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see also Fourier transform All-poles model 573. see also Maximum entropy method (MEM) All-zeros model 573. see also Periodogram. see also Simplex, method of Nelder and Mead. by wavelets 601f., 791 see also Fitting. Autonomous differential equations 735f.. Bairstow’s method 371, 376f.. Best-fit parameters see also Fitting. for differential equations 707f.. 506, 522 Brent’s method. Broyden’s method 380, 389ff., 393...
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Abramowitz, M., and Stegun, I.A. Stoer, J., and Bulirsch, R. Kahaner, D., Moler, C., and Nash, S. Johnson, L.W., and Riess, R.D. Ralston, A., and Rabinowitz, P. Isaacson, E., and Keller, H.B. (3.1.1) There are N terms, each a polynomial of degree N − 1 and each constructed to be zero at all of the x i except one, at...
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3.2 Rational Function Interpolation and Extrapolation 111. 3.2 Rational Function Interpolation and Extrapolation. We de- note by R i(i+1)...(i+m) a rational function passing through the m + 1 points (x i , y i. In specifying a rational function interpolating function, you must give the desired order of both the numerator and the denominator.. A rational function approximation, by contrast,...
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Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN Copyright (C by Cambridge University Press.Programs Copyright (C by Numerical Recipes Software. h will never be zero, since this was tested in the initial- izing loop.. Stoer, J., and Bulirsch, R. Cuyt, A., and Wuytack, L. Equations (3.3.1) and (3.3.2) are a special case of the general...
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Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN Copyright (C by Cambridge University Press.Programs Copyright (C by Numerical Recipes Software. Suppose that you have decided to use some particular interpolation scheme, such as fourth-order polynomial interpolation, to compute a function f (x) from a set of tabulated x i ’s and f i ’s. Then...
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3.5 Coefficients of the Interpolating Polynomial. Occasionally you may wish to know not the value of the interpolating polynomial that passes through a (small!) number of points, but the coefficients of that poly- nomial. A valid use of the coefficients might be, for example, to compute simultaneous interpolated values of the function and of several of its derivatives (see. 5.3),...
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Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN Copyright (C by Cambridge University Press.Programs Copyright (C by Numerical Recipes Software. x n ) from an n-dimensional grid of tabulated values y and n one-dimensional vec- tors giving the tabulated values of each of the independent variables x 1 , x 2. An important concept is...
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Carnahan, B., Luther, H.A., and Wilkes, J.O. Isaacson, E., and Keller, H.B. Stoer, J., and Bulirsch, R. Ralston, A., and Rabinowitz, P. Dahlquist, G., and Bjorck, A. Kahaner, D., Moler, C., and Nash, S. Forsythe, G.E., Malcolm, M.A., and Moler, C.B. Davis, P., and Rabinowitz, P
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Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN Copyright (C by Cambridge University Press.Programs Copyright (C by Numerical Recipes Software. Although not explicitly discussed here, you ought to be able to figure out how to do cubic spline quadrature using the output of the routine spline in § 3.3. (Hint: Integrate equation 3.3.3 over x...
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Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN Copyright (C by Cambridge University Press.Programs Copyright (C by Numerical Recipes Software. At the closed end of the integration, use the weights from the former equations. Our starting point is equation (4.1.11), the extended trapezoidal rule. There are two facts about the trapezoidal rule which make it...
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Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN Copyright (C by Cambridge University Press.Programs Copyright (C by Numerical Recipes Software. To order Numerical Recipes books,diskettes, or CDROMsvisit website http://www.nr.com or call North America only),or send email to [email protected] (outside North America).. 4.3 Romberg Integration. We can view Romberg’s method as the natural generalization of the...
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Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN Copyright (C by Cambridge University Press.Programs Copyright (C by Numerical Recipes Software. qromb, in such circumstances, takes many, many fewer function evaluations than either of the routines in § 4.2. For example, the integral. For our present purposes, an integral will be “improper” if it has any...
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Dahlquist, G., and Bjorck, A. Stoer, J., and Bulirsch, R. The idea of Gaussian quadratures is to give ourselves the freedom to choose not only the weighting coefficients, but also the location of the abscissas at which the function is to be evaluated: They will no longer be equally spaced. it will turn out that we can achieve Gaussian quadrature...
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Piessens, R., de Doncker, E., Uberhuber, C.W., and Kahaner, D.K. Stoer, J., and Bulirsch, R. Johnson, L.W., and Riess, R.D. Carnahan, B., Luther, H.A., and Wilkes, J.O. Ralston, A., and Rabinowitz, P. First, the number of function evaluations needed to sample an N-dimensional space increases as the N th power of the number needed to do a one-dimensional integral. If...
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Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN Copyright (C by Cambridge University Press.Programs Copyright (C by Numerical Recipes Software. Permission is granted for internet users to make one paper copy for their own personal use. To order Numerical Recipes books,diskettes, or CDROMsvisit website http://www.nr.com or call North America only),or send email to [email protected] (outside...
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The purpose of this chapter is to acquaint you with a selection of the techniques that are frequently used in evaluating functions. The purposes of this chapter and the next are thus mostly in harmony, but there is nevertheless some tension between them: Routines that are clearest and most illustrative of the general techniques of this chapter are not always...
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5.10 Polynomial Approximation from Chebyshev Coefficients 197. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN Copyright (C by Cambridge University Press.Programs Copyright (C by Numerical Recipes Software. To order Numerical Recipes books,diskettes, or CDROMsvisit website http://www.nr.com or call North America only),or send email to [email protected] (outside North America).. 5.10 Polynomial Approximation from Chebyshev Coefficients. You...
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k=0 g k x k , where x and y are related by (5.8.10), i.e., the interval − 1 <. 1 is mapped to the interval a <. 5.11 Economization of Power Series. In our example, with x = (2π) 2 , the first term smaller than 10 − 7 is x 13 /(27. This then approximates the error of...
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Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN Copyright (C by Cambridge University Press.Programs Copyright (C by Numerical Recipes Software. If you are summing very large numbers of slowly convergent terms, pay attention to roundoff errors! In floating-point representation it is more accurate to sum a list of numbers in the order starting with the...
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5.3 Polynomials and Rational Functions 173. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN Copyright (C by Cambridge University Press.Programs Copyright (C by Numerical Recipes Software. To order Numerical Recipes books,diskettes, or CDROMsvisit website http://www.nr.com or call North America only),or send email to [email protected] (outside North America).. Thompson, I.J., and Barnett, A.R. 1973, in Pad...