imaginary parts of functions

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Parts of Speech - Coming to Terms

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Terms of Endearment - More Parts of Speech

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Integration of Functions part 1

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... 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

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... 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 value134Chapter 4. Integration of Functions Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING ... 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 Functions Sample page from...

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Tài liệu Description of Functions doc

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... 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

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... computation of ﬁrst derivatives.• 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 §10.1.• Forone-dimensional minimizationwithcalculation of the...

Tài liệu Minimization or Maximization of Functions part 2 docx

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... 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 ... 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 ... those of the so-called golden mean or golden section,whose supposedly aesthetic properties hark back to the ancient Pythagoreans. Thisoptimal method of function minimization, the analog of the...

Tài liệu Minimization or Maximization of Functions part 3 pdf

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... are acceptable but404Chapter 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

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... 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

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... 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 0-521-43108-5)Copyright ... towardsthe low point. An appropriatesequenceof such stepswill alwaysconverge to a minimum of the function.described in §10.8,alsomakes useof thegeometrical conceptofasimplex. Otherwiseit is completely...

Tài liệu Minimization or Maximization of Functions part 6 pptx

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... 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

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... 136Chapter 4. Integration of Functions Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992 ... 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, ... 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

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... 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...

Tài liệu Integration of Functions part 5 docx

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... 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 §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 ... 146Chapter 4. Integration 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 Integration of Functions part 6 pptx

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... 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 ... {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...

Tài liệu Integration of Functions part 7 pptx

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... 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 ... 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...