series of functions uniform convergence

... Integration of Functions 4.0 Introduction Numerical integration,which is also called quadrature, has a history extending back to the invention of calculus and before. The fact that integrals of elementary functions ... (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...

Ngày tải lên: 07/11/2013, 19:15

... READING: Abramowitz, M., and Stegun, I.A. 1964, Handbook of Mathematical Functions , Applied Mathe- matics Series, Volume 55 (Washington: National Bureau of Standards; reprinted 1968 by Dover Publications, ... 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...

Ngày tải lên: 07/11/2013, 19:15

... control of inputs to the analysis . ã Quality control of information is it correct, complete, and adequate? ã Quality control of reasoning is it fair and unbiased? ã Quality control of analysis ... process of dissecting the inputs to a decision. It clarifies the degree of understanding of issues involved and helps to ensure that choice selection is reasoned and balanced. It is the basis of risk management. We ... related to the four steps of the decision making process and finally to the three types of error (risk) that occur in each of the four steps. The paper concludes with an example of how methodical decision...

Ngày tải lên: 10/12/2013, 14:15

... mode). S Correct the program. Software limit switches Function The software limit switches serve as limitations of the maximum traversing range of each in- dividual axis. 2 software limit switch pairs ... monitoring functions 2-24 SINUMERIK 802D, 802D base line 6FC5 697-2AA10-0BP2 (11.03) (DF) 2.2 Motion monitoring functions 2.2.1 Contour monitoring Function The principle of functioning of the contour ... (1st software limit switch minus) MD 36110: POS_LIMIT_PLUS (1st software limit switch plus) MD 36120: POS_LIMIT_MINUS2 (2nd software limit switch minus) MD 36130: POS_LIMIT_PLUS2 (2nd software...

Ngày tải lên: 13/12/2013, 01:16

... computation of rst derivatives. ã You must choose between methods that require storage of order N 2 and those that require only of order N,whereNis the number of dimensions. For moderate values of N ... then the new bracketing triplet of points is (a, b, x); 396 Chapter 10. Minimization or Maximization of Functions Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN ... America). interpolationsof Brent’s method are of no advantage, and you might wish to use the simplest form of golden section search, as described in Đ10.1. ã Forone-dimensional minimizationwithcalculation of the...

Ngày tải lên: 15/12/2013, 04:15

... this way is that, for most functions, the final square root is a number of order unity. Therefore, as a rule of thumb, it is hopeless to ask for a bracketing interval of width less than √  times ... size of the preceding interval. This is comparable to, but not quite as good as, the 0.50000 that holds when finding roots by bisection. Note that the convergence is linear (in the language of Chapter ... in the larger of the two segments (z is positive only if w<1/2). But where in the larger segment? Where did the value of w itself come from? Presumably from the previous stage of applying our...

Ngày tải lên: 15/12/2013, 04:15

... are acceptable but 404 Chapter 10. Minimization or Maximization of Functions Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright (C) 1988-1992 ... the value already evaluated only by an amount of order the roundofferror. Therefore in the code below you will find several tests and modifications of a potential new point, imposing this restriction. ... p = -p; q=fabs(q); 402 Chapter 10. Minimization or Maximization of Functions Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright (C) 1988-1992...

Ngày tải lên: 15/12/2013, 04:15

... consideration of multidimensional minimization, that is, finding the minimum of a function of more than one independent variable. This section stands apart from those which follow, however: All of the ... 408 Chapter 10. Minimization or Maximization of Functions Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright (C) 1988-1992 ... d’s to an out -of- bracket value.d2=d1; if (dw != dx) d1=(w-x)*dx/(dx-dw); Secant method with one point. if (dv != dx) d2=(v-x)*dx/(dx-dv); And the other. Which of these two estimates of d shall we...

Ngày tải lên: 15/12/2013, 04:15

... consideration of multidimensional minimization, that is, finding the minimum of a function of more than one independent variable. This section stands apart from those which follow, however: All of the ... tetrahedron. (The simplex method of linear programming, 412 Chapter 10. Minimization or Maximization of Functions Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN ... towards the low point. An appropriatesequenceof such stepswill alwaysconverge to a minimum of the function. described in Đ10.8,alsomakes useof thegeometrical conceptofasimplex. Otherwise it is completely...

Ngày tải lên: 15/12/2013, 04:15

... number of unknown parameters in f is equal to the number of free parameters in A and b,whichis 1 2 N(N+1),whichweseetobeoforderN 2 . Changing any one of these parameters can move the location of ... members of a set of vectors, they are said to be a conjugate set. If you do successive line minimization of a function along a conjugate set of directions, then you don’t need to redo any of those ... derivative matrix of the function is called the Hessian matrix of the function at P. 412 Chapter 10. Minimization or Maximization of Functions Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC...

Ngày tải lên: 15/12/2013, 04:15

... READING: Abramowitz, M., and Stegun, I.A. 1964, Handbook of Mathematical Functions , Applied Mathe- matics Series, Volume 55 (Washington: National Bureau of Standards; reprinted 1968 by Dover Publications, ... even powers of h occur in the error series of (4.2.1). This fact is not, in general, shared by the higher-order quadrature rules in Đ4.1. For example, equation (4.1.12) has an error series beginning ... 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...

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: ã 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

... 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 Đ4.1, the integral of a function was approximated by the sum of its functional values at a set of equally spaced points, multiplied by certain ... endpoints. It is assumed that choose triples the number of steps on each call, and that its error series contains only even powers of the number of steps. The routines midpnt , midinf , midsql , midsqu , midexp ,...

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, ... numerical stability is greatly improved if, instead of using powers of x as a set of basis functions to represent the p j ’s, one uses some other known set of orthogonal polynomials π j (x), say. Roughly ... READING: Abramowitz, M., and Stegun, I.A. 1964, Handbook of Mathematical Functions , Applied Mathe- matics Series, Volume 55 (Washington: National Bureau of Standards; reprinted 1968 by Dover Publications,...

Ngày tải lên: 15/12/2013, 04:15

... Multidimensional Integrals Integrals of functions of several variables, over regions with dimension greater than one, are not easy. There are two reasons for this. First, the number of function evaluations ... Current values 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. 164 Chapter 4. Integration of Functions Sample ... 307. Abramowitz, M., and Stegun, I.A. 1964, Handbook of Mathematical Functions , Applied Mathe- matics Series, Volume 55 (Washington: National Bureau of Standards; reprinted 1968 by Dover Publications,...

Ngày tải lên: 15/12/2013, 04:15

Ngày tải lên: 15/12/2013, 04:15

Ngày tải lên: 15/12/2013, 04:15

Ngày tải lên: 15/12/2013, 04:15

Ngày tải lên: 15/12/2013, 04:15

Ngày tải lên: 15/12/2013, 04:15