constraint based integration of deep

... Reader Copyright(C) by Foxit Software Company,2005-2008 For 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 19 less accurate results, using higher orders (15 was tested) increased the computation times, but not the accuracy of the results. For integration ... + 1 3 x, x ∈ x. Taylor Model Based Integration of ODEs · August 18, 2006 7 For 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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Ngày tải lên: 05/09/2013, 15:28

... at the 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 the Institute of Technology from ... a degradation of the accuracy and the richness 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 the University of Evry. In 2000, he received the Habilitation à Diriger des Recherches diploma. He is the Head of the Electrical Engineering Department of the Institute of Technology...

Ngày tải lên: 23/10/2013, 15:15

... (4.0.3) Chapter 16 of this book deals with the numerical integration of differential equations. 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 use the methods of Chapter 16. The quadrature methods in this chapter are based, in one way or another, on the obvious device of adding up the value of the integrand ... integrand at a sequence of abscissas within the range of integration. The game is to obtain the integral as accurately as possible with the smallest number of function evaluations of the integrand....

Ngày tải lên: 07/11/2013, 19:15

... f 3 + ···+f N−1 + 1 2 f N  +O  (b−a) 3 f  N 2  (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 holding a and b fixed ... amount that is the product of some numerical coefficient times h 3 times the value 134 Chapter 4. Integration of Functions Sample 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. 130 Chapter 4. Integration of Functions Sample page...

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Ngày tải lên: 23/11/2013, 14:22

... America). Chapter 16. Integration of Ordinary Differential Equations 16.0 Introduction Problems involving ordinary differential equations (ODEs) can always be reduced 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 combining the information from several Euler-style steps (each involving one evaluation of the right-hand ... 708 Chapter 16. Integration of Ordinary Differential Equations Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright...

Ngày tải lên: 15/12/2013, 04:15

... 710 Chapter 16. Integration of Ordinary Differential Equations Sample 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 the midpoint of the interval. Then use the value of both x and y at that midpoint to compute the ... have different coefficients of higher-order error terms. Adding up the right combination of these, we can eliminate the error terms order by order. That is the basic idea of the Runge-Kuttamethod....

Ngày tải lên: 15/12/2013, 04:15

... 136 Chapter 4. Integration of Functions Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright (C) ... fixed limits a and b, one can double the number of intervals in the extended trapezoidal rule without losing the benefit of previous work. The coarsest implementation of the trapezoidal rule is to average ... function at its endpoints a and b.Thefirst stage of refinement is to add to this average the value of the function at the halfway point. The second stage of refinement is to add the values at the 1/4...

Ngày tải lên: 15/12/2013, 04:15

... 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 the following 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...

Ngày tải lên: 15/12/2013, 04:15

... falls off exponentially. Then we want a change of variable that maps e −x dx into (±)dt (with the sign chosen to keep the upper limit of the new variable larger than the lower limit). Doing the integration ... it,j; b=sqrt(bb-aa); a=0.0; if (n == 1) { The rest of the routine is exactly like midpnt and is omitted. 142 Chapter 4. Integration of Functions Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING ... Quadratures and Orthogonal Polynomials In 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...

Ngày tải lên: 15/12/2013, 04:15

... i.e., the computation of the coefficients a j , b j in (4.5.6); (ii) the determination of the zeros of p N (x),and the computation of the associated weights. For the case of the “classical” orthogonal polynomials, ... around the midpoint of the range of integration, there are actually only five distinct values of each: float qgaus(float (*func)(float), float a, float b) Returns the integral of the function func between a and b , ... { 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 orthogonal polynomials with the weight function W (x)=−log x on the interval (0, 1). A suitable set of π j ’s is the...

Ngày tải lên: 15/12/2013, 04:15