The D1-triangulation of Rn for simplicial algorithms for computing solutions of nonlinear equations

Tilburg University

The D1-triangulation of Rn for simplicial algorithms for computing solutions of

nonlinear equations

Chuangyin, D.

Publication date:

1991

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Link to publication in Tilburg University Research Portal

Citation for published version (APA):

Chuangyin, D. (1991). The D1-triangulation of Rn for simplicial algorithms for computing solutions of nonlinear

equations. (Reprint Series). CentER for Economic Research.

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a:'~ ~~R ~,~J~fl~Ir

CB

", o~~R~

I!IIIIIIIIIIIIII~hIINII~InIY~lllllllllllll

,.5,

~

The D,-Triangulation of IR~

for Simplicial Algorithms

for Computing Solutions

of Nonlinear Equations

by

Chuangyin Dang

Reprinted from Mathematics of

Operations Research,

Vol. 16, No. 1, 1991

Q5~ ';~

Reprint Series

~J~,~;,

CENTER FOR ECONOMIC RESEARCR Research Staff

ftelmut Bester

Eric van Damme

Frederick vcui der Ploeg

Board

Helmut Bester

Eric van Damme, director Arie Kapteyn

Frederick van der Ploeg Scientific Council Eduard Bomhoff Willem Ruiter Jacques Drèze Theo van de Klundert Simon Kuipers Jean-Jacques Laffont Merton Miller Stephen Nickell Pieter Ruys Jacques Sijben Residential Fellows Joseph Greenberg Jan Magnus F.mmanuel Petrakis Larry Samuelson Jonathan Thomas Doctoral Students Roel Beetsma Hans Bloemen Chu arigyin Dang Frank de Jong Pieter Kop Jansen

Erasmus University Rotterdam Yale University

Université Catholique de Louvain 'filburg University

Groningen University

Université des Sciences Sociales de Toulouse University of Chicago University of Oxford Tilburg University Tilburg University McGill University Tilburg University

University of California at Los Angeles University of Wisconsin

University of Warwick

Address: }logeschoollaan 225, P.O. Box 90153. 5000 LE Tilburg, The Netherlands

The D, -Triangulation of R"

for Simplicial Algorithms

for Computing Solutions

of Nonlinear Equations

by

Chuangyin Dang

Reprinted from Mathematics of

Operations Research,

Vol. 16, No. 1, 1991

MATHEMATIIx OF OPERATIONS RFSEARCFI Vul. 16. NY. 1, February 19VI

Yruu~d in US.A.

THE DI-TRIANGULATION OF 98" FOR SIMPLICIAL

ALGORITHMS FOR COMPUTING SOLUTIONS

OF NONLINEAR EQUATIONS'

We present a new Iriangulation of R", which is called the DI-lriangulation, for computing zero pnints or fixed pnints ot nonlinear mappings. The DI-Iriangulation subdivides the unit cube and is based on very elemenlary pivot ruks. We compare the DI-triangulation to several well-known triangulations of R" which triangulate the unit cube. According to several measures of efficiency the new triangulation is superior, such as Ihe number of simplices in the unit cube, the diameter of a triangulation, the average Jirectional density, and the sur(ace density.

I. Introduction. There are now a number of simplicial algorithms for computing zero points or fixed points using triangulations of R", for example, Merrill's homotopy restart method [5] and van der Iaan and Talman's variable dimension simplicial restart algorithms without an extra dimension [4J. The other variable dimension algorithms have been introduced by Wright [9] and by Kojima and Yamamoto [3]. Allgower and Georg's paper [1] is an excellent survey of this field.

It has been accepted by now that the efficiency of the various simplicial homotopy and restart algorithms for solving equations is influenced in a critical manner by the triangulation employed. To evaluate and design triangulations for these algorithms, Todd, and Saigal, Solow and Wolsey established several measures in [6J and [7J, such as the number of simplices in the unit cube, the diameter of a triangulation, the average directional density, and the surface density. Eaves and Yorke [2J showed that the average directional densiry and the surface density are equivalent.

'fo improve the efficiency of simplicial fixed point algorithms, we construct a new triangulation of R" and show Ihat according to these measures it is the best of the well-known triangulations of R", which subdivide the unit cube.

In ~2 the Dt-triangulation is introduced. We describe the pivot rules of the D,-triangulation in ~3. The number of simplices in the unit cube, the diameter, and the surface density are calculated in ~~4, 5, and 6, respectively.

2. The Dt-triangulation of R" Let yo, yt, , yk be a set of vectors in R". If they are a(iincly independent, then we call their convex hull, v, a k-simplex and write

Q- ~Yo,Yt,...,yk~ -conv{yo,yt,...,yk}.

A simplex r is called a face of a simplex o if all vertices of r are vertices of a. If dim r- dim Q- I, we call r a facet of P. ln addition, if y is the vertex of a which is not a vertez of r, r is called the facet of o oppositc y.

'Receivcd June 10, 1989; revised September 4, 1989.

AAfS 1980 suhject dasstficutit"1. Primary: 65KOS. Second'rry: 90Cg9. IAOR 1971 suh~cct dasslficutio". Main: Programming.

OR~AIS Inder 1978 subjecr dassiJication. Primary: 433 Malhematics~Convexity.

Kry words. Simplicial Algorithms, Triangulations, Unil Cube, Diametcr, Surface Densily, Average

Direc-tional Dcnsity

l48

0364-765 X ~91 ~ I G01 ~0148~501.25

SIMPLICIAL ALGORITHMS FOR COMPUTING SOLUTIONS 149 L.et C be a convex subset of R" and let dim C - m. We call G a triangulation of C if

(1) G is a collection of m-simplices, (2) C a _~oEr~r

(3) for any a', a2 e G, o' n v2 is either empty or a common face of both o' and o2,

(4) each x E C has a neighborhood meeting only a finite number of simplices of G. We denote the collection of j-simplices that are faces of simplices of G by G~, for ja0,l,.. ,m.

For ease of notation, let N~(1, 2, ..., n}, let D"` -( y e R"I all components of y are even), and for i~ 1, 2, ..., n, let u' be the ith unit vector in R".

As follows, we construct the simplices of a new triangulation of R". We assume n ' 2.

DEF)NtT)ON 2.1. I.et s denote a sign vector in R" such that s; E{-1, f 1) for all i e N. L.et 0~ p~ n- 1 be an integer. L.et -rr a(~r(1), ~rr(2), ...,-rr(n)) be a permu-tation of the n elements of N such that -rr(p) c..- c a(n) if p~ 1 and rrll) ~

.-- C~r(n)ifp~0. L.etyeD~`.IfpaO,letyo-yand

yk ~ y f s~(k,u~`k'.

LEMMA 2.1. L.et y~, y', ..., y" be obtained fiom Definition 2.1. T!)en y~, y', ..., y" are a,~nely independent.

PROOF. If p~ 0, then let

I a yI - yo a ..tu

z s„(~)u ,

z i I ,~(z) ~(I)

z s y - y ~ Sa(z)u - s„(uu ,

n a~,n -yn-I a(n) n(n-I)

Z ~ S~n)!I - Sp("-))U .

Obviously, z', ..., z" are linearly independent. If p~ 1, then let

Ik ~ yk - yk- 1 a-S~ k un(k)~

O k a 1,2,...,p - 1,

(n~

zn zY~ -Yn-) '- L sw(k)!f'(k), and

k-ptl

k k k-1 tr(k) a(k-I)

150 CHUANGYIN DANG

Suppose that z(, ..., z" are linearly dependent. Then there exists a q s

(qi....,q„)r ~ ll such that qiz( t... tq„z" ~ 0. If p~ n- 1, il is ncccssary that

q( -... - 9n-z - 0, -qn-( f 9" - 0 and q„ ~ 0. We conclude that ql -.. ~~

q„ a 0. lf p ~ n 1, we must have that q( ~. ~ qp) 0, yPt ( a 0, qk -qk . I- q~ a 0 for k- p t 1, ..., n- 1, and q„ - qp - 0. Therefore, q~ s q„, q„ - t ~ 29"~ 9n2 : 3qn...4o.z a(n (p f 1))q„, and qot2 t qo ~ 0. Hence, (n -( p f 1) t 1)q„ ~ 0. Since p C n- 1, we have q-( ~ qz -.-. a q„ ~ 0. Thus [he

hypothesis is incorrect, i.e., z', ..., z" are linearly independent. Therefore,

y~, y), ..., y" are affinely independent. The proof is completed. o

Let y", y', ..., y" be obtained from Definition 2.1. Then their convex hull is an n-simplex by Lemma 2.1, which is denoted by DI( y, -rr, s, p). Let D( be the collection of all such simplices D I( y, ~rr, s, p).

LEMMA 2.2. _{U o e Di~ ~~"'}

PHOOF. Let x be an arbitrary point of (Y". For each i e N, let

~ x; J if ~ x; J is even, ~` f 1 if ~ x; J is even,

y; and s; `

~x; J f 1 otherwise, - 1, otherwise.

We have 0~ diag(s~,...,s„Xx - y) ~ u, where u a(1,..., 1)T. Let 'rr' be a permu-tation of N such that

O~ ss'(I)(xsr'O) - yn'(I)) ~ ... ~ sv'(")(x,„.(„) - y,".(n)) ~ 1. 1( E;'-IS;(x; - y,) 5 1, let

qj - sa'(I)(xir'(I) - Yv'(1))s.. iqn ' SoYn)~xv'(") -Yir'(n)~~

and q~ - 1- E~-(q~. Obviously, q~ ~ 0 for all j and F~-aq~ ~ 1. Let a-(1,2,...,n), p- 0, yo z y, and yk ~ y t skuk for k~ 1,2,...,n. It is easily seen that x- Ej.((qi y~, where q„ ~ qo and, for j~ 1, .. ., n, qi - qh with h the index for which ~r'(I)) - j. Thus x E D)( y, ~r, s, p).

lf E,"-)s;(x; - y;) 3 1, then we show that there exists an integer 1~ p~ n- 1 such that the following system has a nonnegative solution:

i-I

~ 4i - S,.'ti)~x,.'(i) - y,.'ci)), Í~ 1,..., p- 1, i-o

P~-`1

Lr qi { qk a SvYk)~xtr'(k) -Yu'(k))i k a p~.- ~ni

i-0

SIMPLICIAL ALGORITHMS FOR COMPUTING SOLUTIONS I51

In fact, rewriting the system, we obtain

9n a s1l'(IJ(XA'(I) -YA~)))f

4i-I -S4'(I)(XA'(I)-Y1flI))

-S,r'(I-I)(xrYl-I) - Yv'lI-1))r

qP-I - -s„'(v-I)(Xt~(v-O -Yn'(v-I))

l-2, . ,P - 1, n , } ~ ~ SA(I)(xA'(I) - y,.'(i)) - l

l

Let No z( 0,1, ... , n}. If q;,-2 ~ 0 for p- n- 1, it is clear that q~ ~ 0 for all j e N~; otherwise, there exists a po, 1~ pa ~ n- 2, such that

[n~ 1

-S„qnn-I)(XwYnn-I) - y,rYvo-1)) f~ L Ss'U)(x~'(i) -Yx'(ÍI) - 1 I (n -Po) ~ 0

I -vo I

and

r [n-~ 1 ,

-S,~YOO)(xa'(OO) -Ye'(CO)) f I L Ss'(i)(xa'(i) -Ya'(i)) - 1

1

-11-vnt )

Hence,

( n

S~'(Ou)(xa'lPU)-Ya'(Po)) } I1 - ~ S,~'(i)(xw'(!)-yn'(i))~ (n -Pli)

1 -Pn

1) ~ 0.

~ S„Inu)(X,.Ynn) - y,r'(On)) f~1 - S„'(no)(x,.Ynn) - y,rTno)) -(n - po - 1)s~.(vn)(X,r'(vn) - Y,r'(vu)) - 1~~(n -Po) - 0.

Therefore, by taking p equal to p~, q~ ~ 0 for all j e N,,.

Let 1~ p ~ n- 1 be such that the system above has a nonnegative solution and let a be such that Tr(k) ~~r'(k), k a 1,2,...,p - l, and 'rr(p) C''' C~rr(n).

l.e t

Y~-Yfs,

y:k Yk- (- S,r(kfuv(k)s k- 1, , P- I,

152 CHUANGYIN DANG

Let q~ be obtained from thc systcm, for j- 0, I, ..., n. Then it is easily sccn that

x- F;-„q;Y'. whcre q„ ~ rl,í and, for j a I,.. ,n, 9; - qí, with h thc indcx fur

which ~rr'(h) ~~rr( j). Thus x e D,( y, ~rr, s, p).

From these results, the Icmma follows immediately. o

LEMMA 2.3. For any o~ and aZ e D„ o' tl a2 is eilher empty or a commun face of botb Q~ and o~.

PROOF. L.CI x e FY" be arbitrary. By Lemma 2.2, we may assume that x E o for some

QL ~Ya,Y), . ,Y"~ aDi(Y,~,s,P),

i.e., x s E,".~q; y', with q; ~ 0 for all i and E"-cq; ~ 1. Then x lies in a face of rr whose vertices are y~ for j e J:z (j e N„Iq; ~ 0}. We show below how each y', j E J, can be generated from x independent of y, ~rr, s, and p. Thus these vertices

are found for any simplex of D) containing x. For each i E N, let

r; -and ~x;f if~x;Jiseven, {~x;Jfl if~x;Jisodd, tl ifx;-r;~0, t;- 0 ifx;-r;a0, -1 ifx;-r;c0. Let w~ E;'. it;(x; - r;). Further, let

r;ft; ifiaÍ, Y~{ti} ~ _{r; otherwise,}

for i- 1,...,n, and let y(t~) ~(Y~(t;),...,Y"(t;))T. Then

{Y(ti),...,Y(t"),r} a {Y'I1 EJ} if w C 1, and

{Y(ti),...,Y(t„)} `{r) ~ (Y'll EJ) if w - I.

Supposc that w~ I. Lct 'I'i, ... , Tx bc subscts uf N such that (J Á- ~ I;, ~ N and fur c~ich I~ k ~c g, t;(x; - r;) - r~(x~ - r~) if r e Tk and j e Tk and for any I~ e ~

j~ g, t;(x; - r;) C t~(x~ - r~) if i e T~ and j E TI. Let T„ ~ 0. Lct !(k) e T;~ for

k- 0, ... , g. Since w~ l, there exist unique 0~ v ~ g and q~ 0 such that

trt,.)(x;a.) - r;(~.)) f (1 - IT,.,)I -... - IT`1~9

f IT }(I~

.f ...

SIMPLICIAL ALGORITHMS FOR COMPLfrtNG SOLLrT10NS 1S3

and

t~til(x~~it - rKi1) - t;t„t(x;t,.l - r;t,.t) - q~ 0, j ~ u t 1,...,g.

For 0~ k ~ u, Ict for i - l, ..., n,

s~r;ft; ifi~TouTIU-.~uTk,

y'(Tk ) r; otherwise,

and let y(Tk) ~(yt(Tk),..., y"(Tk))T. For u f 1~ k ~ g, let for each j e Tk,

r;tti ifiaj,

Y;(1) s _{r; otherwise,}

and for all i, let y(j) a(YI(j),...,Y"(j))T. Let

-( g- 1 _{if t;cn( x~cn - r;cil) - t~~~.~( x;c„~ - r;~,.l) - q- 0 for j - 8,}

g otherwise. 8 (1

If q - 0, then

r l 1

{Y(Tr)10 6 k c u} u( U {Y(j)I1 E Tit)

₁

k-~,t t

and if q~ 0, then

(Y(Tk)10 6 k S u} u ~ U(Y(j)I1 E Tk}

₁

,k-~~tl

From these results, we obtain the proof of the lemma. o TlieoaeM 2.4. Dt is a triangulation of R".

PROOF. IJet x e R" be arbitrary. it is clear that x is only contained in a finite number of simplices of DI. Using Lemma 2.1, Lemma 2.2, and Lemma 2.3, we complete the prcxif of the theorem. o

The DI-triangulation of R~ is illustrated in Figure 1.

3. The pivot rules of the DI-triangutation. L.et o [ y", y', ..., y"] -DI( y, -rr, s, p) be given. We wish to obtain the unique n-simplex

á~ ~yo.yl~ ..,v"] aDl(v,~,s,n),

containing all vertices of a except y'. Table 1 shows how y, á, s, and p depend on y,

-rr, s, p, and i. From this table it is easy to obtain each vertex yk, k S 0, 1,..., n, of

ó, and in particular its new vertex.

4. Comparison of the numbers ot simplices i n the unit cube. Let I" -(x E R"~0 5 x 6 u) be the unit cube in R".

THeoReM 4.1. The number ojsimpliees of the DI-triangulation in the unit cube is equal to

154 CHUANGYIN DANG

Flcuar I. DI-Triangulalion of Ihe Unit Cube in R'.

TA[3LE I

Thr Piuor Rulrs of thr Dr-Triangularion

n i Y s n P U 0 y s ~r p t 1 0 i ~ l y s- 2s„t;lu't'1 ir p I 0 y s 1r O- 1 2 G P U Y s- 2s~tllu'ul tr p Gn-I 2 c n i c i y s (,r(i)...,.(i t u. n c n- t c P- I Tr(i)...srtn)) (n(1)....,n(P - 2), 2GP P-1 y s n(p)...ir(Í). P-1 Gn - I n(p- 1),n(1 t U, ... ,n(n))' (,r( I ). . . . , rrl P - 1). 1GP i~P-I y s a(i).rr(p)...., ptl

t n- 1 a(i - I), ali t U,

...,T(lt))

n- ~ n- ~ y t ZS,~nlYsiwl J- ZS~twtY.(wl ,~

P

II - ~ II y t ZJ~Iw-Ilu~lw-11 J-_{ZS~w-114~tw-1) ~ P}

SIMPLICIAL ALGORITHMS FOR COMPUTING SOLUTIONS 1SS

PROOF. _{Let Q~ (DI( Y, ~, s, P)I Y~ fi, s~(1, 1, ... , 1)T}.}

From Definition 2.1, in Q, there is only one simplex for which p- 0, one simplex for which p a 1, and n!~(n - q t 1)! simplices for which p L q, 2~ q~ n- 1. Thus

IQI - 1 t 1 f n!~(n - 1)!f n!~(n - 2)!f .-. fn!~2! z 2 t n f n(n - 1) t.. - i-n(n - 1) .. - 4~ 3. Since U„EQo - 1", the proof of the theorem follows immedialely. o

For the definitions of the KI-, !I- and HI-triangulations, we refer to [8].

THEOREM 4.2. The number of simplices in 1" of Freudenthal's KI-triangulation, that of Tucker's JI-triangulation, and that of Saigal's HI-triangtdation is n!.

THEOREM 4.3. If n~ 3, then d" c n!. As n goes to infinity, d"~n! conuerges to e-2.

PROOF. For n a 3, we have d; c 3!, since d3 a 5 and 3! s 6. Suppose d" - I ~

(n - 1)!. Thus nd"-I ~ n!. From

nd"-I~n(n-1)fn(n-1)(n-2)t..-fn(n-l)--.4.3f2n zd"t(n-2),

we obtain d" c n!, since n~ 3. By the induction principle, the conclusion d" ~ n! for n ~ 3 follows directly. Furthermore,

d"~n!- l~(n - 1)!t 1~(n - 2)!t ..~ t1~2!f 2~n!,

so d"~n! converges to e- 2 as n goes to infinity. O

From these results, we obtain that the number of simpliccs of the DI-triangulation is the smallest for these triangulations.

5. The diameter of the DI-triangulation. l.et G be a triangulation of F8" such that its restriction to 1", GI1" -(o c 1"lo e G), triangulates I" and all vertices of

GII" are vertices of I". L.et r and r' be two facets of G in the boundary of 1", dl".

Let o~, ol, ..., o," be a sequence of simplices of G such that o; and o; -1 are adjacent, for i~ 1, 2, ..., m. lf r is a facet of oo and r' a facet of o,", then we say that the sequence of o~, a„..., o," is a path of length m t 1 from r to r'. We define the distance between r and r' to be the minimum length of a path between r alid r'. The diameter of G is the mazimal distance between any two facets in the boundary. It is dcnoted by diam(G).

THEOREM 5.1.

diam(KI) - 1 t n(n - 1)~2 - O(nZ), diam(!I) - diam(KI),

diam(HI) ~(n3 - n f 6)~6 ~ O(n~), and

l56 CHUANGYIN DANG

QROOF. L.et o-[Yo, Y~, ., Y"] ~ Ki(0, ~rr) and r z[ ya, .. , Y"- ~]. where a~ (1,2,...,n). L.eI

Q~ [Yo,Y~. - ,Y"~ aK,(O,ir) and r a ~yo, . , y"-~~,

where á s( n, n- 1,..., 1). Let a,,...,o",-, in GII" be such that rr,-, and a; are adjacent for i- 2, ... , m- 1, o and rr, are adjacent, and also o,„-, and ir. lt is easily seen that the smallest m is equal to n(n - 1)~2. The distance between r and T is obviously the greatest of all distances between two facets in d!". Therefore, diam(K,) z n(n - 1)~2 f 1.

Since J,I1" is the same as K,I1", diam(J,) ~ diam(K,).

Let a-[Yo, Y~, , Y"] v H~(yz~~ a) and r-[Y~, ., Y"], where yzx a (1, 0, .. . , 0)T and ,r s (1, 2, . . . , n). Let

ó- [Y~,Y~, - ,Y"~ ~HI~Yzi,ir] and Ta [yo, - .Y'~-~],

where yzr ~ (1,...,1)Tand ~rr - (n,n - 1,...,1).

Let o„ . .. , o,„-, be a sequence such that a;-, and a; are adjacent for i s

2, ..., m- 1, tr and rr, are adjacent, and also v,„ - , and á. Then the smallest m is

equal to (n3 - n t 6)~6 - 1. Thus the distance between r and z is (n~ - n-~ 6)~6. This means diam(H,) ~ 0(n3).

Finally, let o a[Ya, Y~, .-, Y"] ~ Di(Y, ~, s, P) and r-[Y ~. Yz, . --, Y"], where y-U,s~(1,...,1)T,P-n-l,andzr-(1,2,...,n).l.et

0 1 n I n

Qa ~Y ,Y , --,Y ~ ~DI(Y,~,S,P) and T~ Y ,.--,Y

where y- 0, s~(1, 1, ..., 1)T, p a n- 1, and -Fr ~(n, n- 1, ..., 3, 1, 2). L.et

o„ ..., o,„ -, be a sequence such that rr and o„ o;-, and v; for i s 2, ..., m- 1,

and ó and a,„-, are adjacent. Then the smallest m is equal to 2n - 4. The distance bctwcen r and r is obviously the greatest of all distances between two facets in á!". Therefore, diam(D,) ~ 2n - 3.

From thcse results, the thcorem follows immediately. a

6. The average directional density and surface density. From Eaves and Yorke [Z], we know that for a triangulation the average directional density and surface density are equivalent. We calculate below the surface density and obtain the average directional dcnsity from the surface density.

First we calculate the surface density of the D,-triangulation. Let

O~ - [O,U , ..,UI "], UI ~ [u,u ,...,uI "]s tTZ - [u,u - tlI,UZ, . .,u"],...,0"-1

- [U,lr - U~,...,U - U~ - Uz - ... - u"-z,u"-~,u"].

The volumc of a simplcx a is denoted by V(a). The surface arca of a simplcx rr is dcnotcd by SA(rr). Let

SIMPLICIAL ALGORITHMS FOR COMPUTING SOLUTIONS 1SÍ

be the facets of oo. Then

n ~ SA(o") 3 ~ V~r~~ - nV~rn) t V~r~~. ,-o Clearly, V~r~~ - (1,(n - 1)!)~det[ul,uZ,...,u"]I - l~(n - 1)!, and

V~róÍ - (lr(n - 1)!)Idet[u~Jn ,uZ - ul,...,u" - ul~~ ~ ~r(n - 1)!,

so SA(o") -(n t~)~(n - 1)!. Since V(o~) 3 l~n!, we obtain that

SA(o")~V(o~) z n(n f ~].

Fork-2,...,n-l,let

[U,U -U',...,U -UI - ... -Uk-I,Uk,...,UÍ-I~Uiil . ..,U"],

j a k, . . n, ró z [u - u',...,u - ul - ... - uk-I,uk, . ,u"], and

rj -[u,u-ul,...,u-ul-...-u1-',u-u'-...-uitl,...,

u- u' -... - uk-I,uk,...,u"], _{j: 1,2,.. ,k - 1,} denote the facets of ok. Then

k-1

158 Further V(r~~ ~ (1~(n - 1)!) CHUANGYIN DANG det - 0 0 ... 0 1 . 1 1 .~. 1 0 ~ 1 1 ..- I 0 ' 1 1 ". 1 0 ' 0 0 "' 1 0 0 0 1 1 0 1 0 0 0 0

óó

-~ -~ .:: -~ -~-~

SIMPLICIAL ALGORITHMS FOR COMPUTING SOLlIr10NS ]S9 Moreover, 1 1 ... 1 ~ ... f 0 1 -~- 1 1 -.. 1 V(ak) ~ (l~n!) det 0 0 ..- 1 1 .-. 1 0 0 -.- 0 0 --- 1 0 0 .-- 0 1 .-- OJ 3 (n - k)~n!. Hence, SA(ok)~V(Qk) s n(n - k ~- (n - k f 1)((n - k f 1)2 - 3(n - k f 1) f 3)~~Z t(k-2)(n-k)trf((n-kfl)2 -(n - k f 1) f ])~~2)r(n - k).

Tp ~ ~U1, . ,U~~, T~ ~ ~U,U~, ..,U~~,

TZ 6 ~U,UI,U3,...,U~~s...,T~ a ~U,U',...,U~-~~

be the facets of o'. Then

I.et 91 a . Then SA(ol) - nV(r,',) t V(Tá). '- 9„-1 3 (n2 - 3n f 3)-1~2 and _{q„ z -(n - 2)(n2 - 3n f} 3)-1i2. 0 1 ... 1 ql 1 0 .-- 1 qZ 1 1 --. 0 q„-, 1 1 ... 1 q~ ~( n 2- 3n f 3)~~2~( n- 1) !, V(T,',) - (1~(n - 1)!) det

160 Triangulatiun ti i(!i ) Hence, CHUANGYIN DANG TADLE 2

Compurison oJrlu Kt-, l~-, anJ Dt-Triangulurions

Numtkr uf Simplices Diameter uf a Average Directional in a Unit Cube Triangulaliun Density

n. O(n') n(2 t ln - 1)t~)K„

n t n(n - 1) t--- tn(n - 1) O(n) SD(D~)g„ -..4.3t2

SA(ol )~V(ot ) - n~n(n2 - 3n t 3)t~Z t nt~2~,(n - 1).

From the above results we obtain that the surface density of the Dt-triangulation equals

SD(Dt) - max{SA(Q')~V(o')~i - 0, 1,...,n - 1). Lct

K„ - r(,t~z)~((n - i)r(1~2)r((n - I)~2)).

From (2J we know that the average directional density of a triangulation is g„ timcs its surface density. Hence, the average directjonal densjty of the Dt-triangulation is cyual to

ADD(Dt) - SD(Dt)g,,.

It is well known that both the average directional density of the K,-triangulation and the one of the JI-triangulation are equal to n(2 t(n - 1)~)g,,. It is obvious that we have that ADD(Dt) c ADD(Kt) ~ ADD(Jt), and that ADD(Dt)~ADD(KI) convcrges to 1 as n goes to infinity. Thus, the average directional density of the UI-triangulation is smallcr than the one of the Kt- or the Jt-triangulation. Table 2 summarizes thc results above.

Acknowledgement. The author would like to thank Dolf Talman for his remarks on an earlier version of chis paper, and Gerard van der Laan, He Xuchu and Chen

Kaizhou for their encouragement.

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