线性投入产出方程的分解算法与非线性投入产出方程的存在性结果
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摘要
基于经典(矩阵型)投入产出分析,本文首先考虑经典的投入产出方程。为此,将矩阵分析中的矩阵分解,优化理论以及算法等方法应用于投入产出方程,从而得到相关的近似解。
进而研究一类非线性连续型条件Leontief投入产出方程,其主要可解性问题由存在性与连续性组成。对此,应用ky-Fan零点存在定理,并采用从特殊到一般的思想,得到了关于这类方程的可解性结果,并给出了相关的经济意义。
全文共分六个章节:
第一章,简要的概述了投入产出分析模型的产生和发展,回顾了经典的投入产出方程与非线性投入产出分析的研究现状。
第二章,应用广义逆矩阵方法对经典和外贸型投入产出方程进行求解,并用matlab语言进行实例计算。
第三章,当经典或外贸型投入产出矩阵是对称正定矩阵时,相关的投入产出方程被考虑并用共轭梯度法进行处理。
第四章,引入了基于Galerkin原理的GMERS算法,并应用它们对于经典和外贸型投入产出方程。
第五章,研究了一类非线性投入产出方程,应用Ky-Fan零点存在定理和集值映射方法获得了包含存在性与连续性的可解性结果。
第六章,回顾了本文研究的一些方法和结果,展示了一些尚待解决的问题。
Based on the classical (matrix type) input-output analysis, the classical input-outputequations are first considered. To this end, applying the matrix decomposition methods,optimization theory and associated algorithms in matrix analysis, the correspondingapproximatesolutionsareobtained.
Then a class of nonlinear continuous type conditional Leontief input-output equation isconsidered, and the main solvabilityproblems composed of existence andcontinuityare studied.Byapplyingtheky-Fanzeroexistencetheorem,andusingtheideafromparticulartogeneral,thesolvability results with respect to this equation are gotten, and also, the relevant economicsignificanceareprovided.
Thispaperisdividedintosixsections.:
In the first chapter, a brief emergence and development regarding the input-output modelarereviewedfortheclassicalinput-outputequationandthenonlinearinput-outputanalysis.
In the second chapter, the classical and the foreign-trade-type input-output equations aresolvedwiththegeneralizedinversematrixapproach,andcalculatedwiththematlablanguage.
When the classic or the foreign-trade-type input-output matrix is symmetric positivedefinite,thecorrespondinginput-outputequationsareconsideredinChapterIII, andtackledbytheconjugategradientmethod.
In Chapter IV, the Gmers algorithm based on Galerkin principle is introduced, and areappliedtodealwiththeclassicalandforeign-trade-typeinput-outputequations.
A class of nonlinear input-output equation is studied in Chapter V, and the correspondingsolvability results composed of existience and continuity are obtained by applying Ky-Fancriticalpointtheoremandtheset-valuedanalysismethod.
In the final chapter, the methods and results in this article are reviewed, and someunsolvedproblemsarealsopresented.
进而研究一类非线性连续型条件Leontief投入产出方程,其主要可解性问题由存在性与连续性组成。对此,应用ky-Fan零点存在定理,并采用从特殊到一般的思想,得到了关于这类方程的可解性结果,并给出了相关的经济意义。
全文共分六个章节:
第一章,简要的概述了投入产出分析模型的产生和发展,回顾了经典的投入产出方程与非线性投入产出分析的研究现状。
第二章,应用广义逆矩阵方法对经典和外贸型投入产出方程进行求解,并用matlab语言进行实例计算。
第三章,当经典或外贸型投入产出矩阵是对称正定矩阵时,相关的投入产出方程被考虑并用共轭梯度法进行处理。
第四章,引入了基于Galerkin原理的GMERS算法,并应用它们对于经典和外贸型投入产出方程。
第五章,研究了一类非线性投入产出方程,应用Ky-Fan零点存在定理和集值映射方法获得了包含存在性与连续性的可解性结果。
第六章,回顾了本文研究的一些方法和结果,展示了一些尚待解决的问题。
Based on the classical (matrix type) input-output analysis, the classical input-outputequations are first considered. To this end, applying the matrix decomposition methods,optimization theory and associated algorithms in matrix analysis, the correspondingapproximatesolutionsareobtained.
Then a class of nonlinear continuous type conditional Leontief input-output equation isconsidered, and the main solvabilityproblems composed of existence andcontinuityare studied.Byapplyingtheky-Fanzeroexistencetheorem,andusingtheideafromparticulartogeneral,thesolvability results with respect to this equation are gotten, and also, the relevant economicsignificanceareprovided.
Thispaperisdividedintosixsections.:
In the first chapter, a brief emergence and development regarding the input-output modelarereviewedfortheclassicalinput-outputequationandthenonlinearinput-outputanalysis.
In the second chapter, the classical and the foreign-trade-type input-output equations aresolvedwiththegeneralizedinversematrixapproach,andcalculatedwiththematlablanguage.
When the classic or the foreign-trade-type input-output matrix is symmetric positivedefinite,thecorrespondinginput-outputequationsareconsideredinChapterIII, andtackledbytheconjugategradientmethod.
In Chapter IV, the Gmers algorithm based on Galerkin principle is introduced, and areappliedtodealwiththeclassicalandforeign-trade-typeinput-outputequations.
A class of nonlinear input-output equation is studied in Chapter V, and the correspondingsolvability results composed of existience and continuity are obtained by applying Ky-Fancriticalpointtheoremandtheset-valuedanalysismethod.
In the final chapter, the methods and results in this article are reviewed, and someunsolvedproblemsarealsopresented.
引文
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[3]钟契夫,陈锡康.投入产出分析.中国财政经济出版社,北京,1987.
[4]潘吉勋,张顺明,经济均衡的数学原理[M].长春:吉林大学出版社,1997.
[5]张金水。可计算非线性动态投入产出模型[M]。北京:清华大学出版社,2000.
[6]CesariL,KannanR,WeinbergerHF,NonlinearAnalysis.AcademicPress,1978.
[7] Yingfan Liu, Some Existence Theorems on a class of Set-valued Type Input-outputEquationwithRestiction.Journal ofNanjingUniversityMathematical Biquarterly. 2005,Vol.22,No.1.
[8]JPAubin,HFrankowska.Set-ValuedAnalysis[M].Boston,Basel,Berlin:Birkhauser,1990.
[9]Istratescu Vasile I,Fixed Point Theory--An Introduction[M]. Dordrechet, Holland/Boston,USA:DReidelPublishingCompany,1981.
[10]AubinJP,OptimaandEquilibria-AnIntroductiontoNonlinearAnalysis[M].NewYork:Springer-Verlag/Beijing:WorldPublishingCorporation,1998
[11]CXKang.Input-Occupanty-OutAnalysis[M].PaperPresentedattheNinthInternationalConferenceonInput-outputTechniques,Kesxtely,Hungery,Septemper4-9,1989
[12] Rockafellar R T, Convex Analysis[M]. Princeton, New Jersey: Princeton UniversityPress,1970.
[13]Yingfan Liu, Some Results on a class of Generalized Leontief ConditionalInput-OutputInequalities.Acta MathematicaeSinica.March,2005,Vol.28,No.2.
[14]柏杨,刘颖范.一类关于单值型的投入产出方程的可解性定理.科学研究月刊,2007.1,总第二十五期,3-5.
[15]II Vasile. Fixed Point Theory-an Introduction. Dordrecht, Holland/ Boston: D.ReidelPublishingCompany,1981
[16]LYFan,CXHong.Someresultsoncontinuoustypeconditionalinput-outputequation.AppliedMathematicsandMechanics[J],2004,25(3),348-358
[17]YingfanLiu&TianpingChen,SolvabilityTheoremonaclassofConditionalInput-OutputEdition.ChineseAnnalsof Mathematics,2004(A),25(6)(inChinese).
[18]JPAubin,HFrankowska.Set-ValuedAnalysis[M].Boston,Basel,Berlin:Birkhauser,1990
[19]蔡大用白峰杉.现代科学计算[M].科学出版社,2002.
[20]徐树方.矩阵计算的理论与方法[M].北京大学出版社,1995.
[21]魏木生.广义最小二乘问题的理论和计算[M].科学出版社,2003.
[22]张静.求矩阵的广义逆[J].内蒙古大学学报,2005,24(6):379-382.
[23]吴有为.求广义逆矩阵的初等变换法[J].数学通报,1992,(5):26-27.
[24]新饭田宏[日]著,林贤郁,齐舒畅译.投入产出分析入门[M].中国统计出版社,1990.
[25] Moonen M, Dooren PVan,Vandewalle J. Asingular value decomposition, updatingalgorithmforsubspacetracking[J].SIAMJ.MatrixAna.Appl.13(1992):1015-1038.
[26]徐树方.矩阵计算的理论与方法[M].北京大学出版社,1995.
[27]马昌凤,林伟川,现代数值计算方法,科学出版社,2008年6月第一版.
[28]蔡大用.高等数值分析[M].清华大学出版社,1997.
[29]钱伟长,变分法及有限元(上册),科学出版社,1980年8月第一版.
[30]同济大学计算数学教研室,现代数值计算,人民邮电出版社,2009年10月第一版.
[31]SaadY.Krylovsubspacemethodsforsolvingunsymmetriclinearsystems[J].J.Comput.Appl.Math.37(1981):105-126.
[32]Sad Y. Preconditioning technique for nonsymmetric and indefinite linear systems [J]. J.Comput.Appl.Math.24(1985):89-105.
[33]Saad Y. Practical use of polynomial preconditionings for the conjugate gradient methods[J].SIAMJ.Sci.StatistComput.6(1985):865-881.
[34]SaadY.AnalysisofaugmentedKrylovsubspacemethods[J].SIAMJ.MatrixAnal.Appl.18(1997):435-449.
[35]Yingfan Liu, Xiaohong Chen, Some Results on Continuous Type ConditionalInput-Output Equation. AppliedMathematics andMechanics (English Edition).2004Vol.25,No.3:358-366.
[36]吴孝灵,刘颖范.关于一类投入产出方程的拓扑度方法.南京大学学报数学半年刊,2008,25(1):76-85.
[37]张国娟,刘颖范.一类条件投入产出方程微分与拓扑度方法.科学研究月刊,2007.6,总三十期,140-141.
[38] Xiaoling Wu, Yingfan Liu. Some solvability results on a class of set-valued typeinput-outputequation[J].MathematicsinEconomics,2007,24(3):283-290(inChinese).
[39]Yingfan Liu, Qinghong Zhang. The Rogalski-Cornet theorem and a Leontief typeinput-outputinclusion.NonlinearAnalysis:TMA,2008,69(2):325-433.
[40]Yingfan Liu, Qinghong Zhang. A Leontief-Type Input-Output Inclusion. MathematicalProblemsinEngineering.Volume2010,ArticleID572451,13pages.
[41]AubinJP,EkelandI,AppliedNonlinearAnalysis.NewYork:JohnWiley,Sons,1984.
[42]Aubin J P, Mathematical Methods of Game and Economic Theory. North-Hollland,Amsterdam,1979.
[2] W Leontief. Quantitative input-output relations in the economic system of the UnitedStateds.ReviewofEconcmicandstatistics,18(1936).
[3]钟契夫,陈锡康.投入产出分析.中国财政经济出版社,北京,1987.
[4]潘吉勋,张顺明,经济均衡的数学原理[M].长春:吉林大学出版社,1997.
[5]张金水。可计算非线性动态投入产出模型[M]。北京:清华大学出版社,2000.
[6]CesariL,KannanR,WeinbergerHF,NonlinearAnalysis.AcademicPress,1978.
[7] Yingfan Liu, Some Existence Theorems on a class of Set-valued Type Input-outputEquationwithRestiction.Journal ofNanjingUniversityMathematical Biquarterly. 2005,Vol.22,No.1.
[8]JPAubin,HFrankowska.Set-ValuedAnalysis[M].Boston,Basel,Berlin:Birkhauser,1990.
[9]Istratescu Vasile I,Fixed Point Theory--An Introduction[M]. Dordrechet, Holland/Boston,USA:DReidelPublishingCompany,1981.
[10]AubinJP,OptimaandEquilibria-AnIntroductiontoNonlinearAnalysis[M].NewYork:Springer-Verlag/Beijing:WorldPublishingCorporation,1998
[11]CXKang.Input-Occupanty-OutAnalysis[M].PaperPresentedattheNinthInternationalConferenceonInput-outputTechniques,Kesxtely,Hungery,Septemper4-9,1989
[12] Rockafellar R T, Convex Analysis[M]. Princeton, New Jersey: Princeton UniversityPress,1970.
[13]Yingfan Liu, Some Results on a class of Generalized Leontief ConditionalInput-OutputInequalities.Acta MathematicaeSinica.March,2005,Vol.28,No.2.
[14]柏杨,刘颖范.一类关于单值型的投入产出方程的可解性定理.科学研究月刊,2007.1,总第二十五期,3-5.
[15]II Vasile. Fixed Point Theory-an Introduction. Dordrecht, Holland/ Boston: D.ReidelPublishingCompany,1981
[16]LYFan,CXHong.Someresultsoncontinuoustypeconditionalinput-outputequation.AppliedMathematicsandMechanics[J],2004,25(3),348-358
[17]YingfanLiu&TianpingChen,SolvabilityTheoremonaclassofConditionalInput-OutputEdition.ChineseAnnalsof Mathematics,2004(A),25(6)(inChinese).
[18]JPAubin,HFrankowska.Set-ValuedAnalysis[M].Boston,Basel,Berlin:Birkhauser,1990
[19]蔡大用白峰杉.现代科学计算[M].科学出版社,2002.
[20]徐树方.矩阵计算的理论与方法[M].北京大学出版社,1995.
[21]魏木生.广义最小二乘问题的理论和计算[M].科学出版社,2003.
[22]张静.求矩阵的广义逆[J].内蒙古大学学报,2005,24(6):379-382.
[23]吴有为.求广义逆矩阵的初等变换法[J].数学通报,1992,(5):26-27.
[24]新饭田宏[日]著,林贤郁,齐舒畅译.投入产出分析入门[M].中国统计出版社,1990.
[25] Moonen M, Dooren PVan,Vandewalle J. Asingular value decomposition, updatingalgorithmforsubspacetracking[J].SIAMJ.MatrixAna.Appl.13(1992):1015-1038.
[26]徐树方.矩阵计算的理论与方法[M].北京大学出版社,1995.
[27]马昌凤,林伟川,现代数值计算方法,科学出版社,2008年6月第一版.
[28]蔡大用.高等数值分析[M].清华大学出版社,1997.
[29]钱伟长,变分法及有限元(上册),科学出版社,1980年8月第一版.
[30]同济大学计算数学教研室,现代数值计算,人民邮电出版社,2009年10月第一版.
[31]SaadY.Krylovsubspacemethodsforsolvingunsymmetriclinearsystems[J].J.Comput.Appl.Math.37(1981):105-126.
[32]Sad Y. Preconditioning technique for nonsymmetric and indefinite linear systems [J]. J.Comput.Appl.Math.24(1985):89-105.
[33]Saad Y. Practical use of polynomial preconditionings for the conjugate gradient methods[J].SIAMJ.Sci.StatistComput.6(1985):865-881.
[34]SaadY.AnalysisofaugmentedKrylovsubspacemethods[J].SIAMJ.MatrixAnal.Appl.18(1997):435-449.
[35]Yingfan Liu, Xiaohong Chen, Some Results on Continuous Type ConditionalInput-Output Equation. AppliedMathematics andMechanics (English Edition).2004Vol.25,No.3:358-366.
[36]吴孝灵,刘颖范.关于一类投入产出方程的拓扑度方法.南京大学学报数学半年刊,2008,25(1):76-85.
[37]张国娟,刘颖范.一类条件投入产出方程微分与拓扑度方法.科学研究月刊,2007.6,总三十期,140-141.
[38] Xiaoling Wu, Yingfan Liu. Some solvability results on a class of set-valued typeinput-outputequation[J].MathematicsinEconomics,2007,24(3):283-290(inChinese).
[39]Yingfan Liu, Qinghong Zhang. The Rogalski-Cornet theorem and a Leontief typeinput-outputinclusion.NonlinearAnalysis:TMA,2008,69(2):325-433.
[40]Yingfan Liu, Qinghong Zhang. A Leontief-Type Input-Output Inclusion. MathematicalProblemsinEngineering.Volume2010,ArticleID572451,13pages.
[41]AubinJP,EkelandI,AppliedNonlinearAnalysis.NewYork:JohnWiley,Sons,1984.
[42]Aubin J P, Mathematical Methods of Game and Economic Theory. North-Hollland,Amsterdam,1979.