series of functions uniform convergence

Integration of Functions part 1

... Integration of Functions 4.0 IntroductionNumerical integration,which is also called quadrature, has a history extendingback 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 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...

Integration of Functions part 2

Danh mục: Kỹ thuật lập trình

... 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 byDover Publications, ... f3+···+fN−1+12fN+O(b−a)3f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 holdinga and b ﬁxed ... amount that is the product of some numerical coefﬁcient times h3times the value 134Chapter 4. Integration of Functions Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING...

Tài liệu Expert Reference Series of White Papers pptx

Danh mục: Quản trị mạng

... 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 riskmanagement.We ... related to the four steps of the decision making process and finally tothe three types of error (risk) that occur in each of the four steps.The paper concludes with an example of how methodical decision...

Tài liệu Description of Functions doc

Danh mục: Kĩ thuật Viễn thông

... mode).S Correct the program.Software limit switchesFunctionThe software limit switches serve as limitations of the maximum traversing range of each in-dividual axis.2 software limit switch pairs ... monitoring functions 2-24SINUMERIK 802D, 802D base line6FC5 697-2AA10-0BP2 (11.03) (DF)2.2 Motion monitoring functions 2.2.1 Contour monitoringFunctionThe 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...

Tài liệu Minimization or Maximization of Functions part 1 pptx

Danh mục: Kỹ thuật lập trình

... computation of rst derivatives.ã You must choose between methods that require storage of order N2andthose 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); 396Chapter 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 wishto use the simplest form of golden section search, as described in Đ10.1.ã Forone-dimensional minimizationwithcalculation of the...

Tài liệu Minimization or Maximization of Functions part 2 docx

Danh mục: Kỹ thuật lập trình

... this way is that, for most functions, the ﬁnal square root is a number of order unity. Therefore, as a rule of thumb, itis hopeless to ask for a bracketing interval of width less than√ times ... size of the preceding interval. This is comparable to, but notquite as good as, the 0.50000 that holds when ﬁnding roots by bisection. Note thatthe 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...

Tài liệu Minimization or Maximization of Functions part 3 pdf

Danh mục: Kỹ thuật lập trình

... are acceptable but 404Chapter 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 alreadyevaluated only by an amount of order the roundofferror. Therefore in the code belowyou will ﬁnd several tests and modiﬁcations of a potential new point, imposing thisrestriction. ... p = -p;q=fabs(q); 402Chapter 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...

Tài liệu Minimization or Maximization of Functions part 4 ppt

Danh mục: Kỹ thuật lập trình

... 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 ... 408Chapter 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- bracketvalue.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...

Tài liệu Minimization or Maximization of Functions part 5 docx

Danh mục: Kỹ thuật lập trình

... 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, 412Chapter 10. Minimization or Maximization of Functions Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN ... towardsthe low point. An appropriatesequenceof such stepswill alwaysconverge to a minimum of the function.described in Đ10.8,alsomakes useof thegeometrical conceptofasimplex. Otherwiseit is completely...

Tài liệu Minimization or Maximization of Functions part 6 pptx

Danh mục: Kỹ thuật lập trình

... number of unknown parameters in f is equal to the number of freeparameters in A and b,whichis12N(N+1),whichweseetobeoforderN2.Changing any one of these parameters can move the location of ... members of a set of vectors, they are saidto be a conjugate set. If you do successive line minimization of a function alonga conjugate set of directions, then you don’t need to redo any of those ... derivative matrix of thefunction is called the Hessian matrix of the function at P. 412Chapter 10. Minimization or Maximization of Functions Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC...

Tài liệu Integration of Functions part 3 docx

Danh mục: Kỹ thuật lập trình

... 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 byDover 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 ... 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...

Tài liệu Integration of Functions part 4 docx

Danh mục: Kỹ thuật lập trình

... 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: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...

Tài liệu Integration of Functions part 5 docx

Danh mục: Kỹ thuật lập trình

... 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 Functions Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING ... Quadratures and OrthogonalPolynomialsIn 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 thatchoosetriples the number of steps on each call, and that its error series contains only even powers of the number of steps. The routinesmidpnt,midinf,midsql,midsqu,midexp,...

Tài liệu Integration of Functions part 6 pptx

Danh mục: Kỹ thuật lập trình

... 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, ... 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 someother known set of orthogonal polynomials πj(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 byDover Publications,...

Tài liệu Integration of Functions part 7 pptx

Danh mục: Kỹ thuật lập trình

... Multidimensional IntegralsIntegrals of functions of several variables, over regions with dimension greaterthan one, are not easy. There are two reasons for this. First, the number of functionevaluations ... 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 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 byDover Publications,...