Research of optimization to solve nonlinear equation based on granular computing

In this article,a real number is defined as a granulation and the real space is transformed into real granular space[1].In the entironment,solution of nonlinear equation is denoted by granulation in real granular space.Hence,the research of whole optimization to solve nonlinear equation based on granular computing is proposed[2].In classical case,we solve usually accurate solution of problems.If can't get accurate solution,also finding out an approximate solution to close to accurate solution.But in real space,approximate solution to close to accurate solution is very vague concept.In real granular space,all of the approximate solutions to close to accurate solution are constructed a set,it is a granulation in real granular space.Hence,this granulation is an accurate solution to solve problem in some sense,such,we avoid to say vaguely "approximate solution to close to accurate solution".We introduce the concept of granulation in one dimension real space.Any positive real number a together with moving infinite small distance ε will be constructed an interval [a-ε,a+ε],we call it as granulation in real granular space,denoted by ε(a)or [a].We will discuss related properties and operations[3] of the granulations.Let one dimension real space be R,where each real number a will be generated a granulation,hence we get a granular space R based on real space R.Obviously,R∈R.Infinite small number in real space R is only 0,and there are three infinite small granulations in real number granular space R:[0],[ε] and [-ε].As the graph in Fig.1 shows.In Fig.1,[-ε] is a negative infinite small granulation,[ε] is a positive infinite small granulation,[0] is a infinite small granulation.[a] is a granulation of real number a generating,it could be denoted by interval [a-ε,a+ε] in real space [3-5].Fig.1 Real granulations [0] and [a] Let f(x)=0 be a nonlinear equation,its graph in interval [-3,10] is showed in Fig.2.Where-3≤x≤10 Relation ρ(f| |,ε)is defined as follows:(x1,x2)∈ρ(f| |,ε)iff |f(x1)-f(x2)| < ε Where ε is any given small real number.We have five approximate solution sets on the nonlinear equation f(x)=0 by ρ(f| |,ε)∧|f(x)|[a,b]max,to denote by granulations [(xi1+xi2)/2],[(xi3+xi4)/2],[(xi5+xi6)/2],[(xi7+xi8)/2] and [(xi9+xi10)/2] respectively,where |f(x)|[a,b]max denotes local maximum on x∈[a,b].This is whole optimum on nonlinear equation in interval [-3,10].We will get best optimization solution on nonlinear equation via computing f(x)to use the five solutions denoted by granulation in one dimension real granular space[2,5].

Extended abstract： In this article, a real number is defined as a granulation and the real space is transformed into real granular space. In the entironment, solution of nonlinear equation is denoted by granulation in real granular space. Hence, the research of whole optimization to solve nonlinear equation based on granular computing is proposed. In classical case, we solve usually accurate solution of problems. If can＇t get accurate solution, also finding out an approximate solution to clause to accurate solution. But in real space, approximate solution to close to accurate solution is very vague concept. In real granular space, all of the approximate solutions to close to accurate solution are constructed a set, it is a granulation in real granular space. Hence, this granulation is an accurate solution to solve problem in some sense, such, we avoid to say vaguely ＂approximate solution to close to accurate solution＂. We introduce the concept of granulation in one dimension real space. Any positive real number a together with moving infinite small distance ε will be constructed an interval [α-ε,α-ε], we call it as granulation in real granular space, denoted by ε（α） or [a]. We will discuss related properties and operations of the granulations. Let one dimension real space be R, where each real number a will be generated a granulation, hence we get a granular space R^＊ based on real space R. Obviously, R∈R^＊ . Infinite small number in real space R is only 0, and there are three infinite small granulations in real number granular space R^＋ ： [0], [ε] and [-ε]. As the graph in Fig. 1 shows. In Fig. 1, [-ε] is a negative infinite small granulation,[ε] is a positive infinite small granulation, [0] is a infinite small granulation. [α] is a granulation of real number a generating, it could be denoted by interval [α-ε,α＋ε] in real space [3-5]. Let f（x）=0 be a nonlinear equation, its graph in interval [-3,10] is showed in Fig. 2 Where - 3≤x≤ 10 Relation p（f｜｜ ,ε） is defined as follows： （x1,x2） ∈p（f｜｜,ε） iff ｜f（x1）-f（x2）] 〈ε Where e is any given small real number. We have five approximate solution sets on the nonlinear equation f（x）= 0 by ρ（f｜｜ ,ε） ∧｜f（x） [a,b]max to denote by granulations [（xi1 ＋xi2 ）/2], [（xi3 ＋xi4 ）/2], [（xi5 ＋xi6 ）/2], [（xi7 ＋xi8 ）/2] and [（xi9 ＋xi10 ）/2] respectively, where ｜f（x） ｜[a,b]max denotes local maximum on x ∈ [a,b]. This is whole optimum on nonlinear equation in interval [- 3,10]. We will get best optimization solution on nonlinear equation via computing f（x） to use the five solutions denoted by granulation in one dimension real granular space[2,5].