- Email: [email protected]
The connection between two multiplication algorithms
0. M. Makmov
218
In conclusion I thank L. N. Sretenskii, A. G. Sveshnikov and S. Ya. Sekerzh-Zen’kovich for valuable comments and discussions. Translated by J. Berry REFERENCES
1.
SRETENSKII, L. N., Theory
of the wave motions of a liquid (Teoriya volnovykh dvizhenii zhidkosti),
ONTI, Moscow-Leningrad, 2.
1936.
NOBLE, B., The Wiener-Hopfmethod (Metod Vinera-Khopfa),
Izd-vo in. lit;, Moscow, 1962.
SHORT COMMUNICATIONS THE CONNECTION BETWEEN TWO MULTIPLICATION ALGORITHMS* 0. M. MAKAROV Sevastopol (Received 22 March 1973) (Revised version 17 July 1973)
IT IS shown that Karatsubi’s algorithm, applied to the problem of calculating the product of matrices, is transformed into Strassen’s algorithm. In problems of the multiplication of matrices and the multiplication of multidigit numbers the algorithms given in [l] and [2], respectively, which require the performance of a smaller number of multiplication algorithms than the generally accepted algorithms, are well known. It will be shown below that an algorithm of the type of [2], applied to the problem of calculating the product of matrices, is transformed into Strassen’s algorithm [I] . 1. Let it be required to calculate the polynomial G given by
(1)
G=(Aoy+A,) (Boy+B,). We represent (1) in the matrix form
(2) where G=g,iy2+gziy+g31,
g,,=AoBo,
g,,=AoB,+A,Bo,
gzi=AjBi.
We associate with each multiplication operation a natural number (ordinal number): AoBo=l,
AoB,=2,
A,B,,=3,
A,B,=4.
(3)
The matrix Ilgi,ll is then given by IEH=l12:31Vh, vjkhid M&t.mat. Fiz, 15, 1, 221-231, 1975.
(4)
219
Short communications
We introduce the matrices
u g11
1
gz1
2 3 4
a1
(5)
We represent the generally accepted algorithm for the multiplication of polynomials (2) in the form
(6) In order to find the value of the element gil from the expression (6) we have to add those numbers which stand in the same row as the element gjl. For example, the element g2 1 is given by gzi=2+3. In order to fmd the number 1 it is necessary to multiply the algebraic sum of those elements Bi of the matrix Sl which have in their rows the number I, by the algebraic sum of those elements Ai of the matrix S2 which have in their rows the number 1. For example, for the number 2 we obtain ~=BA. It is easy to check that with these rules for calculating the elementsgil expressions (6) and (2) give the same algorithm for the multiplication of polynomials (the generally accepted one). Using the algorithm of [2] to calculate the polynomial G, we obtain G= VoBo)y2+1(Ao+Ai)
(Bo+B,)
-do~o-A~B,]~+A,B,.
(7)
By the algorithm (7) the calculation of the polynomial G requires the performance of three multiplication algorithms. We represent the algorithm (7) in the form (6):
In [3 ] the possibility of passing from the algorithm for the multiplication of polynomials to a matrix multiplication algorithm is demonstrated. For this we transform expression (6) to the form (9) We write (9) in the matrix form
The passage from (6) to (9) consists of the replacement of the expression SiS2=Ss by Performing a similar operation in (8), we obtain
SIPS=&.
(11) Expression (11) calculates the vector IIA~A~II of expression (10) by the algorithm of [2].
220
0. M. Mahmov
2. We will show that the algorithms (8) and (11) define the matrix multiplication algorithm of
PI ’ Suppose it is required to evaluate the matrix C given by
We represent the generally accepted matrix multiplication algorithm in the form (6): an 1 a12 2 nsl 5
3’ 4) 7 ’
bll bzl bu
1a22 6 8
1 2 3
5 6 I =
1 3 5
Cl1 Cl2
cal
bzz 4 8
2 4 6
(13)
’
C%Z7 8
In expression (13) we replace the number 8 by 1: a11
1
1am 2 an 5 a22 6
3 bn 4 bzl 7 o bl2 ? IIb22
1 2 3 4
5 6 7 1
(14)
=
The values of the elements cl 1 1 and c2 2 1 are given by ~~1'=~+2=(aiitazz)(bii+bzz)+aizbz~,
~221=1+7=(a~i+azz)
(bij+bm)
+az,b,z.
Comparing cl 1 and c22 with cl 1 1 and c22 1 respectively, we obtain (15) We write 9=aiibii,
lO=aiibzz,
12=a&ii,
13=azh,
Il=a&t, 14=aiibz2.
Then (15) can be written in the form Efi=cpl~-fO-11-13,
c22=c22’-9-12-i4.
(16)
Taking into account (14) and (16), expression (13) assumes the form 1 am 2 a21 5 a22 6 a11
3 10 - 14 9 bu 1 4 bzl 2 7 o blz 3 1 12 -11 13 bzz 4
5 9 --12 11 6 7 = 1 13 - 10 14
Cl1 1 2 10 - 13 11 T 3 4 ~2~56 c22 I 1 12 -9 14
Cl2
(17)
We extract from (17) the subsystem a11 10 a12 2 a21
0
a22
0
3 4 '
bll b bzl bzz
0 2 3 4
Applying to (18) the algorithm (1 l), we obtain
-
Cl1 c1a =- czl 10 cza
10 3 0 0
2 4 *
(18)
221
Short communications
all
4
3
aI%
2
4
bu blz
0 2 %I 3 bzz 4
a21
a22
c
Cl1
2
-4
Cl2 1: c21 c2t
-3
2
3 0 0
4
(19)
-
Replacing in (17) the subsystem (18) by (19), we obtain 417 12 -1444 -11 13 ~111 aza @a 2 at2 6 39 5
76 15 9 13 -1214 211 --.3
baz bl?# bzl bll 4321
o
= es1 612 3 Cl1 51 642
-412 -1314-9 11
(20) Caa71
We similarly extract the subsystems (we omit the first columns in the matrices)
37’5
0 5 -1 2 0 6 0 0 = 5 6 I’ 5 7 07 7 12 612 0 0 3-14 0 3 4 4 0 7 C 0 7 03 7 24 4 14 0
’
A
,
The ~gorithms (8) and (11) transfo~
0
‘)
7
3
y2
-4
0
-5
0
65 6
0 0
34-7 4
9
7
03
6 0
7= 4
0 -65 26 0 0
-3
2
=5
6’ 0
In (20) we replace successively the subsystems (21) by (22): 1 3 2 5 6
4 7 1
1 2
3 4 7 1
-‘5 6
9
-14
4
12
-11
13
i 2 -5
--4
-7
5
(21)
(21) to the form
0 3
-73 75 0 0 57037=5
7
--11
-6
-14 3 -7 4 7 12 -11 1
1 3 2 4 7 -5 13 6 1
4
-7
4
13
-14
5
-11
4
-2
0 3 4 0 ) I 3(
(259
222
0.M. Makurov
13-7 2 4 ‘5 7 ,6 1 1 2 3 I4
5 6 7 1 1 2
‘3 4
‘3
‘5
I
13-7 2 4 7
I -5
5 9
-2
-11
--12
5 6 7 1 5 6 7
4
12
7
-3
2
5 6 7 1
4 1 2
-6 -3
13 7
14 -6
-3
2 11
,4 1
I 5
I 2
11
14
4
61
-3 13
-1 2
Y31 71 4 5 62
-‘5
4
5
7
-2
7
-12
13
5 6 7
7
1
2
I1
14
2
-6
-
11 +
14
-3
-1314
il3
-6 1
-3
-412
4
-13
1 3
2 4 6
-4
7
1
-5
1 3
2 4 6
-4
6
A7 1
-5
3
11
-13
12
1
11
-3
2
1
-4
2
-4
6
-5
-t
11 -
7
1
-5
14
3
11 , 7
i-5
3
After all the substitutions we fmally obtain am aa1 2 a18 anl3-7 65
47 1
5
-2 4 c, Ybla bzz bzl hn 4321
I15 6
27
-3 - 6 =
ca czl 3571 CPa Cl1
462I
-5 - 4
36
01
cii=l+2-4+6,
m=3+4,
cz,=5+6,
c22=7+1-5+3,
(23)
where I=(ai~+azz)(bl,+bzz), 2=(a~~-a~4 4= (ati+ad bz2, 5= (azi+ad bii, 7=(a~-ad (bit+&).
3=aii (b,z (bzi+bzz), 6=a2~(bzi--bji),
bzz),
The algorithm (23) is the algorithm of [l] for calculating the matrix C of the expression (12). Translated by J. Berry
223
Short communications
REFERENCES 1.
STRASSEN, V., Gaussian elimination is not optimaL Numer. Math., 13,4, 354-356, 1969.
2.
KARATSUBA, A. and OFMAN, Yu., The multiplication of multidigit numbers by automata. Dokl. Akad. Nauk SSSR, 145,2,293-294,1962.
3.
PAN, V. Ya., On schemes for computing the products of matrices and the inverse of a matrix. Usp.mat. Nauk, 27,5, 249-250, 1972.
REPRESENTATION OF CAUCHY TYPE INTEGRALS WITH A POWER SINGULARITY BY SPECIAL FUNCTIONS* M. A. SHESHKO Minsk (Received 10 April 1973) (Revised version 14 September 1973)
FORMULAS are obtained for the representation in the whole plane in terms of Jacobi functions of the second kind and rapidly convergent power series. Integrals of the Cauchy type with a power singularity along an open contour have numerous applications: in the mechanics of continuous media [I], in the solution of boundary value problems of the theory of analytic functions, for example Riemann’s problem [2], and in other problems. In the study of such integrals most attention is paid to their behaviour in the neighbourhood of the limits of the integration. The necessities of computational mathematics require a knowledge of the behaviour of the integrals considered in the whole plane of the complex variable. In the present note we obtain formulas representing Cauchy-type integrals in the whole plane of the complex variable in terms of special functions. The formulas obtained can be used to evaluate the integrals. 1. We will consider the Cauchy-type integral 1
Qt.,=& P(X)--cpwdx
s
2
I
c[-1,
I],
x-z
--L
-I -I, Re B> -1. We define the function p(x) as where p(z) = (I-x)~(I+x)~, follows: (l--x)= is a single-valued branch in the plane with a cut connecting the points 1 and 00along the negative semi-axis, for the function (1+x) b the cut connects the points -1, 00 and also runs along the negative semi-axis; cp(5) is a complex-valued function whose r-th derivative satisfies a Holder condition with exponent ~1,Re a+p.L>O, Re p+p>O.
It is known [2,3], that the integral Q (z) retains a power singularity in the neighbourhood of the points z=*l. In order to isolate this singularity we construct a Hermite interpolation polynomial Q2r+l(x) of degree 2r + 1 by the conditions Q&(*1)
= q+‘(*l),
k=O, 1,. . . , r,
and write the integral (1) in the form Q,(z) =F(z) +H(z),
where *Zh. vychisl Mat. mat. Fiz., 15, 1,231-234, 1975.
218
In conclusion I thank L. N. Sretenskii, A. G. Sveshnikov and S. Ya. Sekerzh-Zen’kovich for valuable comments and discussions. Translated by J. Berry REFERENCES
1.
SRETENSKII, L. N., Theory
of the wave motions of a liquid (Teoriya volnovykh dvizhenii zhidkosti),
ONTI, Moscow-Leningrad, 2.
1936.
NOBLE, B., The Wiener-Hopfmethod (Metod Vinera-Khopfa),
Izd-vo in. lit;, Moscow, 1962.
SHORT COMMUNICATIONS THE CONNECTION BETWEEN TWO MULTIPLICATION ALGORITHMS* 0. M. MAKAROV Sevastopol (Received 22 March 1973) (Revised version 17 July 1973)
IT IS shown that Karatsubi’s algorithm, applied to the problem of calculating the product of matrices, is transformed into Strassen’s algorithm. In problems of the multiplication of matrices and the multiplication of multidigit numbers the algorithms given in [l] and [2], respectively, which require the performance of a smaller number of multiplication algorithms than the generally accepted algorithms, are well known. It will be shown below that an algorithm of the type of [2], applied to the problem of calculating the product of matrices, is transformed into Strassen’s algorithm [I] . 1. Let it be required to calculate the polynomial G given by
(1)
G=(Aoy+A,) (Boy+B,). We represent (1) in the matrix form
(2) where G=g,iy2+gziy+g31,
g,,=AoBo,
g,,=AoB,+A,Bo,
gzi=AjBi.
We associate with each multiplication operation a natural number (ordinal number): AoBo=l,
AoB,=2,
A,B,,=3,
A,B,=4.
(3)
The matrix Ilgi,ll is then given by IEH=l12:31Vh, vjkhid M&t.mat. Fiz, 15, 1, 221-231, 1975.
(4)
219
Short communications
We introduce the matrices
u g11
1
gz1
2 3 4
a1
(5)
We represent the generally accepted algorithm for the multiplication of polynomials (2) in the form
(6) In order to find the value of the element gil from the expression (6) we have to add those numbers which stand in the same row as the element gjl. For example, the element g2 1 is given by gzi=2+3. In order to fmd the number 1 it is necessary to multiply the algebraic sum of those elements Bi of the matrix Sl which have in their rows the number I, by the algebraic sum of those elements Ai of the matrix S2 which have in their rows the number 1. For example, for the number 2 we obtain ~=BA. It is easy to check that with these rules for calculating the elementsgil expressions (6) and (2) give the same algorithm for the multiplication of polynomials (the generally accepted one). Using the algorithm of [2] to calculate the polynomial G, we obtain G= VoBo)y2+1(Ao+Ai)
(Bo+B,)
-do~o-A~B,]~+A,B,.
(7)
By the algorithm (7) the calculation of the polynomial G requires the performance of three multiplication algorithms. We represent the algorithm (7) in the form (6):
In [3 ] the possibility of passing from the algorithm for the multiplication of polynomials to a matrix multiplication algorithm is demonstrated. For this we transform expression (6) to the form (9) We write (9) in the matrix form
The passage from (6) to (9) consists of the replacement of the expression SiS2=Ss by Performing a similar operation in (8), we obtain
SIPS=&.
(11) Expression (11) calculates the vector IIA~A~II of expression (10) by the algorithm of [2].
220
0. M. Mahmov
2. We will show that the algorithms (8) and (11) define the matrix multiplication algorithm of
PI ’ Suppose it is required to evaluate the matrix C given by
We represent the generally accepted matrix multiplication algorithm in the form (6): an 1 a12 2 nsl 5
3’ 4) 7 ’
bll bzl bu
1a22 6 8
1 2 3
5 6 I =
1 3 5
Cl1 Cl2
cal
bzz 4 8
2 4 6
(13)
’
C%Z7 8
In expression (13) we replace the number 8 by 1: a11
1
1am 2 an 5 a22 6
3 bn 4 bzl 7 o bl2 ? IIb22
1 2 3 4
5 6 7 1
(14)
=
The values of the elements cl 1 1 and c2 2 1 are given by ~~1'=~+2=(aiitazz)(bii+bzz)+aizbz~,
~221=1+7=(a~i+azz)
(bij+bm)
+az,b,z.
Comparing cl 1 and c22 with cl 1 1 and c22 1 respectively, we obtain (15) We write 9=aiibii,
lO=aiibzz,
12=a&ii,
13=azh,
Il=a&t, 14=aiibz2.
Then (15) can be written in the form Efi=cpl~-fO-11-13,
c22=c22’-9-12-i4.
(16)
Taking into account (14) and (16), expression (13) assumes the form 1 am 2 a21 5 a22 6 a11
3 10 - 14 9 bu 1 4 bzl 2 7 o blz 3 1 12 -11 13 bzz 4
5 9 --12 11 6 7 = 1 13 - 10 14
Cl1 1 2 10 - 13 11 T 3 4 ~2~56 c22 I 1 12 -9 14
Cl2
(17)
We extract from (17) the subsystem a11 10 a12 2 a21
0
a22
0
3 4 '
bll b bzl bzz
0 2 3 4
Applying to (18) the algorithm (1 l), we obtain
-
Cl1 c1a =- czl 10 cza
10 3 0 0
2 4 *
(18)
221
Short communications
all
4
3
aI%
2
4
bu blz
0 2 %I 3 bzz 4
a21
a22
c
Cl1
2
-4
Cl2 1: c21 c2t
-3
2
3 0 0
4
(19)
-
Replacing in (17) the subsystem (18) by (19), we obtain 417 12 -1444 -11 13 ~111 aza @a 2 at2 6 39 5
76 15 9 13 -1214 211 --.3
baz bl?# bzl bll 4321
o
= es1 612 3 Cl1 51 642
-412 -1314-9 11
(20) Caa71
We similarly extract the subsystems (we omit the first columns in the matrices)
37’5
0 5 -1 2 0 6 0 0 = 5 6 I’ 5 7 07 7 12 612 0 0 3-14 0 3 4 4 0 7 C 0 7 03 7 24 4 14 0
’
A
,
The ~gorithms (8) and (11) transfo~
0
‘)
7
3
y2
-4
0
-5
0
65 6
0 0
34-7 4
9
7
03
6 0
7= 4
0 -65 26 0 0
-3
2
=5
6’ 0
In (20) we replace successively the subsystems (21) by (22): 1 3 2 5 6
4 7 1
1 2
3 4 7 1
-‘5 6
9
-14
4
12
-11
13
i 2 -5
--4
-7
5
(21)
(21) to the form
0 3
-73 75 0 0 57037=5
7
--11
-6
-14 3 -7 4 7 12 -11 1
1 3 2 4 7 -5 13 6 1
4
-7
4
13
-14
5
-11
4
-2
0 3 4 0 ) I 3(
(259
222
0.M. Makurov
13-7 2 4 ‘5 7 ,6 1 1 2 3 I4
5 6 7 1 1 2
‘3 4
‘3
‘5
I
13-7 2 4 7
I -5
5 9
-2
-11
--12
5 6 7 1 5 6 7
4
12
7
-3
2
5 6 7 1
4 1 2
-6 -3
13 7
14 -6
-3
2 11
,4 1
I 5
I 2
11
14
4
61
-3 13
-1 2
Y31 71 4 5 62
-‘5
4
5
7
-2
7
-12
13
5 6 7
7
1
2
I1
14
2
-6
-
11 +
14
-3
-1314
il3
-6 1
-3
-412
4
-13
1 3
2 4 6
-4
7
1
-5
1 3
2 4 6
-4
6
A7 1
-5
3
11
-13
12
1
11
-3
2
1
-4
2
-4
6
-5
-t
11 -
7
1
-5
14
3
11 , 7
i-5
3
After all the substitutions we fmally obtain am aa1 2 a18 anl3-7 65
47 1
5
-2 4 c, Ybla bzz bzl hn 4321
I15 6
27
-3 - 6 =
ca czl 3571 CPa Cl1
462I
-5 - 4
36
01
cii=l+2-4+6,
m=3+4,
cz,=5+6,
c22=7+1-5+3,
(23)
where I=(ai~+azz)(bl,+bzz), 2=(a~~-a~4 4= (ati+ad bz2, 5= (azi+ad bii, 7=(a~-ad (bit+&).
3=aii (b,z (bzi+bzz), 6=a2~(bzi--bji),
bzz),
The algorithm (23) is the algorithm of [l] for calculating the matrix C of the expression (12). Translated by J. Berry
223
Short communications
REFERENCES 1.
STRASSEN, V., Gaussian elimination is not optimaL Numer. Math., 13,4, 354-356, 1969.
2.
KARATSUBA, A. and OFMAN, Yu., The multiplication of multidigit numbers by automata. Dokl. Akad. Nauk SSSR, 145,2,293-294,1962.
3.
PAN, V. Ya., On schemes for computing the products of matrices and the inverse of a matrix. Usp.mat. Nauk, 27,5, 249-250, 1972.
REPRESENTATION OF CAUCHY TYPE INTEGRALS WITH A POWER SINGULARITY BY SPECIAL FUNCTIONS* M. A. SHESHKO Minsk (Received 10 April 1973) (Revised version 14 September 1973)
FORMULAS are obtained for the representation in the whole plane in terms of Jacobi functions of the second kind and rapidly convergent power series. Integrals of the Cauchy type with a power singularity along an open contour have numerous applications: in the mechanics of continuous media [I], in the solution of boundary value problems of the theory of analytic functions, for example Riemann’s problem [2], and in other problems. In the study of such integrals most attention is paid to their behaviour in the neighbourhood of the limits of the integration. The necessities of computational mathematics require a knowledge of the behaviour of the integrals considered in the whole plane of the complex variable. In the present note we obtain formulas representing Cauchy-type integrals in the whole plane of the complex variable in terms of special functions. The formulas obtained can be used to evaluate the integrals. 1. We will consider the Cauchy-type integral 1
Qt.,=& P(X)--cpwdx
s
2
I
c[-1,
I],
x-z
--L
-I
It is known [2,3], that the integral Q (z) retains a power singularity in the neighbourhood of the points z=*l. In order to isolate this singularity we construct a Hermite interpolation polynomial Q2r+l(x) of degree 2r + 1 by the conditions Q&(*1)
= q+‘(*l),
k=O, 1,. . . , r,
and write the integral (1) in the form Q,(z) =F(z) +H(z),
where *Zh. vychisl Mat. mat. Fiz., 15, 1,231-234, 1975.