• <tr id="yyy80"></tr>
  • <sup id="yyy80"></sup>
  • <tfoot id="yyy80"><noscript id="yyy80"></noscript></tfoot>
  • 99热精品在线国产_美女午夜性视频免费_国产精品国产高清国产av_av欧美777_自拍偷自拍亚洲精品老妇_亚洲熟女精品中文字幕_www日本黄色视频网_国产精品野战在线观看 ?

    New Iteration Method for a Quadratic Matrix Equation Associated with an M-Matrix

    2021-01-07 01:23:16GUANJinrui關(guān)晉瑞SONGRuying宋儒瑛ZUBAIRAhmed
    應(yīng)用數(shù)學(xué) 2021年1期

    GUAN Jinrui(關(guān)晉瑞),SONG Ruying(宋儒瑛),ZUBAIR Ahmed

    (1.Department of Mathematics,Taiyuan Normal University,Jinzhong 030619,China;2.Institute of Mathematics and Computer Science,University of Sindh,Sindh 76080,Pakistan)

    Abstract: In this paper,we consider numerical solution of a quadratic matrix equation associated with an M-matrix,which arises in the study of noisy Wiener-Hopf problems for the Markov chain.The solution of practical interest is the M-matrix solution.By a simple transformation,this quadratic matrix equation is transformed into an M-matrix algebraic Riccati equation.We propose a new iteration method for this equation and then give the convergence analysis of it.Numerical experiments are given to show that the new iteration method is feasible and effective than some existing methods in some cases.

    Key words: Quadratic matrix equation; M-matrix; Algebraic Riccati equation; Iteration method

    1.Introduction

    In this paper,we consider a quadratic matrix equation (QME)

    where E,F(xiàn) ∈Rn×n,E is a diagonal matrix and F is an M-matrix.The study of equation(1.1) is motivated by noisy Wiener-Hopf problems for Markov chains.See [5,8] for more background details.

    Under some conditions,it was proved in [5] that the equation (1.1) has an M-matrix solution,which is of practical interest.In addition,by transforming it into an equivalent M-matrix algebraic Riccati equation,a fixed-point iteration method and Newton method have been developed for solving the QME in [5].However,the fixed-point iteration method converges too slowly while the Newton method is too expensive at each iteration.So they are not very efficient for this problem.In this paper,our main aim is to develop an efficient method for solving the QME (1.1).

    In the following,we first review some basic results of M-matrix and M-matrix algebraic Riccati equation.

    Let A = (aij) ∈Rn×n.If aij≤0 for all ij,then A is called a Z-matrix.A Z-matrix A is called an M-matrix if there exists a nonnegative matrix B such that A = sI -B and s ≥ρ(B) where ρ(B) is the spectral radius of B.In particular,A is called a nonsingular M-matrix if s >ρ(B) and singular M-matrix if s=ρ(B).

    The following lemmas can be found in [1,11].

    Lemma 1.1Let A ∈Rn×nbe a Z-matrix.Then the following statements are equivalent:

    1) A is a nonsingular M-matrix;

    2) A-1≥0;

    3) Av >0 for some vectors v >0;

    4) All eigenvalues of A have positive real part.

    Lemma 1.2Let A,B be Z-matrices.If A is a nonsingular M-matrix and A ≤B,then B is also a nonsingular M-matrix.In particular,for any nonnegative real number α,B =αI +A is a nonsingular M-matrix.

    Lemma 1.3Let A be an M-matrix,B ≥A be a Z-matrix.If A is nonsingular or irreducible singular with AB,then B is also a nonsingular M-matrix.

    Lemma 1.4Let A,B be nonsingular M-matrices and A ≤B,then A-1≥B-1.

    M-matrix algebraic Riccati equation is of the form

    where A,B,C and D are real matrices of sizes m×m,m×n,n×m and n×n respectively.MARE appears in many branches of applied mathematics,such as transport theory,Markov chains,stochastic process,and so on.See [2-3] and the references therein for details.For the MARE (1.2),the solution of practical interest is its minimal nonnegative solution.The following basic result is obtained in [3-4].

    Lemma 1.5For the MARE (1.2),if

    is a nonsingular M-matrix or an irreducible singular M-matrix,then (1.2) has a minimal nonnegative solution S.If K is a nonsingular M-matrix,then A-SC and D-CS are also nonsingular M-matrices.If K is irreducible M-matrix,then S >0 and A-SC and D-CS are also irreducible M-matrices.

    When K is an irreducible singular M-matrix,there exist unique,up to a multiplicative constant,u >0 and v >0 such that uTK =0,Kv =0 and uTv =1.Partition the vectors u and v according to the block structure of the matrix M asLetwe have the following result.[3]

    Lemma 1.6If K is an irreducible singular M-matrix and S is the minimal nonnegative solution of the MARE (1.2).Then

    (i) when μ<0,D-CS is singular and A-SC is nonsingular;

    (ii) when μ>0,D-CS is nonsingular and A-SC is singular;

    (iii) when μ=0,both D-CS and A-SC are singular.

    Efficient methods for solving the MARE (1.2) include the Schur method,the fixed-point iteration,the Newton iteration,the doubling algorithms and etc.[2-3,6-7,9-10]

    2.A New Iteration Method

    In this section,we first briefly introduce the fixed-point iteration method in[5]for solving the QME (1.1),and then propose a new iteration method for solving (1.1).

    By introducing X =αI -Y,the equation (1.1) can be transformed into

    It was proved in [5] that when α satisty

    α2I-αE-F is a nonnegative matrix.In addition,if F is a nonsingular M-matrix,then

    is a nonsingular M-matrix,and if F is an irreducible singular M-matrix,then K is an irreducible singular M-matrix.

    By the above analysis and the theory of MARE,the following results were obtained in[5].

    Theorem 2.1If F is a nonsingular M-matrix,then (1.1) has exactly one M-matrix as its solution and the M-matrix is nonsingular.If F is an irreducible singular M-matrix,then(1.1) has M-matrix solutions and all elements of each M-matrix solution are nonzero.In addition,let u,v be positive vectors such that Fv =0 and uTF =0,then

    1) if uTEv ≤0,then (1.1) has exactly one M-matrix as its solution and the M-matrix is singular.

    2) if uTEv >0,then (1.1) has exactly one nonsingular M-matrix as its solution but may also have singular M-matrices as its solutions.

    For solving the QME(1.1),a fixed-point iteration method was proposed in[5]as follows.

    Fixed-point iteration method:

    Convergence analysis showed that the sequence {Yk} in (2.4) is monotonically increasingly and converges to the minimal nonnegative solution of the equation (2.1).In addition,the convergence rate of the fixed-point iteration method is linear for the noncritical case,and is sublinear for the critical case.However,experiments in [5] showed that the fixed-point iteration needs a large numbers of iterations to converge,though at each iteration it is very cheap.

    In the following,we will propose a new iteration method for computing the minimal nonnegative solution of the equation (2.1).

    First,write the equation (2.1) as

    (2αI -E-Y)Y =α2I-αE-F.

    Then we can get the following iterations

    (2αI -E-Yk)Yk+1=α2I-αE-F.

    Thus the new iteration method can be stated as follows.

    New iteration method:

    Compared with the fixed-point iteration method (2.4),the new iteration method (2.5) is a little expensive,since at each iteration a matrix inverse is required to compute.However,the new iteration method may need less iterations than the fixed-point iteration method,which will be confirmed by the numerical experiments.So it is feasible.

    3.Convergence Analysis

    In the following,we give convergence analysis of the new iteration method (2.5).

    The following lemma can be concluded from Theorem 2.1,Lemma 1.5 and Lemma 1.6.

    Lemma 3.1Let the parameter α satisfy (2.2).If F in (1.1) is a nonsingular Mmatrix,then the equation (2.1) has a unique minimal nonnegative solution Sα,and αI-Sα,αI-E-Sαare nonsingular M-matrices.If F in(1.1)is an irreducible singular M-matrix,then the equation(2.1)has a unique minimal nonnegative solution Sα,and αI-Sα,αI-E-Sαare irreducible M-matrices.In addition,let u,v be positive vectors such that Fv =0,uTF =0,then

    1) If uTEv =0,then both αI-Sαand αI -E-Sαare singular M-matrices;

    2) If uTEv <0,then αI -Sαis nonsingular and αI -E-Sαis singular;

    3) If uTEv >0,then αI -Sαis singular and αI -E-Sαis nonsingular.

    ProofWhen F is a nonsingular M-matrix or an irreducible singular M-matrix,equation(2.1) has a unique minimal nonnegative solution Sαby Lemma 1.5.When F is nonsingular,then both αI -Sαand αI -E-Sαare nonsingular M-matrices.

    When F is irreducible singular,take= (uT(αI -E),uT)Tand= (vT,αvT)T,then we haveTK =0,K=0 and μ=-uTEv.By Lemma 1.6,the conclusion follows.

    Theorem 3.1Let F in (1.1) be a nonsingular M-matrix or an irreducible singular M-matrix and the parameter α satisfy (2.2).Then the sequence {Yk} generated by (2.5) is well defined,converges to Sα,and satisfy

    where Sαis the unique minimal nonnegative solution of the equation (2.1).

    ProofWe first prove (3.1) by induction.

    When k = 0,we have Y1= (2αI -E)-1(α2I -αE -F).Since α satisfies (2.2),we have 2αI >E and α2I - αE - F ≥0.It is clear that 0 = Y0≤Y1.Since Sαis the minimal nonnegative solution of (2.1),it holds that (2αI -E-Sα)Sα=α2I -αE-F.By Lemma 3.1,we know αI-E-Sαis an M-matrix,and hence,by Lemma 1.2,2αI-E-Sαis a nonsingular M-matrix.Thus Sα= (2αI - E - Sα)-1(α2I - αE - F).It is evident(2αI-E)-1≤(2αI -E-Sα)-1,thus Y1≤Sα.

    Suppose that the assertions (3.1) hold for k =l-1,i.e.0 ≤Yl-1≤Yl,Yl≤Sα.

    Since Yl≤Sα,we know 2αI -E -Yl≥2αI -E -Sαis a nonsingular M-matrix by Lemma 1.3.Thus Yl+1is well defined.From 0 ≤Yl-1≤Yl≤Sαand Lemma 1.4,we know(2αI-E-Yl-1)-1≤(2αI -E-Yl)-1≤(2αI -E-Sα)-1.We have

    and

    Hence the assertions (3.1) hold for k = l+1.Thus we have proved by induction that the assertions (3.1) hold for all k ≥0.

    Since {Yk} is nonnegative,monotonically increasing and bounded from above,there is a nonnegative matrix Y such that limk→∞Yk=Y.From (3.1) we know Y ≤Sα.On the other hand,taking the limit in(2.5),we know Y is a solution of(2.1),thus Sα≤Y.Hence Sα=Y.

    Theorem 3.2Let F in (1.1) be a nonsingular M-matrix or an irreducible singular M-matrix and the parameter α satisfy (2.2).Then the convergence rate of (2.5) is given by

    where λ=ρ(Sα) and δ is the minimum eigenvalue of the M-matrix αI -E-Sα.

    ProofWe have

    Hence

    Taking limit on both side and noting thatwe have

    Since αI -E-Sαis an M-matrix and Sαis a nonnegative matrix,we can easily verify that

    where δ is the minimum nonnegative eigenvalue of αI-E-Sαand λ is the Perron eigenvalue of Sα.Thus the conclusion (3.2) holds.

    Corollary 3.1If F in (1.1) is a nonsingular M-matrix,then the convergence rate of(2.5)is linear.If F is an irreducible singular M-matrix,then for the cases 2)and 3)in Lemma 3.1,the convergence rate of (2.5) is linear; for the case 1) the convergence rate of (2.5) is sublinear.

    ProofWhen F is a nonsingular M-matrix,αI -Sαand αI -E-Sαare nonsingular M-matrices.Hence α >λ and δ >0.Thus<1,the convergence rate of (2.5) is linear.

    If F is an irreducible singular M-matrix,and the cases 2) and 3) happen,then one of αI-Sαand αI-E-Sαis a nonsingular M-matrix.We have<1,the convergence rate of (2.5) in this case is still linear.

    If F is an irreducible singular M-matrix and the case 1) happens,then αI -Sαand αI -E -Sαare both singular M-matrices.Thus we have α = ρ(Sα) and δ = 0.Hence=1,the convergence rate of (2.5) is sublinear in this case.

    Corollary 3.2The optimal parameter of the new iteration method (2.5) is given by

    ProofLet α1>0,α2>0 be two parameters that satisfy (2.2) and α1>α2.Then the convergence factor of the new iteration method (2.5) are respectively

    where λ1=ρ(Sα1),λ2=ρ(Sα2) and Sα1,Sα2are the minimal nonnegative solutions of (2.1)respectively.Since α1I-S1=α2I-S2,we have S1=(α1-α2)I+S2and λ1=(α1-α2)+λ2.Thus

    and the conclusion follows.

    4.Numerical Experiments

    In this section,numerical experiments are given to show the feasibility and effectiveness of the new iteration method.We compare the new iteration method (NM) with the fixedpoint iteration method(FP)in[5],and present numerical results of each experiment in terms of iteration numbers (IT),CPU time (CPU) in seconds and residue (Res),where the residue is defined to be

    In our implementations,all iterations are run in MATLAB (R2012a) on a personal computer and are terminated when the current iterate satisfies

    Example 4.1[5]Consider the equation (1.1) with

    The numerical result is summarized in Tab.4.1.

    Tab.4.1 Numerical Results of Example 4.1

    Example 4.2[5]Consider the equation (1.1) with

    where we take a=2,b=1.The numerical result is summarized in Tab.4.2.

    Tab.4.2 Numerical Results of Example 4.2

    Example 4.3Consider the equation (1.1) with coefficient matrices defined as

    E =diag(1:n); F =rand(n,n); F =diag(Fe)-F,

    where e=(1,1,··· ,1)T.For different sizes of n,we list the numerical results in Tab.4.3.

    Tab.4.3 Numerical Results of Example 4.3

    From Tabs.4.1-4.3,we can conclude that the new iteration method needs less iteration number and CPU time than the fixed-point iteration method.Hence it is feasible and effective.

    亚洲精品乱码久久久v下载方式| xxx大片免费视频| 精品99又大又爽又粗少妇毛片| 99热网站在线观看| 亚洲经典国产精华液单| 下体分泌物呈黄色| 国产黄片视频在线免费观看| 久久人人爽人人爽人人片va| 亚洲综合色惰| 男的添女的下面高潮视频| 久久毛片免费看一区二区三区| 日本黄色片子视频| 少妇猛男粗大的猛烈进出视频| 十分钟在线观看高清视频www | 丝袜在线中文字幕| 欧美激情国产日韩精品一区| 熟女人妻精品中文字幕| 大片免费播放器 马上看| 欧美性感艳星| 99久久中文字幕三级久久日本| 免费观看在线日韩| 国产欧美日韩精品一区二区| 亚洲国产精品成人久久小说| 国产真实伦视频高清在线观看| 在线亚洲精品国产二区图片欧美 | 99久久人妻综合| 午夜福利视频精品| 女人精品久久久久毛片| 高清视频免费观看一区二区| 国产亚洲av片在线观看秒播厂| 亚洲精品久久午夜乱码| 一级毛片 在线播放| 啦啦啦在线观看免费高清www| av又黄又爽大尺度在线免费看| 精品一区二区免费观看| 多毛熟女@视频| 91在线精品国自产拍蜜月| 国产亚洲一区二区精品| 国产成人精品福利久久| 2022亚洲国产成人精品| 久久久久视频综合| 国产亚洲最大av| 免费观看无遮挡的男女| 18禁在线无遮挡免费观看视频| 国产精品嫩草影院av在线观看| 日本欧美国产在线视频| 欧美成人精品欧美一级黄| 国产成人精品一,二区| 欧美老熟妇乱子伦牲交| 人人妻人人澡人人爽人人夜夜| 人妻夜夜爽99麻豆av| 亚洲美女黄色视频免费看| 极品人妻少妇av视频| a级毛片免费高清观看在线播放| 久久 成人 亚洲| 女性被躁到高潮视频| 一级av片app| 一级二级三级毛片免费看| 亚洲av电影在线观看一区二区三区| 国产成人aa在线观看| 国产亚洲精品久久久com| 亚洲高清免费不卡视频| 日本猛色少妇xxxxx猛交久久| 国产高清三级在线| 午夜影院在线不卡| 丝袜脚勾引网站| 久久午夜综合久久蜜桃| 少妇的逼好多水| 3wmmmm亚洲av在线观看| 99热国产这里只有精品6| 国产成人精品无人区| 成人综合一区亚洲| 又爽又黄a免费视频| 少妇丰满av| 夫妻性生交免费视频一级片| 久热久热在线精品观看| 在现免费观看毛片| 免费久久久久久久精品成人欧美视频 | 五月开心婷婷网| tube8黄色片| 国产精品99久久99久久久不卡 | 国产精品福利在线免费观看| 免费看光身美女| 91精品一卡2卡3卡4卡| 黄色视频在线播放观看不卡| 久久人人爽av亚洲精品天堂| 国产亚洲欧美精品永久| 日本wwww免费看| 22中文网久久字幕| 少妇精品久久久久久久| 亚洲人与动物交配视频| 2021少妇久久久久久久久久久| 久久午夜福利片| www.色视频.com| 亚洲精品日韩在线中文字幕| 亚洲成人av在线免费| 久久精品国产亚洲av天美| 亚洲国产精品成人久久小说| 久久99精品国语久久久| 久久精品国产a三级三级三级| 免费不卡的大黄色大毛片视频在线观看| 蜜臀久久99精品久久宅男| 在线观看人妻少妇| 国产淫片久久久久久久久| 午夜av观看不卡| 亚洲成人手机| 精品久久久久久久久av| 两个人的视频大全免费| 日韩视频在线欧美| 国产精品久久久久久久久免| 最近最新中文字幕免费大全7| 成人毛片a级毛片在线播放| 看十八女毛片水多多多| 久久久久久久国产电影| 国产精品熟女久久久久浪| 五月玫瑰六月丁香| 国产女主播在线喷水免费视频网站| 久久av网站| 日韩欧美精品免费久久| 久久久久久久久大av| 国产免费福利视频在线观看| 赤兔流量卡办理| 韩国高清视频一区二区三区| 在线观看免费高清a一片| 久久精品国产自在天天线| 日日啪夜夜撸| 插逼视频在线观看| 午夜视频国产福利| 人妻 亚洲 视频| 赤兔流量卡办理| 亚洲在久久综合| 五月开心婷婷网| 国产免费福利视频在线观看| 一级毛片黄色毛片免费观看视频| 日本免费在线观看一区| 99视频精品全部免费 在线| 在线观看免费视频网站a站| 不卡视频在线观看欧美| 欧美老熟妇乱子伦牲交| 成人亚洲欧美一区二区av| 美女中出高潮动态图| 春色校园在线视频观看| 丰满少妇做爰视频| 18禁裸乳无遮挡动漫免费视频| 男女啪啪激烈高潮av片| 午夜视频国产福利| 九九爱精品视频在线观看| 在线免费观看不下载黄p国产| 丰满少妇做爰视频| 国产午夜精品久久久久久一区二区三区| 久久亚洲国产成人精品v| 黑人巨大精品欧美一区二区蜜桃 | 国产在线男女| 高清欧美精品videossex| 精品久久国产蜜桃| 天天躁夜夜躁狠狠久久av| av福利片在线观看| 日本欧美视频一区| 中文在线观看免费www的网站| 欧美精品亚洲一区二区| 高清av免费在线| 人妻 亚洲 视频| 乱人伦中国视频| 一级毛片我不卡| 欧美国产精品一级二级三级 | 纵有疾风起免费观看全集完整版| 国产欧美日韩精品一区二区| 亚洲不卡免费看| 亚洲国产最新在线播放| 男女国产视频网站| 美女主播在线视频| 三级国产精品欧美在线观看| 午夜福利,免费看| 亚洲高清免费不卡视频| 国产精品人妻久久久影院| 国产黄色免费在线视频| 两个人免费观看高清视频 | 亚洲av福利一区| 亚洲不卡免费看| 免费黄色在线免费观看| 日韩伦理黄色片| 你懂的网址亚洲精品在线观看| 国产淫语在线视频| 国产成人aa在线观看| 一二三四中文在线观看免费高清| 精品国产乱码久久久久久小说| 日本91视频免费播放| 少妇熟女欧美另类| 免费av中文字幕在线| 少妇的逼好多水| 人妻夜夜爽99麻豆av| 一个人看视频在线观看www免费| 亚洲美女搞黄在线观看| 欧美精品亚洲一区二区| 黑人高潮一二区| 免费av中文字幕在线| av国产久精品久网站免费入址| 国产视频内射| 国产爽快片一区二区三区| 黄色欧美视频在线观看| 97超碰精品成人国产| 看十八女毛片水多多多| 亚洲精品日韩在线中文字幕| 五月玫瑰六月丁香| 国产精品免费大片| 久久ye,这里只有精品| 日日摸夜夜添夜夜爱| 在线观看免费高清a一片| 亚洲综合精品二区| 99热6这里只有精品| 精品亚洲成a人片在线观看| h日本视频在线播放| 欧美日韩在线观看h| 久久综合国产亚洲精品| 性色av一级| 美女视频免费永久观看网站| 如何舔出高潮| 纯流量卡能插随身wifi吗| 91精品一卡2卡3卡4卡| 交换朋友夫妻互换小说| 天天躁夜夜躁狠狠久久av| 99久久精品热视频| 日本av手机在线免费观看| 美女视频免费永久观看网站| 少妇精品久久久久久久| 性高湖久久久久久久久免费观看| 视频区图区小说| 少妇裸体淫交视频免费看高清| 麻豆乱淫一区二区| 亚洲无线观看免费| 日本色播在线视频| 男女边吃奶边做爰视频| a级毛色黄片| 日韩成人伦理影院| 日日啪夜夜撸| 日日撸夜夜添| 午夜免费鲁丝| 男人和女人高潮做爰伦理| 国产乱人偷精品视频| 人人妻人人看人人澡| 人妻夜夜爽99麻豆av| 欧美bdsm另类| 亚洲怡红院男人天堂| 亚洲一级一片aⅴ在线观看| 在线观看国产h片| 午夜免费鲁丝| 国产成人a∨麻豆精品| 国产亚洲精品久久久com| 国产乱来视频区| av专区在线播放| 精品酒店卫生间| 亚洲美女视频黄频| 日韩不卡一区二区三区视频在线| 啦啦啦在线观看免费高清www| 日日摸夜夜添夜夜爱| 老司机影院成人| 欧美日韩亚洲高清精品| h视频一区二区三区| 久久毛片免费看一区二区三区| 久久鲁丝午夜福利片| 国产精品久久久久久久电影| 男女免费视频国产| 久久久亚洲精品成人影院| 又大又黄又爽视频免费| 国产极品粉嫩免费观看在线 | 亚洲国产av新网站| 国产精品免费大片| 人人妻人人添人人爽欧美一区卜| 亚洲av免费高清在线观看| 国产无遮挡羞羞视频在线观看| 免费黄频网站在线观看国产| 亚洲欧美成人精品一区二区| 欧美精品人与动牲交sv欧美| 免费高清在线观看视频在线观看| 国产国拍精品亚洲av在线观看| 女性被躁到高潮视频| 熟妇人妻不卡中文字幕| 亚洲精品国产av蜜桃| 精品亚洲成a人片在线观看| 女性生殖器流出的白浆| 国产成人精品无人区| 在线观看免费日韩欧美大片 | 一级,二级,三级黄色视频| 欧美精品国产亚洲| 亚洲情色 制服丝袜| 最后的刺客免费高清国语| 国产亚洲av片在线观看秒播厂| 久久久久国产网址| 国产永久视频网站| 亚洲国产av新网站| 国产av国产精品国产| 老司机影院成人| 久久久国产欧美日韩av| 男女免费视频国产| 日韩精品有码人妻一区| 亚洲av男天堂| 日日摸夜夜添夜夜爱| 日韩亚洲欧美综合| 欧美成人午夜免费资源| 中文字幕人妻丝袜制服| 亚洲av男天堂| 国产精品久久久久久av不卡| 丝袜脚勾引网站| 中文字幕免费在线视频6| 人妻 亚洲 视频| 精品国产一区二区三区久久久樱花| 日韩av在线免费看完整版不卡| 国产男女内射视频| 乱系列少妇在线播放| 中文字幕免费在线视频6| 精品人妻熟女毛片av久久网站| 国产黄色免费在线视频| 天美传媒精品一区二区| 亚洲精品日本国产第一区| 在线观看人妻少妇| 久久人妻熟女aⅴ| 国产精品三级大全| 99久久精品国产国产毛片| 黑人巨大精品欧美一区二区蜜桃 | 日韩精品有码人妻一区| 美女cb高潮喷水在线观看| 一级毛片aaaaaa免费看小| 两个人的视频大全免费| 少妇人妻久久综合中文| videossex国产| 亚洲精品456在线播放app| 国产探花极品一区二区| 91午夜精品亚洲一区二区三区| 国国产精品蜜臀av免费| 在线 av 中文字幕| 成年女人在线观看亚洲视频| 中文字幕人妻丝袜制服| av又黄又爽大尺度在线免费看| xxx大片免费视频| 日韩免费高清中文字幕av| 欧美精品高潮呻吟av久久| 国产免费福利视频在线观看| 自拍欧美九色日韩亚洲蝌蚪91 | 成人综合一区亚洲| 国产免费福利视频在线观看| 在线观看三级黄色| 久久国产亚洲av麻豆专区| av在线app专区| 中文字幕人妻丝袜制服| av一本久久久久| 中文字幕免费在线视频6| 一级毛片黄色毛片免费观看视频| 九色成人免费人妻av| 妹子高潮喷水视频| 午夜免费观看性视频| 天天操日日干夜夜撸| 2022亚洲国产成人精品| 国产精品国产三级国产av玫瑰| 中文字幕精品免费在线观看视频 | 纵有疾风起免费观看全集完整版| 丰满人妻一区二区三区视频av| 水蜜桃什么品种好| 国产免费又黄又爽又色| 精品熟女少妇av免费看| 777米奇影视久久| 在线 av 中文字幕| 午夜影院在线不卡| 纵有疾风起免费观看全集完整版| 激情五月婷婷亚洲| 另类精品久久| 亚洲国产日韩一区二区| 2022亚洲国产成人精品| 久久午夜综合久久蜜桃| 欧美日韩av久久| 亚洲美女视频黄频| 日韩成人av中文字幕在线观看| 亚洲av在线观看美女高潮| 久久久亚洲精品成人影院| 国精品久久久久久国模美| 99久久人妻综合| 肉色欧美久久久久久久蜜桃| 2018国产大陆天天弄谢| 亚洲成人一二三区av| 亚洲,欧美,日韩| www.色视频.com| 亚洲人与动物交配视频| 97超碰精品成人国产| 日本午夜av视频| 国产在视频线精品| 久久精品国产a三级三级三级| 制服丝袜香蕉在线| 天美传媒精品一区二区| av在线app专区| 日本色播在线视频| av女优亚洲男人天堂| 精品人妻熟女毛片av久久网站| 成人国产av品久久久| 51国产日韩欧美| 国产成人freesex在线| 色吧在线观看| 狂野欧美白嫩少妇大欣赏| av视频免费观看在线观看| 久久久久精品久久久久真实原创| 国产精品久久久久成人av| 国产成人freesex在线| 亚洲内射少妇av| 熟妇人妻不卡中文字幕| 欧美亚洲 丝袜 人妻 在线| 亚洲,一卡二卡三卡| 欧美97在线视频| 少妇人妻一区二区三区视频| 夜夜骑夜夜射夜夜干| 国产精品不卡视频一区二区| 日韩不卡一区二区三区视频在线| 日韩欧美精品免费久久| 午夜福利在线观看免费完整高清在| 色婷婷久久久亚洲欧美| 寂寞人妻少妇视频99o| 成年人午夜在线观看视频| 久久热精品热| 成人二区视频| 精品一区二区免费观看| 晚上一个人看的免费电影| 纵有疾风起免费观看全集完整版| 夫妻午夜视频| 久久久久久久久久久丰满| 高清黄色对白视频在线免费看 | 免费人成在线观看视频色| 如日韩欧美国产精品一区二区三区 | 国产成人精品福利久久| 国产精品三级大全| 精品亚洲乱码少妇综合久久| av在线观看视频网站免费| 插阴视频在线观看视频| 久久亚洲国产成人精品v| 色吧在线观看| 如何舔出高潮| 男人舔奶头视频| 日韩免费高清中文字幕av| 麻豆成人av视频| 日韩三级伦理在线观看| 久久久久久久国产电影| 只有这里有精品99| 人妻少妇偷人精品九色| 日韩人妻高清精品专区| 婷婷色av中文字幕| 草草在线视频免费看| 少妇精品久久久久久久| 伦理电影免费视频| 亚洲在久久综合| 美女福利国产在线| 国产成人精品无人区| 少妇熟女欧美另类| 亚洲精品日韩av片在线观看| videos熟女内射| 一级片'在线观看视频| 亚洲精华国产精华液的使用体验| 久久 成人 亚洲| 99视频精品全部免费 在线| 中文乱码字字幕精品一区二区三区| a级毛片在线看网站| 青春草国产在线视频| 国产欧美日韩综合在线一区二区 | 街头女战士在线观看网站| 国产 一区精品| av又黄又爽大尺度在线免费看| 国产欧美日韩一区二区三区在线 | 少妇猛男粗大的猛烈进出视频| 精品国产露脸久久av麻豆| 下体分泌物呈黄色| av免费观看日本| 超碰97精品在线观看| 亚洲精品,欧美精品| 97精品久久久久久久久久精品| 波野结衣二区三区在线| 久久综合国产亚洲精品| 免费人成在线观看视频色| 国产伦精品一区二区三区视频9| 亚洲国产欧美日韩在线播放 | 欧美精品亚洲一区二区| 国产成人精品久久久久久| 王馨瑶露胸无遮挡在线观看| 久久鲁丝午夜福利片| 综合色丁香网| 久久av网站| av天堂久久9| 18+在线观看网站| 国产精品人妻久久久影院| videos熟女内射| 亚洲国产av新网站| 久久久久久久大尺度免费视频| 久久久亚洲精品成人影院| 国产片特级美女逼逼视频| 国产成人精品久久久久久| 2021少妇久久久久久久久久久| 在线天堂最新版资源| 最后的刺客免费高清国语| 99视频精品全部免费 在线| 欧美日韩国产mv在线观看视频| 亚洲成色77777| 久久97久久精品| 中文字幕人妻丝袜制服| 下体分泌物呈黄色| 少妇被粗大猛烈的视频| 黄色欧美视频在线观看| 综合色丁香网| 日本-黄色视频高清免费观看| 韩国av在线不卡| 一区二区av电影网| 成人午夜精彩视频在线观看| 国产av国产精品国产| 免费观看av网站的网址| 欧美丝袜亚洲另类| 丰满人妻一区二区三区视频av| 国产精品人妻久久久久久| 国产精品久久久久久av不卡| 精品人妻偷拍中文字幕| 国产精品蜜桃在线观看| 啦啦啦在线观看免费高清www| 欧美激情国产日韩精品一区| 日韩免费高清中文字幕av| 日日啪夜夜撸| 极品少妇高潮喷水抽搐| 国产精品一区二区在线不卡| 黑丝袜美女国产一区| 国产精品一区www在线观看| 精品久久久久久久久亚洲| 一级爰片在线观看| 日本av手机在线免费观看| 亚洲第一区二区三区不卡| 国产精品一二三区在线看| 菩萨蛮人人尽说江南好唐韦庄| 两个人免费观看高清视频 | 免费观看在线日韩| 国产日韩欧美视频二区| 日日撸夜夜添| 日韩在线高清观看一区二区三区| 观看美女的网站| 国产精品一区二区三区四区免费观看| 亚洲国产最新在线播放| freevideosex欧美| videos熟女内射| 9色porny在线观看| 日本-黄色视频高清免费观看| 欧美3d第一页| 欧美精品国产亚洲| 国产精品一区二区三区四区免费观看| 欧美日韩av久久| 亚洲欧美成人综合另类久久久| 日韩欧美一区视频在线观看 | 亚洲av日韩在线播放| 亚洲国产欧美日韩在线播放 | 久久精品国产亚洲av涩爱| 日本欧美国产在线视频| 精品久久久久久久久亚洲| 久久精品熟女亚洲av麻豆精品| 国产高清有码在线观看视频| 我的老师免费观看完整版| 晚上一个人看的免费电影| 国产精品免费大片| 黄色日韩在线| 天堂俺去俺来也www色官网| 久久久久视频综合| 99re6热这里在线精品视频| 日韩亚洲欧美综合| 日本午夜av视频| 精品一品国产午夜福利视频| 国产免费视频播放在线视频| 亚洲精品亚洲一区二区| 在线观看人妻少妇| 中国国产av一级| 天堂中文最新版在线下载| 丰满饥渴人妻一区二区三| 一级毛片 在线播放| 欧美激情国产日韩精品一区| av不卡在线播放| 日韩伦理黄色片| 久久久欧美国产精品| 能在线免费看毛片的网站| 国产精品无大码| 久久精品熟女亚洲av麻豆精品| av卡一久久| 免费人妻精品一区二区三区视频| 丝袜喷水一区| 久久97久久精品| 日韩av在线免费看完整版不卡| 新久久久久国产一级毛片| 观看美女的网站| 中文天堂在线官网| 乱人伦中国视频| 久久人人爽av亚洲精品天堂| 日本黄大片高清| 欧美成人午夜免费资源| 高清av免费在线| www.av在线官网国产| 国语对白做爰xxxⅹ性视频网站| 国产av码专区亚洲av| 狂野欧美白嫩少妇大欣赏| 婷婷色综合大香蕉| 一级爰片在线观看| 久久影院123| 三级经典国产精品| 在线 av 中文字幕| 国产精品偷伦视频观看了| 成年av动漫网址| 亚洲av福利一区| 欧美+日韩+精品| 国语对白做爰xxxⅹ性视频网站| 一级,二级,三级黄色视频| 晚上一个人看的免费电影| 18禁裸乳无遮挡动漫免费视频| av视频免费观看在线观看| av国产精品久久久久影院| 肉色欧美久久久久久久蜜桃| 水蜜桃什么品种好| 亚洲熟女精品中文字幕| 十八禁高潮呻吟视频 | 美女中出高潮动态图| 美女内射精品一级片tv| 亚洲欧美精品专区久久|