Integration of Functions part 2

... h  3 2 f 2 +f 3 +f 4 +···+f N 2 + 3 2 f N−1  +O  1 N 2  (4.1.15) Equations (4.1.8) and (4.1. 12) give  x N x 1 f(x)dx = h  23 12 f 2 + 7 12 f 3 + f 4 + f 5 + ···+f N−3 + 7 12 f N 2 + 23 12 f N ... O(h 3 f  )(4.1.8)  x 1 x 0 f(x)dx = h  23 12 f 1 − 16 12 f 2 + 5 12 f 3  + O(h 4 f (3) )(4.1.9)  x 1 x 0 f(x)dx = h  55 24 f 1 − 59 24 f 2 + 37...

Ngày tải lên: 07/11/2013, 19:15

... http://www.nr.com or call 1-800-8 72- 7 423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America). Chapter 4. Integration of Functions 4.0 Introduction Numerical integration, which is ... RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0- 521 -43108-5) Copyright (C) 1988-19 92 by Cambridge University Press.Programs Copyright (C) 1988-19 92 by Numer...

Ngày tải lên: 07/11/2013, 19:15

... < √ |b|  2| f(b)| b 2 f  (b) (10.1 .2) The reason for writing the right-hand side in this way is that, for most functions, the final square root is a number of order unity. Therefore, as a rule of thumb, ... bracketing triplet of points is (a, b, x); 398 Chapter 10. Minimization or Maximization of Functions Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC...

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 ... 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 approach to t...

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) 1988-1...

Ngày tải lên: 15/12/2013, 04:15

... c j ,d j ,e j ,andf j are given by c j =2( j+1)(j+α+β+ 1)(2j + α + β) d j =(2j+α+β+1)(α 2 −β 2 ) e j =(2j+α+β)(2j + α + β + 1)(2j + α + β +2) f j =2( j+α)(j + β)(2j + α + β +2) (4.5.15) We now give individual ... used. 0.67940956 82, 0.8650633666,0.973906 528 5}; static float w[]={0.0,0 .29 5 524 224 7,0 .26 926 67193, 0 .21 90863 625 ,0.1494513491,0.0666713443}; xm=0.5*(b+a); xr=0.5*...

Ngày tải lên: 15/12/2013, 04:15

... SSSR , vol. 154, pp. 28 3 28 6 (in Russian). [11] Patterson, T.N.L. 1968, Mathematics of Computation , vol. 22 , pp. 847–856 and C1–C11; 1969, op. cit. , vol. 23 , p. 8 92. [ 12] Piessens, R., de Doncker, ... z 2 (x, y). Inotherwords,findthe numbers x 1 and x 2 , and the functions y 1 (x),y 2 (x),z 1 (x, y),andz 2 (x, y) such that I ≡  dx dy dzf(x, y, z) =  x 2 x 1 dx  y...

Ngày tải lên: 15/12/2013, 04:15

... 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 series converge for all ... 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 o...

Ngày tải lên: 15/12/2013, 04:15

... transfer IV. 2. Pragmatic transfer of request strategies and Proficiency IV.3. Pragmatic transfer of request strategies and Gender Chapter V: Conclusion and Implications 14 V.1. Summary V .2. Implications V.3. ... TABLE OF CONTENTS Chapter I: Introduction 1 Chapter II: Literature review 2 Chapter III: Methodology 5 Chapter IV: Presentation and Discussion of results 8 IV. 1. Contex...

Ngày tải lên: 07/11/2012, 14:47