-TR-87・64 ーSE

lSE−TR−87−64
ノ×∴毎.
乏 紋蹴鴇
ノ ハ ゴしま ト
し/’
ご!フ
、
\/応∴.\/
\曳ン〆
Hyperplane vs. Multicolor Vectorization of lncomplete
LU Preconditioning for the Wilson Fermion on the Lattice
Yoshio Oyanagi
August 5, 1987
1SE・一TRL87‘64
August 5, 1987
H.yperplane’ Vs. Multicolor Ve’ctoriza’tibn of’ lhcomplete’
LU Preconditio’ning for the Wilson Fermion’
[email protected]
[email protected]
Ybshi o’ Oy’anagi
lnSti’tute’ef・1nformatio’n Sciences. University of Tsukuba
Saku・a」mu』ra・N・i iha・i−gUn・lba由‘ki冷0らJAPAN
We combare’the hyperplane’and 16−celor vecti6ri;
zations of minimal residual and conjugate residual
methods with an incomplete LU preconditioning to
solve the lattice Dirac equation in the Wilson for−
mulation. The performance was assessed in term$ of
the Euclidean diStance from the true・ solution for
various hopping pq,r.ameter$ o.n.a.q.ueng,hed gauge qon−
figdra’ 煤f [email protected]=5.5 on a’n 84 ’ =f 狽狽奄モ?D’ @Th’ @16“color
?f
P’
’vectorization re’ 曹浮堰fre’ [email protected].“4 timesi’ thbre it’
?窒≠狽奄盾窒獅
.than .the. hyperplane.,vecterization. Even wi,th the
best. cheice of th,e. accele.r..atign.pq.ramgtert the
latter method is preferabl” [email protected]
the computing tinie for一 brie it’
[email protected]
пf
奄??狽狽秩fs 6y’a
factor of twb・ or mo.re.
(submitted to Journal of lnformdtion processing)
1. lnt.roduct.ion
The nunierical simulations of the lattice gauge theoryi the
quantum chromodynamics on the lattice[1]. has proven to be useful
for’ @extrqc.t ing quant. ’t tot.ive pr.edigtio.ns abqut the .hadron physics
from the first principles.. The lattice gauge the.ory is. fo,rmulat−
ed on a feur−dimensional hypercubic lattice. ln the Wilson
formulation[1]. the fermion (quark) field 3P’ioc(×) is a 12
component complex quantity allocated on a lattice site and has
two indices: one for the Dirac index (,jFlt 2. 3. 4)t the other
for the co16r (ex=1. 2. 3). On the other hand. the gauge (gluon)
field U(×.y) is a 3×3 unitary matrix with unit determinant and is
defined on a /ink, i.e. the. side between the nearest neighbor
lattice sites’ x and y・ 丁he ro.w$ and .columns cor、respond to the
color indices. The recipiocity requires U(y.×) = U(×.y)’t’
D
ln the numerical simulation of the lattice gauge theoryi
m。・t。f the cgm叫e「timr is spept in splving the Di・ac equati。n
on the lattice.
A 3P =b
(D
where A is a large sparse non−Hermitiari matrix which depends on
the background gauge field.U and a parameter jV called ”hopping
parameter” as described in the fbllowing section. The hoppi’ng
parameter is a measure of couplin’g be.tween t.he qugrk ,fie.ld’
№?高垂盾
ne,nts at the nearest neighbor sites .and i.s rglated tp the mass of
the. quark mh in such a way that m. is proportional to 1/iG 一一 1!rG .
q . q ”一 ”一H’ ” T’ H C
When rc approaches iC一. the equation becomes nearly singular and
c
difficult to solve.
一2一
Since/ the size o’f’ thg coef・ficien’t matrix A is very large.
the amount of’work and storage req’uired in direct methods such as
Gauss elim.inati・oh is nearly prohibitlive excePt for s出ち目
Iattices> such as 44.. The standard procedure so far ha’s been the
t’ ’t’
conjugate gradient (CG) method[2] f・or A’A or AA’. which’gives the
exact solution in finite steps if the round一・off error is absent.
ln the previous paper[3] we presented a. fast method based on the
conjugate residual (CR) method and an incomplete LU (ILU) decom−
position and proposed a hyperplane’vecterization of the ILU
precondit’io’ning. Recently a different vectorization based on a’
16rcolor. classificat.ion・ appeared in the 1 iteratur’ ?m4) and based
upon this vectorization,. they・claimed that the ILU precondition−
ing was worse than other preconditioning methods especially’ when
the hopping parameter fG is close to the c.r.itical value rcc.
In the present paper・we w日」 compare the hyperplane and 16−
color vectorizations of an incomPlete LU preconditioning as
applied to the minimal residual (MR) as well as the conjugate
residual (CR) metheds. For the t.wo ways of vectorization. we
want to assess the performance as a function of the hopping
parameter tc on a quenched gauge configuration at i3=s.s on an s4
1attice. w.e measured the error l13Pt,一 A−lbll. where・’{P. is the’vatue
’ ’ 1 ” T I
of 3P’ at the i−th iterationi as a function of i in’the two cases.
The CPU time is not a good measure of the performance. since it
critically depends on the arghitecture,and especially on the
fine−tuning of the code[3].
ln the next section we will describe the Wilson termiOh on
the lqttice; section 3 discusses the Conjugate residual methods;
sec’ [email protected] .incompiete LU
−3一
degQmpositibn; se.ction ,5 d・i s, cus$es the・v.ectori,zation’ of
m”ltipli.q at ・i Qn and lLU・P・eC。ndi・ti。nin99f.th・W日・。n fe・mi。n
matrix; section 6 gives a brie・f descrjptioh of 16−color
vector.izat.ion; section.7 disgusses numerical results.
2. W日、son Lattice Fermi.ons
The、coefficient matrix A is defined as』a block matトi「x
{A(x’y)} Where xゴand.y・rePresent generic Iattice sites,on a fou・r−
dimen・i・nal’ ?凾吹E・cubi・lattice・。f a・bit・a・y・size nlxn2xn3xn・4・
The si・te x i$specifieq by f・u唱r lnte9e・r−・c。。・dinates(xτ’x2’』x3’
×4)whe「e xμ=・1’2・・…nμ(μ=し2・3’4)・
Each block A(・》く’・y)、、has a struct.ure due to the・ internal
degrees of.freedom・of the quark:f、ieId. lt is a 12x12 complex
matriメ! Ψhose, rows (and co.lumns ;a翫so) are specified』by士he pair
of Dir.ac..(、i,j=1, 2, 3,’4). and cbl・or・indicels (α,β=1・,:2”3)三
1:n、gre.a.ter.detail the Wilson fermion matrix is・given as’
f。ll。.W・s:、 、 、... ._,= ’噛 ■・・ ..、 ’ ・ .・・
A(x・y)iα.jβ = δij・δαβ・∴ ・・・・ ・・… if y =・・×・
・A(x’y)iα,j・g3・r毎(1.一γ¥j)U(x’y)αβ if.・y=x+n
A(x’y)iα.jβ・・∴κ(い・γ¥j)u(xぞy)αβ if、y=×一fi ・t・(2)
Alx・y)・iα,」β,=… 、・ ・th・・wise
ム
Herey=x.土μmeans that the site y lies ne×・t to the silte・x in
the、 P。sitiV・(neg・tive)μr・di.・ec・t.i。n..「Thr 4x4 c。th P 1−e×ma・t・ic・esγμ
ar俘 t,he pirgc’s、γ ma㌻rices、 defined・by l . : ・ t一・
一4. 一
O O .O .一T・“ ・ ・ .IO O O.
.II・i , IOD 一:i ・.Ol ・: ・・
D.1・1一.D ・O O
?”f189 一6’ 81.72k18 9,ilo 81 .. 73・.” 1? Oo 8 6’1・. .7’‘=・ 18 6 一?・81.(3)
iD O OI 1一一1 0 0 01 IO −i O OI IO O O −1
Wd n6te that o rilゾ6nd・dlemdnt in ・each
valUe’
t.o W is’nonzero and 七h e‘』
奄刀fe i’ ther土1◎r」二i.
Since A’is related to the discr’etized’version of the DiraC
equation in the Continu’ 浮香D the ’matrix A as a whole has’ a
structure similar to the matrix generated by discretizatioh bf:
eM・iptic or parabolic partial differential equation; that is, an
of.f・一diaggnal block A(×.y) is non,一zero o’ 氏Cly when × and y. ar,.e
adjacent with each ether.
3,.,C.g.n.ju.gate Residual..Meth.o.d.s. . .,.
A class of iter’ative .me.thods .for splvi.ng the system.gf .
lipear eqyat.io.ns b.y decreasing t,h. q n.orm of .t.hg r..esidual,. yeq,to.r
llA’yt’ 一 bl12 has been proposed[5−6]. The algorithm consist,s pf.
i’t,erative steps. star,tin.g. with.
r”= b一 AV. p= r’
and repeating
.oc = (r.’Ap) / (Ap. Ap>
v = v +一 ctp’
r弓 r一αAp
update p
till convergence is achieved.’ @He’ [email protected]{ficlent ct is
so deterin’ined’as:’to minimize the n’brm ’of the n:d’w
謳・一αAp畦.”・ ノ・ ”
r5一
r‘
[email protected]
[email protected]w dire6tion
vector p. The simplest choice would be to Set p= r. ln this
cas6 only three vectors ‘ytt. r and q=Ar have to be stored in the
memory; We. call this algorithm the .mini.mal re$’tdual (MR) methed.
The convergence’ would be faster if the co’efficient rpatrix A
is approximately proportional to a unit matrix. More precisely.
if we denote ri the residuq.1 veqtor at the i−th iteration. we
have [6] ,
呂1≦1一綴), (4)
1
max
provided H=(At + A)/2, the Hermitian part of A. is positive−
definite. Here)t..,一and){ ’deno’te the minimum and maximum
mln max
eigenvalues. respectively. ln the algorithm. the positivity of H
is crucial. lf the positivity is lost (e.g. for rc 〉 rc ). the
c
coefficientαbecomes zero or very sma口and theψis no longer
ithproved.
ln the case of the CG method for positive symmetric l inear
eguation Ax=b. the conjugacy.of the correc・tion vectors {pi} with
respec’ [email protected]bilinear
form (x. b ・一 Ax). By only making pi+1 conjugate to pi. the
former is automatically conjugate to ali preceding correction
vect。「s PH’p卜2・…’P1・ln。・・case the c・nj・9acy・f th・
correction vectors with respect to AtA also plays an impertant
・。le in・educing l旧12・The c。nj・9qcy’h・weve・・is n。t passed。n
conjugate
to the new direction vectors. so that we can make p
i+1
0nly to’ @the correction vector.s stored in the memery. We call
this class of algorithm as conjugate residual (CR) method・.
一6一
The simplest choice (CR(1) me’thod) is t6 mak.e ’pi+1 conjugate
only to the previods vector pi. More spe’cifica”y. we sltart with
r・= b 一 A3P’. p = r. q =・ Ap
and repeating
ct = (q, r) / (q. q)
v=v+ ctp
r = r 一 ctq
s= Ar
P= 一・(qt. s) / (q. q)
p=r+ Pp
qごS+βq
unt’i l the convergence is.attained. One step of thd CR(1) thethod
entails one matrix multiplicationt three inner products’ ahd four
vector addition with scalar multiplication. that is it has one
more inner product and two more vector additions as compared with
the MR. The memory necessary to imp’ [email protected](1) is larger.
sjnce it has twe more working vectors p and s. Although the
upper bound for the ratio of residuals (4) is valid for the CR
methods. the residual for the CR decreases faster than that for
the MR.
4. 置ncomP!ete LU De60mPositi◎n
Since the relative reduction of the norm’ of the reSidual in
the MR and CR methods is bounded by (4). one can impiov’e the
convergenc6 by trapsforming A to a matrix which iS approximately
equal to the unit matrix (or its scalar multiple).
一7一
け
lfwe hav.・、amat・ix.A’which・i.s a gPgd apP「.。× i、.mat.ign lt。 A’
1ρ、「ca㎡e綱云二1A i.・“c 1. ・.s ・’・∫t・the吋.即ll醸hanlitρel.口f
バ
the solution of the equation A × = y requires relatively small
amount of comPutation’ it would be easier .to s.ol.ve the equation
ズAψ=ズb. 「’1⑤
in stead of eq.(1). This observation is at the basi’s’‘of ahy
む
preconditioning techni−que. The more the A resembles A’ the
better wilil [email protected]. On the
の
other hand, the solution of A x = y should not be.幽too time−
exhaustin9. since we have to solve it once ih each‘step.
Some years ago Meijer.ink 弓nd,van der Vorst[7] proposed・an
incomplete LU factorization for the「matri× originated in partia.I
differential β「quatjons, which approximately decomposes.A as.
A、= L.U .一 N, .・ . ・ (6)
wh・・e L・nd U a.・e l・w・・and uPPe・t・iangula・matrices and N i・
the err…f,d・c・mP6・iti・n.・By su;t年bly ch・・、sing th・n・h−ze・・、
entries of L and U, one can make L and U as sparse as the
original A. If the error N 幽is smalI. the factorized form LU
む
plays the role of A in eq.(5).
ln the case of the Wilson fermions on the lattice, it can
re、a.d日.y be shown[3] tりat the block.triangular sPlittin9.of』A’
L(x・y》.=A(xぞy) ., ... ・、 .(x≧y)
=、O tt, ・, 、 1,(x<Y)・
(7)
一8一
R(×.y) = O
(x, 〉 y’),
= A(×.y)
(× s y)
provides ’ ≠獅k @incomple’ 狽?f Lu de’composition
due t’ [email protected]
projection
。Pe,at。,s(1止γμ)∵th’a ’t is
L R = A 十 O(rc 2).
(8>
We use the syinb’ ol R ihstead’of U ebr the right (u’ 垂垂?秩j triangu’
matr’ 奄?D s’ince we hav’ [email protected]
奄窒?刀f
撃≠
[email protected].
The inequality of × and y’ ls to be un’dei’Stbod accord’ing to’
the site number × (denoted by IX in the program) in the usual
manner.
× = 〈((×4−1)Xn3 + x3−1)Xn2’+ ×2−1)Xn l + ×1. ’ (9)
The si’te numbe’「「uns f・・mll t・n’ wh:・・e’n’ @= nln2.n3n4 i’s ’th・℃・一t bl
number of thb lattice sites. T’ [email protected]
iLUCR methods .is given’
奄氏@[3].
We f・und[3・8]th・ガthe c・nv、e.・gence・at’e』is fし・th・・imp,。ved
by a Gustafsson−type acceleration[71.’ This is a trick of
r’
[email protected]
[email protected]LR by crc.
c being an appropriate constant. This accelerati6n can be
underst。。d as apP・。xim、tin96A by LR.丁he err。, N=LR−cA nbw
has non’zero diagonal entries
N(×.×) = 一(c一’1) 1 ’ ’ ’ (loa) ’
as we目 a s’
[email protected]
’一9一
N(x,y)=c2κ2(1一γμ)(1+7レ)U(x,x+ρ)U†(y.y一あ (10b)
ハレ ハ ヲ
f◎・th・next−nea・est Pai・s(x’y)with y=×+μ一v・When the 9 .a ”ge・
field U as weU as the fermion fieldψare nearly aligned’the
effect of those two errors tends to cancel with each other’ so
that(LR)一1A is effectively cl。se, t。 a c。nstant m。ltiPle。f a
unit matrix. We f◎und the best choice of c is 1.1・》1.3. Unlike
the acceleration parameter ω in the SOR method, the number of
ite’rations needed to fulfill a convergencg criterion does not
critically dePend on the choice of c.
5. Vectorization
we n◎w need to discuss how to carry out the computation 6n a
,
vector processor. Th6 vectorization of MR and CR methods offers
no problem since they consist of vector operations and matri、x
multlplication of a vector. We show in Fig. 1 the core of the
c。d。 which gives,=Aq. The array。 Q(1×’ C i.α)and R(IX, i,α)
・e
[email protected]㌻he Di「.a c and c。1。「
indice』. Th。 hnk c。nnecting the site l×and its nea,est
neighbor in the positiveμ一direction is numbered as LL = 4*IX − 4
+μ, so that the link number LL runs from l to 4*N. IGA》(i,μ)
9ives th・Di・ac inde×jf。・which(γμ)ij=GAM(i’μ)≠0・丁he
array elements NRRq×,μ) and NLL(IX,μ) give the site number which
Iies ne×t to IX in the positive and negativeμ一directions
respectively. Due to the periodic boundary conditiont NRR(IX,μ)
cannot be given in terms of a Iine『諭r form of IX. The hopping
parameter is denoted by HK. Since there is no data dependency in
=10_
the innermost loop DO 10.’ [email protected]
compile’ [email protected][email protected]he
Vectorization of the’selution’of’the triangul’ [email protected]: q
and R s = ’ 吹D These equatiens are solvbd recurSively ih terms of
the forward and’backward’substitutions. as
do×= し n
p(.) . q(,) 一XsilL(×.y) p(Y)
y=1
and.
do x = n. 1. 一1
n
s(×) = p(×) 一一 X R(×.y) s(y).
y=×+1
The al.gorithm is hard to vectorize since the previous
variables are referred to in the・loops. ln erder that the
vectorized code may produce the same’ @results as the scal’ar
qomputers. we have to fi.nd a subset of, lattice sites which are
independent with ea’ [email protected]
concur.rently. Th.is cannot be done by dividing the lattice into
sublattices with doubled lattice spacing. For example p(1.1.1.3)
depends on p(1.1.1i1) via p(1.1.1t2).
It is eas日y seen in the forward substitution, p(x) depends
on p(y) if and only if there exists at least one sequence of
latti・ce sites. ×=z(1). z(2). z(3)....・. z(S)=y for which z(i)
and z(i+i). are adjacent with each ether and z(i) 〉 z(i+i). one
can prove by induction that p(×1. ×2i ×3. ×4) is depgndent on
P(Yl’ Y2’ Y.3’ Y4) if and only if xlz>yl,c holds for a11 i,t. This
−11一
・tate脚㌻・isρ、1・ρya.lid wりen・wρtake the.P・・ri。dic(。.r_..・. 自
antiPe・i。di・)b。unda・y・・。nditi。ns i.ntg qρ・・unt・・
ワ
ln。・・p・evi・u$Paperl[3]w・p・esented a hype・plane
vect。 D・ izati。nでwhich Was。「i9 i. nal(yρr。P。sed Yga・rs ag。 f・・」LL・IAC
lY[!・].and l・t・・rと・i. ved f・・.. th・.1LU pr・剛itゆnin9・f pa・tial
differential equations[11]・ Thi$ apPrgach is based upon the
observation that the sites lying on a P−th hyperplane defined by
×1+x2+x3+×4=P呂。。nst・・. ・』、一 ・(11)
are independent of each other and that if P(x) dePends on P(y)‘
then y hes on a hyperplane with sma日er p・ We、can、start.with
p=4 and increment the constant..p after eac.h stgp, untiI p reaches
its ma×im・m value NP=n1+n2+n3+n4・
A sljght・.modification of t・he Program in Fi9. 1 yields a
solver of L p = q’ 、which is sho・wn in Fi9. 2. Here t・he soIution P
is ◎ver.written on q in lorder t.o save the storage.・ The site
numbers of』those lat.・tice sites whose ne.arest neighbor site in the
positiveμ一direction has smaller site number than・themselves’
(i.e. connect、ed in the matri× L) are reordered accordin ’9 to t・he「
hyp。,Plane numb。, IP and NNLR(IXP,μ)c。nta」ns the lXP−t・h・。f su6h・
site numbers. The largest I×P’on the lP−th hyperpIane is given
in NBLR(IP,μ). In the same way, the lattice s・ites whose nearest
neigh・bor si.te in the negative・μ一direction has smaller‘site humber
are stored iれ NNLL(*,k).. We not6, NBLR(NP,μ)+NBLL(MP,μ):= h for・
anyμ. Sin’ce t・he c’omp日er cannot ・identify the independency of
the。pe「ations。n Q in th・・1。gP・DO 10・hd DO 20・w・have t6.1
a comp口er directive. ・Fig. 2 『hows the one for HITAC S810’.
一12一
垂tt
6.Quasi−vecto’r’
奄嘯≠煤f
奄盾?堰fby’Mult“
奄モ盾撃潤f
[email protected]
’: ’’
・ ’”
’・Re’ce’nt’IY P. Ross’i, C.T;H. Davies・ahd’ G.P. L6page[41
imP量・mdnted thるILU Pr6c。hd「由。nin9’i n tb,ms。f a 16−c。1。r
sublatti’c’ [email protected]
モ狽潤
r’
奄嘯≠狽奄Un”’L’ They expr’ ?唐唐?пf
[email protected]
[email protected]
the’1attice as
×= 2y +n . (12)
with
Y= ’(Yii y2) y3. y4)’ . i. s yk s npt/2
and
nt’(nt.n2.nd.”4) n’pt=oer 1’.
a.nd treated all the sitgs labele.d by different y but identical n
simultaneously.
Although the sites wi.th the same 17 are not connected direct−
ly. they・..ar.e not independ.en.t as we saw in the previeus sectio.n.
so that thi・ Ds methpd does not give t,he same result qs the original
ILY p.recondition,ing. lt should, be r.egarded as.a different solver
based on the vector iteration in the sense ef Schendel[12].
We sh。w「in Fi9・3・th・・elative err。・e。f th・16 c・1・・ILリ
precQnditioning for a gomplex random right.hand vector’b (both
「eal and・・imaginarY p9・ts a・r…m・1・and。m numb・.・s・and n.。・m・日zed
as llbl12=P. The .error e is defined by
e=HUrlL認b−U−IL口1 b ll。/ll U白1ビ、1 b、 2 (13)
一1 3 一一
where Li61 and ui61 represeht the 16−color vectorization of L’1 and
u−1. As is expectdd. the error is small fg’ [email protected], whereas it
gets worse when rc be・comes larger. One may・think that since the
ILU preconditioning itself is.an appreximation. exact identity of
numerical algorithm may not be necessary in the vectorization.
We will see the effect in the next section”
7. Summary and Results
ln in Fi9.4th、 Euc口dean n。,m。f the err・。, llψ,一A−1 b』皿i。
墾
given as a function of i. the iteration number. for the ILUMR
algorithm vectorized by the hyperp!ane and 16−color methods.
The acceleration parameter c i6 set to 1.0 (no acceleration).
The gauge configuration was taken from a quenched simulation at
f3=s.s on an s4 lattice. The hoppihg parameters are rc=o.17. o.ls.
O.181 and O.183. The critical value rc for which the pion mass
c
vanishes in t「his configuration is O.1844.土0.0009[8】. The
righthand side b is b complex gaussian random vector des6ribed in
the previous section. The initial. value 3Pto is set equal to b.
We do not plot the residua1 llb−A3Pil12t since it would require an
extra computation. The vect6r r in ILUCR or ILUMR algorithm is a
−1
(b−A3Pt,) and depends on the vectorization
modified residual (LR)
t
and acceleration of ILU preconditioning.
We ’also show in Table 1 the number of’ iterations until ll’Yb,・“
1
A‘一lb” 〈 lo−4 is attained for the various choices of c for ILUMR
and ILUCR methods in both hyperplane (column h) and 16−color
(column m) vectorizations. For all values of hopPing parameters
the 16−color version requires 2 N 4 times more iterations. From.
一14一
the result in Table 1i the optimum・ value of c in 16−color
vgctoriz,ation is larger than that in the hyPerplane method. Even
if we pompare the best’choices of c for each case. the difference
is large. especially for the range of rc of our interest. ’which is
close to rc.. The criti’cal slowing down is more strikihg in the
c
multicolor vectorization.
This result can be understood’in terms of the velocity of
information running across the lqttice. ln the 16−color
vectorization’of ILU preconditioning. an element of the ’residUal
vector influences only the elements of tYtt in a hypercube which
lies・next to it. The velocity is only V7’ per iteration in the
lattice unit. This feature exhibits a striking contrast to the
hyperplane vectorization. in which any element of the residual
[email protected]attice site. albeit
incomplete.
On the other hand, the multicolor vectorization posesses
some computational’advantages. For one thing. the vector length
in the 16−color vectorization is n/16. which is in practical
cases larger than n/2(nl+n2+n3+n4−3). the ,average vector length
in the hyperplane vectorization. Moreover the access to the
memory is more regular in the former method. $o that the
execution time for one iteration is shorter in the multicolor
vectorization than in the hyperplane one. especially on a v’
?モ狽盾
machine with slow memory access. This fact may cover the
shortage of the former that more iterations are necessary than
the latter. It is also to be noted tha’ [email protected]
can be easily i帥1。頃ent。d。n a‘ @highly pa,a口el array。f
processors with distributed memory. We conclude, however. that
−15一
the’ hyperplane vectorization is superior than th,e multicolbr
vectorizqtion. unless the execution of the former is, at, least.
twice (or more) faster than that of the latter.
Acknowledgement
Th.e numerical calculation for the .present wprk ・was car’ried
out on Hl丁AC s−810/10 at KEK’ Nat,ional.Lab◎ratory.fo.r High Energy
Physics. We thank the .Theory Division of KEK for its warm hospi・一
tality. We are indebted to M. Fukugita. A. Ukawa. Y, lwasaki. M.
Mori. M. Natori. K. Murata and U. Ushiro for stimulating discus−
sions. We are alse greatful to S. Fujino and−K. Takeda.for
informing us of Lamport’s paper[8]. Our work was suppor.ted in
part by the Grants“ih−Aid for Scientific Research of Ministry of
Education. Science and Culture (No. 61540142).
References
1. K. G. Wilsoni Phys. Rev. DJ−Q. (1974). 2445−2459.
2. M. R.’
gestenes and E. Stiefel. J.. Res. Nat. Bur. Standards
49. (1952). 409.
3. Y. Oyanqgi. Comput. Phy.s, Com,mun. A2. (1986). 333−343.
4. P. Rossi. C.. T. H. Davies and G. P. Lepage) University of
Calif.・・nia・’$an Dieg・・ep・・t UCSD二PTH 87/・8.
5. P. Concus and G. H. Gplub. in : Lecture Notes in Economics
and Mathomatical Systems. vol. 134. eds. R. Glowinski and J.
L. Lions .(Springer−Verl,ag. Berli.n. 1976) p. 56−65.・
一一一
@1,6一
P・K=、鴨.V.i ns,g mf f in,:P.「?r・、F。u「th SymP・・n rese・v・ir
Simuiati。n, S。c. P6t,。leum.En9. AIME(.1976)P.149’・、・
の
6. S. L. Eisenstat’ H. C. Elman and M. H. SchuIts, SlAM Jξ
Numer. Anal. 2Ω, (1983), 345一・357.
7.J.『A. Meij・rink and H. Z. van』 [email protected].』「Math. C。mPりt. at’
(1977), 148−162.
8・M・F・kugita’Y・Oyanagi and A・Ukawa’Ph著sl Rrv・D鱒’2’
(1987) in press・ . ’
9. 1. Gustafsson, BIT⊥a, (1978)’ 1.42.
1r4・Llm t/ gr.㌣.C・mm・rf、AC叩:’qr74)・83−93・
U寸Y・U鱒・・IM☆Ni6hlkrtalland F・N櫛・i’Hitachi、ry・r・n鰻’
(脚’、5r7二5rr・(in・Japane$e)・
’i 2.U.9Schende旧nt・r。du6ti。n lρNum6ricat Meth。ds f◎r Para濯1、l
Computers, trans. by B. W. ConoiIy’ E三llis Horwood⊥im’ited’
Chichester, 1984, p.35. 9 ・ ,
Figure ra噸r\1匹1
臼g.1’
D・
`、1 痰窒U幽門幽f。,IR:.、A’ Q−
Fi9.2A.Pr・9’amf。rQ←L−IQ ’■’
Fi9.3Relative err。r。f 16=c。1。r Prec。nditi。ning as a functi。n
of the hopping barameter.‘
Fi9・4丁he err・・IIψi−A−1・b ll as a functi・n・f the n・mb…f
iterations i for 16−color prec◎nditioning (upper curve)
and hyperplane preconditioning (lower curve).
一17一
COMPLEX U(4*N.3,3),Q(N.4,3).R(N.4,3),QQ1.QQ2.QQ3.GAM(4,4).GM
lNTEGER IGAM(4.4),NRR(N,4).NLL(N,4)
c
DO 5 IALPHA=1,3
DO 5 1=1.4
DO 5 lX=1.N
5 R(1×.1.IALPHA)=Q(IX,1,JALPHA)
c
DO 10 MU=1.4
J=lGAM(1.MU)
GM=GAM(1.MU)
DO 10 1×=1.N
NR=NRR(1×.MU)
LR=4*1× 一一 4+MU
QQI=Q(NR,1,1) 一 GM*Q(NR.J,1)
QQ2=Q(NR.1,2) 一 GM*Q(NR.J.2)
QQ3=Q(NR,し3) 一 GM*Q(NR,J,3)
R(1×.1.1)=R(1×.1.1) ・一一 HK*(U(LR.1.1)*QQI + U(LR.1.2)*QQ2 +
1 U(LR,1.3)*QQ3)
R(IX.1.2)=R(lX.1.2) 一 HK*(U(LR.2.1)*QQI + U(LR.2.2)*QQ2 +
R(lX.1,3)=R(lX,1.3) 一 HK*(U(LR.3.1)*QQI + U(LR.3.2)*QQ2 +
c
NL=NLL(1×.MU)
LL=4*NL−4+MU
QQI=Q(NL.1.1) + GM*Q(NL.J.1)
QQ2=Q(NL,1.2) + GM*Q(NL,J..2)
QQ3=Q(NL.1.3) + GM*Q(NL.J.3)
R(IX.1.1)=R(IX.1.1) 一 HK*(CONJG(U(LL.1.1))*QQI +
1 CONJG(U(LL.2.1))*QQ2 + CONJG(U(LL.3.1))*QQ3)
R(IX.1.2)=R(1×,1,2) 一 HK*(CONJG(U(LL.1.2))*QQI +
2 CONJG(U(LL.2.2))*QQ2 + CONJG(U(LL.3.2))*QQ3)
R(1×.1.,3)=R(1×.1.3) 一一・ HK*(CONJG(U(LL.1.3))*QQI +
3 CONJG(U(LL.2.3))*QQ2 + CONJG(U(LL.3.3))*QQ3)
10 CONTINUE
Fig. 1 A program for R=AQ
COMPLEX U(4*N.3.3,).Q(N.4.3),QQ1.QQ2.QQ3.GAM(4.4).GM
lNTEGER IGAM(4.4),NRR(N.4).NLL(N,4).NBLR(NP,4).NBLL(NP.4).
1 NNLR(N.4).NNLL(N.4)
c
DO 30 IP=4.NP
DO 30 MU=1.4
DO 30 1=1.4
J’=’lGAM(1.MU)
GM=GAM(1.MU)
*VOPTION INDEP(Q)
DO 10 IXP=NBLR(1P−1.MU)+1,NBLR(lP.MU)
1×=NNLR(IXP.MU)
NR=NRR(1×.MU)
LR=4*1×一4+MU
QQI=Q(NR.1,1) 一 GM*Q(NR.J,1)
QQ2=Q(NR.1.2) 一一 GM*Q(NR.J,2)
QQ3=Q(NR.Ii3) 一 GM*Q(NR.J.3)
Q(1×.1.1)=Q(IX,1.1) + HK*(U(LR.1.1)*QQI
十
1 U(LR.1.2)*QQ2 + U(LR.1.3)*QQ3)
Q(IX.1.2)=Q(1×,1.2) + HK*(U(LR.2.1)*QQI
十
2 U(LR,2.2)*QQ2 + U(LR.2,3)*QQ3)
Q(IX.1.3)=Q(IX.1.3) + HK*(U(LR.3.1)*QQI
十
3 U(LR.3.2)*QQ2 + U(LR.3.3)*QQ3)
10 CONTINUE
c
*VOPT l ON
1NDEP(Q)
DO 20 1XP=NBLL(lP−1.MU)+1.NBLL(lP.MU)
1×=NNLL(1XP.MU)
NL=NLL(1×.MU)
LL=4*NL−4+MU
QQI=Q(NL.1.1) + GM*Q(NL.J,1)
QQ2=Q(NL.i.2) + GM*Q(NL.J.2)
QQ3=Q(NL.1.3) + GM*Q(NL.J,3)
Q(IX.1.1)=Q(1×,1.1) + HK*(CONJG(U(LL.1.1))*QQI
十
1 CONJG(U(LL.2.1))*QQ2 + CONJG(U(LL.3.1))*QQ3)
Q(IX/1,2)=Q(IX,1,2) + HK*(CONJG(U(LL,1,2))*QQ1
十
2 CONJG(U(LL,2..2)・)*QQ2 + CONJG(U(LL.3.2))*QQ3)
Q(1×.1,3)=Q(1×.1.3) + HK*(CONJG(U(LL,1.3))*QQI
20
30
3 CONJG(U(LL.2.3))*QQ2 + CONJG(U(LL.3,3))*QQ3)
CONTINUE
CONTINUE
Fig. 2
A
一1
program for Q 一e一 L ’Q
十
O.4
O.3
M
o
k
の
。
an
O.2
.一
p
tiO
・H
防
−
o
O.1
o.o
O.14
O.18
O.1 6
rc
O.2
rc 一一一一一 O.180
rc :O.170
1oO
1oO
lo−1
lo−1
10−2
10一一2
1o−3
10 一一 3
10一陶4
10−4
ρ
マ
く
1
コ や
昏
o
20
10
O 50
30 40 50 60
!00 150 200 250 300
オ
面
rc = O.1 8 1
rc 一一一一 O.183
loO
1oO
lo−1
lo−1
1o−2
10一一2
1o−3
1o−3
1o−4
1o−4
畠
N
ρ
〒
<
1
バ
昏
o
100
300
200
i
400
o
200
400 600
i
800 1000
rc
(m
a)2
T
1LUMR
O.170
0.83
h
m
57
O.180
0.24
O.181
0.18
O.183
0.02
h m
h ’m
h m
105 265
134 361
270 1045
c=1 .0
31
c=1 .1
25
52
79 227
97 312
178 838
’c =1 .2
ee
48
鎚 201
&1. 274
⊥3旦 687
’c =1 .3
25
45
78 一183
94 244
147 574
c=1 .4
38
43
168
153 221
394 495
c=1 .5
80
皿
157
204
一.一 432
c=1 .6
皿
坦
一LgS
一一一
@400
c=1 .7
45
160
204
一一一
@鋤2.
c=1.8
191
239
一一一
@450
c=1 .9
294
386
一一一一一
@811
m
h m
h m
;83
100 245
191 660
1LUCR(1)
h
m
117
h
46
82
23
43
69
165
82 219
149 543
c=1.2
忽
41
鍾 150
エ旦 197
L3−L 452
c=1.3
24
39
74
139
89
180
139 388
c=1 .4
36
鉦
118
130
149
167
257 345
c=1 .5
74
鉦
123
157
一一一
c=1 .6
鉦
119
L5−1.
cFl .7
40
“一2,
152
c=’1 .8
138
169
一一一
c=1 .9
202
254
一一一一
c=1 .0
26
c=1 .1
inT
一一一
一.一..
@316
@297
@292
@319
@466
Fll罵齢llnlf(ll瀦Sall・繍。ltl:ll藷)一v翻i二all;ls.
The bar (”一’) denotes the failure of convergence. We underline the best
cho’ice of c in each case.
INST I TUTE OF I NFORIVIAT I ON SC I ENCES AND ELECTRON I CS
UN IVERS I IY OF TSUKUBA
SAKURA−IVIURA, N I I HAR I・一GUN, r BARAKI 305 JAPAN
REPORT DOCUIVIENTATION PAGE
REPORT NUIVII3ER
工SE−TR−87−64
TITLE
Hyperplane vs. Multicolor Vectorization of 工ncomplete
LU Preconditioning for the Wilson Fermion on the Lattice
Aしm{)R(S)
Yoshio Oyanagi
REPORT DATE
August 5, 1987
D4AIN(瀦正GORY
NLIIV[BER OF PAGES
22
CR CATEGOR I ES
KEY woanS
Lattiee gauge theory, Wilson fermions, veetorization,
conjugate residual method, preconditioning
ABSTRACT
We compare the hyperplane and 16−color vectorization of
血inilnaユ residua玉 and conj ugate residual methods with an incomp−
lete LU preconditioning to solve the lattiee Dirac equation in
the Wilson formulation. The performance was assessed in terms
of the Euelidean distance form the true solution for various
hopping pai?amete]?s on a quenched configuration at B=5.5 on an 8“
lattice. The 16−color vector’i・zatton requires 2−4 times more
iterations than ’the hyperplane vectorization. Even with the
best choice of the acceleration parameter, the latter method is
preferqble than the former unless the co’mputing time for one.
iteration differes by a factor to two or more.
SUPPLEiVIENTARY NOTES