Several Classes Of Numerical Quadrature Formulas And Their Intelligent Algorithms

With the deep research of numerically computing theory and methods and with the rapid development of computer technology,the computer algorithms theory plays a more and more important role in numerically scientific computation.When solving many mathematical models in practice,computing definite integrals is mandatory.In the field of computer,applications of computing integrals include graph handling,web covering and connecting problems,PIN controllers,continuously simulating system,meshless numerical simulation system,and the like.Because the prime functions of an amount of integrands are very complicated or generally difficult to be acquired explicitly,or because integrands are just a group of discrete data obtained by experiments,one usually computes integrals for these integrands numerically and approximately.This method is called numerical integration and the approximately computing formulas are called quadrature formulas for definite integrals.These approximate methods can be stimulatingly carried out by computer algorithms.The topic of numerically integrating methods can be traced back to ancient times.In recent centuries,especially after the 16th century,many scholars proposed various numerically integrating methods.The classical numerically integrating methods include trapezoid formula and its complex analogues,parabola(Simpson)integral formula and its complex analogues,the Newton-Cotes type formulas,the Gauss type formulas,and the like.In practice,due to huge computing amount of the higher Newton-Cotes type formulas and Gauss type formulas,their applications are of limitations.Currently,along speedy advancement of many directions in society,in the process of scientific research and practice,integrands become more and more complicated and diverse.Moreover,classical quadrature formulas cannot be used to deal with the case that integrands are functions who are almost everywhere differentiable and whose non-differentiable points are the first class of discontinuous ones.The intelligent algorithms cannot be used to handle the case that integrands are of higher derivatives.Even so,plenty of practical workers occasionally ignore these special conditions and mistakenly apply known quadrature formulas.Therefore,it is necessary to find out new numerically computing methods.This has been a main investigating direction in this area.The main results and innovative points of this dissertation are as follows.(1)In current textbooks and monographs,the trapezoid and parabola formulas appear separately,error estimation needs respectively existence of the second and fourth derivatives of integrands,and integrands in recently literature require continuous derivative of the first order in integrating intervals unless at finite points.In this dissertation,the author unifies the classical trapezoid and parabola integral formulas and their complex analogues and estimates errors under conditions that integrands have the first derivatives almost everywhere and discontinuous points of the first kind.These formulas not only require weak conditions but also have unified forms.Therefore,these formulas are more suitable for practical environments.(2)In order to increase accuracy of different classes of estimations for integrals,basing on the specification that the coefficient in the middle term of classical complex trapezoid formula is 2,the author constructs three new complex quadrature formulas whose coefficient in the middle term can take 1,3,and 4.Under conditions that integrands are differentiable almost everywhere and have the first class discontinuous points of the first kind,the author estimates errors one by one.By comparing,the author finds that the computing amount is same as the complex trapezoid formula and that the accuracy of the new formula is higher frequently.When considering complex quadrature formula whose coefficient in the middle term is 1 for convex functions,the estimating accuracy is theoretically higher.(3)In many practical cases,integrands in mathematical models have higher derivatives.At this time,if one still use classical quadrature formulas,the properties of integrands cannot be cultivated sufficiently,or computing amount may be increased,or error estimation has lower accuracy.Therefore,it is necessary to construct new quadrature formulas for highly differentiable integrands.For this,under conditions that integrands are highly differentiable,the author constructs 2 new quadrature formulas and their 3 complex formulas.Under conditions that f(m)(x)is differentiable almost everywhere and has discontinuous points of the first kind,the author estimates errors.After analyzing and comparing,the authors discover that,when higher order derivatives at end points are not larger,this kind of quadrature has higher accuracy of error estimation.This demonstrates that this research is much significant in theory and practices.The author also define a class of generalized convex functions:the m-?-convex and(m,?)-?-convex functions.Hereafter,the author studies applications of some generalized convex functions to error estimations of newly constructed complex quadrature formulas.By simulating examples,the author demonstrates advantages of applying convex functions to error estimations.(4)In order to increase accuracy of approximate values of integrals,aiming at specification of non-equidistance complex quadrature formulas(non-equidistance random problem),the author optimizes non-equidistance intervals.This optimization usually is solved by intelligent algorithms.In this dissertation,the author does firstly by intelligent algorithms(for example,genetic algorithms,artificial fish swarm algorithms,and particle swarm optimization)to optimize non-equidistance intervals,then he do by making use of newly constructed complex quadrature formulas.Through simulating experiments,the author finally verifies accurate problem for approximating values of quadrature formulas.To search out better intelligent algorithms for realizing numerically integrating algorithms nowadays is an important direction in current field and has better applicability in future.All newly constructed quadrature formulas and numerically integrating algorithms on intelligent algorithms in this dissertation are realized through MatLab 2012 program simulation.The computing conclusions include complex quadrature of the unified form of trapezoid and parabola formulas,3 complex quadrature formulas with the same series as complex trapezoid formulas,and 3 complex quadrature formulas for integrands with higher derivatives.Among them,some quadrature formulas have higher accuracy than classical formulas.Under the same intervals,numerical integration methods on intelligent algorithms promote computing efficiency and accuracy.