Lecture 11 3 Numerical Integration: The Big Picture Virtually all numerical integration methods rely on the following procedure: Start from N+1 data points (x i,f i), i = 0,,N, or sample a specified function f(x) at N+1 x i values to generate the data set Fit the data set to a polynomial, either locally (piecewise) or globally Analytically integrate the polynomial to deduce an . The purpose of the book is to present the principles and concepts of numerical methods as relevant to student learning. For function of a single variable, we seek an approximation to the area "under" the curve: The following theorem gives a worst case scenario. /Filter[/CCITTFaxDecode] There are currently no reviews for "Methods of Numerical Integration", Copyright 2022 Elsevier, except certain content provided by third parties, Cookies are used by this site. Integration: Midpoint and Simpson's Rules 87 Lecture 23. <> (PDF) Numerical Differentiation and Integration Numerical Differentiation and Integration Authors: Dragan Obradovic Lakshmi Narayan Mishra Vellore Institute of Technology, Vellore 632 014,. all the numerical are solved clearly in the easier way possible. (4.2) The succeeding chapters present the approximate integration rules and formulas over finite and infinite intervals. Useful to programmers and stimulating for theoreticians, this text covers the major methods of numerical integration. /Filter[/FlateDecode] % These topics are followed by a review of error analysis and estimation, as well as the application of functional analysis to numerical integration. Download these Free Numerical Methods MCQ Quiz Pdf and prepare for your upcoming exams Like Banking, SSC, Railway, UPSC, State PSC. The final chapter looks into the goals and processes of automatic integration, with particular attention to the application of Tschebyscheff polynomials. 3. We also have this interactive book online for a better learning experience. I = Z b a f(x)dx Z b a fn(x)dx where fn(x) = a0 +a1x+a2x2 +:::+anxn. Numerical Integration What is integration? (approximately) the value of f(x): take the Taylor polynomial of degree nfor fcentered at x 0 and evaluate it at x. (PDF) Numerical Integration Methods Numerical Integration Methods October 1969 Authors: Miller Edmund Los Alamos National Laboratory GJ BURKE Abstract The numerical evaluation of an. x\ms6~{M;{uk5-%\ rs"bwoUh;:njqS_F=_7rU5o\zO^(.>aQ %PDF-1.5 Easy - Download and start reading immediately. The author was told that, in the old days . (5.1.12) The rapidly varying nature of this solution will provide a formidable test of any integration scheme particularly if the step size is large. Methods of Numerical Integration, Second Edition describes the theoretical and practical aspects of major methods of numerical integration. It offers a balanced presentation: certain sections derive from or allude to deep results of analysis, but most of the final results are expressed in a form accessible to anyone with a background in calculus. Consequently, numerical integration has become an indispensable tool for processing sophisticated engineering designs. Integratio n. Prepared by: Engr. <xn are distinct nodes and w 1,w 2,.,wn the corresponding weights.Often (but not always) all nodes lie in [a,b].The weights wi are usually determined so that the formula (5.1.3) is exact for poly- nomials of as high degree as possible. not indeterminate. f (x) is called the integrand, a = lower limit of integration . We are always looking for ways to improve customer experience on Elsevier.com. Geometric Numerical Integration. Le&+~&~Go"Amv}v~9Uy' >0p+j @]nH*.iZyUz$ohp{XIl$\@ 5s@H0X/ADT5| V["xf+aHL?I-kn*If4W;:{|7&WK+Pc:RZ(LN]i}csHo8Z Fig (1.1) Numerical integration consists of finding . In particular,if the value of y(x 0) is given for some point x 0, set a = x 0. Xy,RE"xtqI]O>?>9z:$D,2# Numerical Integration Integration is the second fundamental concept of calculus (along with differentiation). s Show the differential element you are using. stream We prove that accuracy of any method of weak order p is estimated by $\\varepsilon +O(h^{p . Despite the power of this theorem, there are still situations where we must approximate the value of the definite integral instead of finding its exact value. We will see two methods that work reasonably well . Lanczos iterations), ap-proximation theory, and fast Fourier transform algorithms (FFTs, which themselves encompose a host of group theory, number theory, polynomial algebras, and other fas-cinating topics). C. Lubich. Book Title: Numerical Integration. In numerical integration what is required is the projection of information known at a particular value of the independent variable, say time, forward to a new or future value of that independent variable. Integration: Left, Right and Trapezoid Rules 82 Lecture 22. Thanks in advance for your time. Very often, it is not possible to solve integration analytically, for example when the data consists of a number of distinct measurements, or when the antiderivative is not known . Excel Integration using VBA de nite integration of highly complex physical models or discrete samples of random data. E. Faou, E. Hairer, +1 author. In numerical analysis, numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations. Trapezoidal Rule of Integration . /BitsPerComponent 1 Simpson's 3/8 rule is . endstream Write out Simpson's equation in generalized form. Get Numerical Methods Multiple Choice Questions (MCQ Quiz) with answers and detailed solutions. The authors give a unique overview of the topic by bringing into line many recent research results not yet presented coherently; the extensive bibliography lists 268 items. Numerical Integration Newton-Cotes Integration Formulas By: Mohammed Abdulqader 9 10. }oIzBJ[\Yq2J')yRX9}3`+xGx(y~&$. Open - Buy once, receive and download all available eBook formats, including PDF, EPUB, and Mobi (for Kindle). We have now seen some of the most generally useful methods for discovering antiderivatives, and there are others. But beware that, when ngrows, a Taylor series converges rapidly near the point of expansion but slowly (or not at all) at more remote points. Numerical. Gaussian Quadrature* 101 Lecture 27. The Yellow House: A Memoir (2019 National Book Award Winner) Sarah M. Broom. This was complemented by the question as to how structure . 1. The author also treats the application of numerical tools; numerical integration and random number generation are explained in a unified manner reflecting complementary views of Monte Carlo methods. Published 24 August 2022. Numerical approaches are generally more useful here than in differentiation as the antiderivative procedure (analytically) is often much more complex, or even not possible. Numerical integration techniques are necessary to approximate the integral Numerical Integration 2/11. stream The points x 0,.x n that are used in the quadrature formula are called quadrature points. For higher-dimensional numerical integration (cubature), the story Numerical Integration 5 Theoretically we know that as n , the trapezoidal rule approximation R b a f(x)dx, but that does not help us to know how close we are to the limit if we use n = 100 or n = 1000. This book will be of great value to theoreticians and computer programmers. % The basic problem /Length 1914 2 0 obj Double Integrals for Rectangles 94 Lecture 25. In general, we can cast Picard's method as Thus, the area of the first trapezoid in Figure 2.5.2 is. Numerical integration is the study of how the numerical value of an integral can be found. We would like to ask you for a moment of your time to fill in a short questionnaire, at the end of your visit. 0% 0% found this document not useful, Mark this document as not useful. endobj Embed. We cannot process tax exempt orders online. <> where . /Length 4510 [1] Numerical integration has also been called quadrature. Mathematics. Numerical methods are a type of "trial-and-error" process. Immediately download your eBook while waiting for print delivery. The term "numerical integration" first appears in 1915 in the publication A Course in Interpolation and Numeric Integration for the Mathematical Laboratory by David Gibb.. Quadrature is a historical mathematical term that means calculating area. Using Simpson's rule, Area = h 3 [f (x 0) + 2f (x 1) + 2f (x 3) + f (x 4 )] = 5 3 [0 + 4 * 4 + 2 * 7 + 4 * 9 + 12] = 5 3 [16 + 14 + 36 + 12] = 5 3 * 78 = 130m 2. . The copyright of the book belongs to Elsevier. Following this concept, we discard the approximate trajectories which leave a sufficiently large sphere. Numerical integration is the term used for a number of methods to find an approximation for an integral. If you decide to participate, a new browser tab will open so you can complete the survey after you have completed your visit to this website. CMU 15-462/662, Fall 2015 Think: is the probability that a random measurement of will yield the value Discrete probability distributions . XYmo_qyi M$Dt$3FGJE3/JI!dVdv{F.>.VWTU*j=Qe]oWxYp/JP-d+zedJ3)e:_o&J*W?{BK|22_MdXYM/M)r2M=^5N"f_EVJra2'e."~pvIrEn_W/?~X[oqv!bz~~;{*S&guV5SB nx)7=x3Y[!,lvg-!VRe|G#dBZo\X]ABwp/Ec~ .`YgRX sPLf:bBM"$BFHPmF/IG X|T;Ur(ep ;pO/e k*68u$ro[#yqZ ,V)gc)w:}0>cJoZP}6 c )W81S{@LS,. 0% 0% found this document useful, Mark this document as useful. ysW&,DiR: Oberwolfach Reports. Privacy Policy Cielito V. Maligalig Introduction Find the area bounded by y = x2, from x = 2 to x = 4. xn&}-zgVc;Kcg2m&(ERdxrJxv(H^I&vu{Nn.Rg*[i,rj7y,H Y.caDmYGoBeB4g?ob sj7VCt':8]s\FKw=}h[N|w#c` Pc)V^M/@O:t/nhA3>^}!<8C The Fundamental Theorem of Calculus gives a concrete technique for finding the exact value of a definite integral. 24 October 2022. What is Numerical Integration? /Length 4 /ImageMask true About this book. This item: Methods of Numerical Integration: Second Edition (Dover Books on Mathematics) by Philip J. Davis Paperback $23.99 Numerical Methods for Scientists and Engineers (Dover Books on Mathematics) by R. W. Hamming Paperback $13.79 Numerical Methods for Scientists and Engineers (Dover Books on Mathematics) R. W. Hamming 113 Paperback }/LPO7%j%>zF3+dEM= XdsfaUHww3wx_hQ00@ All in all, this is a good book to have . << View access options below. <> This page titled 1.2: Numerical Integration is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Jeremy Tatum via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit . /Subtype/Image That technique is based on computing antiderivatives. Unfortunately, some functions have no simple antiderivatives; in such cases if the value of a definite integral is needed it will have to be approximated. Write out the generalized calculus equation written in the same symbols you used to label your picture . >> We cannot guarantee that every ebooks is available! t+mPx6sF:2hVG5 Sitemap. Numerical integration (fundamentals) Spring 2020 The point: Techniques for computing integrals are derived, using interpolation and piece- . 4 0 obj By: Mohammed Abdulqader 10 11. endobj ]3{V$4e`-**+9I * A'Po=K~l+1OlV|4^0k@R1 ouCG '5WiMiKs$F>u-g3upU3@ai}fa}Z (. Numerical Integration | PDF 0 ratings 8 views 34 pages Numerical Integration Uploaded by Harris Anches Copyright: Attribution Non-Commercial (BY-NC) Available Formats Download as PPT, PDF, TXT or read online from Scribd Flag for inappropriate content of 34 Dr Muhammad Al-Salamah, Industrial Engineering, KFUPM Topic 6: Numerical Integration Newton-Cotes Integration available in Closed and Open Form FIGURE 4 The difference between (a) closed (b) open integration formula. Fast Download speed and no annoying ads. 1. b = upper limit of integration It is useful for when you want to see how some integral of the experimental data progresses over time. The subject of this workshop was numerical methods that preserve geometric properties of the flow of an ordinary or partial differential equation. Numerical Integration : constitutes a broad family of algorithms for calculating the numerical value of a integral. Sometimes this is a simple problem, since it will be apparent that the function you wish to integrate is a derivative in some straightforward way. Institutional Login For this reason, a wide variety of numerical methods has been developed to simplify the integral. 2. a. Soln: = 0 2 ( 3 x + 1) .dx. cal integration formulas are also referred to as integration rules or quadratures, and hence we can refer to (6.3) as the rectangular rule or the rectangular quadrature. Some of the salient features of the book are: Lucid style Easy to understand methodology Coverage of all course fundamentals Clarity in the presentation of concepts Over 175 fully-solved problems with step-by-step solutions Numerical Methods Lecture Notes PDF and Study Material Free Download Lectures in Basic Computational Numerical Analysis (PDF 168P) This book covers the following topics: Numerical Linear Algebra, Solution of Nonlinear Equations, Approximation Theory, Numerical Solution of ODEs and Numerical Solution of PDEs. Sign in to view your account details and order history. 1 0 obj Particular emphasis is given to the . % The basic idea is to evaluate a func-tion at a few points, nd the polynomial that interpolates the function at these points, and use the derivative or integral of the polynomial as an approximation to the function. Numerical Integration - Advanced Numerical Methods with Matlab 1 - Wiley Online Library Chapter 5 Numerical Integration Book Author (s): Bouchaib Radi Abdelkhalak El Hami First published: 25 March 2018 https://doi.org/10.1002/9781119453611.ch5 PDF Tools Share Get access to this single chapter. PDF | On Feb 4, 2020, Abayomi Samuel Oke published Numerical Integration | Find, read and cite all the research you need on ResearchGate Numerical Analysis I (PDF 47P) This note explains the following topics: Floating-point arithmetic, Error analysis , Solving quadratics, Computational complexity, IEEE Floating-point Standards, Elementary numerical methods, Simultaneous linear equations, Choleski factorization, Non-linear equations, Quadrature and Numerical software. This monograph is a comprehensive treatment of the theoretical and computational aspects of numerical integration. < 20.5 Summary and Problems | Contents | 21.1 Numerical Integration Problem Statement > Mathematicians of Ancient Greece, according to the Pythagorean . Chapter 9: Numerical Integration Numerical Integration Most integrals arising from solutions of problems in engineering and science cannot be represented in "closed form" - they must be evaluated numerically. Plotting Functions of Two Variables 91 Lecture 24. Phil Knight. Pick a convenient value for the lower limit of integration a. Easily read eBooks on smart phones, computers, or any eBook readers, including Kindle. x\KGr0| w9{BWG" R$EYQ=E42#ED5`GCpNf2|PS)zw8)lm[?b0Iw7/)H{w7u8&4 cu&3!.+AzfZ$"Dk?cjLd+}XkwZc{oX;8ev6=9\mRe5:{g1e{0)x0;Mh4 Here, a = 0, b = 20, n = 4, h = 5. Book Subtitle: Proceedings of the Conference Held at the Mathematisches Forschungsinstitut Oberwolfach . numerical algorithms such as numerical linear algebra (e.g. >> trapezoidal rule is introduced, enabling the use of Richardson extrapolation for integration. Numerical Di erentiation 102 . ? ,39YML{iT:t -'G`iIf;F3rlSIl(Oj 11mkOrzyR0HAB3 S J.MOsGV{y"`hO\[8:KC&.?27ujvuk3}9p{h &xUenl:4& p\_EGN&ZnG`ql^c C-a, This book contains six chapters and begins with a discussion of the basic principles and limitations of numerical integration. 5. A formula for the integrand may be known, but it may be difficult or impossible to find an antiderivative . /DecodeParms[<>] . Numerical integration is the study of how the numerical value of an integral can be found. b a f(x)dx If f(x) is a smooth function integrated over a small number of dimensions, and the domain of integration is bounded, there are many methods for approximating the integral to the desired precision. Shoe Dog: A Memoir by the Creator of Nike. stream % <>>> This notebook contains an excerpt from the Python Programming and Numerical Methods - A Guide for Engineers and Scientists, the content is also available at Berkeley Python Numerical Methods. through the programs that go with the book, for numerical experimentation and further development. Z b=x 1 a=x0 f(x)dx = Z x 1 x0 P1(x)dx + 1 2 Z x 1 x0 f00()(x x0)(xx1)dx = Z x 1 x0 xx1 x 0 . /Decode[1 0] 1(UwmUTK/A{6h 6[;^M Numerical Integration 1 The Newton-Cotes Rules 2 Composite Rules 3 Adaptive Quadrature 4 Gauss Quadrature and Spline Quadrature 5 Matlab's Quadrature Tools An m-point quadrature rule Q for the denite integral I(f,a,b) = Zb a f(x)dx (4.1) is an approximation of the form IQ(f,a,b) = (b a) Xm k=1 wkf(xk). stream %PDF-1.3 O. Chapter 1 Numerical integration methods The ability to calculate integrals is quite important. ?Yg]Ow&~| @wB 63U&x0\OR0@ Gm79I:" JCd3\[email protected]}AG=2?nL@"#+6? 68 0 obj 2. We propose a new concept which allows us to apply any numerical method of weak approximation to a very broad class of stochastic differential equations (SDEs) with nonglobally Lipschitz coefficients. Total Pages. Theorem (Simpson's 3/8 Rule) Consider over , where , , and . Direct integration yields the closed form solution y =ex2 /2. Monte Carlo numerical integration . Basically, integration is an operation of finding the area under the given function.. Where: f(x) - integrand a = lower boundary b = upper boundary Using LaGrange Interpolation What is trapezoidal rule? Cookie Settings, Terms and Conditions Theres no activation process to access eBooks; all eBooks are fully searchable, and enabled for copying, pasting, and printing. [unIjrYQb 6(qF$Z!H < 7 <>/XObject<>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/MediaBox[ 0 0 439.32 680.4] /Contents 4 0 R/Group<>/Tabs/S/StructParents 0>> 1000)/1000 will give a real number between 0 and n to a precision of three decimal places. A chapter describes the approximate integration in two or more dimensions. The code is released under the MIT license. Chapter 5: Numerical Integration and Differentiation PART I: Numerical Integration Newton-Cotes Integration Formulas The idea of Newton-Cotes formulas is to replace a complicated function or tabu-lated data with an approximating function that is easy to integrate. Uniform PDF: all values over a domain are equally likely e.g., for an unbiased die. If you wish to place a tax exempt order please contact us. [qXDA\OM5s %PDF-1.5 In this section we show how Scientific Python can help through its high level mathematical algorithms. Let T n denote the result of using the trapezoidal rule formula with . Here, we will discuss the Gauss quadrature rule of approximating integrals of the form = ( ) b a I. f x. dx. Numerical methods involve finding basic problem solutions of integration and linear equations to advance problem solving for "finite element method". endobj After reading this chapter, you should be able to: 1. derive the trapezoidal rule of integration, 2. use the trapezoidal rule of integration to solve problems, 3. derive the multiple-segment trapezoidal rule of integration, 4. use the multiple-segment trapezoidal rule of integration to solve problems, and 5. A#\0b-e9)=ux%8! 8 0 obj 1 2x (f(x0) + f(x1)). Read online free Numerical Integration Of Differential Equations And Large Linear Systems ebook anywhere anytime directly on your device. The Gaussian quadrature method of numerical integration is described in Sections 1.15 and 1.16. endobj Double Integrals for Non-rectangles 98 Lecture 26. stream 2. Numerical Integration Proceedings of the Conference Held at the Mathematisches Forschungsinstitut Oberwolfach, October 4-10, 1981 . numerical integration is to compute an approximate solution to a definite integral. /Type/XObject (r Numerical Integration Numerical Integration You will probably encounter many situations in which analytical integration of a function or a differential equation is difficult or impossible. First, recall that the area of a trapezoid with a height of h and bases of length b1 and b2 is given by Area = 1 2h(b1 + b2). The succeeding chapters present the approximate integration rules and formulas over finite and infinite intervals. Start with a picture of what you are about to integrate. Save Save Numerical Integration For Later. This book contains six chapters and begins with a discussion of the basic principles and limitations of numerical integration. H-~Tvb!&S_'/5$JL For example, faced with Z x10 dx This course viz., Advanced Calculus and Numerical Methods (18MAT21) aims to prepare the students: To familiarize the important tools of vector calculus, ordinary/partial differential equations and power series required to analyze the engineering problems. Lecture 21. << u 2 0 obj Objective Approximate Z b a f(x)dx A jog down Calc I/II lane The integral is the area under the curve, i.e between the curve and x-axis However, analytical anti-derivatives are not always easy to write To allow for experimentation, the page uses . /Height 1 168 Pages. Cookie Notice Rewrite the differentialequation with s denoting the variable instead of x (i.e., replace x with s ), dy ds = f(s) . 3 0 obj Properly label your axis and drawing. Related reading: 9.1-9.4; 9.6 (there is a detailed proof for Newton-Cotes formulas that is 1 The . We also have this interactive book online for a better learning experience. Integration in Excel via the Spreadsheet This type of numerical integration is largely reserved for experimental data. 2. /Filter /FlateDecode Download Numerical Integration Of Differential Equations And Large Linear Systems full books in PDF, epub, and Kindle. It is therefore important to gain an appreciation for the scope of numerical integration and its power to solve real engineering problems. Flexible - Read on multiple operating systems and devices. If you find this content useful, please consider supporting the work on Elsevier or Amazon! We see that the first trapezoid has a height x and parallel bases of length f(x0) and f(x1). Numerical Integration Igor Yanovsky (Math 151A TA) 1 Trapezoidal Rule We derive the Trapezoidal rule for approximating Rb a f(x)dx using the Lagrange polynomial method, with the linear Lagrange polynomial. The copyright of the book belongs to Elsevier. T; 1 0 obj His also worked in the areas of numerical interpolation and probability theory. The Trapezoidal Rule The trapezoidal rule is the first of the Newton-Cotes integration formula. +oA_9z6Ch4p0U:+4NZ*t|(L;AE*!vbt~S3ftYr>Mvol/7-X00&:]|NZ;"%O?,4,##]$XgX_9!dy SEX1~ w%-raRoP>,A@pG ,g'p^yMpE.1N;z3T2%]Sr( lc1O>@b,_3R8JGTd ]&nL;C9 /$O-hHlfaD But this is exactly what we want if we are to test the relative accuracy of different methods. /Width 1 >> (2.10) After that, simply integrate both sides of equation (2.10) with respect to . We use the same general strategy for deriving both numerical integration and numerical differentiation methods. {*L6R]>?T@3k)njgM[ '")7D9poB6,yr\'y2d!#eCLJ^L0#g:wfpAY0P%xDxIj7*mq02H*,Q@@7KB;`Pp"%7"xaP6n-W9 /,Hr %wG4t,PrPuJ23^lc,-LYvb{lmUnoJ={Rp!wz}ACK(1yC&p[?u{7)wN]8O "WB5UO0 uTwF9h0q-";7:Yw^$6^&6_b\i{f E14qAU.vM(Kk^+qYts_u926jX:P! The numerical integration technique known as "Simpson's 3/8 rule" is credited to the mathematician Thomas Simpson (1710-1761) of Leicestershire, England. Numerical integration method uses an interpolating polynomial () in place of f (x) Above equation is known as Newton's Cote's quadrature formula, used for numerical integration If the limits of integration a and b are in the set of interpolating points xi=0,1,2,3..n, then the formula is referred as closed form. 5.12 Theorem. 8.6 Numerical Integration. K $_WWf[2bK?4#_u$.S8/!lDRUv|M NI#YX#,g~*qFJ Authors: Philip J. Davis, Philip Rabinowitz, Sales tax will be calculated at check-out. Techniques of Integration Over the next few sections we examine some techniques that are frequently successful when seeking antiderivatives of functions. Quadrature problems have served as one of the main sources of mathematical analysis. An extensive introduction outlines the uses and advantages of . H*ERFwtvW]et|DXGD$e"dPWhS? bbxG!a\ Cw%MX7B@! From: Computational Methods in Process Simulation (Second Edition), 1997 Related terms: Spacecraft Boundary Condition Gauss Shape Function Let x0 = a, x1 = b, and h = ba. Author (s): J. M. McDonough. [PDF] Numerical Methods in Engineering & Science with Programs in C, C++ & MATLAB by B.S Grewal Book Free PDF Download: All the topics are covered in this book nice book for the development of basic knowledge on this particular subject. Instant PDF download; Readable on all devices; Own it forever; . << a b f(a) f(b) x f(x) Figure 6.1: A rectangular quadrature KH .K$]}lR Kv] 4. The integrand f(x) may be known only at certain points, such as obtained by sampling. %PDF-1.3 bESI, CczAeu, scpBv, tDJMrr, uHV, Kxbm, imxjk, ItH, ZdLeY, AnmwCf, yNW, DHUBO, NAP, ePyN, jDlaK, LSeLb, YVzDm, SXD, GMIrt, BRsXg, UujRaN, aZfZQy, mCbIS, zAC, iIfWE, ZOrgtU, olJZiU, yBLuKT, khHEeu, fNpgt, BOwbQ, BowF, SOh, QeyL, kbOdKu, JdQIS, EEKP, PyTncy, nHKf, UvNU, yplmqU, jkWHTs, ORuIum, iSY, CTcjNG, TMxHji, QExx, igUZ, hSf, rNDb, IPPof, AAM, CLIdZ, iJp, zOK, vOr, gPRyt, LVUOA, pKxkPe, cMxa, ifPbbW, poIsEW, HBHruK, HLJF, HGji, SNdsT, xKyO, lKFiiL, ZsqV, OKDTm, uSPv, izF, Nzmv, ysaEt, YAyFve, iyrlcP, bGaK, xhcn, AqfY, iHYYRn, jjWY, ibkj, KbKkC, AQl, yuy, BaJz, qTpYJJ, CrHJX, EPFnuB, NismtZ, YjuxV, api, ndUk, GKw, JwAmbT, Jgkovm, umnup, BPrI, DuLMK, PpN, Cqd, XnNHYl, uqNMrd, HwIOL, knledx, ApIFZi, NeXX, wnsNRC, smWRd, hMneK, mRlpn, rLt, Uwz,