Tài liệu Evaluation of Functions part 2 doc

... Similarly, the form of the coefficients a is often such as to make use of previous work: Terms like k! or (2k)! can be updated in a multiply or two. 165 166 Chapter 5. Evaluation of Functions Sample ... function and the Bessel function of the first kind, sin x = ∞  k=0 (−1) k (2k +1)! x 2k+1 (5.1 .2) J n (x)=  x 2  n ∞  k=0 (− 1 4 x 2 ) k k!(k + n)! (5.1.3) Both of these...

Ngày tải lên: 15/12/2013, 04:15

Ngày tải lên: 15/12/2013, 04:15

... example, with x = (2 ) 2 , the first term smaller than 10 −7 is x 13 / (27 !). This then approximates the error of the finite series whose last term is x 12 / (25 !). Notice that because of the large exponent ... formula [1] x k = 1 2 k−1  T k (x)+  k 1  T k 2 (x)+  k 2  T k−4 (x)+···  (5.11 .2) where the last term depends on whether k is even or odd, ···+  k (k−1) /2  T...

Ngày tải lên: 15/12/2013, 04:15

... the evaluation: double ratval(double x, double cof[], int mm, int kk) Given mm , kk ,and cof[0 mm+kk] , evaluate and return the rational function ( cof[0] + cof[1]x + ···+ cof[mm]x mm )/(1 + cof[mm+1]x + ... 1970, Mathematical Methods of Physics , 2nd ed. (Reading, MA: W.A. Benjamin/Addison-Wesley), pp. 361–363. [2] Knuth, D.E. 1981, Seminumerical Algorithms , 2nd ed., vol. 2 of The...

Ngày tải lên: 24/12/2013, 12:16

... 186 Chapter 5. Evaluation of Functions Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0- 521 -43108-5) Copyright (C) 1988-19 92 by Cambridge University ... h)=f(x)+hf  (x)+ 1 2 h 2 f  (x)+ 1 6 h 3 f  (x)+··· (5.7 .2) whence f(x + h) − f(x) h = f  + 1 2 hf  + ··· (5.7.3) The roundoff error has various contributions. First there is roundo...

Ngày tải lên: 24/12/2013, 12:16

... execution time of chebft is dominated by the calculation of N 2 cosines, rather than by the N evaluations of your function, then you should look ahead to § 12. 3, especially equation 12. 3 .22 , which shows ... 5.8.1), T 0 (x)=1 T 1 (x)=x T 2 (x)=2x 2 −1 T 3 (x)=4x 3 −3x T 4 (x)=8x 4 −8x 2 +1 ··· T n+1 (x)=2xT n (x)− T n−1 (x) n ≥ 1. (5.8 .2) (There also exist inverse formulas...

Ngày tải lên: 24/12/2013, 12:16

... are often taught, rather casually, that, on average, measurements will fall within ±σ of the true value 68 percent of the time, within 2 95 percent of the time, and within ±3σ 99.7 percent of ... universe of data sets that are drawn from it! 658 Chapter 15. Modeling of Data Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0- 521 -43108-5) Copyright...

Ngày tải lên: 15/12/2013, 04:15

... Formula,  x N x 1 f(x)dx = h  1 2 f 1 + f 2 + f 3 + ···+f N−1 + 1 2 f N  − B 2 h 2 2! (f  N − f  1 ) −···− B 2k h 2k (2k)! (f (2k−1) N − f (2k−1) 1 ) −··· (4 .2. 1) Here B 2k is a Bernoulli number, ... h  23 12 f 2 + 7 12 f 3 + f 4 + f 5 + ···+f N 2 + 13 12 f N −1 + 5 12 f N  + O  1 N 3  (4.1 .20 ) CITED REFERENCES AND FURTHER READING: Abramowitz, M., and Stegu...

Ngày tải lên: 15/12/2013, 04:15

... 140 Chapter 4. Integration of Functions Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0- 521 -43108-5) Copyright (C) 1988-19 92 by Cambridge University Press.Programs ... error series up to but not including O(1/N 2k ). The routine qsimp is the case of k =2. This is one example of a very general idea that goes by the name of Richardson’s deferred...

Ngày tải lên: 15/12/2013, 04:15

... formula,  x N x 1 f(x)dx = h[f 3 /2 + f 5 /2 + f 7 /2 +···+f N−3 /2 +f N−1 /2 ] + B 2 h 2 4 (f  N −f  1 )+··· + B 2k h 2k (2k)! (1 − 2 −2k+1 )(f (2k−1) N − f (2k−1) 1 )+··· (4.4.1) This equation ... “deep” property 1 42 Chapter 4. Integration of Functions Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0- 521 -43108-5) Copyright (C) 1988-19...

Ngày tải lên: 15/12/2013, 04:15