constraint based integration of deep
06 ON TAYLOR MODEL BASED INTEGRATION OF ODES
... ReaderCopyright(C) by Foxit Software Company,2005-2008For Evaluation Only.Taylor Model Based Integration of ODEs · August 18, 2006 9 of the interval remainder terms are nondecreasing. Often, these diameters ... Model Based Integration of ODEs · August 18, 2006 19less accurate results, using higher orders (15 was tested) increased the computation times, but not theaccuracy of the results. For integration ... +13x, x ∈ x.Taylor Model Based Integration of ODEs · August 18, 2006 7For illustration, we use order n = 3 and step size h = 0.1 in the Taylor model integration of (4.1).All numbers are displayed...
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Sensor-based navigation of a mobile robot in an indoor environment
... atthe University of Paris XI, Orsay in 1981.Since 1986, he is Professor at the Uni-versity of Evry and he was the Head of Electrical Engineering Department of theInstitute of Technology from ... a degradation of the accuracy and therichness of the information.Some constraints are added to the intrinsic draw-backs of these methods caused by:• the imprecision or lack of knowledge in ... Assistant Professor at theUniversity of Evry. In 2000, he receivedthe Habilitation à Diriger des Recherchesdiploma. He is the Head of the ElectricalEngineering Department of the Institute of Technology...
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Integration of Functions part 1
... (4.0.3)Chapter 16 of this book deals with the numerical integration of differentialequations. In that chapter, much emphasis is given to the concept of “variable” or“adaptive” choices of stepsize. ... the form of (4.0.2)–(4.0.3) and usethe methods of Chapter 16.The quadrature methods in this chapter are based, in one way or another, on theobvious device of adding up the value of the integrand ... integrand at a sequence of abscissaswithin the range of integration. The game is to obtain the integral as accuratelyas possible with the smallest number of function evaluations of the integrand....
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Integration of Functions part 2
... f3+···+fN−1+12fN+O(b−a)3fN2(4.1.11)Here we have written the error estimate in terms of the interval b − a and the number of points N instead of in terms of h. This is clearer, since one is usually holdinga and b fixed ... amount that is the product of some numerical coefficient times h3times the value134Chapter 4. Integration of FunctionsSample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING ... down as an additional power of N (since it is used only twice,not N times). This makes the resulting formula of degree one less than Simpson.130Chapter 4. Integration of FunctionsSample page...
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Tài liệu Integration of Ordinary Differential Equations part 1 doc
... America).Chapter 16. Integration of OrdinaryDifferential Equations16.0 IntroductionProblems involving ordinary differential equations (ODEs) can always bereduced to the study of sets of first-order ... description of each of these types follows.1. Runge-Kutta methods propagate a solution over an interval by combiningthe information from several Euler-style steps (each involving one evaluation of theright-hand ... 708Chapter 16. Integration of Ordinary Differential EquationsSample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright...
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Tài liệu Integration of Ordinary Differential Equations part 2 pptx
... 710Chapter 16. Integration of Ordinary Differential EquationsSample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright ... stable(see §16.6 below).Consider, however, the use of a step like (16.1.1) to take a “trial” step to themidpoint of the interval. Then use the value of both x and y at that midpointto compute the ... have different coefficients of higher-ordererror terms. Adding up the right combination of these, we can eliminate the errorterms order by order. That is the basic idea of the Runge-Kuttamethod....
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Tài liệu Integration of Functions part 3 docx
... 136Chapter 4. Integration of FunctionsSample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) ... fixedlimits a and b, one can double the number of intervals in the extended trapezoidalrule without losing the benefit of previous work. The coarsest implementation of the trapezoidal rule is to average ... function at its endpoints a and b.Thefirststage of refinement is to add to this average the value of the function at the halfwaypoint. The second stage of refinement is to add the values at the 1/4...
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Tài liệu Integration of Functions part 4 docx
... RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Permission ... while qsimp requires 8 calls (8 times as many evaluations of the integrand) and qtrap requires 13 calls (making 256 times as many evaluations of the integrand).CITED REFERENCES AND FURTHER READING:Stoer, ... “improper” if it has any of thefollowing problems:• its integrand goes to a finite limitingvalue at finite upper and lower limits,but cannot be evaluated right on one of thoselimits(e.g., sin...
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Tài liệu Integration of Functions part 5 docx
... falls offexponentially. Then we want a change of variable that maps e−xdx into (±)dt (withthe sign chosen to keep the upper limit of the new variable larger than the lowerlimit). Doing the integration ... it,j;b=sqrt(bb-aa);a=0.0;if (n == 1) {The rest of the routine is exactly likemidpntand is omitted.142Chapter 4. Integration of FunctionsSample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING ... Quadratures and OrthogonalPolynomialsIn the formulas of §4.1, the integral of a function was approximated by the sum of its functional values at a set of equally spaced points, multiplied by certain...
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Tài liệu Integration of Functions part 6 pptx
... i.e., the computation of the coefficients aj, bjin (4.5.6); (ii) the determination of the zeros of pN(x),andthe computation of the associated weights. For the case of the “classical” orthogonalpolynomials, ... around the midpoint of the range of integration, thereare actually only five distinct values of each:float qgaus(float (*func)(float), float a, float b)Returns the integral of the functionfuncbetweenaandb, ... {sig[k][l]=sig[k-1][l+1]+(alpha[l-1]-a[k-2])*sig[k-1][l]-b[k-2]*sig[k-2][l]+beta[l-1]*sig[k-1][l-1];}a[k-1]=alpha[k-1]+sig[k][k+1]/sig[k][k]-sig[k-1][k]/sig[k-1][k-1];b[k-1]=sig[k][k]/sig[k-1][k-1];}free_matrix(sig,1,2*n+1,1,2*n+1);}As an example of the use of orthog, consider the problem[7] of generating orthogonalpolynomials with the weight function W (x)=−log x on the interval (0, 1). A suitable set of πj’s is the...
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Tài liệu Integration of Functions part 7 pptx
... entirelydifferent approaches to doing the problem. The questions are: Is the shape of theboundary of the region of integration simple or complicated? Inside the region, isthe integrand smooth and ... Currentvalues of x and y, and the pointer to your function func, are passed “over the head” of the intermediate calls through static top-level variables.164Chapter 4. Integration of FunctionsSample ... North America).inner integration yxouter integration Figure 4.6.1. Function evaluations for a two-dimensional integral over an irregular region, shownschematically. The outer integration routine,...
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- chulalongkorn university in thailand this smalllab kit was created as a result of the research project entitled chemistry laboratory based on chemical safety and pollution minimization sponsored by thai research fund rdg 3072543 one of the outcomes of t
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