... Young Asians are not so as their American counterparts.A. Rome B. Roman C. romantic D. romanticize75. All of my students appreciate the ______ of English learning. A. importance B. important ... differently 10. All of us were _____ that he was successful in the final exam. A. surprising B. surprisingly C. surprise D. surprised 11. One recent ________ in medicine is the development of laser in ... achievements B. achievement C. achiever D. achieved12. You have to be aware of the damage humans are doing to quicken the ________ of wildlife.A. extinct B. extinctive C. extinctions D. extinction13....
... 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...
... 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 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...
... 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...
... 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...
... 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...
... 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...
... 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...
... 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...
... 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...
... 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...
... 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...
... 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...
... Multidimensional IntegralsIntegrals offunctionsof 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...