MULTIPLICATIVE COMPLEXITY OF DIRECT SUM OF QUADRATIC ...
TECHNION - Israei 'Institute of Technology
Technion - Computer Science Department - Tehnical Report CS0575 - 1989
Computer Science Department
MULTIPLICATIVE COMPLEXITY OF DIRECT SUM OF QUADRATIC SYSTEMS by
Nader H. Bshouty Technlca1 Report July 1989
#575
Technion - Computer Science Department - Tehnical Report CS0575 - 1989
MULTIPLICATIVE COMPLEXITY OF DIRECT SUM OF QUADRATIC SYSTEMS Nader H. Bshouty Department of Computer Science Technion - Israel Institute of Technology Haifa 32000 Israel
ABSTRACI', We consider the quadratic complexity of certain sets of quadratic forms. We study classes of direct sums of quadratic fonnl. For these classes of problems we show that the complexity of one direct sum is the sum of the complexity of the summands and that every minimal quadratic algorithm for computing the direct sums is a direct-sum algorithm. Key Words: multiplicative complexity. direct sum, quadratic algorithms. bilinear algorithms. algebras.
1. INTRODUCTION
LetF be a field. and let x
=(Xl' •.. ,x l lI
and z
=(Zl • ..• • zJc l
be vectors of indetenninates.
Let Q x = (Q 1•... , QJc ) be a vector of quadratic fonns on X I, ... , XII over F. A quadratic algorithm that computes the quadratic system QXz with multiplicative complexity Lis L
QXz= 1: Q j(z)bj (x)cj(x).
(1)
j ..1
where Qj • b j and Cj are linear forms of the corresponding variables. The minimal L such that equation (1) holds is.denoted by L (Qx z ) and then the quadratic algorithm is said to be minimal. It is known, cf. [Sll. that when F is an infmite field then L (QXz) is the 'complexity of QXz by means of sttaight-line algorithms.
Technion - Computer Science Department - Tehnical Report CS0575 - 1989
·2· Let Y
= (y I, ... ,Y". l
of b~linear forms on
be a veco of indeterminates and let QI"
= (Q' I' ... ,Q'Ie )
be a vector
x and y. A bilinear algorithm that computes the bilinear system Q X" z with multipli-
cative rank ~ is
QI,yZ=
't
1: ai (z)bj(x)cj(Y),
(2)
j,.. 1
where aj ,bi and ci are linear forms of the corresponding variables. The minimal ~ such that equation (2) holds is denoted by ~ ( Q x,y z) . It is known that
I
cf. [Jll .
The direct sum of two quadratic systems Q r z and Q ~ z (respectively, two bilinear systems Q r-' z
Q fl,'1zl + Qr-,2 ~) where Xj ,Yi ,Zj , i = 1 ,2 are distinct vectors of indeterminates. It is obvious that (
L (QrzEBQ~z)S; L (Qrz)+L (Q~z), ~(Qr,y zEBQf z)S ~(Qr·1z)+~(Q~" z).
'
When equality holds (in the second inequality) for every bilinear system Q~,y z then we say that Q f1 Z satisfies the direct sum conjecture, cf. [SIl, (in short DSC), We say that Q
Q
r l
,1
1
rZI is separable (respectively, l
zl satisfies the direct sum conjecture strongly, cf. [W3l, in short DSCS) if for every quadratic sys-
tern Q f ~ (respectively, every bilinear system Q -r,y2~, every qrinimal quadratic algorithm (respec-
£1
£2
Qr l zl +Qf~ = 1:a j(ZI )bj(XI )Cj(Xl)+ 1:a'i(~)b'j(X2)C'i(X2) j~1 j=1 (respectively, ~
~
Qr l ,11 zl +Qf,'2~ = 1:aj(zl )bj(xl )Cj(Yl)+ 1:a'j(~)b'j(X2)C'j(Y2) ), i-I i=1 where the frrst summand is a minimal quadratic algorithm (respectively, bilinear algorithm ) for Q f zl (respectively, Q
r
lo11
zl) and the second is a minimal quadratic algorithm (respectively, bilinear algorithm)
forQf~ (respectively, Qf·12ZV . L I
Obviously, quadralic algorithms can also compute bilinear systems, Q".1z =
1:aj (z) bl jel
(x, y) CI (x, y).
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l:i.
QI" Z
We defme row rank
=~f-l (XTAiY)Zi =xT ~t.lAiZi )Y=XT Q (z)y.
QI.1z
(respectively, col rank
QIJZ) to be the
dimension of the linear space
spanned by the rows of Q (z) (respectively, columns of Q (z». Define rank
QIJZ =
max (col rank QIJ z ,row rank QI"
Z ).
Denote by L (Q IJ) the linear space spanned by the entries of Q IJ and dim Q IJ its dimension. In the sequel we shall need the following definitions: Definition I • Let ?/J3 ( 't , , ) denote tM collection ofbilinear systems Q 1,1z such that: TMre exist integers sandt such that tM following conditions hold:
rank «Ql" .. •Q", Qjl' ...• Qj•.) i) ~ t. (i = (z 1•...• ZI
Definition
l
in all the definitions).
n . Let QIJZ be a bilinear system.
We say that Q' E L (QI.1) is active. if, for every base
{Q 1••..• Qk} of L (QIJ) that contains Q' , there exist Qi2' ...• Qi. such that
rank «Q' ,Qi2•
••• •
Qi)i)~ t.
Definition ill , Let 9{JJ. (, ) denote tM collectioh of bilinear systems Q IJz in 9{JJ ( 0,' ) which satisfy thefollowing conditions: (i)
For every base {Q 1••••• Qk} of L (Q1,1) there exists an active element Qi , such that for every non-active element Qi 2 and every / 1'/2 E F '/1 *0, we have / lQi l +/ ~i2 is active.
(ii)
For every base {Q 1••..• Qk } of L (Q IJ) there exist s active elements Qi ... ,Qi. such that "
rank «Qh' ... •Qi)i)
~
t.
Our main results are Theorem 1. IfQI"z E '}fJJ (0,.,) thenfor every quadratic system QIZ we have
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Theorem 2. IfQ"'J zE ?/}J (1.0) then Q ...·yz is separable. Theorem 3 . If Q"'J Z E 9IfJ3 ( 1 • r ) .r ~ 1 , then for every quadratic system Q"'z we have
L(QX'YzEB QX z ) ~ dim QXJ +t -8 + 1+L(Q"'z) = L (QxJ z ) +L (Qxz)-(r -1). Theorem 4 . If Q"'J Z E ?/}J. ( 0). then Q "'Jz is separable. Theorem 5 . If Q"'J Z E tJ,fJJ. ( r) • r ~ 1. then for every quadratic system Q Xz we have L (Q"'JzEB Q"'z) ~ dim Q ....Y+ t -8 + 1 +L (Q'Sz) = L (Q ....Yz ) +L (Q"'z) - (r -1). The set ?f}J ( 1 ,0) contains a one bilinear form. bilinear systems Q ....Yz that satisfy
L (Qx,yz ) = rank QX'Y z or L (Qx,y z ) = dim Q....Y. bilinear systems defined by polynomial multiplication and their dual systems and bilinear systems defmed by the product of two polynomials modulo a squarfree polynomial. The set
9.f/J ( 1. 1), contains bilinear systems QX.yz that satisfy
L (Q ....Yz )=
rank Q"'J z + 1 or
L (Q ....Yz ) = dim Q ....Y + 1. bilinear systems defined by the product of two quatemions and bilinear systems defined by the product X Y and Y X of two 2 x 2 matrices. The set
9IfJ3.
(0) contains bilinear systems defined by the product of two polynomials modulo a
third one.- bilinear systems Q ....Yz that satisfy L (Q"'J z ) ~ dim QXJ + 1. bilinear systems QXJz with
row rank Q"'Jz=m, col rank Q ....Yz=n and dim Q ....Y~nm-3, bilinear systems defined by the product X Y and Y X of two quatemions and bilinear systems defmed by the product of two triangular
2 x 2 matrices. The set tJ,fJJ. (1). contains bilinear systems QX'Y~ that satisfy L (QxJz)~
dim Q ....Y +2. bilinear
systems Q x.Yz that satisfy rank Q xJz ~ 3 • bilinear systems Q XJz with row rank Q x·yz = m •
col rank QX'Yz = n and dim QX'Y = n m -4 or n m -5, the bilinear system defmed by the cross product of two
3~dimensional
vector. the bilin~ system defined by the product of two element in the Lie
algebra of 2 x2 matrices and the bilinear system defmed by the multiplication of two 3 x 3. triangular matrices.
Remark. Theorem 1-5 are also true for bilinear algorithms. Let Q = (A l' ... ,Ak
)
be a vector of n x m matrices with entries from F and let x T Q y =
Technion - Computer Science Department - Tehnical Report CS0575 - 1989
- 5(xTA lY,' .. , x TAA:: y). The dual systems of (xTQ y) is (xT QD z) and (yT QT x) defmed by (xT Q y)z = (xT QD z)y = (yT QT x)z.
Our main result for bilinear algorithms is: Theorem «) • ut (xT Q y) z be a btliMar system. TMn (i)
If (xT Q y)z satisfie~ the DSCS then so does each of the dual systems (xT QD z) y and (yT QT x) Z.
(ii)
Iffor every bilinear system (xT Q '.Y)z we have !1(.( (xT Q y) z ED (xT Q' y) z) ~ c +!1(.( (xT Q' y) z). Thenfor every bilinear systems: (xTQ 1z)y and (yT Q2X)Z. we have !1(.«xTQD z)yEB(xT Q 1z) y)~ c + !1(.«xT Q l~) y), !1(.( (yT QT x)zEB (yT Q2 X ) z) ~
Let
~
c +!1(.( (yTQ2 X ) z).
be an associative algebra of dimension k with a ,u,nity element and let {a l' ... , aA::} be a
base of ~. We denote by Q ~.Y
=(Q 1, ... , QA:: ) the vector of bilinear forms defmed by the product of
two elements in ~, i.e,
It has been shown that L (Q ~.Y z) does not depend on the chooses base, cf. [FZ]. A beautiful result of Alder and Strassen~ cf. [ASt] Slates: If ~ is an associative algebm of dimension
k then for any quadmtic system Q Xz we have L(Q~,y zEB QXz) ~
2k -t
(~)+L (Ql(z),
where t (~ ) is the number of two-sided maximal ideals of ~. If L ( Q}Y z) = 2 k - t (~) (respectively, !1(.( Q ~.Y z) = 2 k - t (~ ) then we say that the algebra is of minimal complexity (respectively, minimal rank).
Denote the radical of ~, i.e the maximal (two-:.sided) nilpotent ideal contained in call the algebra ~ local if ~ I rad
~
is a division algebra. We call ~ clean if ~ I rad
of division algebras. For direct sum of algebras
~
~,
~
by rad
~
. We
is a fmite product
=~ 1 X ... X~l let C JI. (respectively, R JI) be the
number of ~i which are not of minimal complexity (respectively, of minimal rank).
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Our main results in complexity of atlgebras are:
Coronary I Lei JiI be a clean algebra. TMnfor every quadratic syslem QIz- we have
Corollary II
=Jill X
Lei JiI
... XJilI be a direci sum of division algebras.
plexity (i.e each JiI;
jj'
If Ji1
is an algebra of minimal com-
algebra ofminimal complexily). Then Q ~.y is separable.
Corollary III Lei JiI
=F
l a] / (p (a»
where p (a)
E
F I a] is a polynomial. T~n Q ~.y is separable.
Corollary IV
Lei JiI be a clean algebra. If JiI is an algebra of minimal ranlc lhen Q}' z satisfies lhe DSCS. Corollaries (I), (II) and (III) have been Proved by Feig and Winograd, cf. [FW], for bilinear algorithms. Characterizalion of division
alg~bras
Ji1 of minimal complexity are sLu.died in [Gr3] and [FeiJ, it has
been proved lhat division algebras of minimal complexity are simple field eXLellsion of F with
IF I
~2
dim JiI-2. No results are known about flon-division algebras of minimal complexity. Charac-
terization of commutative algebras, local alge.bras, and clean algebras of minimal rank ovec a close field are given in [GH1], [BCJ and.(HMo), respectively
In sectiOfl,S 2 and 3 we prove some preliminary results needed foe the proof of the theorems. In section 4 we prove Theorems 1,2 and 3 and corollaries I and II, In section 5 we prove Theorems 4 and 5 and corollary 111. In section 5 we prove 'Theorem 6. In 5e(;tion 6 we prove corollary IV All the results in inear algorithms
SecUOfl
2,3 and 4 are for quadratic algorithms. They ar(, readiJy generalized for bil-
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2. PRELIMINARY RESULT In this section we develop some lower bound techniques needed for the proof of our results.
Let QX =(Q 1"
•• , Qk) be a vector
of quadratic forms on x =(Xl, ... ,X" lover F. Every qua-
dratic form is
"" QI = 'LLPI,jjXjXj =x A1x j..lj=i T
where Al is an upper n
Xn
triangular matrix. We derme Q = (A 1, ... ,Ak ) and the characteristic matrix
of Q is Q (z) = r.f"'IAj Zj. Obviously, we have x T Q (z) x = Q Iz.
For a vector of quadratic forms QX we define L (QX) to be the linear space spanned by the entries of QX and dim Q II its dimension. For the bilinear system Q lI,1z = x T Q (z)y we derme row rank Q X,1 Z (respectively. col rank QX,1 z) to be -the dimension of the linear space spanned by the rows (respectively, columns) of Q (z). Finally, we let
rank QX,1z = max (col rank QX,yz, row rank QlI,1z). The following Proposition is frequently used in this paper £1
£z
j..l
j=1
Proposition 1 . Let Q fz = r.aj(z)bj(x)cj(X) and Qf" z = x T Q2(Z)y = r.aj'(z)bj'(x,y)c;'(x,y) ~
minimal quadratic algorithms. Then
(ii)
Interchanging some bj ' with c;' we have
dimL (b{(O,y), ... ,b£z'(O,y)} Proof. Let Q
~
col rank Qf" z.
r= (Q 1, ... , Qk)' If Q 1~O then Q f z depends on zl and therefore there exists
ajo( z) = r.f=1 'A.j Zj
depends
on
zl'
i.e,
'A.1~ O~
Then substituting
Z1 =
I (z 2•... 'Zk ) =
- (11 'A. 1 ) r.t.2'A.jZj we obtain
(Q2-( ~I 'A. 1 )Q l' ... , Qk -('A.,t I 'A. 1 ) Q 1) z =
£1
'L Qj(z)bj(x)Cj(X) j-=1 j~o
where z= (z2, ...• z,t ) and
Technion - Computer Science Department - Tehnical Report CS0575 - 1989
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dj(i)=aj(z)
'ZI=I(Z20 •••• 'a)=aj(z)-
5·
1
~: ajo(z).
where 5j.l is the coefficient of z 1 in aj (z). By induction ,(i) follows. Let Q2(Z)=[Q~I)(z), ... ,Q~m)(z)] where Q~j)(z) is the i-th column of Q2(z). If Q~I)(z);J!: 0 then Q~'Y z = x TQ2(z)y depends on Yl and therefore there exist b'jo (<X' e'j) depends on
Yl. i.e. b'jo(x,y) =
where Aj ,5j
E
/I
m
j .. l
j=1
1: AjXj + 1: 5jYj
F and 51 ;J!:O. Then substituting Y1
/I
m.
i-I
j-2
=1'(x,Y2, ... ,Ym) =-( 1/51 ) (1: AjXj + 1: 5jYj )
we obtain
£1
= 1:a'j(z)~(X,Y)Ci(X,Y) i=1 jt'jo
for someK (z) where y = (Y2,' .. ,Ym
l
and
~(X,y) = b;'(x,y) I )l1=1'(X,)'20 .••• )1.)
,
Cj(x,y) = e'j(x,y) 1)11 =l'(x,)'a, ... ,)I.)'
By induction, (ii) follows. • Remark. In what follows the results are true for row rank in as much as col rank. The results are proved either for row rank or col rank for our convenience. Allover the paper we assume that al(z), ... ,.aA;(Z),
k = dimQr, (respectively,
b 1(O,y), ... , bm ( O,y). m = col rank Q ~.Y z) are linearly independent Proposition 1 implies the following Lemma 1 • ([Fil,[Wl]) We have
L(QXz) ~ dimQx, L(xT Q(z)y) ~ rank QX,Jz.
r·
Lemma 2 • Let x, y , u , v , z· be distinct vectors of indeterminates and let Q u z = x T Q 1( z)u be a bi!inear system. Then
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£(XT Q I(Z)U+XTQ2(Z)y+yT Q3(Z)y) ~ col rank Qf··z+£(yT Q3(Z)y), v and hencefor the bilinear systems Qr'-z u TQ I(Z)X and Q z u TQ I(Z)v we have
r· =
=
£(uT QI(Z)X+XTQ2(Z)y+yT Q3(z)y) ~ row rank Qr~z+£(yTQ3(z)y), £ (u T QI(Z)v+yT Q3(z)y)
~ rank Qr·v z+£ (yT Q3(z)y).
Proof. By the same substitution in u as in the proof of Proposition 1 we obtain an algorithm for XTAI(z)X+XTA 2(z)Y+XT Q2(Z)y+yT Q3(Z)y, for some A I (z) and A z( z). with multiplicative complexity £' = £ (xTQ I(z) u + xTQ 2( z) y + yTo. 3( z) y) - col rank Q
r·
B
z.
Now substituting x = 0 in the algorithm • we obtain a quadratic algorithm for yT Q 3( z)y. Therefore £' ~ £ (yT Q 3( z)y) and the proof is followed.
Ii
The last two results of this section are well known. Lemma 3, is used in the literature to obtain
low~r
bounds for the complexity of bilinear systems. cf. [W2]. [W3] and. [KB]. Fo," the sake of completeness a proof is given which illustrate our methods. Lemma 4 is trivial and we shall refer to it iJf the sequel. Lemma 3 • /f for every non-singular matrix N there exist sentries Qi\,"" Q;. of QXN such that
L
QXz = Lai(Z)bi(x)Ci(X), i-I
By proposition 1 there exist k = dimQx independent ai(z) and therefore w.l.o.g. there exist a nonsingular matrix N satisfying a I(N z j = ZI' ...• a" (N z) = z" flnd hence " QXNz= LZjbj(x)cj(x)+
i=1
LL
aj(Nz)bi(x)Ci(X),
i=k+1
Assume w.l.o.g. that the first entries Q I, ... , Q., ofQxN satisfy
Then. substituting z.J+I
= ... =z" =0 we get s
(QI.· .. ,Qs ,0, ... ,O)i= LZjbj(x)cj(x)+ j-I
L
L i=k+1
aj
(Ni)bj(x)cj(x),
Technion - Computer Science Department - Tehnical Report CS0575 - 1989
- 10where i = (ZI •.•• • z" ,0•...•
ol _Therefore L -Ie +s ~ t. and this completes the proof.
•
and let N and K be n X n' and Ie x Ie' matrices ofranJc. n and Ie. respectively. Then x T Q(z)x
L
=1:aj(z)bj(x)Cj(x)
(3)
j=1
is a minimal qlUJdratic algorithm for Q (z) if and only if L
iTN T Q (Ki)Ni= 1:aj (Ki)bj (Ni)Ci(N i)
(4)
i ..1
is a minimal qlUJdratic algorithmfor NT Q (K i)N. Proof. Obviously. (3) implies (4). There exist a matrices N- and K- such that N N-
=/" ,and K K- =/Jc
where /" is the n x n identity matrix. Substituting N-x for i and K-z for i in (4) we obtain (3). •
3. SEPARABLE ALGORITHMS In this section we shall give conditions on the quadratic algorithms that compute the direct sum of two quadratic system~ Q IZ and Q2Z, to be separable to two minimal-quadratic algorithms, the flISt for Q IZ and the second for Q 2z.
Let
and
~ = (zr+I' . - .• zr+s l be vectors of indeterminates and let x = (xi, xll and z =(zi, i[)T. For Q F and Qf
, two vectors of quadratic forms, we say that the minimal quadratic algorithm L
QXz= Q;'ZI +Qf~ = 1:ai (z)bj(x)cj(x) j=1 A
is a separable algorithm if there exists a ~t / ~ {I. _..• L ,} = N L such that the following conditions hold: (i)
Qflzl= 1:ai(z)bj(x)cj(x) ,QrZ2= 1:aj(z)bi (x)Ci(X). iel
(ii)
The first quadratic algorithm- in (i) (respectively, the second) is minimal quadratic algorithm for
Q (iii)
if I
r zl (respectively, Qfzv. l
The terms ai(z).bj(x),Cj(x) ,i E / (respectively, i ~ /) are linear forms of zl and Xl (respectively, z2 and XV-
Technion - Computer Science Department - Tehnical Report CS0575 - 1989
- 11We say that Q f' zl is separable if for every quadratic 'system rithm for Q f' zl + Q
r
Q Zz every minimal quadratic algo-
r Zz is separable.
The following lemma proves that condition (i) is sufficient-for an algorithm to be separable. (The
teetmique we use in our proof was used in [AFW] for bilinear algorithms). Lemma 5 •If there exist a set I ~ N L such that
Qf'Zl = Laj(z)bj(x)cj(x), j
eI
then the algorithm is separable.
Proof. Obviously,
QfZz=Qxz -Qf'Zl
= Laj(z)bj(x)cj(x). j
If II I
> L(Qi'Zl) then, since INL-I
,I
I~L(QI2z2),wehave
L(QXz )= I I I + I NL-I I> L(Qtz1)+L(QrZ2)' A contradiction. Therefore I I I = L ( Q f' zl ) and (ii) follows.
Assume that for some i 0 E I ,ajo (z) = AZio +d'jo( z) depends on zio (Le. A;t 0) where j 0 > r, Le ajo( z) depends on Zz. Then substituting zio
Qi'Zl
=-( 1 / A) d'jo( z) in the first algorithm'we obtain
= L
d'j(z)bj(x)cj(x)
j e 1-( jo)
where d'j (z) = aj ( z) 1'10 __ ( 1/ A.) d.( z)" Therefore L ( Q i' zl ) S I I I - 1. A contradiction. Interchanging the roles aj and bj , (iii) follows. •
Lemma 6 • If the algorithm is not separable then there exist aj (z) depending on zl and z2' and b j(x) or Cj(x)
depending on xl and x2'
Proof. Assume that each aj (z ) depends either on zl or z2' Let I = (i I aj ( z) depends on -zl }. Then by substituting Zz = 0 we have
Qi'Zl
= Laj(z)bj(x)cj(x), j
eI
By lemma 5 the proof is completed. A similar proof holds forbj(x) and Cj(x). • Lemma 7 • Let x,z,i,i,N and K be as in lemma 4. If xTQ (z)x is separable then iTN T
Q (K i)N i is separable.
Technion - Computer Science Department - Tehnical Report CS0575 - 1989
- 12·
x
Proof. Assume that T N T Q (K i) IV
x is not separable. Then by lemma 6 there exist a minimal qua-
dratic al,goritlun x T NT Q (~i)N x+QUV
L
=LQj(i, u)bj (x, V)Cj(x, v) j ..l
=
such that Q 1(i , u ) depends on i and u. Then substituting x N-x and using lemma 4 we have that L
xT Q (K i)x+Qu v = LQj(i. u)bj(N-x, v)cj(N-x, v), j.l
is a minimal quadratic algorithm. Note that Q 1(i •u) still depends on i and u and therefore x T Q (K i) x is not separable. By lemma 6 there exists a (w.1.o.g) bjo (N-x , v.) depending on x and v. Substituting i = K-z and using lemma 4 again we have that L
x T Q(z)x+Quv = LQj(K-z, u) bj(N-", v)cj(N-x, v) j ..l
is a minimal quadratic algorithm. Since bjo (N-x, v) depends on x and v then by lemma 6 we have a con-
tradiction to the fact that x T Q( z)x is separable. • Remark. For bilinear systems xT Q(z)y Z=(ZI•... • zkl and matrices
=QX'Yz 'with x =(Xl •••. ,x
lI
l . y = (Yl • ... •Ylftl and
N ,M ,K of rank n ,m ,Ie ,respectively, lemma 4 and 7 hold for
xT Q (z)y and xT N T Q (K z) M y. 'Otis follows because T T
xlV
0 Q (K z) ·N 0 x ' OM1[1 y' T[ J 0 0 l[
T T N 0 Q(Kz)My=(x ,y) OM
[
Thus we can assUme all over the paper that-
row rank QX'Yz =n • col rank QX'Yz = m ,dim Q'S'Y = Ie. An equivalent condition for the seperabelity of algorithms is given in temma 8 below. We need the
following defmition
r zl +QfZ2 we say"that a non-singular matrix N does not mix Q r with if the entries of Q fl N are either in L (Q ,r or in L (Q f). We recall that L (Q t ) is the linear
Definition 1 • For QXz = Q Qf
t
t
t
)
space spanned by the entries of Q ft .
Lemma 8 • Iffor every non-singullJr matrix N such that QX N z = L zjbj,(x)Cj;(x)+ L Qj(N z)bj(x)cj(x), j;el
j"l
Technion - Computer Science Department - Tehnical Report CS0575 - 1989
·13 where 1 = {h, ... ,jr+8} is a subset of {I, ... , L} and 11 I = r +s , the matrix N does not mix Q F with Qr. Then the qUlldTatic algorithm is separable.
Proof. Assume that the quadIatic algorithm is not separable. Then (w.l.o.g) at(z) = at,l (Zt )+a 1.2(~) where a t.l (Zt) ':j:. 0 and a t,2 (z2) ':j:. O. LetN be a non-singular matrix such that (w.l.o.g) L
r~
QXN Z = ~zibi(X)Ci(X)+ i,.,1
Since N does not mix Q
~
aj(N x)bj(x)cj(x).
j=r~+t
r\ with Qr then N is of the fonn N1
for some permutation matrix
E.,
Z.-\(I)
OJ
N = [ 0 N E. 2
where Q-'N = a 1(N
E.
E.
t
1
= (QrW I.QrN 2 ). Since al(N z) =Zt. we have
z) = a 1,1 (N t Zl )+a-t.2 (N 2~)
and therefore a 1,1 (N 1Zt ) = 0 or a 1,2 (N 2~) = O. Since N is non-singular. Nt and N 2 are non-singular
and therefore al.l (zl) = 0 oral,2(~) =0. A contradiction. •
4. DIRECT SUM OF SOME CLASSES
In this section we define some classes of quadratic systems and prove that they are separable. Definition 2 • Let ?{JJ ('t • r ) denote the collection ofbilinear systems QXoYz such that: There exist integers
sandt such that thefollowing conditions hold: (i)
For every non-singular matrix N and for every 't entries Qj\, ...• Qi, of Qx.1N there exist S -'t entries Qit' ... , Qj..... ofQx'1N such that
rank (ii)
« Qj\, ... , Qi,. Qh' ... , Qj..... ) i) ~ t.
L (QX.1 z ) = dimQx'1 +t -s +r.
By lemma 3 if QX.1z satisfies (i) then L (QxoYz ) ~ dim Qx.1 + t
Remark. Defmition 2 is equivalent to definitiOn I in the introduction.
-s .
Technion - Computer Science Department - Tehnical Report CS0575 - 1989
-14 Our main results in this section are Theorem 1 .IfQ 'SJZe 'J,fJ3 (0 , r ) then for every quadratic system Q'Sz we have L(Q'S·JzEBQ'Sz).~
Theorem 2 • IfQ'SJ ze
dimQX,J +t -8 +L(Q'Sz) = L (Q'SJz)+L (Q'Sz)_ r .
9tfJJ (I ,0) then Q'SJz is separable.
Theorem 3 • If Q'SJZ e !J.fjJ ( 1, r ) ,r ~ 1 then for every quadratic system Q'Sz we have L(Q'SJzEB Q'Sz) ~ dimQX,J +t -8 + 1+L(Q'Sz) = L (Q'SJz)+L (Q'Sz)- (r -1).
Proof of Theorem 1 •
=Q IZ 1+ ... +Qkzk e !J.fjJ (0, r) and QXaz2 =Qk+lzk+l+ ... +Qk+k,Zk+.t' be any quadratic system, set Q'S =(Q'StJt , Q'S2), x = (xf, y f, xll and z = (zf, zll. Let N be any nonLet Q'ShJtzl
singular matrix and let Q'SN =(Q{+Q(, ... , Q'A:+k,+Q"k+.t')
where Q'I'
,Q'k+k,e L (Q'ShJt) and Q" I' ... ,Q"k+k,e L (Q'S2). Since N in a non-singular there
,jk' such that {Q"h"" ,Q"ja'} is a base for L (Q'S2). Since Q'StJtzl e !J.fjJ (O,r) there
existh, exist Q'i t ,
•••
,Q'i. such that
Then for A
(Q ' I + Q"I ' Q'J + Q") J = (Q' it + Q"it' ... , Q' i. + Q"i.' Q'h + Q"h' ... , Q'ja'+ Q") ja'
we have
£
«Q', +Q", )" +(Q'/ +Q"/ )z,) = £ «Q', ,Q'/) [ ~] + (Q""Q"/ ) [~ )
~
rank «Q', ,Q'/)
Since Q'/' is a base for L (Q'S2),
Now using l€?mma 3 and 4 we have
[~]
)+ £ «Q", ,Q"/)
[~]
) (Lemma 2)
Technion - Computer Science Department - Tehnical Report CS0575 - 1989
- 15L(QX z )
=L(Qx N z) ~ (k +k')+(t +L (Q x2z2 ))-(s +k') =k +t -s +L (QXa~) . •
Proof of Theorems 2 and 3 • Let x and z be as in the proof of theorem 1. Let L
QXz = :Eaj(z)bj(X)Ci(X) i=1
be any minimal quadratic algorithm for QXz. Let N be any non-singular matrix such that
QX N z = :E Zibjj(X)Cjj(x)+ :E aj(N z)bj(x)cj(x), ftES
where S = {j I'
. . . , A+k'}
j~S
is a subset of {1, . . . , L} and rS I = k
+ k '. If every matrix N
that satisfies
the above equation does not mix QXIJI with QXa then by lemma 8 the two theorems are proved. Assume
that some N mix Q XIJI with Q X2 then there exists an entry Q'h + Q "h of Q x N such that Q ' h ' Q "iJ
Q'he L (Q X2) and Q"he L (Q XIJI). Since N is a non-singular matrix there exist h, that Q"iJ"" ,Q"ji' is a base for L (QXa). Since QXIJI Z1 e
'JlP (l,r)
* 0,
... ,A, such
there exist Q'i l , ' " ,Q'i._1
such that
Then for t.
(Q ' /'+ Q"/', Q'J + Q") J = (Q' il + Q" ii' ... , Q'i.-I + Q"i.-I' Q'h + Q"h' ... , Q'A· + Q") ji' we have (as in the proof of Theorem 1) x
L «Q'/,+Q"/, )ZI +(Q'J +Q"J )z2) ~ t +L (Q 2z2 ). Substituting zi = 0 for i ~ I' U J in the quadratic algorithm of Q x N z we obtain a quadratic algorithm for (Q'/,+Q"/, )zl +(Q'J +Q"J )~ with L -(k +k' -(s +k' -1)) multiplication. Then by lemma 4 we have
L(Qx z ) = L(QxN z) ~ (k +k')+(t +L (QXa~))_(s+k' -1) = k +t -s + 1+L (Qx~). If L (QXIJI Zl) = k + t
-s we have a contradiction and therefore QXIoYI Z1 is separable. On the other hand
In the sequel, Qx'Yz denptes a bilinear system and QXz denotes any quadratic system. Corollary 1 • Let QX,Y be a vector ofone bilinear form. Then QXJz is separable.
Technion - Computer Science Department - Tehnical Report CS0575 - 1989
- 16-
CoroJIary2 .If L (Q:l J z ) = rankQ:l J z or L (Q:I,yz)::;: dlmQ:l J then QX,yz is separable. Proof. If L (Q:I,yz) = rank QX,yz then for every non-singular matrix N we have rank (QX,y N z )
= rank (Q:I,yz)
and therefore Q:I,yze
9tfJJ ( 1,0). If
L (QX,yz)
= dimQ:I,y
then for every non-
singular matrixN, any entry Qi of Q:I,yN satisfies rank (QiZi ) ~ 1. Therefore Q:I,yz e
9tfJJ ( 1,0). •
It follows from corollary 2 Coronary 3 .If L(Q:I,yz) ~ rankQ:I,yz+ 1
=lor L(QX,yz) ~ dimQ:I,y + 1 =I then
L (Q:I,yzeQ:lz ) ~ I +L(Q:Iz).
.Coronary 4 • Let J'l be a division algebra. If J'l is a simple field extension and I F I ~. 2 dim J'l - 2 then
Q ~,y z is separable. If J'l is not a simple field extension or I F I
< 2 dim J'l- 2 then
L (Q}Yze Q:lz ) ~ 2 dim J'l +L (Q:lz ).
Proof. For every non-singular matrix N every entry of Q~'YN has rank k = dimJ'l. Therefore
Q ~,y z e
9tfJJ ( 1 ,r ).
De Groote, cf. [Gr3], proved that L( Q ~,y z) = 2 dim J'l-1 if and only if J'l is a
simple field extension and I F I ~ 2 dim J'l - 2. Now the proof follows immediately. •
Remark. Corollary 4 with the results of [Fei] and [W4] classify all the minirilal quadratic algorithms for when
I F I ~2 max
ISiSt
P (a)=pl (a)·· ·Pt (a)e F [a]
deg Pi (a)-2.
For
Cn
the
cyclic
group
are of
sqwufree
and
n
have
order
F [Cn ] =F [a] I (an -1) and ex" -lis squarfree for char F ~.O (mod n ). Corollary 4 implies the following t
Coronary 5 • If J'l Q is the algebra ofquaternions over the real field R and J'l =
Corollary 6 • Let J'l be a clean algebra. Then
Proof. In [ASt] it is shown that
P J'l Q • Then
1=1
we
Technion - Computer Science Department - Tehnical Report CS0575 - 1989
- 17.£:.(Q}YzEBQ;)~2 dim
rad >t+.£:.(Q~'T'adAZEBQ;).
Since >t I rad }I is the direct sum of division algebras then from corollary 4 the proof follows. • Corollary 7 • Let char F "" 2. Let QI,yz be the bilinear system defined by the product X Y and YX of two 2 x2 matrices. Then
.£:. (QI,JzEBQIz) =.£:. (QI,yZ) +.£:. (QI Z ). ~
Q
E
It is known, cf. [Gal, that for char F "" 2 we have .£:. ( QS,yz) S 9, dim Q x,y = 7 and every
L (QI,J) satisfies rank Qz = 2. Therefore QX,Jz E 9{J3 (1,1). •
Corollary 8 • Let QX,Jz be the bilinear system dqrned by the product of two polynomials of degree n. Then (i) If
I F I ~2 n then QI,JZ is separable.
(ii ) If I F I
< 2 n S 2 I F I + 2 then £ (Q"z(jjQ'z)
(iii) If I F
= 3n
+l-ll ~ j I
+£ (Q'z).
I < n 1/3 then
Proof. If I F I ~2n then, cf. [W3],
.£:. (QI,yZ) = dim QX'y = 2n-l.
Using corollary 2 (i) follows. (ii) and (iii) follows from [ABK] and [KB, Lemma 4 and 5]. •
s.
DIRECT SUM OF ?l1J ( 0, r ) .
In this section we define a subset of 9{J3 ( 0, r ) and prove that all quadratic systems in this subset are separable. This subset contains QI'{a) I (I (a» Z for f (a) E F [a]. Definition 3 • Let Q I,yZE ?(JJ ( 0, r ) be a bilinear system. We say that Qj) e L (Q I,,) is active
if for
every non-singular matrix N such that Qj) is one of the entries of Q,S,YN there exist s -1 entries
Technion - Computer Science Department - Tehnical Report CS0575 - 1989
- 18Definition 4. Let '1IfJJ. (r ) denotes the collection ofbilinear system QIJ'Z in '1IfJJ ( O. r ) such that: (i)
For every non-singular matrix N there exist an active entry Q'i. of QX,Y N such that for every non-active entry Q'i20fQI'YN and every I 1.12
(ii)
E
F .11 *Owe have I lQ'i. +12Q'i2is active.
For every non-singular matrix N there exist $ active entries Qi•• ...• Qi. ofQI'YN such that
rank «Qi••...• Q;)i)
~t.
Remark. DefInitions 3 and 4 are equivalent to defInitions IT and min the introduction. When $ = 1 an equivalent definition of '1IfJJ" (r ) is given by the following
Proof • We recall that if Q IJ'Z E 9{JJ (0. r) with $ = 1 then for every non-singular matrix N there exist an entry Qi of QIJ'N such that rank Qizi
rank QjZj
~
~ t.
Hence every active entry is an entry Qj with
t (non-active entry is an entry Qj with rank QjZj
< t).
Assume by contradiction that for some non-singular matrix N and for every active entry Q'i of QIJ'N there exist a non-active entry Q 'j(i) in QIJ' N and/h(i) .Ijii) E F ,INi) (lh(i)Q'j
Then for /
={i
+Ijij~'j(i»
is not active.
fi!! /
(5)
I Q'j is active} we defme QI,y N N' = (Q"I" .. ,Q"k) such that Q'i { Q"; = Q'j + (Ijij/ fh(j»Q!j(j)
Since j (i )
*0 where
i fi!! /. i
E /.
we have {Q" 1. . . . • Q " k} is a ~ for L (Q IJ') and therefore N' is non-singular.
Now, N N' is non-singular. Q"i • i fi!! / is not active and by (5). Q" j • ie/ is not active as well.A contradiction. • In this section we prove the following Theorem 4. IfQI'YZE 9{JJ. (0) then QIJ'Z is separable. Theorem 5 • If Q I.YZE ?f}3. (r) • r ~ 1 then for every quadratic system Q I Z we have L (QI.YzEf) QIZ ) ~ dimQI,y +t -$ + 1+L (QI Z ) = L (QIJ'z)+L (~z) -(r -1).
Proof of Theorems 4 and- 5 •
Technion - Computer Science Department - Tehnical Report CS0575 - 1989
- 19Let QX.'Y· Z1 = Q lzl+' .. +QkZk be in 'J.(JJ* (r) and let QX2zz = Qk+1Zk+l+' .. +Qk+k,zk+k' be
an arbitrary quadratic system. Set QX=(QxIJ .,Q X2),
z=(z{,zll and x=(x['y{,xi)T.
Assume that L
QXz = Laj (z)bj (x)Cj (x) j-l is a minimal quadratic algorithm for QXz. If for an arbitrary non-singular matrix N such that (w.l.o.g) k+k'
QX N z =
L zjbj(x)cj(x)+
L
L
aj(N z)bj(x)cj(x),
(6)
j=k+k'+1
j=1
it does not mix QX.JI with QX\ then, by lemma 8 the two theorems are proved. Assume therefore that
L (Qxz ) ~ k +t -s + 1 +L (QX2~)
("') X For then if Qx.,YI ZI e 'J.(JJ* (r) ,r ~ 1, Theqrem 5 follows and if Q hYI ZI e ?(JJ* (0), then we have a contradiction and Theorem 4 follows. Let
QXN =(Q'I+Q"I"" ,Q'k+k,+Q"k+k')
where Q'j e L (QXhYI) and Q"j e L (QX2), i
=1, ... , k +k'.
Since' QXIJ·Z e ?(JJ* (r ) there exist
Q'j. +Q";. in QX N such that Q'jl is active. Let ZI = {w I Q'w is active} anddeflne Zj
={w I there exists llr(N z) depending on some zp ,p e Zj-l and zw}.
Let Z/ be the fust set that satisfies Z/
~Z/+1
(Obviously, Z/+1
=Z/+2 = .. ').
We now distinguish
between two cases: Case I. There exists i' e Z/ such that Q "i' * O. By the definition of Zj. we can fmd a sequence it, i 2 • ..• , i p of minimal length p such that Q"i"
* O. i q e Zq -Zq-l • q =2, .. . ,p
• i 1 e Z l' We proceed to proof ("') by induction onp.
If i 1 = ip i.e. Q "i. * 0 then since Qi.' is active we have as in the proof of theorem 2 and 3
Technion - Computer Science Department - Tehnical Report CS0575 - 1989
- 20· Assume p
> 1.
Becau~
*" 0 we have Q "i
P is the minimal integer such that Q";,
l =
••• =
Q";'_1 = O.
and by the defInition of Z 1 we have Q'i,• ...• Q'i, are not active. By the defmition of Zq and P. there exists aro(Nz) which depends on zi l
and zi, and does not depend on zi,•... ,zip' i.e.
1. A (z. -~z, -a(z)) I,
II
for zi, in (6) we obtain k+k'
Q N N'z=
L zibi(x)Ci(x)+Zizbro(x)cro(x)
jz1
j ~jz
L ,
1
,...,
L.
+
ai(N N Z)bi(X)Ci(X)+~(Ziz-~zil-a(z))bi,(x)ci,(x),
i
co
k+A:'
i
~
ro
where the i 2 entry of Q N N' is now
~ Q'iz-I ~ Q"iz Q ' i'-I and the ip ' ... , i 3 entries remain Q'i" ... , Q'jJ' respectively. Since Q'jl is active and Q 'jz is not active we have Q 'jl +
~
Q'i, is active. ,Note also that there exists aj (N N'z) depend on Zjl and Zj,. (Actually. if
ai (Nz) depends on zi, and ZjJ then aj (N N'z) depends on Zjl and ZjJ. The aj (N z) that depends on Zj.
j E ZZ-Z2 satisfies aj (N N'z) =aj (N z). So we have now the ~w sequence i 1. i 3 , .•• ,ip thatsatisfies the above conditions with the new sets Zl UZ2 ,Z3' ... •Zp. Assuming that (*) holds for p -1, it follows that it holds for·p. This accomplishes the proof for this case. Case
n . For
every i E Zz. we have Q"i = O. I.e. every ai (N z) depends on either
Zq
,q E Zz or
Let P = {i I aj (N z) that depends on some We now estimate the number of ai (N z) that depends on we obtain an algorithm tha~ computes entries are Q'j,
Zq ,
Zq
,q
E
Zz }.
q E Zz. Substituting Zq = 0 for q '" Zz
Qz where the entries of Q are from L (QXt.YI).
Since all active
i E,ZlaZ, by condition (ii) (in Defmition 5), there exist s active entries
Technion - Computer Science Department - Tehnical Report CS0575 - 1989
- 21-
Q'it.··· .Q'j•• {jt.··· .j~} E Zt such that rank (Q'it•...• Q'j.)i ~t and therefore by lemma 3
we have I Z/ I
+ I P I =£(Qi)~ I Z/ I +t-s.
(7)
Let pC = {k +e' + 1, ... , £ }-P , Zp = (I, ... , k + k' }-Z/. Now substituting in the quadratic algorithm Zq = 0 •for q E Z/ we obtain
r, zjl?i(x)cj(x)+ r, jf!z,
Since Q"i
M, L
:= 0
Qj
(N z)bj(x)Cj(x).
if!P
for i E Z/' the above ~gorithm computes (Q lll:hYI , Q I2) M Z for some non-singular matrix
(Q,IhYI)l;;;;L (QII.1I), i=(zrl"" ,zrw)T ,
£ (Q ,IhYlit ) = dim
i
w =k+k'-I Z/ I and
{'t,-... ,'w} =.Zp. If
(Q 'II.1I) then, by corollary 2. the above algorithm is separable and so is algo-
rithm (6);. If £ (Q,llhYI Z1 ) = dimL(Q,II.1I)+I=k-IZ/I+l.
then, by corollary 3, we·have I Zp I + I pC I ~ £ (Q I2 Z:2)+k -I Z/ 1+1.
(8)
combining this-with (7) we have £(QIz) =£= IZ/ I+izPl+IP 1+lpc I ~£(QI2~)+k+t-s+1. •
The following Corollaries are consequences of'Theore~s 4 and 5. Coronary 9 • Let
j(
=F [a]/ (p (a»
=pI( a)d
where P (a)
Pk. (a) are distinct irreducible polynomials. Let d
= max
l~~
l
...
Pk. (a)~' and PI (a) , ... ,
deg Pi ( a)d;.
(i)
If I F I ~ 2 d - 2 then Q ~.Y z is separable.
(ii)
If I F I < 2 d - 2 then for every quadratic system Q I Z we have £ (Q~.1zmQIz) ~2deg P (a)-~ +sp + £ (QI Z )
where sp is the number ofPi ( a) that satisfy I F I
nto, (1988). [Orl]
H. F. De Groote, On the complexity of quatemion multiplication, Information Processing Letter 3
(197S) 177-J79. [G12]
H. F. De Groote, On varieties of optimal algorithms for the computatiOn of bilinear mapp~g m. Optimal algorithm for the computation of x
y and y x where x ,y e M 2 (K ), Theoretical Com-
pllter Science 7 (1978),239-249. [Gr3]
H. F. De Groote, Characterization of division algebras of minimal rank and the structure of their algorithm varieties, SIAM J. Comput.12 (1983),101-117.
[Gr4]
H. G. De Groote, Lectures on the complexity of biiinear problems. J.N CompUJ. Sci. 245, Springer, Berlin 1987.
[GHl]
/
H. F. De Groote,1. Heintz, Commutative algebra of minimal rank, LiMar Algebra and its Appli-
catiOns, 55(1983), 37-68. [GH2]
H. F. De Groote, 1. Heintz, A lower bound for the bilinear complexity of some semisimple Lie algebras, Algebraic algorithms and error correcting codes. Proc. AAECC-3 , Grenoble 1985. J.N
CompUJ. Sci. 229, (1986),211-222. [HM]
J. Hopcroft, 1. Munsinski, Duality applied CompUJ. 2 (1973), 159-173.
to the complexity of matrix multiplication,
SIAM J.
Technion - Computer Science Department - Tehnical Report CS0575 - 1989
- 34[HMo] J. Heintz, J. Morgenstern, On associative algebras of minimal rank, Proc. ofthe AAECC-2 Conference (Grenoble 1984). [n]
J. Ja'Ja' On the complexity 'of bilinear forms with commutativity, SIAM. J.Comput 9,4, (1980), 713-728.
[IT I]
J. Ja' Ja', J: Takche, On the validity of the direct sum conjecture, SIAM. J.Comput 15, 4, (1986), 1004-1020.
[KB]
M. Kaminski, N. H. Bshouty, Multiplicative complexity of polynomial multiplication over fInite freld, Proceedings 28th Annual Symposium on Foundations of Computer Science, (1987).
[LSW] A. Lempel, G. Seroussi, S. Winograd, On the Complexity of Multiplication in Finite Fields,
Theoret. Comput. Sci. 22 (1983), 285-296. [Mi]
R. Mirwald, The algorithmic structure of sl (2, k ), Algebrq{c algorithms and error correcting codes. AAECC-3 Grenoble 1985. IN Comput. Sci. (l986), 274-287.
[SI]
V. Strassen, Vermeidung von Divisionen, J. Reine Angew. Math. 264 (1973), 184-202.
[WI]
S. Winograd, On the Number of Multiplications Necessary to Compute Certain Functions, Comm. Pure andAppl. Math. 23 (1970),165-179.
[W2]
S. Winograd, On multiplication of 2>
Technion - Computer Science Department - Tehnical Report CS0575 - 1989
Computer Science Department
MULTIPLICATIVE COMPLEXITY OF DIRECT SUM OF QUADRATIC SYSTEMS by
Nader H. Bshouty Technlca1 Report July 1989
#575
Technion - Computer Science Department - Tehnical Report CS0575 - 1989
MULTIPLICATIVE COMPLEXITY OF DIRECT SUM OF QUADRATIC SYSTEMS Nader H. Bshouty Department of Computer Science Technion - Israel Institute of Technology Haifa 32000 Israel
ABSTRACI', We consider the quadratic complexity of certain sets of quadratic forms. We study classes of direct sums of quadratic fonnl. For these classes of problems we show that the complexity of one direct sum is the sum of the complexity of the summands and that every minimal quadratic algorithm for computing the direct sums is a direct-sum algorithm. Key Words: multiplicative complexity. direct sum, quadratic algorithms. bilinear algorithms. algebras.
1. INTRODUCTION
LetF be a field. and let x
=(Xl' •.. ,x l lI
and z
=(Zl • ..• • zJc l
be vectors of indetenninates.
Let Q x = (Q 1•... , QJc ) be a vector of quadratic fonns on X I, ... , XII over F. A quadratic algorithm that computes the quadratic system QXz with multiplicative complexity Lis L
QXz= 1: Q j(z)bj (x)cj(x).
(1)
j ..1
where Qj • b j and Cj are linear forms of the corresponding variables. The minimal L such that equation (1) holds is.denoted by L (Qx z ) and then the quadratic algorithm is said to be minimal. It is known, cf. [Sll. that when F is an infmite field then L (QXz) is the 'complexity of QXz by means of sttaight-line algorithms.
Technion - Computer Science Department - Tehnical Report CS0575 - 1989
·2· Let Y
= (y I, ... ,Y". l
of b~linear forms on
be a veco of indeterminates and let QI"
= (Q' I' ... ,Q'Ie )
be a vector
x and y. A bilinear algorithm that computes the bilinear system Q X" z with multipli-
cative rank ~ is
QI,yZ=
't
1: ai (z)bj(x)cj(Y),
(2)
j,.. 1
where aj ,bi and ci are linear forms of the corresponding variables. The minimal ~ such that equation (2) holds is denoted by ~ ( Q x,y z) . It is known that
I
cf. [Jll .
The direct sum of two quadratic systems Q r z and Q ~ z (respectively, two bilinear systems Q r-' z
Q fl,'1zl + Qr-,2 ~) where Xj ,Yi ,Zj , i = 1 ,2 are distinct vectors of indeterminates. It is obvious that (
L (QrzEBQ~z)S; L (Qrz)+L (Q~z), ~(Qr,y zEBQf z)S ~(Qr·1z)+~(Q~" z).
'
When equality holds (in the second inequality) for every bilinear system Q~,y z then we say that Q f1 Z satisfies the direct sum conjecture, cf. [SIl, (in short DSC), We say that Q
Q
r l
,1
1
rZI is separable (respectively, l
zl satisfies the direct sum conjecture strongly, cf. [W3l, in short DSCS) if for every quadratic sys-
tern Q f ~ (respectively, every bilinear system Q -r,y2~, every qrinimal quadratic algorithm (respec-
£1
£2
Qr l zl +Qf~ = 1:a j(ZI )bj(XI )Cj(Xl)+ 1:a'i(~)b'j(X2)C'i(X2) j~1 j=1 (respectively, ~
~
Qr l ,11 zl +Qf,'2~ = 1:aj(zl )bj(xl )Cj(Yl)+ 1:a'j(~)b'j(X2)C'j(Y2) ), i-I i=1 where the frrst summand is a minimal quadratic algorithm (respectively, bilinear algorithm ) for Q f zl (respectively, Q
r
lo11
zl) and the second is a minimal quadratic algorithm (respectively, bilinear algorithm)
forQf~ (respectively, Qf·12ZV . L I
Obviously, quadralic algorithms can also compute bilinear systems, Q".1z =
1:aj (z) bl jel
(x, y) CI (x, y).
Technion - Computer Science Department - Tehnical Report CS0575 - 1989
-3-
l:i.
QI" Z
We defme row rank
=~f-l (XTAiY)Zi =xT ~t.lAiZi )Y=XT Q (z)y.
QI.1z
(respectively, col rank
QIJZ) to be the
dimension of the linear space
spanned by the rows of Q (z) (respectively, columns of Q (z». Define rank
QIJZ =
max (col rank QIJ z ,row rank QI"
Z ).
Denote by L (Q IJ) the linear space spanned by the entries of Q IJ and dim Q IJ its dimension. In the sequel we shall need the following definitions: Definition I • Let ?/J3 ( 't , , ) denote tM collection ofbilinear systems Q 1,1z such that: TMre exist integers sandt such that tM following conditions hold:
rank «Ql" .. •Q", Qjl' ...• Qj•.) i) ~ t. (i = (z 1•...• ZI
Definition
l
in all the definitions).
n . Let QIJZ be a bilinear system.
We say that Q' E L (QI.1) is active. if, for every base
{Q 1••..• Qk} of L (QIJ) that contains Q' , there exist Qi2' ...• Qi. such that
rank «Q' ,Qi2•
••• •
Qi)i)~ t.
Definition ill , Let 9{JJ. (, ) denote tM collectioh of bilinear systems Q IJz in 9{JJ ( 0,' ) which satisfy thefollowing conditions: (i)
For every base {Q 1••••• Qk} of L (Q1,1) there exists an active element Qi , such that for every non-active element Qi 2 and every / 1'/2 E F '/1 *0, we have / lQi l +/ ~i2 is active.
(ii)
For every base {Q 1••..• Qk } of L (Q IJ) there exist s active elements Qi ... ,Qi. such that "
rank «Qh' ... •Qi)i)
~
t.
Our main results are Theorem 1. IfQI"z E '}fJJ (0,.,) thenfor every quadratic system QIZ we have
Technion - Computer Science Department - Tehnical Report CS0575 - 1989
-4-
Theorem 2. IfQ"'J zE ?/}J (1.0) then Q ...·yz is separable. Theorem 3 . If Q"'J Z E 9IfJ3 ( 1 • r ) .r ~ 1 , then for every quadratic system Q"'z we have
L(QX'YzEB QX z ) ~ dim QXJ +t -8 + 1+L(Q"'z) = L (QxJ z ) +L (Qxz)-(r -1). Theorem 4 . If Q"'J Z E ?/}J. ( 0). then Q "'Jz is separable. Theorem 5 . If Q"'J Z E tJ,fJJ. ( r) • r ~ 1. then for every quadratic system Q Xz we have L (Q"'JzEB Q"'z) ~ dim Q ....Y+ t -8 + 1 +L (Q'Sz) = L (Q ....Yz ) +L (Q"'z) - (r -1). The set ?f}J ( 1 ,0) contains a one bilinear form. bilinear systems Q ....Yz that satisfy
L (Qx,yz ) = rank QX'Y z or L (Qx,y z ) = dim Q....Y. bilinear systems defined by polynomial multiplication and their dual systems and bilinear systems defmed by the product of two polynomials modulo a squarfree polynomial. The set
9.f/J ( 1. 1), contains bilinear systems QX.yz that satisfy
L (Q ....Yz )=
rank Q"'J z + 1 or
L (Q ....Yz ) = dim Q ....Y + 1. bilinear systems defined by the product of two quatemions and bilinear systems defined by the product X Y and Y X of two 2 x 2 matrices. The set
9IfJ3.
(0) contains bilinear systems defined by the product of two polynomials modulo a
third one.- bilinear systems Q ....Yz that satisfy L (Q"'J z ) ~ dim QXJ + 1. bilinear systems QXJz with
row rank Q"'Jz=m, col rank Q ....Yz=n and dim Q ....Y~nm-3, bilinear systems defined by the product X Y and Y X of two quatemions and bilinear systems defmed by the product of two triangular
2 x 2 matrices. The set tJ,fJJ. (1). contains bilinear systems QX'Y~ that satisfy L (QxJz)~
dim Q ....Y +2. bilinear
systems Q x.Yz that satisfy rank Q xJz ~ 3 • bilinear systems Q XJz with row rank Q x·yz = m •
col rank QX'Yz = n and dim QX'Y = n m -4 or n m -5, the bilinear system defmed by the cross product of two
3~dimensional
vector. the bilin~ system defined by the product of two element in the Lie
algebra of 2 x2 matrices and the bilinear system defmed by the multiplication of two 3 x 3. triangular matrices.
Remark. Theorem 1-5 are also true for bilinear algorithms. Let Q = (A l' ... ,Ak
)
be a vector of n x m matrices with entries from F and let x T Q y =
Technion - Computer Science Department - Tehnical Report CS0575 - 1989
- 5(xTA lY,' .. , x TAA:: y). The dual systems of (xTQ y) is (xT QD z) and (yT QT x) defmed by (xT Q y)z = (xT QD z)y = (yT QT x)z.
Our main result for bilinear algorithms is: Theorem «) • ut (xT Q y) z be a btliMar system. TMn (i)
If (xT Q y)z satisfie~ the DSCS then so does each of the dual systems (xT QD z) y and (yT QT x) Z.
(ii)
Iffor every bilinear system (xT Q '.Y)z we have !1(.( (xT Q y) z ED (xT Q' y) z) ~ c +!1(.( (xT Q' y) z). Thenfor every bilinear systems: (xTQ 1z)y and (yT Q2X)Z. we have !1(.«xTQD z)yEB(xT Q 1z) y)~ c + !1(.«xT Q l~) y), !1(.( (yT QT x)zEB (yT Q2 X ) z) ~
Let
~
c +!1(.( (yTQ2 X ) z).
be an associative algebra of dimension k with a ,u,nity element and let {a l' ... , aA::} be a
base of ~. We denote by Q ~.Y
=(Q 1, ... , QA:: ) the vector of bilinear forms defmed by the product of
two elements in ~, i.e,
It has been shown that L (Q ~.Y z) does not depend on the chooses base, cf. [FZ]. A beautiful result of Alder and Strassen~ cf. [ASt] Slates: If ~ is an associative algebm of dimension
k then for any quadmtic system Q Xz we have L(Q~,y zEB QXz) ~
2k -t
(~)+L (Ql(z),
where t (~ ) is the number of two-sided maximal ideals of ~. If L ( Q}Y z) = 2 k - t (~) (respectively, !1(.( Q ~.Y z) = 2 k - t (~ ) then we say that the algebra is of minimal complexity (respectively, minimal rank).
Denote the radical of ~, i.e the maximal (two-:.sided) nilpotent ideal contained in call the algebra ~ local if ~ I rad
~
is a division algebra. We call ~ clean if ~ I rad
of division algebras. For direct sum of algebras
~
~,
~
by rad
~
. We
is a fmite product
=~ 1 X ... X~l let C JI. (respectively, R JI) be the
number of ~i which are not of minimal complexity (respectively, of minimal rank).
Technion - Computer Science Department - Tehnical Report CS0575 - 1989
- 6-
Our main results in complexity of atlgebras are:
Coronary I Lei JiI be a clean algebra. TMnfor every quadratic syslem QIz- we have
Corollary II
=Jill X
Lei JiI
... XJilI be a direci sum of division algebras.
plexity (i.e each JiI;
jj'
If Ji1
is an algebra of minimal com-
algebra ofminimal complexily). Then Q ~.y is separable.
Corollary III Lei JiI
=F
l a] / (p (a»
where p (a)
E
F I a] is a polynomial. T~n Q ~.y is separable.
Corollary IV
Lei JiI be a clean algebra. If JiI is an algebra of minimal ranlc lhen Q}' z satisfies lhe DSCS. Corollaries (I), (II) and (III) have been Proved by Feig and Winograd, cf. [FW], for bilinear algorithms. Characterizalion of division
alg~bras
Ji1 of minimal complexity are sLu.died in [Gr3] and [FeiJ, it has
been proved lhat division algebras of minimal complexity are simple field eXLellsion of F with
IF I
~2
dim JiI-2. No results are known about flon-division algebras of minimal complexity. Charac-
terization of commutative algebras, local alge.bras, and clean algebras of minimal rank ovec a close field are given in [GH1], [BCJ and.(HMo), respectively
In sectiOfl,S 2 and 3 we prove some preliminary results needed foe the proof of the theorems. In section 4 we prove Theorems 1,2 and 3 and corollaries I and II, In section 5 we prove Theorems 4 and 5 and corollary 111. In section 5 we prove 'Theorem 6. In 5e(;tion 6 we prove corollary IV All the results in inear algorithms
SecUOfl
2,3 and 4 are for quadratic algorithms. They ar(, readiJy generalized for bil-
Technion - Computer Science Department - Tehnical Report CS0575 - 1989
·7-
2. PRELIMINARY RESULT In this section we develop some lower bound techniques needed for the proof of our results.
Let QX =(Q 1"
•• , Qk) be a vector
of quadratic forms on x =(Xl, ... ,X" lover F. Every qua-
dratic form is
"" QI = 'LLPI,jjXjXj =x A1x j..lj=i T
where Al is an upper n
Xn
triangular matrix. We derme Q = (A 1, ... ,Ak ) and the characteristic matrix
of Q is Q (z) = r.f"'IAj Zj. Obviously, we have x T Q (z) x = Q Iz.
For a vector of quadratic forms QX we define L (QX) to be the linear space spanned by the entries of QX and dim Q II its dimension. For the bilinear system Q lI,1z = x T Q (z)y we derme row rank Q X,1 Z (respectively. col rank QX,1 z) to be -the dimension of the linear space spanned by the rows (respectively, columns) of Q (z). Finally, we let
rank QX,1z = max (col rank QX,yz, row rank QlI,1z). The following Proposition is frequently used in this paper £1
£z
j..l
j=1
Proposition 1 . Let Q fz = r.aj(z)bj(x)cj(X) and Qf" z = x T Q2(Z)y = r.aj'(z)bj'(x,y)c;'(x,y) ~
minimal quadratic algorithms. Then
(ii)
Interchanging some bj ' with c;' we have
dimL (b{(O,y), ... ,b£z'(O,y)} Proof. Let Q
~
col rank Qf" z.
r= (Q 1, ... , Qk)' If Q 1~O then Q f z depends on zl and therefore there exists
ajo( z) = r.f=1 'A.j Zj
depends
on
zl'
i.e,
'A.1~ O~
Then substituting
Z1 =
I (z 2•... 'Zk ) =
- (11 'A. 1 ) r.t.2'A.jZj we obtain
(Q2-( ~I 'A. 1 )Q l' ... , Qk -('A.,t I 'A. 1 ) Q 1) z =
£1
'L Qj(z)bj(x)Cj(X) j-=1 j~o
where z= (z2, ...• z,t ) and
Technion - Computer Science Department - Tehnical Report CS0575 - 1989
-8-
dj(i)=aj(z)
'ZI=I(Z20 •••• 'a)=aj(z)-
5·
1
~: ajo(z).
where 5j.l is the coefficient of z 1 in aj (z). By induction ,(i) follows. Let Q2(Z)=[Q~I)(z), ... ,Q~m)(z)] where Q~j)(z) is the i-th column of Q2(z). If Q~I)(z);J!: 0 then Q~'Y z = x TQ2(z)y depends on Yl and therefore there exist b'jo (<X' e'j) depends on
Yl. i.e. b'jo(x,y) =
where Aj ,5j
E
/I
m
j .. l
j=1
1: AjXj + 1: 5jYj
F and 51 ;J!:O. Then substituting Y1
/I
m.
i-I
j-2
=1'(x,Y2, ... ,Ym) =-( 1/51 ) (1: AjXj + 1: 5jYj )
we obtain
£1
= 1:a'j(z)~(X,Y)Ci(X,Y) i=1 jt'jo
for someK (z) where y = (Y2,' .. ,Ym
l
and
~(X,y) = b;'(x,y) I )l1=1'(X,)'20 .••• )1.)
,
Cj(x,y) = e'j(x,y) 1)11 =l'(x,)'a, ... ,)I.)'
By induction, (ii) follows. • Remark. In what follows the results are true for row rank in as much as col rank. The results are proved either for row rank or col rank for our convenience. Allover the paper we assume that al(z), ... ,.aA;(Z),
k = dimQr, (respectively,
b 1(O,y), ... , bm ( O,y). m = col rank Q ~.Y z) are linearly independent Proposition 1 implies the following Lemma 1 • ([Fil,[Wl]) We have
L(QXz) ~ dimQx, L(xT Q(z)y) ~ rank QX,Jz.
r·
Lemma 2 • Let x, y , u , v , z· be distinct vectors of indeterminates and let Q u z = x T Q 1( z)u be a bi!inear system. Then
Technion - Computer Science Department - Tehnical Report CS0575 - 1989
- 9.-
£(XT Q I(Z)U+XTQ2(Z)y+yT Q3(Z)y) ~ col rank Qf··z+£(yT Q3(Z)y), v and hencefor the bilinear systems Qr'-z u TQ I(Z)X and Q z u TQ I(Z)v we have
r· =
=
£(uT QI(Z)X+XTQ2(Z)y+yT Q3(z)y) ~ row rank Qr~z+£(yTQ3(z)y), £ (u T QI(Z)v+yT Q3(z)y)
~ rank Qr·v z+£ (yT Q3(z)y).
Proof. By the same substitution in u as in the proof of Proposition 1 we obtain an algorithm for XTAI(z)X+XTA 2(z)Y+XT Q2(Z)y+yT Q3(Z)y, for some A I (z) and A z( z). with multiplicative complexity £' = £ (xTQ I(z) u + xTQ 2( z) y + yTo. 3( z) y) - col rank Q
r·
B
z.
Now substituting x = 0 in the algorithm • we obtain a quadratic algorithm for yT Q 3( z)y. Therefore £' ~ £ (yT Q 3( z)y) and the proof is followed.
Ii
The last two results of this section are well known. Lemma 3, is used in the literature to obtain
low~r
bounds for the complexity of bilinear systems. cf. [W2]. [W3] and. [KB]. Fo," the sake of completeness a proof is given which illustrate our methods. Lemma 4 is trivial and we shall refer to it iJf the sequel. Lemma 3 • /f for every non-singular matrix N there exist sentries Qi\,"" Q;. of QXN such that
L
QXz = Lai(Z)bi(x)Ci(X), i-I
By proposition 1 there exist k = dimQx independent ai(z) and therefore w.l.o.g. there exist a nonsingular matrix N satisfying a I(N z j = ZI' ...• a" (N z) = z" flnd hence " QXNz= LZjbj(x)cj(x)+
i=1
LL
aj(Nz)bi(x)Ci(X),
i=k+1
Assume w.l.o.g. that the first entries Q I, ... , Q., ofQxN satisfy
Then. substituting z.J+I
= ... =z" =0 we get s
(QI.· .. ,Qs ,0, ... ,O)i= LZjbj(x)cj(x)+ j-I
L
L i=k+1
aj
(Ni)bj(x)cj(x),
Technion - Computer Science Department - Tehnical Report CS0575 - 1989
- 10where i = (ZI •.•• • z" ,0•...•
ol _Therefore L -Ie +s ~ t. and this completes the proof.
•
and let N and K be n X n' and Ie x Ie' matrices ofranJc. n and Ie. respectively. Then x T Q(z)x
L
=1:aj(z)bj(x)Cj(x)
(3)
j=1
is a minimal qlUJdratic algorithm for Q (z) if and only if L
iTN T Q (Ki)Ni= 1:aj (Ki)bj (Ni)Ci(N i)
(4)
i ..1
is a minimal qlUJdratic algorithmfor NT Q (K i)N. Proof. Obviously. (3) implies (4). There exist a matrices N- and K- such that N N-
=/" ,and K K- =/Jc
where /" is the n x n identity matrix. Substituting N-x for i and K-z for i in (4) we obtain (3). •
3. SEPARABLE ALGORITHMS In this section we shall give conditions on the quadratic algorithms that compute the direct sum of two quadratic system~ Q IZ and Q2Z, to be separable to two minimal-quadratic algorithms, the flISt for Q IZ and the second for Q 2z.
Let
and
~ = (zr+I' . - .• zr+s l be vectors of indeterminates and let x = (xi, xll and z =(zi, i[)T. For Q F and Qf
, two vectors of quadratic forms, we say that the minimal quadratic algorithm L
QXz= Q;'ZI +Qf~ = 1:ai (z)bj(x)cj(x) j=1 A
is a separable algorithm if there exists a ~t / ~ {I. _..• L ,} = N L such that the following conditions hold: (i)
Qflzl= 1:ai(z)bj(x)cj(x) ,QrZ2= 1:aj(z)bi (x)Ci(X). iel
(ii)
The first quadratic algorithm- in (i) (respectively, the second) is minimal quadratic algorithm for
Q (iii)
if I
r zl (respectively, Qfzv. l
The terms ai(z).bj(x),Cj(x) ,i E / (respectively, i ~ /) are linear forms of zl and Xl (respectively, z2 and XV-
Technion - Computer Science Department - Tehnical Report CS0575 - 1989
- 11We say that Q f' zl is separable if for every quadratic 'system rithm for Q f' zl + Q
r
Q Zz every minimal quadratic algo-
r Zz is separable.
The following lemma proves that condition (i) is sufficient-for an algorithm to be separable. (The
teetmique we use in our proof was used in [AFW] for bilinear algorithms). Lemma 5 •If there exist a set I ~ N L such that
Qf'Zl = Laj(z)bj(x)cj(x), j
eI
then the algorithm is separable.
Proof. Obviously,
QfZz=Qxz -Qf'Zl
= Laj(z)bj(x)cj(x). j
If II I
> L(Qi'Zl) then, since INL-I
,I
I~L(QI2z2),wehave
L(QXz )= I I I + I NL-I I> L(Qtz1)+L(QrZ2)' A contradiction. Therefore I I I = L ( Q f' zl ) and (ii) follows.
Assume that for some i 0 E I ,ajo (z) = AZio +d'jo( z) depends on zio (Le. A;t 0) where j 0 > r, Le ajo( z) depends on Zz. Then substituting zio
Qi'Zl
=-( 1 / A) d'jo( z) in the first algorithm'we obtain
= L
d'j(z)bj(x)cj(x)
j e 1-( jo)
where d'j (z) = aj ( z) 1'10 __ ( 1/ A.) d.( z)" Therefore L ( Q i' zl ) S I I I - 1. A contradiction. Interchanging the roles aj and bj , (iii) follows. •
Lemma 6 • If the algorithm is not separable then there exist aj (z) depending on zl and z2' and b j(x) or Cj(x)
depending on xl and x2'
Proof. Assume that each aj (z ) depends either on zl or z2' Let I = (i I aj ( z) depends on -zl }. Then by substituting Zz = 0 we have
Qi'Zl
= Laj(z)bj(x)cj(x), j
eI
By lemma 5 the proof is completed. A similar proof holds forbj(x) and Cj(x). • Lemma 7 • Let x,z,i,i,N and K be as in lemma 4. If xTQ (z)x is separable then iTN T
Q (K i)N i is separable.
Technion - Computer Science Department - Tehnical Report CS0575 - 1989
- 12·
x
Proof. Assume that T N T Q (K i) IV
x is not separable. Then by lemma 6 there exist a minimal qua-
dratic al,goritlun x T NT Q (~i)N x+QUV
L
=LQj(i, u)bj (x, V)Cj(x, v) j ..l
=
such that Q 1(i , u ) depends on i and u. Then substituting x N-x and using lemma 4 we have that L
xT Q (K i)x+Qu v = LQj(i. u)bj(N-x, v)cj(N-x, v), j.l
is a minimal quadratic algorithm. Note that Q 1(i •u) still depends on i and u and therefore x T Q (K i) x is not separable. By lemma 6 there exists a (w.1.o.g) bjo (N-x , v.) depending on x and v. Substituting i = K-z and using lemma 4 again we have that L
x T Q(z)x+Quv = LQj(K-z, u) bj(N-", v)cj(N-x, v) j ..l
is a minimal quadratic algorithm. Since bjo (N-x, v) depends on x and v then by lemma 6 we have a con-
tradiction to the fact that x T Q( z)x is separable. • Remark. For bilinear systems xT Q(z)y Z=(ZI•... • zkl and matrices
=QX'Yz 'with x =(Xl •••. ,x
lI
l . y = (Yl • ... •Ylftl and
N ,M ,K of rank n ,m ,Ie ,respectively, lemma 4 and 7 hold for
xT Q (z)y and xT N T Q (K z) M y. 'Otis follows because T T
xlV
0 Q (K z) ·N 0 x ' OM1[1 y' T[ J 0 0 l[
T T N 0 Q(Kz)My=(x ,y) OM
[
Thus we can assUme all over the paper that-
row rank QX'Yz =n • col rank QX'Yz = m ,dim Q'S'Y = Ie. An equivalent condition for the seperabelity of algorithms is given in temma 8 below. We need the
following defmition
r zl +QfZ2 we say"that a non-singular matrix N does not mix Q r with if the entries of Q fl N are either in L (Q ,r or in L (Q f). We recall that L (Q t ) is the linear
Definition 1 • For QXz = Q Qf
t
t
t
)
space spanned by the entries of Q ft .
Lemma 8 • Iffor every non-singullJr matrix N such that QX N z = L zjbj,(x)Cj;(x)+ L Qj(N z)bj(x)cj(x), j;el
j"l
Technion - Computer Science Department - Tehnical Report CS0575 - 1989
·13 where 1 = {h, ... ,jr+8} is a subset of {I, ... , L} and 11 I = r +s , the matrix N does not mix Q F with Qr. Then the qUlldTatic algorithm is separable.
Proof. Assume that the quadIatic algorithm is not separable. Then (w.l.o.g) at(z) = at,l (Zt )+a 1.2(~) where a t.l (Zt) ':j:. 0 and a t,2 (z2) ':j:. O. LetN be a non-singular matrix such that (w.l.o.g) L
r~
QXN Z = ~zibi(X)Ci(X)+ i,.,1
Since N does not mix Q
~
aj(N x)bj(x)cj(x).
j=r~+t
r\ with Qr then N is of the fonn N1
for some permutation matrix
E.,
Z.-\(I)
OJ
N = [ 0 N E. 2
where Q-'N = a 1(N
E.
E.
t
1
= (QrW I.QrN 2 ). Since al(N z) =Zt. we have
z) = a 1,1 (N t Zl )+a-t.2 (N 2~)
and therefore a 1,1 (N 1Zt ) = 0 or a 1,2 (N 2~) = O. Since N is non-singular. Nt and N 2 are non-singular
and therefore al.l (zl) = 0 oral,2(~) =0. A contradiction. •
4. DIRECT SUM OF SOME CLASSES
In this section we define some classes of quadratic systems and prove that they are separable. Definition 2 • Let ?{JJ ('t • r ) denote the collection ofbilinear systems QXoYz such that: There exist integers
sandt such that thefollowing conditions hold: (i)
For every non-singular matrix N and for every 't entries Qj\, ...• Qi, of Qx.1N there exist S -'t entries Qit' ... , Qj..... ofQx'1N such that
rank (ii)
« Qj\, ... , Qi,. Qh' ... , Qj..... ) i) ~ t.
L (QX.1 z ) = dimQx'1 +t -s +r.
By lemma 3 if QX.1z satisfies (i) then L (QxoYz ) ~ dim Qx.1 + t
Remark. Defmition 2 is equivalent to definitiOn I in the introduction.
-s .
Technion - Computer Science Department - Tehnical Report CS0575 - 1989
-14 Our main results in this section are Theorem 1 .IfQ 'SJZe 'J,fJ3 (0 , r ) then for every quadratic system Q'Sz we have L(Q'S·JzEBQ'Sz).~
Theorem 2 • IfQ'SJ ze
dimQX,J +t -8 +L(Q'Sz) = L (Q'SJz)+L (Q'Sz)_ r .
9tfJJ (I ,0) then Q'SJz is separable.
Theorem 3 • If Q'SJZ e !J.fjJ ( 1, r ) ,r ~ 1 then for every quadratic system Q'Sz we have L(Q'SJzEB Q'Sz) ~ dimQX,J +t -8 + 1+L(Q'Sz) = L (Q'SJz)+L (Q'Sz)- (r -1).
Proof of Theorem 1 •
=Q IZ 1+ ... +Qkzk e !J.fjJ (0, r) and QXaz2 =Qk+lzk+l+ ... +Qk+k,Zk+.t' be any quadratic system, set Q'S =(Q'StJt , Q'S2), x = (xf, y f, xll and z = (zf, zll. Let N be any nonLet Q'ShJtzl
singular matrix and let Q'SN =(Q{+Q(, ... , Q'A:+k,+Q"k+.t')
where Q'I'
,Q'k+k,e L (Q'ShJt) and Q" I' ... ,Q"k+k,e L (Q'S2). Since N in a non-singular there
,jk' such that {Q"h"" ,Q"ja'} is a base for L (Q'S2). Since Q'StJtzl e !J.fjJ (O,r) there
existh, exist Q'i t ,
•••
,Q'i. such that
Then for A
(Q ' I + Q"I ' Q'J + Q") J = (Q' it + Q"it' ... , Q' i. + Q"i.' Q'h + Q"h' ... , Q'ja'+ Q") ja'
we have
£
«Q', +Q", )" +(Q'/ +Q"/ )z,) = £ «Q', ,Q'/) [ ~] + (Q""Q"/ ) [~ )
~
rank «Q', ,Q'/)
Since Q'/' is a base for L (Q'S2),
Now using l€?mma 3 and 4 we have
[~]
)+ £ «Q", ,Q"/)
[~]
) (Lemma 2)
Technion - Computer Science Department - Tehnical Report CS0575 - 1989
- 15L(QX z )
=L(Qx N z) ~ (k +k')+(t +L (Q x2z2 ))-(s +k') =k +t -s +L (QXa~) . •
Proof of Theorems 2 and 3 • Let x and z be as in the proof of theorem 1. Let L
QXz = :Eaj(z)bj(X)Ci(X) i=1
be any minimal quadratic algorithm for QXz. Let N be any non-singular matrix such that
QX N z = :E Zibjj(X)Cjj(x)+ :E aj(N z)bj(x)cj(x), ftES
where S = {j I'
. . . , A+k'}
j~S
is a subset of {1, . . . , L} and rS I = k
+ k '. If every matrix N
that satisfies
the above equation does not mix QXIJI with QXa then by lemma 8 the two theorems are proved. Assume
that some N mix Q XIJI with Q X2 then there exists an entry Q'h + Q "h of Q x N such that Q ' h ' Q "iJ
Q'he L (Q X2) and Q"he L (Q XIJI). Since N is a non-singular matrix there exist h, that Q"iJ"" ,Q"ji' is a base for L (QXa). Since QXIJI Z1 e
'JlP (l,r)
* 0,
... ,A, such
there exist Q'i l , ' " ,Q'i._1
such that
Then for t.
(Q ' /'+ Q"/', Q'J + Q") J = (Q' il + Q" ii' ... , Q'i.-I + Q"i.-I' Q'h + Q"h' ... , Q'A· + Q") ji' we have (as in the proof of Theorem 1) x
L «Q'/,+Q"/, )ZI +(Q'J +Q"J )z2) ~ t +L (Q 2z2 ). Substituting zi = 0 for i ~ I' U J in the quadratic algorithm of Q x N z we obtain a quadratic algorithm for (Q'/,+Q"/, )zl +(Q'J +Q"J )~ with L -(k +k' -(s +k' -1)) multiplication. Then by lemma 4 we have
L(Qx z ) = L(QxN z) ~ (k +k')+(t +L (QXa~))_(s+k' -1) = k +t -s + 1+L (Qx~). If L (QXIJI Zl) = k + t
-s we have a contradiction and therefore QXIoYI Z1 is separable. On the other hand
In the sequel, Qx'Yz denptes a bilinear system and QXz denotes any quadratic system. Corollary 1 • Let QX,Y be a vector ofone bilinear form. Then QXJz is separable.
Technion - Computer Science Department - Tehnical Report CS0575 - 1989
- 16-
CoroJIary2 .If L (Q:l J z ) = rankQ:l J z or L (Q:I,yz)::;: dlmQ:l J then QX,yz is separable. Proof. If L (Q:I,yz) = rank QX,yz then for every non-singular matrix N we have rank (QX,y N z )
= rank (Q:I,yz)
and therefore Q:I,yze
9tfJJ ( 1,0). If
L (QX,yz)
= dimQ:I,y
then for every non-
singular matrixN, any entry Qi of Q:I,yN satisfies rank (QiZi ) ~ 1. Therefore Q:I,yz e
9tfJJ ( 1,0). •
It follows from corollary 2 Coronary 3 .If L(Q:I,yz) ~ rankQ:I,yz+ 1
=lor L(QX,yz) ~ dimQ:I,y + 1 =I then
L (Q:I,yzeQ:lz ) ~ I +L(Q:Iz).
.Coronary 4 • Let J'l be a division algebra. If J'l is a simple field extension and I F I ~. 2 dim J'l - 2 then
Q ~,y z is separable. If J'l is not a simple field extension or I F I
< 2 dim J'l- 2 then
L (Q}Yze Q:lz ) ~ 2 dim J'l +L (Q:lz ).
Proof. For every non-singular matrix N every entry of Q~'YN has rank k = dimJ'l. Therefore
Q ~,y z e
9tfJJ ( 1 ,r ).
De Groote, cf. [Gr3], proved that L( Q ~,y z) = 2 dim J'l-1 if and only if J'l is a
simple field extension and I F I ~ 2 dim J'l - 2. Now the proof follows immediately. •
Remark. Corollary 4 with the results of [Fei] and [W4] classify all the minirilal quadratic algorithms for when
I F I ~2 max
ISiSt
P (a)=pl (a)·· ·Pt (a)e F [a]
deg Pi (a)-2.
For
Cn
the
cyclic
group
are of
sqwufree
and
n
have
order
F [Cn ] =F [a] I (an -1) and ex" -lis squarfree for char F ~.O (mod n ). Corollary 4 implies the following t
Coronary 5 • If J'l Q is the algebra ofquaternions over the real field R and J'l =
Corollary 6 • Let J'l be a clean algebra. Then
Proof. In [ASt] it is shown that
P J'l Q • Then
1=1
we
Technion - Computer Science Department - Tehnical Report CS0575 - 1989
- 17.£:.(Q}YzEBQ;)~2 dim
rad >t+.£:.(Q~'T'adAZEBQ;).
Since >t I rad }I is the direct sum of division algebras then from corollary 4 the proof follows. • Corollary 7 • Let char F "" 2. Let QI,yz be the bilinear system defined by the product X Y and YX of two 2 x2 matrices. Then
.£:. (QI,JzEBQIz) =.£:. (QI,yZ) +.£:. (QI Z ). ~
Q
E
It is known, cf. [Gal, that for char F "" 2 we have .£:. ( QS,yz) S 9, dim Q x,y = 7 and every
L (QI,J) satisfies rank Qz = 2. Therefore QX,Jz E 9{J3 (1,1). •
Corollary 8 • Let QX,Jz be the bilinear system dqrned by the product of two polynomials of degree n. Then (i) If
I F I ~2 n then QI,JZ is separable.
(ii ) If I F I
< 2 n S 2 I F I + 2 then £ (Q"z(jjQ'z)
(iii) If I F
= 3n
+l-ll ~ j I
+£ (Q'z).
I < n 1/3 then
Proof. If I F I ~2n then, cf. [W3],
.£:. (QI,yZ) = dim QX'y = 2n-l.
Using corollary 2 (i) follows. (ii) and (iii) follows from [ABK] and [KB, Lemma 4 and 5]. •
s.
DIRECT SUM OF ?l1J ( 0, r ) .
In this section we define a subset of 9{J3 ( 0, r ) and prove that all quadratic systems in this subset are separable. This subset contains QI'{a) I (I (a» Z for f (a) E F [a]. Definition 3 • Let Q I,yZE ?(JJ ( 0, r ) be a bilinear system. We say that Qj) e L (Q I,,) is active
if for
every non-singular matrix N such that Qj) is one of the entries of Q,S,YN there exist s -1 entries
Technion - Computer Science Department - Tehnical Report CS0575 - 1989
- 18Definition 4. Let '1IfJJ. (r ) denotes the collection ofbilinear system QIJ'Z in '1IfJJ ( O. r ) such that: (i)
For every non-singular matrix N there exist an active entry Q'i. of QX,Y N such that for every non-active entry Q'i20fQI'YN and every I 1.12
(ii)
E
F .11 *Owe have I lQ'i. +12Q'i2is active.
For every non-singular matrix N there exist $ active entries Qi•• ...• Qi. ofQI'YN such that
rank «Qi••...• Q;)i)
~t.
Remark. DefInitions 3 and 4 are equivalent to defInitions IT and min the introduction. When $ = 1 an equivalent definition of '1IfJJ" (r ) is given by the following
Proof • We recall that if Q IJ'Z E 9{JJ (0. r) with $ = 1 then for every non-singular matrix N there exist an entry Qi of QIJ'N such that rank Qizi
rank QjZj
~
~ t.
Hence every active entry is an entry Qj with
t (non-active entry is an entry Qj with rank QjZj
< t).
Assume by contradiction that for some non-singular matrix N and for every active entry Q'i of QIJ'N there exist a non-active entry Q 'j(i) in QIJ' N and/h(i) .Ijii) E F ,INi) (lh(i)Q'j
Then for /
={i
+Ijij~'j(i»
is not active.
fi!! /
(5)
I Q'j is active} we defme QI,y N N' = (Q"I" .. ,Q"k) such that Q'i { Q"; = Q'j + (Ijij/ fh(j»Q!j(j)
Since j (i )
*0 where
i fi!! /. i
E /.
we have {Q" 1. . . . • Q " k} is a ~ for L (Q IJ') and therefore N' is non-singular.
Now, N N' is non-singular. Q"i • i fi!! / is not active and by (5). Q" j • ie/ is not active as well.A contradiction. • In this section we prove the following Theorem 4. IfQI'YZE 9{JJ. (0) then QIJ'Z is separable. Theorem 5 • If Q I.YZE ?f}3. (r) • r ~ 1 then for every quadratic system Q I Z we have L (QI.YzEf) QIZ ) ~ dimQI,y +t -$ + 1+L (QI Z ) = L (QIJ'z)+L (~z) -(r -1).
Proof of Theorems 4 and- 5 •
Technion - Computer Science Department - Tehnical Report CS0575 - 1989
- 19Let QX.'Y· Z1 = Q lzl+' .. +QkZk be in 'J.(JJ* (r) and let QX2zz = Qk+1Zk+l+' .. +Qk+k,zk+k' be
an arbitrary quadratic system. Set QX=(QxIJ .,Q X2),
z=(z{,zll and x=(x['y{,xi)T.
Assume that L
QXz = Laj (z)bj (x)Cj (x) j-l is a minimal quadratic algorithm for QXz. If for an arbitrary non-singular matrix N such that (w.l.o.g) k+k'
QX N z =
L zjbj(x)cj(x)+
L
L
aj(N z)bj(x)cj(x),
(6)
j=k+k'+1
j=1
it does not mix QX.JI with QX\ then, by lemma 8 the two theorems are proved. Assume therefore that
L (Qxz ) ~ k +t -s + 1 +L (QX2~)
("') X For then if Qx.,YI ZI e 'J.(JJ* (r) ,r ~ 1, Theqrem 5 follows and if Q hYI ZI e ?(JJ* (0), then we have a contradiction and Theorem 4 follows. Let
QXN =(Q'I+Q"I"" ,Q'k+k,+Q"k+k')
where Q'j e L (QXhYI) and Q"j e L (QX2), i
=1, ... , k +k'.
Since' QXIJ·Z e ?(JJ* (r ) there exist
Q'j. +Q";. in QX N such that Q'jl is active. Let ZI = {w I Q'w is active} anddeflne Zj
={w I there exists llr(N z) depending on some zp ,p e Zj-l and zw}.
Let Z/ be the fust set that satisfies Z/
~Z/+1
(Obviously, Z/+1
=Z/+2 = .. ').
We now distinguish
between two cases: Case I. There exists i' e Z/ such that Q "i' * O. By the definition of Zj. we can fmd a sequence it, i 2 • ..• , i p of minimal length p such that Q"i"
* O. i q e Zq -Zq-l • q =2, .. . ,p
• i 1 e Z l' We proceed to proof ("') by induction onp.
If i 1 = ip i.e. Q "i. * 0 then since Qi.' is active we have as in the proof of theorem 2 and 3
Technion - Computer Science Department - Tehnical Report CS0575 - 1989
- 20· Assume p
> 1.
Becau~
*" 0 we have Q "i
P is the minimal integer such that Q";,
l =
••• =
Q";'_1 = O.
and by the defInition of Z 1 we have Q'i,• ...• Q'i, are not active. By the defmition of Zq and P. there exists aro(Nz) which depends on zi l
and zi, and does not depend on zi,•... ,zip' i.e.
1. A (z. -~z, -a(z)) I,
II
for zi, in (6) we obtain k+k'
Q N N'z=
L zibi(x)Ci(x)+Zizbro(x)cro(x)
jz1
j ~jz
L ,
1
,...,
L.
+
ai(N N Z)bi(X)Ci(X)+~(Ziz-~zil-a(z))bi,(x)ci,(x),
i
co
k+A:'
i
~
ro
where the i 2 entry of Q N N' is now
~ Q'iz-I ~ Q"iz Q ' i'-I and the ip ' ... , i 3 entries remain Q'i" ... , Q'jJ' respectively. Since Q'jl is active and Q 'jz is not active we have Q 'jl +
~
Q'i, is active. ,Note also that there exists aj (N N'z) depend on Zjl and Zj,. (Actually. if
ai (Nz) depends on zi, and ZjJ then aj (N N'z) depends on Zjl and ZjJ. The aj (N z) that depends on Zj.
j E ZZ-Z2 satisfies aj (N N'z) =aj (N z). So we have now the ~w sequence i 1. i 3 , .•• ,ip thatsatisfies the above conditions with the new sets Zl UZ2 ,Z3' ... •Zp. Assuming that (*) holds for p -1, it follows that it holds for·p. This accomplishes the proof for this case. Case
n . For
every i E Zz. we have Q"i = O. I.e. every ai (N z) depends on either
Zq
,q E Zz or
Let P = {i I aj (N z) that depends on some We now estimate the number of ai (N z) that depends on we obtain an algorithm tha~ computes entries are Q'j,
Zq ,
Zq
,q
E
Zz }.
q E Zz. Substituting Zq = 0 for q '" Zz
Qz where the entries of Q are from L (QXt.YI).
Since all active
i E,ZlaZ, by condition (ii) (in Defmition 5), there exist s active entries
Technion - Computer Science Department - Tehnical Report CS0575 - 1989
- 21-
Q'it.··· .Q'j•• {jt.··· .j~} E Zt such that rank (Q'it•...• Q'j.)i ~t and therefore by lemma 3
we have I Z/ I
+ I P I =£(Qi)~ I Z/ I +t-s.
(7)
Let pC = {k +e' + 1, ... , £ }-P , Zp = (I, ... , k + k' }-Z/. Now substituting in the quadratic algorithm Zq = 0 •for q E Z/ we obtain
r, zjl?i(x)cj(x)+ r, jf!z,
Since Q"i
M, L
:= 0
Qj
(N z)bj(x)Cj(x).
if!P
for i E Z/' the above ~gorithm computes (Q lll:hYI , Q I2) M Z for some non-singular matrix
(Q,IhYI)l;;;;L (QII.1I), i=(zrl"" ,zrw)T ,
£ (Q ,IhYlit ) = dim
i
w =k+k'-I Z/ I and
{'t,-... ,'w} =.Zp. If
(Q 'II.1I) then, by corollary 2. the above algorithm is separable and so is algo-
rithm (6);. If £ (Q,llhYI Z1 ) = dimL(Q,II.1I)+I=k-IZ/I+l.
then, by corollary 3, we·have I Zp I + I pC I ~ £ (Q I2 Z:2)+k -I Z/ 1+1.
(8)
combining this-with (7) we have £(QIz) =£= IZ/ I+izPl+IP 1+lpc I ~£(QI2~)+k+t-s+1. •
The following Corollaries are consequences of'Theore~s 4 and 5. Coronary 9 • Let
j(
=F [a]/ (p (a»
=pI( a)d
where P (a)
Pk. (a) are distinct irreducible polynomials. Let d
= max
l~~
l
...
Pk. (a)~' and PI (a) , ... ,
deg Pi ( a)d;.
(i)
If I F I ~ 2 d - 2 then Q ~.Y z is separable.
(ii)
If I F I < 2 d - 2 then for every quadratic system Q I Z we have £ (Q~.1zmQIz) ~2deg P (a)-~ +sp + £ (QI Z )
where sp is the number ofPi ( a) that satisfy I F I
nto, (1988). [Orl]
H. F. De Groote, On the complexity of quatemion multiplication, Information Processing Letter 3
(197S) 177-J79. [G12]
H. F. De Groote, On varieties of optimal algorithms for the computatiOn of bilinear mapp~g m. Optimal algorithm for the computation of x
y and y x where x ,y e M 2 (K ), Theoretical Com-
pllter Science 7 (1978),239-249. [Gr3]
H. F. De Groote, Characterization of division algebras of minimal rank and the structure of their algorithm varieties, SIAM J. Comput.12 (1983),101-117.
[Gr4]
H. G. De Groote, Lectures on the complexity of biiinear problems. J.N CompUJ. Sci. 245, Springer, Berlin 1987.
[GHl]
/
H. F. De Groote,1. Heintz, Commutative algebra of minimal rank, LiMar Algebra and its Appli-
catiOns, 55(1983), 37-68. [GH2]
H. F. De Groote, 1. Heintz, A lower bound for the bilinear complexity of some semisimple Lie algebras, Algebraic algorithms and error correcting codes. Proc. AAECC-3 , Grenoble 1985. J.N
CompUJ. Sci. 229, (1986),211-222. [HM]
J. Hopcroft, 1. Munsinski, Duality applied CompUJ. 2 (1973), 159-173.
to the complexity of matrix multiplication,
SIAM J.
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J. Ja'Ja' On the complexity 'of bilinear forms with commutativity, SIAM. J.Comput 9,4, (1980), 713-728.
[IT I]
J. Ja' Ja', J: Takche, On the validity of the direct sum conjecture, SIAM. J.Comput 15, 4, (1986), 1004-1020.
[KB]
M. Kaminski, N. H. Bshouty, Multiplicative complexity of polynomial multiplication over fInite freld, Proceedings 28th Annual Symposium on Foundations of Computer Science, (1987).
[LSW] A. Lempel, G. Seroussi, S. Winograd, On the Complexity of Multiplication in Finite Fields,
Theoret. Comput. Sci. 22 (1983), 285-296. [Mi]
R. Mirwald, The algorithmic structure of sl (2, k ), Algebrq{c algorithms and error correcting codes. AAECC-3 Grenoble 1985. IN Comput. Sci. (l986), 274-287.
[SI]
V. Strassen, Vermeidung von Divisionen, J. Reine Angew. Math. 264 (1973), 184-202.
[WI]
S. Winograd, On the Number of Multiplications Necessary to Compute Certain Functions, Comm. Pure andAppl. Math. 23 (1970),165-179.
[W2]
S. Winograd, On multiplication of 2>
