Integration of Functions part 2
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...
Integration of Functions part 1
... 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...
Tài liệu Minimization or Maximization of Functions part 2 docx
... <
√
|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...
Tài liệu Integration of Functions part 3 docx
... 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...
Tài liệu Integration of Functions part 4 docx
... 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...
Tài liệu Integration of Functions part 5 docx
... 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...
Tài liệu Integration of Functions part 6 pptx
... 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*...
Tài liệu Integration of Functions part 7 pptx
... 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...
Tài liệu Evaluation of Functions part 2 doc
... 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...
Pragmatic Transfer in Interlanguage Requesting by Vietnamese learners of English part 2
... 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...