integration of rational functions

Ngày tải lên: 13/05/2014, 09:37

Ngày tải lên: 05/08/2014, 15:20

... sequence gm of rational functions whose zeros and poles are related to the endpoints of the intervals removed from the construction of the Cantor set K Motivated by the previous work of Poletsky ... uncountable then our proof of Proposition 4.4 breaks down because we not know if the set T is polar Finally we come to the promised explicit example of a sequence of rational functions that converges ... Determination of the pluripolar hull of graphs of certain holomorphic functions, Ann Inst Fourier (Grenoble) 54 (2004), 2085-2104 [Go] A Gonchar, A local condition for the single valuedness of analytic functions...

Ngày tải lên: 16/10/2015, 14:19

... communes with the spirits of his or her predecessors back across the centuries, as far as the time of Newton, if not farther Alas, times change; with the exception of two of the most modest formulas ... America) of various orders, with higher order sometimes, but not always, giving higher accuracy “Romberg integration, ” which is discussed in §4.3, is a general formalism for making use of integration ... formalism for making use of integration methods of a variety of different orders, and we recommend it highly Apart from the methods of this chapter and of Chapter 16, there are yet other methods for...

Ngày tải lên: 07/11/2013, 19:15

... North America) h 132 Chapter Integration of Functions Simpson’s rule: x3 f(x)dx = h x1 1 f1 + f2 + f3 3 + O(h5 f (4) ) (4.1.4) Here f (4) means the fourth derivative of the function f evaluated ... N2 (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 ... from x0 to x1 , using values of the function f at x1 , x2 , These will be useful building blocks for the “extended” open formulas 134 Chapter Integration of Functions xN f(x)dx = h x1 13...

Ngày tải lên: 07/11/2013, 19:15

... Chapter Integration of Functions We come now to the “deep” fact about the extended trapezoidal rule, equation (4.1.11) It is this: The error of the approximation, which begins with a term of order ... and we can write it as a routine exactly analogous to qtrap above: 140 Chapter Integration of Functions 4.3 Romberg Integration #include #define EPS 1.0e-6 #define JMAX 20 #define JMAXP ... total number of steps; K is the number of points used in the extrapolation float qromb(float (*func)(float), float a, float b) Returns the integral of the function func from a to b Integration...

Ngày tải lên: 15/12/2013, 04:15

... while qsimp requires 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: ... “improper” if it has any of the following problems: • its integrand goes to a finite limiting value at finite upper and lower limits, but cannot be evaluated right on one of those limits (e.g., sin ... we call it impossible No amount of clever algorithmics will return a meaningful answer to an ill-posed problem In this section we will generalize the techniques of the preceding two sections to...

Ngày tải lên: 15/12/2013, 04:15

... 142 Chapter Integration of Functions of having an error series that is entirely even in h Indeed there is a formula, not ... change of variable float midinf(float (*funk)(float), float aa, float bb, int n) This routine is an exact replacement for midpnt, i.e., returns the nth stage of refinement of the integral of funk ... lower limit, one also makes a change of variable If the integrand diverges as (x − a)γ , ≤ γ < 1, near x = a, use the identity 146 Chapter Integration of Functions Similarly, #include ...

Ngày tải lên: 15/12/2013, 04:15

... 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 ... , i.e., the computation of the coefficients aj , bj in (4.5.6); (ii) the determination of the zeros of pN (x), and the computation of the associated weights For the case of the “classical” orthogonal ... numerical stability is greatly improved if, instead of using powers of x as a set of basis functions to represent the pj ’s, one uses some other known set of orthogonal polynomials πj (x), say Roughly...

Ngày tải lên: 15/12/2013, 04:15

... 162 Chapter Integration of Functions If, on the basis of the above guidelines, you decide to pursue the repeated onedimensional integration approach, here is how it works ... integration, plus a method of determining whether a random point is inside or outside the region of integration Monte Carlo integration evaluates the function at a random sample of points, and estimates ... only),or send email to trade@cup.cam.ac.uk (outside North America) outer integration inner integration 164 Chapter Integration of Functions static float xsav,ysav; static float (*nrfunc)(float,float,float);...

Ngày tải lên: 15/12/2013, 04:15

... of integration 15 Particular problems of integration 17 The limitations of the methods of integration 20 Conclusion 22 V The integration of algebraical functions Algebraical functions 22 22 Integration ... Introduction II Elementary functions and their classification III The integration of elementary functions Summary of results IV The integration of rational functions 11 1–3 The method of partial fractions ... function of x III THE INTEGRATION OF ELEMENTARY FUNCTIONS SUMMARY OF RESULTS The integral of a rational function (iv.) is always an elementary function It is either rational or the sum of a rational...

Ngày tải lên: 28/06/2014, 19:20

... shares the benefits of each individual process design model The main objective of this model is to integrate and optimize each process by conducting a detailed study of the benefits of each approach ... treatment plant processes an average of million gallons of wastewater daily The main step in the treatment process is the separation of solids, which accounts for about 2% of wastewater Wastes are separated ... Assistant Professor with the Department of Engineering Management and Systems Engineering, Missouri University of Science and Technology Prior to joining Missouri S&T she was an Assistant Professor...

Ngày tải lên: 05/09/2013, 15:28

... products Assuming 1,000 OFDs, only 37 percent of products will be free of defects if the quality level at each OFD averaged 99.9 percent, and 90 percent of products will be free of defects if the quality ... shifts, resulting in a target production of 480 units per day Work in the cell consists of 20 individual tasks, and each unit of product possesses a total of 100 OFDs In this cell, when the 480-unit ... having a goal of optimizing the performance of a single portion of a company risks violating lean management rules For example, a purchasing manager who is given a reward for cutting costs of component...

Ngày tải lên: 23/11/2013, 14:22

... 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 ... small increments to your functions corresponding to derivatives (right-hand sides of the equations) multiplied by stepsizes In this chapter we consider three major types of practical numerical ... description of each of these types follows Runge-Kutta methods propagate a solution over an interval by combining the information from several Euler-style steps (each involving one evaluation of the...

Ngày tải lên: 15/12/2013, 04:15

... free_vector(dym,1,n); 714 Chapter 16 Integration of Ordinary Differential Equations } CITED REFERENCES AND FURTHER READING: Abramowitz, M., and Stegun, I.A 1964, Handbook of Mathematical Functions, Applied Mathematics ... a graph of the function Then all you need may be a simple driver program that goes from an initial xs to a final xf in a specified number of steps To check accuracy, double the number of steps, ... step in a sequence of steps in identical manner Prior behavior of a solution is not used in its propagation This is mathematically proper, since any point along the trajectory of an ordinary differential...

Ngày tải lên: 15/12/2013, 04:15

... Speaking and Writing These types of integration are diagraphically presented in Table 2.1 Integration Table 2.1 Types of integration Integration of skills Integration of three areas Listening Writing ... follows is a detailed description of the two types of intergration, i.e the integration of skills and the integration of linguistic areas 2.4 .Integration of skills Skill integration is an innovative ... 2.2 The power of extensive reading 21 2.3.1 Definition 23 2.3.2.Types of integration 24 2.4 .Integration of skills .25 2.4.1 Integration of three areas...

Ngày tải lên: 18/12/2013, 10:08

... generally useful stepper routine 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] Let us 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 the three separate Runge-Kutta steps in the procedure requires ... they bound the error of each step individually In some applications you may be unusually sensitive about a “global” accumulation of errors, from beginning to end of the integration and in the...

Ngày tải lên: 24/12/2013, 12:16

... (16.3.4) above You will see that the transition in Chapter to the idea of Richardson extrapolation, as embodied in Romberg integration of §4.3, is exactly analogous to the transition in going from this ... (16.3.2), expressed as a power series in h, the stepsize, contains only even powers of h, 724 Chapter 16 Integration of Ordinary Differential Equations } CITED REFERENCES AND FURTHER READING: Gear, ... method does an excellent job of feeling its way through rocky or discontinuous terrain It is also an excellent choice for quick-and-dirty, low-accuracy solution of a set of equations A second warning...

Ngày tải lên: 24/12/2013, 12:16

... the method a degree of robustness for problems with discontinuities Let us remind you once again that scaling of the variables is often crucial for successful integration of differential equations ... and its limited radius of convergence, out only to the distance of the first pole in the complex plane Rational function fits can remain good approximations to analytic functions even after the ... number of n’s to be tried is called KMAXX For reasons described below we usually take this equal to 8; the 8th value of the sequence (16.4.2) is 16, so this is the maximum number of subdivisions of...

Ngày tải lên: 24/12/2013, 12:16