Cmjv04i04p0216 by Journal of Computer and Mathematical Sciences

J. Comp. & Math. Sci. Vol.4 (4), 216-224 (2013)

Computation of Fixed Point by Using Efficient Algorithm R. S. PATEL1, RITURAJ RUSIA2 and PRATEEKSHA PATEL3 1

Department of Mathematics, Govt. Post Graduate College, Satna, M.P., INDIA. 2 Department of Computer Application, Vindhya Institute of Technology and Science, Satna, M.P., INDIA. 3 B.E., M.Tech (IT) Satna, M.P., INDIA (Received on: July 20, 2013) ABSTRACT This paper introduces the artificial algorithm for the computation of sequence of approximate fixed points of a continuous function defined on Sn to itself as discussed by Todd8 as well as Vertgeim10 by using triangulation K2[M]. To study the process of iteration and the speed by which this algorithm execute has been considered for n=2 but for different values of m. It has also been shown through a diagram how the fixed point arrived through the sequence of the connected components of graph whose nodes are barycentres of the corresponding simplex and whose edges are double lines. The admissible integer labelling has been used to obtain the sequence of approximate fixed points, which always occurred within the completely labelled simplices. A fixed point sum of whose co-ordinates is one is unique otherwise not. Keywords: Artificial algorithm, triangulation, labelling, nodes, barycentre, extra layer.

1. INTRODUCTION Since the appearance of Brouwers fixed point theorem in 1912 stating that a continuous mapping from a simplex into itself has at least one fixed point, there has been considerable effort towards developing an efficient algorithm for computing approximate fixed points for such a mapping. But from 1912 to 1967 most of the

admissible

work in this field remain confine to prove only the existence of a fixed point, for various types of functions in different types of spaces, rather than to have spaces, rather than to have technique to carry out any computational procedure. However in 1967 H. Scarf succeeded in initiating such a algorithm through the systematic pivoting procedure carried on primitive sets defined by him. In

Journal of Computer and Mathematical Sciences Vol. 4, Issue 4, 31 August, 2013 Pages (202-321)

217

R. S. Patel, et al., J. Comp. & Math. Sci. Vol.4 (4), 216-224 (2013)

the following years H. Kuhn proposed an improved method for computing an approximate fixed point based on a technique of triangulation of a given simplex. Both these methods not only determine an approximate fixed point based on Brouwer's fixed point theorem, but were successfully utilized to solve many problems related to optimization, Game theory and equilibrium points in the area of economics e.t.c. The main problem for determination of exact fixed point depends upon : 1. Nature of the problem 2. Types of the Triangulation used 3. The Process of Labelling 4. Selection of Pivoting Point 5. To develop appropriate Algorithm. It has been shown by various authors like Kuhn, Whitney e.t.c. that how the value of approximate fixed point is effected by types of triangulation used that is the efficiency of fixed point is sensitive to triangulation used, Therefore a continuous refinement in a triangulation provided a approved computational techniques for the fixed point algorithm, to calculate a fixed point of a mapping to greatest accuracy. The labelling procedure is very significant to have a better approximation of fixed point, Vertgeim10 has discussed some general labelling rules for the vertices of the simplex. These set of rules will depend upon the nature of triangulation used and help in the process of algorithm. Pivoting point plays very crucial role for determination of fixed point through the given step of the algorithm. This algorithm always terminate whenever a pivoting procedure reaches a completely labelled simplices.

2. NOTATION AND DEFINITIONS The following notations has been used in this paper: R : Set of real numbers Z: Set of all integers. N : Set of positive integers (1, 2, ...............n). No : Set of all integers N ∪ (o) Rn : n dimensionally space, having coordinates indexed 1 through n. Rn+1: n +1 dimensional space, with coordinates indexed 0 through n. π : Group of permutation on (1,2, ...............n) and π+1 group of permutation on (0,1,2 ........n). ui : ith unit vector in Rn+1, jεN and

u=∑ vj:

iεN

jth unit vector in Rn+1,

v=∑

iεN0

j = ∑ No

and

. R+m : Non negative orthant of Rm i.e. (x ε Rm:x ≥ o). Now we will consider some standard definitions and explanations have used in this paper. 2.1 Standard Simplex The standard n dimensional closed simplex Sn is the convex hull of v0,v1...vn i.e.Sn ={xεR+n+1:vTx=1}. sin denotes the face of sn opposite vi i.e.Sin={xεsn:xi=0} and boundary of sn is denoted by ∂s=UiεN sin. Again a j-dimensional simplex or [j-simplex] is the relative interior of the convex hull of j+1 affinely independent

Journal of Computer and Mathematical Sciences Vol. 4, Issue 4, 31 August, 2013 Pages (202-321)

R. S. Patel, et al., J. Comp. & Math. Sci. Vol.4 (4), 216-224 (2013)

points y0,y1,y2,y3....yj>. called its vertices. We write σ=<y0,y1,y2,...yj>. A simplex τ is a face of σ if its vertices are a subset of vertices of the σ. It is convenient to call the closure of a (j-1) dimensional face of the j simplex σ as a facet of σ. Two j simplices are said to be adjacent if they share a common facet. 2.2 Triangulation

0 . . .0  -1 . . .0   0 . . .0   . . . ..  . . . ..   0 . . . -1  0 . . . +1

qj is jth column of Q for j ε N.

A triangulation G of S is a collection of n simplices satisfies the following two condition: 1. The simplices in G together with all their faces form a portion of Sn and 2. Each point of Sn has a neighbourhood meeting only a finite number of simplices. 2.3 (a) PIVOT Rule K2 (m) For a give simplices G and a vertex y of σ the rules for obtaining the simplex of G whose vertices include all vertices of a except y are called the pivot rules of G. Let σ = K2 (y0, π) = <y0....yn> and τ = (z0, ρ> contains all vertices of σ except yi. Then z0 and ρ are obtained from y0, n, and i according to the table given below. 2.3 (b) Mesh The mesh of a triangulation G is supσεεG diamσ. we will use the Euclidian norm through out this paper. n

2.4 Regular Triangulation K2 [m] of S Let Q denote (n+1) x n matrix

-1  +1   0   .  .  0 0 

218

Let K20(m)={yεSn:myiεZ for all iεN0} and π is a permutation of N. let σ=<y0,y1,...yn> is denoted by K2[y0,π}. Where yi=yi-1+m-1 qπ(i), for all iεN. Then the triangulation K2[m] of Sn is the collection of all such K2[y0,π]. K2 triangulation is mostly used to find solution of several types of problems like economics equilibria, game theory e.t.c. 2.5 Standard Integer Labelling Let G be a triangulation of Sn and let K0 denotes the set of all 0-simplices (0 simplices are points] in G. An integer labelling is a function from K0 into No. If xεKo be any arbitrary point then label of x is denoted by l [x]. If f:sn→sn is a continuous function then l [x]=min.{i:fi [x] ≤ xi, xi>0} is standard labelling procedure to label any x εSn If l [x] = j = > xj >0, j=0, 1,2 ..... n where is jth component of x. The such a labelling is called admissible labelling. 2.6 Completely Labelled and Almost Completely Labelled Simplex Suppose we have a triangulation G of Sn with the vertex of G admissible labelled, we

Journal of Computer and Mathematical Sciences Vol. 4, Issue 4, 31 August, 2013 Pages (202-321)

219

R. S. Patel, et al., J. Comp. & Math. Sci. Vol.4 (4), 216-224 (2013)

call the members of G and Gn-1 merely n simplices and n-1 simplices. An n simplex is said to be completely labelled (c.l.) if its vertices carry all labels in N0 while an n or n-1 simplex is said to almost completely labelled (a.c.l.) if its vertices carry all the labels 0,1,2,...(n-1). 2.7 The Graph Γn: The graph Γn of f is the n simplices of a triangulation G with the labels 0,1,2...n1, formed paths .These paths are called the graph Γn of G. The nodes of Γn are represented by heavy dots placed on the barycenter of the corresponding simplex and the edges of Γn are represented by double lines. Two nodes of Γn are adjacent if one is a face of other or if they share an a.c.l. face. The number of adjacent nodes of a simplex

is called its degree. 3. PRELIMINARIES 3.1 Theorem: Each connected component of Γn has one of the following form: [i] A simple circuit whose nodes are a.c.l. but not c.l. n-simplex. [ii] A simple path whose intermediate nodes are a.c.l. but not c.l. n- simplices and each of whose end point is either [a] a completely labelled simplex or [b] An a.c.l. [n-1] simplex in Snn. Fig. 1 [for n= 2] shows how all four forms will arise. Since the number of end points of a path is two, the total number of c.l. nsimplices in snn is even. This proves the inductive steps of Sperner's lemma given below.

(Fig. 1) Journal of Computer and Mathematical Sciences Vol. 4, Issue 4, 31 August, 2013 Pages (202-321)

R. S. Patel, et al., J. Comp. & Math. Sci. Vol.4 (4), 216-224 (2013)

3.2 Sperner's Lemma If G be a triangulation of Sn with each vertex of G has admissible labelling, then there is a simplex in G whose vertices carry all the labels in N0 .This lemma concludes that a completely labelled simplex will provide an approximate fixed point which was used to prove the Brouwer's fixed point theorem. The proof of Sperner's lemma does not suggest any method to find completely labelled simplices, however it has been found that n simplices of G with labels 0,1,2...[n-1] formed a path. Cohen1 gave a proof of Sperner's lemma based on these paths, Cohen proof is inductive. However we still have the problem of how to start the path. Here we will show that using K2[m] how one can start the path of graph Γn .

220

={xεS~n:xi=0 for i=0,1,...(n-1)} we clearly have an admissible labelling, in the sense that no vertex in Si~n has the label i. Also it is apparent that if σεK2~(m) does lie in Sn then each of its vertices has nth coordinate 0 or – m-1 and can not be labelled n. Hence all completely labelled simplices of K2~(m) in fact lie in Sn and give approximate fixed point of f. The following lemma proved in Todd8 specifies the condition for the start of path. 3.4 Lemma Let m=kn for some integer k>0 then there is just one almost completely labelled [n-1]simplex τ* of K2~(n-1) (m) in Sn~n. The barycenter of this almost completely labelled [n-1] simplex is the starting point of Γn .

3.3 The Artificial Start Algorithm

3.5 Algorithm Procedures

If f:Sn →Sn be any given function using the standard labelling we obtain an approximate fixed point of f by finding a completely labelled simplex of K2[M]. To do this we first show how to extend K2(m) to a triangulation of Sn with an extra 'layer' below Snn. By applying the suitable homeomorphism, we obtain triangulation K2~(m) of S~n = {xεRn+1:Σxi=1,xi ≥0, i=0, 1...(n-1) and xn ≥-m-1} where S~n is Sn with an extra layer added to Snn. Each vertex of K2~m lies in Sn or has its last co-ordinates – m-1. If yεk2~m lies in Sn, we label it min{jεN0:fj(y) ≤yj>0}. If y∉sn we label it min.{jεN0:yj=maxkyk}. The latter labelling roughly corresponds to extending the function f to S~n so that f(x)=(n+1)]-1v for xεSn~n ={xεS~n:xn=m-1}. If we let Si~n

For a given mapping f:Sn→Sn pick m=kn for some integer k>0. Triangulate S~n with K2~(m) and label the vertices of K2~(m) as above. Let τ* be the unique almost completely labelled (n-1) simplex of k2~(n-1) (m) lying in sn~n. Step 1: let σl be the unique n-simplex of k2~(m) that has τ* as a facet, let y+ be the vertex of σl that is not a vertex of τ*. set l =1. Step 2: Calculate the label of y+ if it is n, STOP ; σ1 is completely labelled and yields an approximate fixed point of f, otherwise, the label of y+ duplicate that of exactly one other vertex of σl ,say yStep 3: Find the simplex σl +1 that has as vertices all the vertices of σl except y-. let y+

Journal of Computer and Mathematical Sciences Vol. 4, Issue 4, 31 August, 2013 Pages (202-321)

221

R. S. Patel, et al., J. Comp. & Math. Sci. Vol.4 (4), 216-224 (2013)

be the vertex of σl +1 that is not a vertex of σl set l ←l+1and return to step 2. This algorithm

has been used in the main result to have an approximate fixed point for n=2 and m=8.

(Fig. 2)

3.6 Main Result We take n=2. so that the progress of iteration can be seen on a diagram (fig.2). We execute the algorithm using the formal descriptions to show how rapidly and conveniently the operation can be performed using k2. let f:S2→S2 be defined by

.4. 3. 2  f [x] = .1. 4. 5    .5. 3. 3  we will use k2(8) to find approximate fixed point of f. In fact f has just one fixed point x*=(27,32,33)T/92.

Journal of Computer and Mathematical Sciences Vol. 4, Issue 4, 31 August, 2013 Pages (202-321)

222

R. S. Patel, et al., J. Comp. & Math. Sci. Vol.4 (4), 216-224 (2013) l 1

–

2 1

< Y ,Y ,Y Labels k 2 (Y1 , π 1),

> = –

< Y1 , Z 2 , Y 3 > = Labels k 2 (Y1 , π 2),

 .35  .25    .4 

–

 4   4  3  5   4  5      -1  0   0  0 1 ? π 1 = (2, 1)

 3  5    0

 4   3   3  5   6   5      -1  − 1  0 

case i=0

 .3 3 7 5   .2 8 7 5     .3 7 5 

0<i<n

3  6     − 1

i=0

 2  6    0

 .3 2 5   .2 2 5    .3 5 

i=0

 2  5   1 

 .3 125   .3375     .35 

0<i<n

 3  4   1 

 .3 25  .3  —    .375 

i=0

0 1 ? π 2 = (2, 1)

0 1 2 π 4 = (1, 2)

 3   2  2 3 4 5 < Y , Z , Y > =  5   4  5       0  1  1  Labels k 2 (Y3 , π 5),

Yy0

0 1 ? π 2 = (1, 2)

 3   2  2 3 4 5 < Z , Y , Y > =  5   6  5       0   0  1  Labels k 2 (Y3 , π 4),

Label of Y+ 0

0 1 ? π 0 = (1, 2)

 3 3   2 2 3 4 < Z , Y , Y > =  6 5  6      -1) (0  0 Labels k 2 (Y2 , π 3),

 4  4    0

Labels k 2 (Y0 , π 0),

f(y+)

 5   4   4 < Y , Y , Y > =  4   5   4      -1  -1  0 0

Y+

σl

0 1 2 π 5 = (2, 1)

Journal of Computer and Mathematical Sciences Vol. 4, Issue 4, 31 August, 2013 Pages (202-321)

223 l 7

R. S. Patel, et al., J. Comp. & Math. Sci. Vol.4 (4), 216-224 (2013) Y+

f(y+)

 2  2  5  4    1   2 1 ?

 2  4    2

 .3   35     .3 5 

 3   3  2 4 5 6 < Z , Z , Y > =  4   3  4     1   2  2

 3  3    2

σl

< Y 4 , Z5 , Y 6 > = Labels k 2 :z4, π 6), π 6 = (1,2) 8

Labels k 2 :z4, π1), π1 = (2,1) 9

 .3125   .3125     .375 

i=0

 2  3    3

 .2875   .3625     .35 

0<i<1

 3  2    3

 .3   .325    .375

0<i<1

 2  2    4

 .275   .375     .35 

Proof We have m=8=kn=4.2. x=(x0,x1.x0)T be any in S2. We have

.4. 3. 2   x0  f [x] = .1. 4. 5   x1     .5. 3. 3   x2 

case 0<i<n

 3   2  2 6 7 8 < Z ,Y ,Y > =  2   3   2      (3)   (3)  (4) Labels k 2 z6, π10), π10), π10= (1,2)

YY5

 3   2  3 5 7 6 < Z , Y , Y > =  3   3  2      2   3  3 Labels k 2 z5, π9), π9 = (2,1)

 3   2  2 5 6 7 < Z , Y , Y > =  3   4  3      2   2  3 Labels k 2 :z5, π8), π8 = (1,2)

3  4    1  0

Label of Y+ 1

Let f(x) = (f1 (x), f2 (x), f3 (x)). So that f(x) = (.4x0+.3x1+.2x2, .1x0+.4x1+.5x2, .5x0 + .3x1+. 3x2)T .........(1) we label x by min{jεN0:fj (x) ≤xj>0} here N0={0,1,2} For convenience we omit the denominator of 8 in every vertex of k2(m). e.g. if we take x=(3/8,1/2,1/8)T we write it as x=(3,4,1)T.

Journal of Computer and Mathematical Sciences Vol. 4, Issue 4, 31 August, 2013 Pages (202-321)

R. S. Patel, et al., J. Comp. & Math. Sci. Vol.4 (4), 216-224 (2013)

Now f(3,4,1)T=(.325,.3,.375)T from (1) and label of (3,4,1)T is 0 in this way we label each vertex of k2(8). This process of algorithm is shown in the table given below The operations are performed from left to right and top to bottom. For the given function{(27/35,32/33,1) Ts:sεR+} gives the sequence of approximate fixed points of f and x*=(27,32,33)T/92 is just a fixed point of f since f(x*)=x* here x lies in

 3  2  2 σ 11 =  2  3  2      3  3  4 which is a completely labelled simplex of k2(m). REFEENCES 1. Cohen D.I.A.” On the Sperner lemma" J. Comb, Theory 2, 586-587 (1967). 2. Kuhn H. W., "Some combinatorial lemmas in Topology" IBM J. Research and Develop.45,518-524 (1960). 3. Kuhn H.W., "Simplicial approximation of fixed points."Proc. Nat Acad. Society U.S.A.61, 1238-1242 (1968).

224

4. Kuhn H.W. and J.G. Mackinnon, "The sandwich method for finding fixed points. J. Optimisation theory and Applications. 17 (1975). 5. Saigal R. "Investigations into the efficiency of fixed point algorithm" Fixed point; algorithm and applications, S. Karmardian (ed.) Academic press. 6. Scarf. H. "The approximation of fixed points of a continuous mapping. Siam J. Appl. Mathematics. 155, 1328-1343 (1967). 7. Todd. M. J. "On Triangulation for computing fixed points "Maths Programming” (10), 332 (1976). 8. Todd M. J. "The computation fixed point and application "Lecture Notes in economics and mathematical system 124, Springer Verlag Newyork (1976). 9. Vander Lann, G. and A.J.J. Talman "A restart algorithm for computing fixed points with an extra dimention. Math. Programming 17, 74-84 (1979). l0. Vertgeim, B.A."On an Approximate Determination of the Fixed Points of Continuous Mappings" Soviet Math. Dokl. 11, 295-298 (1970).

Journal of Computer and Mathematical Sciences Vol. 4, Issue 4, 31 August, 2013 Pages (202-321)