integration of layout information
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. ... 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. ... North America).Chapter 4. Integration of Functions4.0 IntroductionNumerical integration, which is also called quadrature, has a history extendingback to the invention of calculus and before....
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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
... 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 ... 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 ... 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 ... derivative information onlyat the beginning of that interval (see Figure 16.1.1). That means (and you can verifyby expansion in power series) that the step’s error is only one power of h smallerthan ... 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...
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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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Tài liệu Bài 10: ICA by Minimization of Mutual Information pdf
... Minimization of Mutual Information An important approach for independent component analysis (ICA) estimation, in-spired by information theory, is minimization of mutual information. The motivation of ... can be developed usingmutual information, which is a well-motivated information- theoretic measure of statistical dependence.One of the main utilities of mutual information is that it serves ... 10−4iteration countmutual information Fig. 10.1The convergence of FastICA for ICs with uniform distributions. The value of mutual information shown as function of iteration count.10.5 EXAMPLESHere...
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Tài liệu Integration of Ordinary Differential Equations part 3 doc
... generally useful stepperroutine is this: One of the arguments of the routine will of course be the vector of dependent variables at the beginning of a proposed step. Call that y[1 n].Letus require ... steps (see Figure 16.2.1). How much overhead is this,say in terms of the number of evaluations of the right-hand sides? Each of thethree separate Runge-Kutta steps in the procedure requires ... we willlearn how such informationcan beobtained. Obviously,720Chapter 16. Integration of Ordinary Differential EquationsSample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING...
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Tài liệu Integration of Ordinary Differential Equations part 4 ppt
... 722Chapter 16. Integration of Ordinary Differential EquationsSample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright ... modified midpoint method, which advances a vector of dependent variables y(x) from a point x to a point x + H by a sequence of nsubsteps each of size h,h = H/n (16.3.1)In principle, one could ... above. You will seethat the transition in Chapter 4 to the idea of Richardson extrapolation, as embodiedin Romberg integration of §4.3, is exactly analogous to the transition in going fromthis...
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- bureau of international information programs
- sources of further information
- organization of product information
- constraint based integration of deep
- the representation of inconsistent information
- integration of visual interword linguistic knowledge
- integration of largescale linguistic resources
- use of mutual information
- the contribution of stylistic information
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