higher
realization
of
formation
which
of
than is
of
com-
can be a
is
re-
either
of
(subtractions),
or
by
algorithms is
the number of
an appropriate
de-
by re-
displacements,
construction
in
some-
may or may not
can be reduced
reducing
class
by one number,
number of
the
code,
one
and displacements
one operation the
algo-
and certain
a binary
range
summator.
for
summations
of
generally
non-binary,
determining
the
factor
Mat.
which
4,
of with
No.
5,
is
a control
this
problem
transforms
form of
transfer
a minimum as compared
l’ych.
the
multiplication,
an algorithm
Zh.
of
the
multiplication
the methods
involve
of
l
to
“optimum”
device
of
the
displace-
logical
scheme
mechanism.
Since which
point
since
basically
within
code
of
a minimum number of
interest
the
characteristic
(subtractions)
pend on the actual code value of ducing the number of summations, both at the same time.
leading
of
minimum time
in all
number of
by various
concept base. The “optimum” algorithm according to the technical possibilities
requiring
certain
is
operations
transmission
algorithms
1963)
possibilities
code.
number of
the
October
can be increased
technical
the code with any other what conditional since, puters,
26
the
recording,
the
multiplicand
any other
956-963. 243
code is factor
1964.
code
in which into
algorithm
for
reduced the the
using
the mechanism to into
the
formula-
another
number of summator, the
symbols would be
same symbols
244
V.W.
dMikhelev
for recording the factor. Naturally, the symbols used should conform to the technical possibilities of the type of machine used. All the algorithms studied below assume in particular, the presence of a certain number of discriminators or other devices in the machine for storing or obtaining the multiples of the multiplicand which can be transmitted with a plus or minus sign to the summator. In this case it is advisable + - + to use e.g. the following as symbols: 0, 1. 1. 3. 3 or similar symbols denoting the transfer to the summator of the multiplicand or trebled multiplicand with a plus or minus sign. Using these symbols, the binary cods of factor 0111101101 can be written, e.g. in one of the following forms:
fn the first case to multipiy. seven additions have to be made; in the second, three and one subtraction and in the third, one addition and two subtractions. It can be seen from the example given that even with the according to the algorithm, can be represented same symbols the factor, in different forms by different numbers of operations. As a very simple
example we will
consider
two algorithms which trans+ form a binary code to a code with symbols 0. 1 and i. The first algorithm indicates that the multiplicand is transmitted to the summator only if the value of the factor order is changed to a binary entry (transition This algorithm can be exfrom zero to unity and from unity to zero). pressed in the form of the logical functions Si = (Ui# Qi_1),
sign si = Ui,
where Si denotes the transfer of the factor to the summator on multiplying by the i-th order of the factor and sign Si is the sign of the operation. With sign Si = 0 addition and with sign Si = 1, subtraction is carried out. The meaning of the algorithm is to replace a group of units by two unities with plus and minus signs. It is easy to see that the + number of operations (additions 1 and subtractions 1) in the transformed code will be an average of (a + 1)/2, where n is the length of the factor code. Another algorithm which also uses symbols 0, l and 1 for recording a transformed code, can be expressed in the form of the logical functions si = (ai + Ui_i)Si_i’
sign si = ai+i.
Optimum multiplication
This the
algorithm optimum
was proposed
algorithm
age number of
i. e.
for
operations
a high
These method
value
factor.
orders,
denote
US
than
of
by S; the
the if,
multiplication
operation
by the
i-order
the
the with of
of
the
the
the
the
factor
is
previous
using to
factor i-th
the
general
multiples
the
of
summator
order
factor,
aver-
algorithm.
by an i-th
operation
is
case
to understand
algorithms
transmission
as a result
be seen,
In this
for
possible
class
as will
by an n-order
when multiplying
only
and,
24.5
symbols.
less
make it
a wider
Obviously, out
analogous
1% times
examples
multiplicand
can be carried out,
n.
of
Let
a j-multiple the
of
formulation
[II
by Lehman in using
q when multiplying
two simplest
of
a multiplicand. of
for
algorithms
of
of
order the
factor
in previous
has not
been
carried
i.e.
_
.vi =
where
pi
is
the
-.s.1_lsi_’
function
orders by which with tion can be effected, plicand
for
of
with the
1
-. s’lo. i-k-1
.
resolving
_.
;j,,
’ ’ ’
i-_k__l
the
operation,
s’lii
i-/(-l’
k
is
the
selected multiples of the multiplicand, irrespective of the value of orders of
and k + 1 is
code,
can be effected multiples
-Al Yi_&&
.
_
the
a certain
number of code
of
orders
by which
the multiplicand,
number of
multiplicathe multi-
multiplication using
certain
multiplicand.
The operation which has to be carried out in the i-th order of the factor in order to multiply by the i-th and subsequent orders, depends on the nearest orders of the factor and is carried out only in case of resolving
the
operation .s!,= $j (. . ., ai+l’ aj, ui_l, . . .) <.
The sign the
of
the
operation
is
also
determined
by the
nearest
orders
of
factor sign
Thus a set information
of
logical
on the
Si =
functions operations
q(. . ., cZi+l,(li,Ui_1,. . .).
is
obtained,
in previous
the orders
factor being
orders the
and the
initial
in-
formation. Logical functions y and 9 can be selected by widely different methods. Obtaining the optimum algorithm depends on the method of selection. We will now consider the subclass of algorithms of factor coding which is
246
V.M.
distinguished present, i. e.
by the
,w is
class
of
the
number of
algorithm
changing
sign”
from
condition
the
considering groups
in Ivj a factor
of
of
various
multiples
of
be termed
factor
coding
that
the
of
logical
multiplying
admissibility
units
the
will
implying
the
of
express
that
the
type
oi
f
“i-l
is
$ @i* ai_l)and~,;V~p;V~~V...VV~=”
4) = where
fact
Mikhelev
zeros
are
logical
si = (ai # ai_l)
sf_ 1
4 = $i(. . ., ai+l,
.
functions
made.
Taking
equation .
‘i-k -&
i_k_1
are
for
selected
clearly
number of
Si = (ai #
system . * .
This
“operation
Thus operations
subtraction.
not
*
with
by a maximum possible
of
following
the multiplicand.
for
the
we can
ai_l)<,
this
orders
inside
class:
?&_, . . . q’k_l. . . ij&_,,
. . .)si,
ai, ai_l,
sign si = 9 (. . ., ai+l, ai, ai_i, . . .). We will
now prove
in relation
to the
of
the
average
of
the
multiplicand.
The set are
ak+l
jc <
is
is
example
factor ation
of j<
the
are
optimum
if
of
. . . .OllOlO..
effected “with
multiples
If the selection
leads set of of the
one hand,
number of changing
three
1,
2,
of
the
algrithms
5,
of
. . For a set in the
form
from unity
multiples
jz, . . . , jl
where ak -
an operation
is
1 <
changing
dense.
changing the also follows
to unity.
At the
the sign” +-+ . ..lOlOlO...
operations
or
for
multiples
when changing
on the other, unity
number of
sign” from
multiples of 1 and 5 this + + . ..010050... and the second oper-
can be carried
to
assumptions reduction
of
multiplication and,
of
. . . , Zk - 1, jl,
“with
is dense, it is operation by the
orders
the
multiplicand,
multiplicand
multiples required
from zero
certain
character
“with an operation for multiples is not dense,
operation to
type
set
by changing the
with
the
odd.
of
the
explain,
on increasing
multiples
can be represented is
algorithm
the
operations
The fact that the algorithm not optimum, if the set of
the
of
multiples,
The subclass
f.
sign”
which
factor
of
set
I and all
-
Theorem
the
multiples
a dense
theorems
of
number of
of
termed
the set
the
from unity
possible following out
same time the
the
same set
to organize the method so that
on
by the
maximum possible
is
out
operation to
for
zero.
carried
by
multiplication
Optimum
We note that the
order
always
by
by
the
(k
dense
+ 1)
k orders.
set
of
algorithms
makes
multiples
with
almost
any
We use
the
following
247
it possible
combination
of
selection
to
factor of
multiply
orders
and
multiplicand
multiples: 1)
if
ai+k+I
= 0.
2)
if
ai+k+l
= ai+k+2
then
then
= . . .
aa . . . ap is
the
absence
carried
ples
of
out
out
the
With
real.
can
with
be
sign.
in
Since
the
optimum
“with
We will number
operations
multiples
be termed Theorem
using of
2.
used. the
For
a complete
operations n/(3 the
multiplicand.
For
a given
termined
Let of
us
+ log2
w) for
class
by the
denote
a j-order
of
orders
of
logical
by Pj factor.
of
entry
of
of
is
Is
always begins
from
1 to
to
k + 1 by
Place
the
place
0
the as
k,
following
by changing for
multi-
carried
unity
i.e.
selected
of
operation
replacing
to
j
versa.
4i+k+l,
procedure
set
the
multiplication
by cbsnging
a subclass
of
by
the
sign.
algorithms
sign”. of
the
dependence
algorithm
set
of
on the
the
type
multiples
*with
of
an operation
of
the
n values,
II is
existence
the
3.
5,
average of
multiplicand.
for
changing the
the
the
operation
in
1
sign’ number
multfplier
number
of
2k -
average
n-ordered the
multi-
. . . .
the
by one
where
the
1,
of
number
multiplicand,
multiplication
algorithms
value
vice
takes
belongs the
multiples
large
f
same the
addition
of
is
the
since
following
and takes
nature
set
case also
an optimum
for
binary
operations
(2).
the
case
The dense
necessary
is
of
algorithms set
of
order this
complete
= 0,
(or
Set,
case
the
of
When ei+k
in
the
(1)
The
the
this
involves
changing
for
factor
respective
case
constructed for
now elucidate
of
plicand will
algorithm
an operation
ei+k+k+l
following
of
the
i + k + 1-st number
the
In
In
the
case
in
1 + 1.
operation
of
the
of
OCCUrS Only
a maximum possible the
the 1,
i + k +
k.
dense,
complement
seen,
the
In
= 0;
and Sign Si = 1,
multiple
each
0 to
the
number
easily
is
sign. from
obtaining
operation
= 1 and
code
k + 1 by
selection
Si
multiple).
a suitable
multiplicand
by changing
order
of
the
replacing
this
when changing since
‘Li+k+I
a binary
of In
=
j = 2itktz+1 - ai+k+l .. . ai
where
Is
j = Oi+k.. . Oi and sign
of
multiples
operation
is
de-
function
the For
probability Pj
based
of
the
on the
logical
function
the of
j-tb order the
248
a difference
we have
operation,
Pj = $ (’ The particular teristic
solution takes
2&-
zk-i f
us consider
Let
of
equation
equation
-Pj_l
-
this
the
Pj-2
-
equation
root
to in addition X0 = 1. We have
iik (2; Consequently, Since terms of j
it
value l,‘(k
It of
+ 2);
,*+I _ Z- i
I”+
the
charac-
1
= 0.
i = r-j *
the
initial
:zjq,z -
!I=
iand
or
I z( < 1, since
solution
the
Pj = l/(k
of
independent
is
Thus,
Pj_*)-
of
the
of
theorem
I;{”
contains
I 23I # 1.
homogeneous
length
of
proved.
another
(2 /;I-i)[
equation
Ci is a constant, for Thus, the probability
the is
equation,
the
code
Since
comprises
k
higher Pj for
value a high
the of
and is
equal
k = 1 + logEm,
to then
+ log,m). is
the
j
2).
q En/(3
general
roots
the type ci(xi)j, where can be as low as desired.
of f
the
lZ/ < 1,
the
of
-
equation
1,
1) =
is
or
22”+” _ :k _ which,
*
form
. * * +z+i=O,
the
* ’
easy
to
see
that
multiplicand,
the
with
a dense
number of
but
set
incomplete
operations
of
in the
q lies
multiples range
&
consider
with of
symbols
examples
multiplier
zero
of
the coding + + 0, 1. 1, 3, orders.
to unity;
the
algorithms
algorithm 3. Table
The first
second,
vice
belonging using
to
a given
a threefold
1 shows
all
the
column
corresponds
versa,
from unity
possible to to
zero.
-
I
i
i+l
I
-
i+2
i-l
j
if-l
j
i
/
we i.e.
combinations
transition
TABLE 1
ii_?
class,
multiplicand,
i-l
-
0
0
1
0
;
0
u
0
1
0
1
1
i)
$
I)
1
0
1
1
0
1
0
i
1
0
0
I
i
1
1
0
T
I
I
u
I
from
Ootinrun mu~t~plicaiio~
249
afrrorithms
Operations in the first and second row of the first column cause DO Uncertainty. Subtraction in the third and fourth, apparently, lead to a minimum number of operations irrespective of the code value in i + 3 and subsequent orders. It should be noted that since subtraction is effected in thse orders. in the i t 3 order unity should be added. Multiplication is thus. in fact, carried out by at least three orders. In case of unity has to be effected in the (i + 3) or in the fi + 2) order, correction next order, where the operation has to be carried out, i.e. where the sign is changed. This has been taken into account when determining the operation in the second column. It is carried trebled ation is pressed
easy to see that the operation with a simple multiplicand is out in the case of equality of (i + I) and (i + 2). and with a multiplicand, in the case of inequality. The sign of the operdetermined by the (i f 2) order. Thus, the algorithm can be exin the form _ $4= (di # oi_i) s*_iSi.+ ‘t = (‘i+i # ‘$*s) si, $ = (ai+i = a*+s)si, sign si = u{+~.
This algorithm
is optimum since
the set
The average number of operations factor for this algorithm is
q2++-
1 2 o/y)
(l/Tjn
of multiples
in mUltipliC8tiOn
. s’n
[
(n
- if arc tg VT-
is dense. by one a-th order
arc tg 5v-
. 3
This formula does not take into account the operation in the (II + 1) order, if the previous operation was subtraction. This, however, is not necessary since the known methods of multiPlication by a negative number mean precisely that multiplication by an order following the sign, is not effected. The result of the formula is similar to the proof of theorem 2. In this case. however, the determination of the roots of the characteristic equation can be completed. Here
It is easy to see that q & n/4 directly also from the general
for high values of n which follows theory since this algorithm belong
to the
V.M. Mikhelev
250
subclass of algorithms “with an operation set of multiples Is complete. As an example of the algorithm multiples, an algorithm sidered (see Table 2).
using
with symbols
0,
for
changing
the sign”
and the
a dense but incomplete set of t t - + 1. 1. 5. 3, 3 and 5 may be con-
TABLE 2
i+3
I
i+2
i+l
i-f
i+3
i+.Z
i+r
i-i
I
0
0
0
0
0
0
0
1
0
1
1
0
0
1
1
1
1
0
0
0
1
0
0
1
i
i
1
0
1
1
1
1
In all the rows of the first column in Table 2, in addition to the fourth and fifth, the operation selected mahes it possible to multiply by orders of up to i + 3. In the fourth and fifth rows multiplication is carried out by orders of up to i + 2 inclusive. However, in this case, there is a change of sign in i + 3. In addition, all the rows, in which It is remembered that unity is added to the order following the last, by contain unity in this last order. which multiplication has been effected, This has been taken into account when determining the operation in the second column. It is easy to see that operation with a simple multivlicand (I or 1) is orders and with ity. The treble the equality of plicsnd, in the in the form
effected with the equality of the (i f 1) and (i + 2) treble or quintuvle multiplicand, ia the case of inequalmultivlicand is transferred to the summator in case of the (i + 2) and (i + 3) orders and the quintuple multicase of inequality. Thus, the algorithm can be expressed _ _ Si = (ai # a*_i) si-l ‘i-e’:-,* s; = (uitL=
54.2 1 ‘i,
sf =
‘I$+3 #
tai+$ =
ai+l)
“i,
Optimum
multiplication
sf=(ai+3=ai+l# sign si = (ai+ 1 = ai+ J ai+
For this ations
algorithm,
which is also
251
algorithns
ai +2) %,
V (ai+1# ai+2)‘i+ 3.
optimum, the average number of oper-
is
All the observations in this case also.
concerning
g for
the previous
algorithm
are valid
It follows from theorem 2 that the average number of orders, by which multiplication can be effected in one operation (n/q), is proportional to the logarithm of the number of multiples of the multiplicand (3 + log2 m). Thus, the number of orders has increased much more slowly than the number of multiples, which reflects increased equipment in the multiusing a trebled plying mechanism. From this point of view, multiplication multiplicand, is the most effective. However, in sequential machines a sufficiently large number of multiples can easily be obtained with only one additional summator which makes it possible to use more complicated algorithms.
REFERMCES 1.
Lehman, Y.,
IRE Trans.
Electronic
Comput.,
Sept.,
EC-6, 204-205.
1957.