几类完全模糊线性系统模糊近似解的讨论
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摘要
在数学、物理学、工程计算和统计分析等领域的数学建模中,比较成熟也比较容易计算的是考虑能否将其转化为线性系统.然而,在具体的数学建模过程中经常涉及到参数的不确定性,这种不确定性又往往表现为一个模糊数.因此,涉及模糊数的线性系统,即模糊线性系统的求解问题是模糊数学的重要组成部分.本文对几类完全模糊线性系统的模糊近似解及其求解方法进行了研究和讨论.
首先,扩展了LR-模糊数的概念,给出了广义LR-模糊数的定义,提出了一种四则运算的近似表示.其次,讨论了完全模糊线性系统的非负(正)模糊近似解,给出了其解的判定定理,并对两类特殊的完全模糊线性系统(模糊线性系统和齐次模糊线性系统)解的情况进行了研究,分别给出了解存在的判定定理和通解表示定理.而后,借助于系数矩阵与增广矩阵之间的秩关系,提出了广义相容完全模糊线性系统和广义不相容完全模糊线性系统的概念,分别讨论了其模糊近似解、模糊近似通解、最小二乘模糊近似解、最小二乘模糊近似通解和极小最小二乘模糊近似解.最后,针对对偶完全模糊线性系统,讨论了其模糊近似解,给出了对偶完全模糊线性系统模糊近似解的Cramer规则算法,并对直接算法和Cramer规则算法进行了比较.
In the mathematical modeling of mathematics, physics, engineering computation and statistical analysis, the useful and easily estimated method is that it could be transferred into a linear system in some sense. However, the uncertainty of the parameters is involved in the process of actual mathematical modeling, which is often represented as fuzzy numbers. So the investigation of solving the solutions for fuzzy linear systems whose elements of coefficient matrix or augmented matrix are fuzzy numbers, plays an important role in the fuzzy mathematics theory. In this thesis, the fuzzy approximate solution and its computation methods for some kinds of fully fuzzy linear systems are investigated.
Firstly, the concept of the LR-fuzzy numbers is expanded, and the generalized LR-fuzzy numbers is defined. The approximate expressions of the fuzzy number arithmetic operations are given. Secondly, the non-negative and non-position fuzzy approximate solutions of the fully fuzzy linear system are discussed, and the judge theorem of its solutions is obtained. Furthermore, two special kinds of fully fuzzy linear systems including the fuzzy linear system and the homogenous fuzzy linear system are studied. A judge theorem of its solutions and a general solution representation for the fuzzy linear system and the homogenous fuzzy linear system are shown, respectively. In additions, a generalized consistent and inconsistent fully fuzzy linear system are further defined by using the relation between the ranks of coefficient matrix and augmented matrix. The fuzzy approximate solution, the fuzzy approximate general solution, the fuzzy least squares approximate and the fuzzy minimum least squares approximate solution about them are discussed. Finally, the dual fully fuzzy linear system is defined, its fuzzy approximate solution is discussed, and the Cramer ruler method of dual fully fuzzy linear system is pointed out. Both direct method and Cramer method are also obtained and compared by an example.
首先,扩展了LR-模糊数的概念,给出了广义LR-模糊数的定义,提出了一种四则运算的近似表示.其次,讨论了完全模糊线性系统的非负(正)模糊近似解,给出了其解的判定定理,并对两类特殊的完全模糊线性系统(模糊线性系统和齐次模糊线性系统)解的情况进行了研究,分别给出了解存在的判定定理和通解表示定理.而后,借助于系数矩阵与增广矩阵之间的秩关系,提出了广义相容完全模糊线性系统和广义不相容完全模糊线性系统的概念,分别讨论了其模糊近似解、模糊近似通解、最小二乘模糊近似解、最小二乘模糊近似通解和极小最小二乘模糊近似解.最后,针对对偶完全模糊线性系统,讨论了其模糊近似解,给出了对偶完全模糊线性系统模糊近似解的Cramer规则算法,并对直接算法和Cramer规则算法进行了比较.
In the mathematical modeling of mathematics, physics, engineering computation and statistical analysis, the useful and easily estimated method is that it could be transferred into a linear system in some sense. However, the uncertainty of the parameters is involved in the process of actual mathematical modeling, which is often represented as fuzzy numbers. So the investigation of solving the solutions for fuzzy linear systems whose elements of coefficient matrix or augmented matrix are fuzzy numbers, plays an important role in the fuzzy mathematics theory. In this thesis, the fuzzy approximate solution and its computation methods for some kinds of fully fuzzy linear systems are investigated.
Firstly, the concept of the LR-fuzzy numbers is expanded, and the generalized LR-fuzzy numbers is defined. The approximate expressions of the fuzzy number arithmetic operations are given. Secondly, the non-negative and non-position fuzzy approximate solutions of the fully fuzzy linear system are discussed, and the judge theorem of its solutions is obtained. Furthermore, two special kinds of fully fuzzy linear systems including the fuzzy linear system and the homogenous fuzzy linear system are studied. A judge theorem of its solutions and a general solution representation for the fuzzy linear system and the homogenous fuzzy linear system are shown, respectively. In additions, a generalized consistent and inconsistent fully fuzzy linear system are further defined by using the relation between the ranks of coefficient matrix and augmented matrix. The fuzzy approximate solution, the fuzzy approximate general solution, the fuzzy least squares approximate and the fuzzy minimum least squares approximate solution about them are discussed. Finally, the dual fully fuzzy linear system is defined, its fuzzy approximate solution is discussed, and the Cramer ruler method of dual fully fuzzy linear system is pointed out. Both direct method and Cramer method are also obtained and compared by an example.
引文
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[2]吴从忻,马明.模糊分析基础[M].北京:国防工业出版社,1991.
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[4]张文修.模糊数学基础[M].西安:西安交通大学出版社,1984.
[5]马骥良,于纯海.模糊代数学选论[M].学苑出版社,1989.
[6]王国俊.L-fuzzy拓扑空间论[M].西安:陕西师范大学出版社,1988.
[7]S.S.L.Chang,L.A.Zadeh.Fuzzy Mapping and Control[J].IEEE Trans.Systems Man Cybernet,1972(2)30-34.
[8]M.Mizumoto,K.Tanaka.The Four Operations of Arithmetic on Fuzzy Numbers[J].Systems Comput.Control,1976(7)73-81.
[9]S.Nahias.Fuzzy Variables[J].Fuzzy Sets and Systems,1978(1)97-111.
[10]D.Dubios,H.Prade.Operations on Fuzzy Numbers[J].International Journol of systems Science,1978,9(6):613-625.
[11]D.Dubios,H.Prade.Fuzzy Sets and Systems:theory and Applications[M].New York:Acdamic Press,1980.
[12]D.Dubios,H.Prade.Systems of Linear Fuzzy Constraints[J].Fuzzy Sets and Systems,1980,3(1):37-48.
[13]Wu Congxin,Ma Ming.Embedding Problem of Fuzzy Number Space:Part Ⅰ[J].Fuzzy Sets and Systems,1991(44)33-38.
[14]Wu Congxin,Ma Ming.Embedding Problem of Fuzzy Number Space:Part Ⅱ[J].Fuzzy Sets and Systems,1992(45)198-202.
[15]S.Abbasbandy,A.Jararian,R.Ezzati.Conjugate Gradient Method fbr Fuzzy Symmetric Positive Definete System of Linear Equations[J].Applied Mathematics and Computation,2005(171) 1184-1191.
[16]B.Asady,S.Abbasbandy,M.Alavi.Fuzzy General Linear Systems[J].Applied Mathematics and Computation,2005(169)34-40.
[17]M.Dehghan,B.Hashemi,M.Ghatee.Computational Methods fbr Solving Fully Fuzzy Linear Systems[J].Applied Mathematics and Computation,2006(179)328-343.
[18]M.Dehghan,B.Hashemi.Solution of the Fully Fuzzy Linear Systems Using the Decomposition Procedure[J].Applied Mathematics and Computation,2006(182)1568-1580.
[19]Feng Yuhu.The Solutions of Linear Fuzzy Stochastic Differential Systems[J].Fuzzy Sets and Systems,2003(140)541-554.
[20]Feng Yuhu.Fuzzy Stochastic Differential Systems[J].Fuzzy Sets and Systems,2000(115)351-363.
[21]张跃,王光远.模糊随机动力系统理论[M].北京:科学出版社,1993.
[22]B.Boswell,S.Taylar.An Application of a Fuzzy Random Variable to Volnurability Modeling[J].Fuzzy Sets and Systems,1990(33)19-28.
[23]Wu Congxin,Gong Zengtai.On Henstock Integral of Fuzzy-number-valued Functions (Ⅰ)[J].Fuzzy Sets and Systems,2001(120)523-552.
[24]Gong Zengtai,Wu Congxin,Li Baolin.On the Problem of Characterizing Derivatives for the Fuzzy-valued Functions[J].Fuzzy Sets and Systems,2002(127)315-322.
[25]Gong Zengtai,Wu Congxin.Bounded Variation,Absolute Continuity and Absolute Integrability fbr Fuzzy-number-valued Functions[J].Fuzzy Sets and Systems,2002(129)83-94.
[26]Gong Zengtai.On the Problem of Characterizing Derivatives for the Fuzzy-valued Functions(Ⅱ):Almost Everywhere Differentiability and Strong Henstock Integral[J].Fuzzy Sets and Systems,2004(145)381-393.
[27]Gong Zengtai,Wang Lin.The Numberical Calculus of Expectations of Fuzzy Random Variables[J].Fuzzy Sets and Systems,2007(158)722-738.
[28]巩增泰.模糊有界变差函数及其可导性[J].兰州大学学报:自然科学版,2003,39(3):17-21.
[29]M.Piedman,M.Ming,A.Candel.Fuzzy Linear Systems[J].Fuzzy Sets and Systems,1998,96(2):201-209.
[30]T.Allahviranloo.Numerical Methods fbr Fuzzy System of Linear Equations[J].Applied Mathematics and Computation,2004(155)493-502.
[31]T.Allahviranloo.Successive Overrelaxation Iterative Methods fbr Fuzzy System of Linear Equations[J].Applied Mathematics Computation,2005(162)189-196.
[32]M.Dehghan,B.Hashemi.Iterative Solution of Fuzzy Linear Systems[J].Applied Mathematics and Computation,2006(175)645-674.
[33]Zheng Bing,Wang Ke.General Fuzzy Linear Systems[J].Applied Mathematics and Computation,2006(181) 1276-1286.
[34]Wang Ke,Zheng Bing.Inconsistent Fuzzy Linear Systems[J].Applied Mathematics and Computation,2006(181)973-981.
[35]Wang Ke,Zheng Bing.Symmetric Successive Overrelaxation Methods for Fuzzy Linear Systems[J].Applied Mathematics and Computation,2006(175)891-901.
[36]Zheng Bing,Wang Ke.On Successive Overrelaxation Methods for Rank Deficient Linear Systems[J].Applied Mathematics and Computation,2006(173)951-959.
[37]Zheng Bing,Wang Ke.Symmetric Successive Overrelaxation Methods for Solving the Rank Deficient Linear Least Squares Problem[J].Applied Mathematics and Computation,2005(169)1305-1323.
[38]E.Babolian,J.Biazar,A.R.Vahidi.On the Decomposition Method for System of Linear Equations and System of Linear Volterra Integral Equations[J].Applied Mathematics and Computation,2004(147)19-27.
[39]Rostilav Horcik.Solution of System of Linear Equations with Fuzzy Numbers[J].Fuzzy Sets and Systems,2008(159)1788-1810.
[40]J.J.Buckley,Y.Qu.Solving Systems of Linear Fuzzy Equations[J].Fuzzy Sets and Systems,1991(43) 33-43.
[41]M.Dehghan,B.Hashemi.Iterative Solution of Fuzzy Linear Systems[J].Applied Mathematics and Computation,2006(175)645-674.
[42]R.E.Giachetti,R.E.Young.A Parametric Respresentation of Fuzzy Numbers and Their Arithmetic Operators[J].Fuzzy Sets and Systems,1997(91)185-202.
[43]B.Asady,S.Abbasbandy,M.Alavi.Fuzzy General Linear Systems[J].Applied Mathematics Computation,2005(169)34-40.
[44]S.S.Rao,L.Chen.Numerical Solution of Fuzzy Linear Equations in Engineering Analysis[J].International Journal for Numerical Methods in Engineering,1998,42(5):829-846.
[45]M.Dehghan,B.Hashemi,G.Mehdi.Solution of the Fully Fuzzy Linear Systems Using Iterative Teckniques[J].Chaos,Solutions and Fractals,2007,34(2):316-336.
[46]M.Ming,M.Priedman,A.Kandel.Duality in Fuzzy Linear Systems[J].Fuzzy Sets and Systems,2000(109)55-58.
[47]X.Wang,Z.Zhong,M.Ha.Iteration Algorithms for Solving a System of Fuzzy Linear Equations[J].Fuzzy Sets and Systems,2001(119)121-128.
[48]S.Muzziolis,H.Reynaersts.Fuzzy Linear Systems of the Form A_1x+b_1=A_2x+b_2[J].Fuzzy Sets and Systems,2006(157)939-951.
[49]S.Abbasbandy,M.Otadi,M.Mosleh.Mininal Solution of General Dual Fuzzy Linear Systems[J].Chaos,Solutions and Fractals,2008(37)1113-1124.
[50]巩增泰,刘坤.完全模糊线性系统的模糊近似解[J].兰州大学学报:自然科学版,2009,45(1):78-82.
[51]刘坤,巩增泰,王欣欣.两类特殊的完全模糊线性系统的模糊近似解[J].黑龙江大学学报:自然科学版(已接受).
[52]刘坤,巩增泰.广义完全模糊线性系统解的讨论[J].西北师范大学学报:自然科学版,2008,44(6):6-10.
[53]巩增泰,刘坤.对偶完全模糊线性系统的模糊近似解[J].兰州理工大学学报,2009,35(3).
[54]曹炳元.应用模糊数学与系统[M].北京:科学出版社,2005.
[55]吴昌悫,魏洪增.矩阵理论与方法[M].北京:电子工业出版社,2006.
[56]张贤达.矩阵分析与应用[M].北京:清华大学出版社,2004.
[2]吴从忻,马明.模糊分析基础[M].北京:国防工业出版社,1991.
[3]吴从忻,马明,方锦暄.模糊分析学的结构理论[M].贵州:贵州科技出版社,1994.
[4]张文修.模糊数学基础[M].西安:西安交通大学出版社,1984.
[5]马骥良,于纯海.模糊代数学选论[M].学苑出版社,1989.
[6]王国俊.L-fuzzy拓扑空间论[M].西安:陕西师范大学出版社,1988.
[7]S.S.L.Chang,L.A.Zadeh.Fuzzy Mapping and Control[J].IEEE Trans.Systems Man Cybernet,1972(2)30-34.
[8]M.Mizumoto,K.Tanaka.The Four Operations of Arithmetic on Fuzzy Numbers[J].Systems Comput.Control,1976(7)73-81.
[9]S.Nahias.Fuzzy Variables[J].Fuzzy Sets and Systems,1978(1)97-111.
[10]D.Dubios,H.Prade.Operations on Fuzzy Numbers[J].International Journol of systems Science,1978,9(6):613-625.
[11]D.Dubios,H.Prade.Fuzzy Sets and Systems:theory and Applications[M].New York:Acdamic Press,1980.
[12]D.Dubios,H.Prade.Systems of Linear Fuzzy Constraints[J].Fuzzy Sets and Systems,1980,3(1):37-48.
[13]Wu Congxin,Ma Ming.Embedding Problem of Fuzzy Number Space:Part Ⅰ[J].Fuzzy Sets and Systems,1991(44)33-38.
[14]Wu Congxin,Ma Ming.Embedding Problem of Fuzzy Number Space:Part Ⅱ[J].Fuzzy Sets and Systems,1992(45)198-202.
[15]S.Abbasbandy,A.Jararian,R.Ezzati.Conjugate Gradient Method fbr Fuzzy Symmetric Positive Definete System of Linear Equations[J].Applied Mathematics and Computation,2005(171) 1184-1191.
[16]B.Asady,S.Abbasbandy,M.Alavi.Fuzzy General Linear Systems[J].Applied Mathematics and Computation,2005(169)34-40.
[17]M.Dehghan,B.Hashemi,M.Ghatee.Computational Methods fbr Solving Fully Fuzzy Linear Systems[J].Applied Mathematics and Computation,2006(179)328-343.
[18]M.Dehghan,B.Hashemi.Solution of the Fully Fuzzy Linear Systems Using the Decomposition Procedure[J].Applied Mathematics and Computation,2006(182)1568-1580.
[19]Feng Yuhu.The Solutions of Linear Fuzzy Stochastic Differential Systems[J].Fuzzy Sets and Systems,2003(140)541-554.
[20]Feng Yuhu.Fuzzy Stochastic Differential Systems[J].Fuzzy Sets and Systems,2000(115)351-363.
[21]张跃,王光远.模糊随机动力系统理论[M].北京:科学出版社,1993.
[22]B.Boswell,S.Taylar.An Application of a Fuzzy Random Variable to Volnurability Modeling[J].Fuzzy Sets and Systems,1990(33)19-28.
[23]Wu Congxin,Gong Zengtai.On Henstock Integral of Fuzzy-number-valued Functions (Ⅰ)[J].Fuzzy Sets and Systems,2001(120)523-552.
[24]Gong Zengtai,Wu Congxin,Li Baolin.On the Problem of Characterizing Derivatives for the Fuzzy-valued Functions[J].Fuzzy Sets and Systems,2002(127)315-322.
[25]Gong Zengtai,Wu Congxin.Bounded Variation,Absolute Continuity and Absolute Integrability fbr Fuzzy-number-valued Functions[J].Fuzzy Sets and Systems,2002(129)83-94.
[26]Gong Zengtai.On the Problem of Characterizing Derivatives for the Fuzzy-valued Functions(Ⅱ):Almost Everywhere Differentiability and Strong Henstock Integral[J].Fuzzy Sets and Systems,2004(145)381-393.
[27]Gong Zengtai,Wang Lin.The Numberical Calculus of Expectations of Fuzzy Random Variables[J].Fuzzy Sets and Systems,2007(158)722-738.
[28]巩增泰.模糊有界变差函数及其可导性[J].兰州大学学报:自然科学版,2003,39(3):17-21.
[29]M.Piedman,M.Ming,A.Candel.Fuzzy Linear Systems[J].Fuzzy Sets and Systems,1998,96(2):201-209.
[30]T.Allahviranloo.Numerical Methods fbr Fuzzy System of Linear Equations[J].Applied Mathematics and Computation,2004(155)493-502.
[31]T.Allahviranloo.Successive Overrelaxation Iterative Methods fbr Fuzzy System of Linear Equations[J].Applied Mathematics Computation,2005(162)189-196.
[32]M.Dehghan,B.Hashemi.Iterative Solution of Fuzzy Linear Systems[J].Applied Mathematics and Computation,2006(175)645-674.
[33]Zheng Bing,Wang Ke.General Fuzzy Linear Systems[J].Applied Mathematics and Computation,2006(181) 1276-1286.
[34]Wang Ke,Zheng Bing.Inconsistent Fuzzy Linear Systems[J].Applied Mathematics and Computation,2006(181)973-981.
[35]Wang Ke,Zheng Bing.Symmetric Successive Overrelaxation Methods for Fuzzy Linear Systems[J].Applied Mathematics and Computation,2006(175)891-901.
[36]Zheng Bing,Wang Ke.On Successive Overrelaxation Methods for Rank Deficient Linear Systems[J].Applied Mathematics and Computation,2006(173)951-959.
[37]Zheng Bing,Wang Ke.Symmetric Successive Overrelaxation Methods for Solving the Rank Deficient Linear Least Squares Problem[J].Applied Mathematics and Computation,2005(169)1305-1323.
[38]E.Babolian,J.Biazar,A.R.Vahidi.On the Decomposition Method for System of Linear Equations and System of Linear Volterra Integral Equations[J].Applied Mathematics and Computation,2004(147)19-27.
[39]Rostilav Horcik.Solution of System of Linear Equations with Fuzzy Numbers[J].Fuzzy Sets and Systems,2008(159)1788-1810.
[40]J.J.Buckley,Y.Qu.Solving Systems of Linear Fuzzy Equations[J].Fuzzy Sets and Systems,1991(43) 33-43.
[41]M.Dehghan,B.Hashemi.Iterative Solution of Fuzzy Linear Systems[J].Applied Mathematics and Computation,2006(175)645-674.
[42]R.E.Giachetti,R.E.Young.A Parametric Respresentation of Fuzzy Numbers and Their Arithmetic Operators[J].Fuzzy Sets and Systems,1997(91)185-202.
[43]B.Asady,S.Abbasbandy,M.Alavi.Fuzzy General Linear Systems[J].Applied Mathematics Computation,2005(169)34-40.
[44]S.S.Rao,L.Chen.Numerical Solution of Fuzzy Linear Equations in Engineering Analysis[J].International Journal for Numerical Methods in Engineering,1998,42(5):829-846.
[45]M.Dehghan,B.Hashemi,G.Mehdi.Solution of the Fully Fuzzy Linear Systems Using Iterative Teckniques[J].Chaos,Solutions and Fractals,2007,34(2):316-336.
[46]M.Ming,M.Priedman,A.Kandel.Duality in Fuzzy Linear Systems[J].Fuzzy Sets and Systems,2000(109)55-58.
[47]X.Wang,Z.Zhong,M.Ha.Iteration Algorithms for Solving a System of Fuzzy Linear Equations[J].Fuzzy Sets and Systems,2001(119)121-128.
[48]S.Muzziolis,H.Reynaersts.Fuzzy Linear Systems of the Form A_1x+b_1=A_2x+b_2[J].Fuzzy Sets and Systems,2006(157)939-951.
[49]S.Abbasbandy,M.Otadi,M.Mosleh.Mininal Solution of General Dual Fuzzy Linear Systems[J].Chaos,Solutions and Fractals,2008(37)1113-1124.
[50]巩增泰,刘坤.完全模糊线性系统的模糊近似解[J].兰州大学学报:自然科学版,2009,45(1):78-82.
[51]刘坤,巩增泰,王欣欣.两类特殊的完全模糊线性系统的模糊近似解[J].黑龙江大学学报:自然科学版(已接受).
[52]刘坤,巩增泰.广义完全模糊线性系统解的讨论[J].西北师范大学学报:自然科学版,2008,44(6):6-10.
[53]巩增泰,刘坤.对偶完全模糊线性系统的模糊近似解[J].兰州理工大学学报,2009,35(3).
[54]曹炳元.应用模糊数学与系统[M].北京:科学出版社,2005.
[55]吴昌悫,魏洪增.矩阵理论与方法[M].北京:电子工业出版社,2006.
[56]张贤达.矩阵分析与应用[M].北京:清华大学出版社,2004.