A Novel Class of Symmetric and Nonsymmetric

Journal of Scientific Computing, Vol. 31, No. 3, June 2007 (© 2006) DOI: 10.1007/s10915-006-9110-z

A Novel Class of Symmetric and Nonsymmetric Periodizing Variable Transformations for Numerical Integration Avram Sidi1 Received May 29, 2005; accepted (in revised form) August 25, 2006; Published online December 7, 2006

Variable transformations for numerical integration have been used for improving the accuracy of the trapezoidal rule. Specifically, one first transforms the inte1 gral I [f ] = 0 f (x) dx via a variable transformation x = φ(t) that maps [0, 1] to itself, and 1 then approximates the resulting transformed integral I [f ] = f φ(t) φ (t) dt by the trapezoidal rule. In this work, we propose a new class 0 of symmetric and nonsymmetric variable transformations which we denote T r,s , where r and s are positive scalars assigned by the user. A simple representative of this class is φ(t) = (sin π2 t)r /[(sin π2 t)r + (cos π2 t)s ]. We show that, in case f ∈ C ∞ [0, 1], or f ∈ C ∞ (0, 1) but has algebraic (endpoint) singularities at x = 0 and/or x = 1, the trapezoidal rule on the transformed integral produces exceptionally high accuracies for special values of r and s. In particular, when f ∈ C ∞ [0, 1] and we employ φ ∈ Tr,r , the error in the approximation is (i) O(hr ) for arbitrary r and (ii) O(h2r ) if r is a positive odd integer at least 3, h being the integration step. We illustrate the use of these transformations and the accompanying theory with numerical examples. KEY WORDS: Numerical integration; variable transformations, sinm -transformation, Euler–Maclaurin expansions; asymptotic expansions; trapezoidal rule. MATHEMATICS SUBJECT CLASSIFICATION 2000: 30E15, 40A25, 41A60, 65B15, 65D30, 65D32.

1

Computer Science Department, Technion – Israel Institute of Technology, Haifa 32000, Israel. E-mail: [email protected] 391 0885-7474/07/0600-0391/0 © 2006 Springer Science+Business Media, LLC

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1. INTRODUCTION AND BACKGROUND Consider the problem of evaluating finite-range integrals of the form 1 I [f ] = f (x) dx, (1.1) 0

∈ C ∞ (0, 1) but is not necessarily continuous or differentiable at x = 0

where f and/or x = 1, and may even have different types of singularities at the endpoints. One very effective way of computing I [f ] is by first transforming it with a suitable variable transformation and next applying the trapezoidal rule to the resulting transformed integral. Thus, if we make the substitution x = ψ(t), where ψ(t) is an increasing differentiable function on [0, 1], such that ψ(0) = 0 and ψ(1) = 1, then the transformed integral is 1 I [f ] = (1.2) f(t) dt; f(t) = f ψ(t) ψ (t), 0

and I [f ] can be approximated by applying the trapezoidal rule to namely, by n−1 1 1 Qn [f ] = h f (0) + f (ih) + f (1) ; 2 2 i=1

1 h= . n

1 0

f(t)dt,

(1.3)

If, in addition, ψ(t) is chosen such that ψ (i) (0) = ψ (i) (1) = 0, i = 1, 2, . . . , p, for some sufficiently large p, then the function f(t) is such that f(i) (0) = f(i) (1) = 0, i = 1, 2, . . . , q, for some q. Thus, from the Euler– n [f ] (see, e.g., Davis and Rabinowitz [5] or StoMaclaurin expansion of Q er and Bulirsch [19] or Atkinson [2] or Sidi [10, Appendix D]), it follows n [f ] − I [f ] = o(n−q ) as n → ∞, which means that Q n [f ] approxithat Q mates I [f ] with surprisingly high accuracy even for moderate n. In such n [f ] assumes the simpler form a case, we also have f(0) = f(1) = 0, and Q n [f ] = h Q

n−1 i=1

f(ih);

1 h= . n

(1.4)

Variable transformations in numerical integration, especially in the presence of integrands with endpoint singularities, have been of considerable interest lately. In the context of one-dimensional integration, they are used as a means to improve the performance of the trapezoidal rule, as we have already mentioned. They have also been used by Verlinden et al. [20], in conjunction with extrapolation methods, to improve the performance of the product trapezoidal rule in multiple integration. Recently, they have

Symmetric and Nonsymmetric Periodizing Variable Transformations

393

been used to improve the performance of the Gauss–Legendre quadrature; see, for example, Monegato and Scuderi [7] and Johnston [6]. In the context of multi-dimensional integration, they are used to “periodize” the integrand in all variables so as to improve the accuracy of lattice rules; see, for example, Sloan and Joe [18] and Robinson and Hill [8]. (Lattice rules are extensions of the trapezoidal rule to many dimensions.) They have also been used in conjunction with the product trapezoidal rule in computing integrals over surfaces of spheres in three dimensions and, more generally, on surfaces of bounded sets in R3 ; see Atkinson [3], Atkinson and Sommariva [4], and Sidi [12–14, 17]. Normally, we also demand that ψ(t) be symmetric, in the sense that ψ(1 − t) = 1 − ψ(t), which forces on ψ (t) the symmetry property ψ (1 − t) = ψ (t). Most variable transformations are indeed symmetric in this sense. However, as discussed by Monegato and Scuderi [7], by employing nonsymmetric variable transformations, we can cope with integrands having singularities of different strengths at the endpoints more efficiently. Actually, these authors show exactly how to modify the known symmetric transformations in simple ways to obtain nonsymmetric ones. Following Monegato and Scuderi [7] and Sidi [15], in [16], the author developed a class of nonsymmetric variable transformations, denoted Sp,q , that is defined as follows: Definition 1.1. A function ψ(t) is in the class Sp,q , with p, q > 0 but arbitrary, if it has the following properties: 1. 2.

ψ ∈ C[0, 1] and ψ ∈ C ∞ (0, 1); ψ(0) = 0, ψ(1) = 1, and ψ (t) > 0 on (0, 1). ψ (t) has the following asymptotic expansions as t → 0+ and t → 1−: ψ (t) ∼

∞

i t p+2i

as t → 0+,

i=0

ψ (t) ∼

∞

(1.5) δi (1 − t)q+2i

as t → 1−,

i=0

and 0 , δ0 > 0. Consequently, ψ (t) ∼

∞ i=0

ψ (t) ∼ 1 −

i

t p+2i+1 p + 2i + 1

∞ i=0

δi

as t → 0+,

(1 − t)q+2i+1 q + 2i + 1

(1.6) as t → 1 − .

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3.

For each positive integer k, ψ (k) (t) has asymptotic expansions as t → 0+ and t → 1− that are obtained by differentiating those of ψ(t) term by term k times.

A representative of this class is the sinp,q -transformation defined as in ψp,q (t) =

Θp,q (t) ; Θp,q (t) = Θp,q (1)

t 0

sin π2 u

p

cos π2 u

q

du, p, q > 0. (1.7)

This transformation is the nonsymmetric generalization of the sinm transformation of the author (proposed originally with integer m in [9], and extended to arbitrary values of m recently in [15]) and was proposed with integer p and q in [7]. With integer p and q, it can be computed recursively, as shown in [7]. In [16], the sinp,q -transformation was extended to arbitrary and not necessarily integer p and q to enhance their effectiveness. When either p or q or both are not integers, ψp,q (t) cannot be computed by recursions because initial values are not readily available. In such cases, ψp,q (t) can be computed by summing a quickly converging infinite series representation, as shown in [16]. Also, when p = q = m, the sinp,q transformation becomes the (symmetric) sinm -transformation. Both when p = q and p = q, the sinp,q -transformation can be expressed in terms of the hypergeometric function F (a, b; c; z) =2F1 (a, b; c; z), among others, in the form ψp,q (t) =

1

1 1 1 1 3 2 2 − 2 q, 2 p + 2 ; 2 p + 2 ; S 1 1 1 F 2 − 2 q, 2 p + 21 ; 21 p + 23 ; 1

F

S p+1 ,

S = sin

πt . 2

(1.8)

n [f ] as A complete analysis of the trapezoidal rule approximations Q given in (1.4), with ψ(t) ∈ Sp,q there, was also provided in [16]. In order to motivate the developments of the present work, we need to recall Theorem 4.2 and Corollary 4.3 in [16], which are the main results of [16]. These are given as Theorem 1.2 and Corollary 1.3 below: Theorem 1.2. Let f ∈ C ∞ (0, 1), and assume that f (x) has the asymptotic expansions

f (x) ∼

∞ i=0

ci x γi

as x → 0+;

f (x) ∼

∞ i=0

di (1 − x)δi

as x → 1−.


395

Here γi and δi are distinct complex numbers that satisfy −1 < γ0 γ1 γ2 · · · ; lim γi = +∞, i→∞

−1 < δ0 δ1 δ2 · · · ; lim δi = +∞. i→∞

Assume furthermore that, for each positive integer k, f (k) (x) has asymptotic expansions as x → 0+ and x → 1− that are obtained 1 by differentiating those of f (x) term by term k times. Let I [f ] = 0 f (x) dx, and let us now make the transformation of variable x = ψ(t), where ψ ∈ Sp,q , n [f ] = in I [f ].Finally, us approximate I [f ] via the trapezoidal rule Q let

n−1 i=1 f ψ(ih) ψ (ih), where h = 1/n, n = 1, 2, . . . . Then the following hold: (i)

In the worst case, n [f ]−I [f ]= O hω Q

(ii)

as h→0;

ω=min{(γ0 +1)(p+1), (δ0 +1)(q+1)}.

If γ0 and δ0 are real, and if p = (2k − γ0 )/(γ0 + 1) and q = (2l − δ0 )/(δ0 + 1), where k and l are positive integers, then n [f ]−I [f ] = O hω Q

as h→0;

ω=min{(γ1 +1)(p+1), (δ1 +1)(q+1)},

at worst. Remark. If f (x) = x μ (1 − x)ν g(x), g(x) being infinitely differentiable on [0, 1], then f (x) satisfies the conditions of the theorem. In such a case, if f (x) has full Taylor series at x = 0 and x = 1, we have γi = μ + i and δi = ν + i, i = 0, 1, . . . . Note that this f (x) has an algebraic branch singularity at x = 0 if μ is not a positive integer. Similarly, it has an algebraic branch singularity at x = 1 if ν is not a positive integer. Corollary 1.3. In case f (x) = x μ (1 − x)ν g(x), g(x) being infinitely differentiable on [0, 1], the following hold: (i)

In the worst case, n [f ]−I [f ]=O hω Q

(ii)

as h→0;

ω=min{(μ+1)(p+1), (ν+1)(q+1)}.

If μ and ν are real, and if p = (2k − μ)/(μ + 1) and q = (2l − ν)/(ν + 1), where k and l are positive integers, then n [f ] − I [f ] = O hω Q

at worst.

as h → 0;

ω = min{(μ+ 2)(p+ 1), (ν+2)(q+1)},

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Remark. Note that the results in parts (ii) of Theorem 1.2 and Corollary 1.3 are the best that can be obtained and are made possible by our definition of the class Sp,q transformations, where we have excluded the powers t p+2i+1 and (1 − t)q+2i+1 , i = 0, 1, . . . , from the asymptotic expansions of ψ (t) as t → 0+ and t → 1−. n [f ] in a way Now, in [15], we introduced the concept of quality of Q that is relevant to symmetric variable transformations. Here we modify this concept to make it relevant to nonsymmetric transformations in Sp,q as follows: If ψ (t) ∼ αt p as t → 0+ and ψ (t) ∼ β(1 − t)q as t → 1−, and n [f ] − I [f ] = O(hσ ) as h → 0, the quality of Q n [f ] is the ratio σ/w, if Q n [f ] where w = max{p + 1, q + 1}. Note that the effective abscissas in Q given in (1.4) are xi ≡ ψ(ih) = ψ(i/n) and these cluster near x = 0 and x = 1 in the variable x and that the clustering increases with increasing p and q n [f ]. Because too much clustering simultaneously with the accuracy of Q is not desirable, we would like to get as much accuracy as possible from a given amount of clustering. In other words, we would like the quality of n [f ] to be as high as possible. This is achieved by the variable transforQ mations in Sp,q with special (not necessarily integer) values of p and q. In view of what we have described so far, we now address ourselves to the following question: Are there variable transformations ψ(t) that are not in Sp,q with the properties that (i) they satisfy ψ(t) ∼ t p+1 as t → 0+ and ψ(t) ∼ 1 − δ(1 − t)q+1 as t → 1− just as those in Sp,q do, (ii) are easy to compute, and (iii) Theorem 1.2 and Corollary 1.3 remain unchanged in their presence? In this work, we propose a novel class of variable transformations, which we denote Tr,s , that have these properties in general. The members of Tr,s are analogous to those of Sr−1,s−1 , and although they differ from the latter substantially, they have similar properties. In particular, there are analogues of Theorem 1.2 and Corollary 1.3 that pertain to members of Tr,s and that say that, in general, whatever can be done with variable transformations in Sr−1,s−1 can be done also with those in Tr,s . In addition, we have been able to produce transformations in Tr,s that are easier to compute than those in the classes Sp,q . In the next section, we define the class Tr,s and show how transformations in this class can be determined easily. We provide a remarkably simple and easily computable representative of it, which we denote the T r,s -transformation. When r = s, this transformation, denoted the T r transformation for short, is symmetric. In Section 3, we provide the conn [f ] obtained vergence theory of the trapezoidal rule approximations Q using members of Tr,s . We also show how the performance of our approach can be improved substantially by subtracting from the integrand a simple function with known integral. In Section 4, we provide


397

numerical examples that illustrate the theory. We use the T r,s -transformation in these examples. Finally, in Section 5, we discuss an already existing modification of the class proposed here, and show that it is inferior to the class Tr,s . To distinguish the variable transformations in Tr,s from those in Sp,q , in the sequel, we will denote them by φ(t) instead of ψ(t). 2. THE CLASS Tr,s 2.1. Definition of the Class Tr,s Definition 2.1. A function φ(t) is in the class Tr,s , with r, s > 0 but arbitrary, if it has the following properties: 1. φ ∈ C[0, 1] and φ ∈ C ∞ (0, 1); φ(0) = 0, φ(1) = 1, and φ (t) > 0 on (0, 1). 2. φ(t) has the following asymptotic expansions as t → 0+ and t → 1−: φ (t) ∼

r/2

αi t r+2i +

i=0

φ (t) ∼ 1 −

∞

α˜ i t σi

as t→0+;

α0 > 0, (2.1)

i=0

s/2 i=0

βi (1 − t)s+2i −

∞

β˜i (1 − t)ρi

as t → 1−;

β0 > 0,

i=0

where 2r = σ0 < σ1 < · · · ; 2s = ρ0 < ρ1 < · · · ; 3.

lim σi = ∞,

(2.2)

i→∞

lim ρi = ∞.

i→∞

For each positive integer k, φ (k) (t) has asymptotic expansions as t → 0+ and t → 1− that are obtained by differentiating those of φ(t) term by term k times. In particular, φ (t) has the following asymptotic expansions as t → 0+ and t → 1−: φ (t) ∼ φ (t) ∼

r/2

∞

i=0

i=0

(r + 2i)αi t r+2i−1 +

s/2 i=0

σi α˜ i t σi −1

(s + 2i)βi (1 − t)s+2i−1 +

as t → 0+, (2.3)

∞ i=0

ρi β˜i (1 − t)ρi −1

as t → 1−.

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1. From Definition 2.1, it is clear that φ ∈ Tr,s satisfies

Remarks.

φ (t) =

r/2

αi t r+2i + O t 2r

as t → 0+, (2.4)

i=0

φ (t) = 1 −

s/2

βi (1 − t)s+2i + O (1 − t)2s

as t → 1−,

i=0

and φ (t) =

r/2

(r + 2i)αi t r+2i−1 + O t 2r−1

as t → 0+, (2.5)

i=0

φ (t) =

s/2

(s + 2i)βi (1 − t)s+2i−1 + O (1 − t)2s−1

as t → 1−.

i=0

2.

As we will see in Section 3, what makes class Tr,s transformations effective is the fact that we have excluded the powers t r+2i+1 , i = 0, 1, . . . , r/2 − 1, and (1 − t)s+2i+1 , i = 0, 1, . . . , s/2 − 1, from the asymptotic expansions of φ(t) as t → 0+ and t → 1−, respectively.

To keep the developments below and the notation simple, we give two definitions. Definition 2.2. We denote generically by Rμ (t) any function g(t) that has an asymptotic expansion of the form g(t) ∼

∞

ri t μ+2i

as t → 0+.

i=0

Remark. Note that, in Definition 2.2, we do not require r0 = 0, because such a requirement is not needed in the sequel. By Definition 2.2, we have Rμ (t) + Rμ+2j (t) = Rμ (t),

j = 0, 1, 2, . . . ,

(2.6)

hence ∞ j =0

Rμ+2j (t) = Rμ (t).

(2.7)


399

In addition, [Rμ (t)]α = Rαμ (t)

and Rμ (t)Rν (t) = Rμ+ν (t).

(2.8)

In case μ1 < μ2 < · · · ;

lim μk = +∞,

k→∞

(2.9)

and Rμk (t) ∼ rk,0 t μk

as t → 0+;

rk,0 = 0,

k = 1, 2, . . . ,

(2.10)

there holds Rμk+1 (t) = 0, t→0+ Rμk (t) lim

k = 1, 2, . . . ,

(2.11)

that is, the sequence {Rμk (t)}∞ k=1 is an asymptotic scale as t → 0+. If Rμ (t) can be differentiated term by term, we have d Rμ (t) = Rμ−1 (t). dt

(2.12)

If Rμ (t) can be integrated term by term, we have t Rμ (u) du = Rμ+1 (t).

(2.13)

0

We will make free use of all this in the sequel. Definition 2.3. We say that a function g(t) belongs to the set Kμ , μ > 0, if g(0) = 0,

g ∈ C ∞ (0, 1),

g(1) = 1;

g (t) > 0

for t ∈ (0, 1),

and if g(t) has asymptotic expansions as t → 0+ and as t → 1− given as g(t) ∼

∞

(0)

gi t μ+2i

as t → 0+;

(0)

g0 > 0,

i=0

g(t) ∼

∞

(1)

gi (1 − t)2i

as t → 1−;

(1)

g0 = 1,

i=0

and if, for each k = 1, 2, . . . , the kth derivative of g(t) has asymptotic expansions as t → 0+ and t → 1− that are obtained by differentiating those of g(t) term by term k times.

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Sidi

Remark. Note that the function g(t) = t p , p > 0, even though it shares some of the properties of functions in Kp , does not belong to the set Kp . The following conclusions can easily be drawn from Definition 2.3: (for t → 0+) Rμ (t) . g ∈ Kμ ⇒ g(t) = R0 (1 − t) (for t → 1−) g ∈ Kμ ⇒ u(t) = [g(t)]ν ∈ Kμν

if ν > 0.

g1 ∈ Kμ , g2 ∈ Kν ⇒ u(t) = g1 (t)g2 (t) ∈ Kμ+v . We will make use of these properties of functions in the sets Kμ shortly. 2.2. Construction of Functions in Tr,s We now propose one simple way of constructing some functions in Tr,s : Let u(t) and v(t) be such that u(0) = 0, v(0) = 0,

u(1) = 1;

u ∈ C ∞ (0, 1),

v(1) = 1;

∞

v ∈ C (0, 1),

u (t) > 0

v (t) > 0

for t ∈ (0, 1), for t ∈ (0, 1),

Thus, u(t) > 0 and v(t) > 0 for 0 < t 1 as well. Now set φ(t) =

u(t) . u(t) + v(1 − t)

It is easy to see that u(t) + v(1 − t) > 0 for t ∈ [0, 1], and that φ(0) = 0,

φ(1) = 1,

φ (t) =

u (t)v(1 − t) + u(t)v (1 − t) >0 [u(t) + v(1 − t)]2

for 0 < t < 1.

That is, φ(t) is an increasing function of t for t ∈ (0, 1) and satisfies 0 φ(t) 1 for 0 t 1, and hence is a valid variable transformation. Note that we also have φ(t) = 1 −

v(1 − t) , u(t) + v(1 − t)

so that, when u(t) = v(t), there holds φ(1 − t) = 1 − φ(t); that is, φ(t) becomes a symmetric transformation. We now choose u(t) and v(t) such that u ∈ Kr and v ∈ Ks . Then w0 (t) =

u(t) = Rr (t) = O(t r ) = o(1) v(1 − t)

as t → 0+,


401

hence, for all small t > 0, φ(t) has the convergent expansion φ(t) =

∞

∞

k=1

k=1

w0 (t) Rkr (t) = (−1)k−1 [w0 (t)]k = 1 + w0 (t)

that is also a valid asymptotic expansion as t → 0+. Similarly, w1 (t) =

v(1 − t) = Rs (1 − t) = O (1 − t)s = o(1) u(t)

as t → 1−,

hence, for all t < 1 but close to 1, φ(t) has the convergent expansion φ(t) = 1 −

∞

∞

k=1

k=1

w1 (t) (−1)k−1 [w1 (t)]k = 1 − Rks (1 − t) =1− 1 + w1 (t)

that is also a valid asymptotic expansion as t → 1−. Invoking the properties of the functions Rμ (t), it is easy to check that φ ∈ Tr,s by Definition 2.1. Summarizing, we have u ∈ Kr , v ∈ Ks ⇒ φ(t) =

u(t) ∈ Tr,s . u(t) + v(1 − t)

Remark. Recall that the function g(t) = t p , p > 0, does not belong to the set Kp . Consequently, the rational transformation φ(t) = t r /[t r + (1 − t)s ], defined as explained in the first paragraph of this subsection, and that was first proposed in [7], is not in Tr,s . 2.3. Construction of Functions in K μ What remains now is the construction of functions in Kμ as inexpensively as possible. Following Definition 2.1, we listed several conclusions pertaining to the class Kμ . One of these conclusions is that if g ∈ Kμ , then g ν ∈ Kμν , which means that, if we know one single function w ∈ Kσ for some arbitrary σ , we can use it to generate functions u ∈ Kμ for any μ simply via u(t) = [w(t)]μ/σ . Such functions are already known to us from previous work: functions in the class Sm , m being an odd integer, are in Km+1 , as can be verified by the definition of the class Sm . Recall that if ψ ∈ Sm , m a positive integer, then ψ ∈ C ∞ [0, 1];

ψ(0) = 0, ψ(1) = 1;

ψ (t) > 0 on (0, 1),

and ψ(t) has asymptotic expansions as t → 0+ and t → 1− of the forms

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ψ (t) ∼

∞

i t m+2i+1

as t → 0+,

i=0

ψ (t) ∼ 1 −

∞

i (1 − t)m+2i+1

as t → 1−,

i=0

and, for each positive integer k, ψ (k) (t) has asymptotic expansions as t → 0+ and t → 1− that are obtained by differentiating those of ψ(t) term by term k times. (Thus, ψ ∈ Sm,m as well.) Choosing two transformations, (t) ∈ Sk and (t) ∈ Sl , where k and l are odd positive integers, we set u(t) = [(t)]r/(k+1)

and v(t) = [ (t)]s/(l+1) .

A nice feature of this construction is that we can choose (t) and (t) to be simply ψk−1 (t) and ψl−1 (t), namely, the sink−1 and sinl−1 transformations, respectively, which are readily available at a low computational cost, see [9]. For example, 2 1 ψ1 (t) = 21 (1 − cos πt) = sin π2 t , ψ3 (t) = 16 (8 − 9 cos π t + cos 3π t). 2.4. The T r,s - and T r -Transformations Choosing (t) = (t) = ψ1 (t) = (sin π2 t)2 in the preceding subsection, we obtain a remarkably simple and readily computable transformation in the class Tr,s . We then have u(t) = (sin π2 t)r and v(t) = (sin π2 t)s , and recalling that sin[ π2 (1 − t)] = cos π2 t, we finally obtain r sin π2 t (2.14) φ(t) ≡ φr,s (t) = r s . sin π2 t + cos π2 t Therefore, φ

(t) ≡ φr,s (t) =

π 2 π 2 π π r−1 π s−1 s sin 2 t + r cos 2 t cos 2 t sin 2 t r s 2 . 2 sin π2 t + cos π2 t (2.15)

We will call φr,s (t) the T r,s -transformation. Clearly, when r = s, we have u(t) = v(t), hence the T r,r -transformation φr,r (t) is symmetric. We denote φr,r (t) by φr (t) for short and call it the T r -transformation. We use the T r,s - and T r -transformations in our numerical examples in Section 4.


403

3. ANALYSIS OF THE TRAPEZOIDAL RULE WITH CLASS Tr,s TRANSFORMATIONS In this section, we analyze the behavior of the transformed trapezoin [f ] given in (1.4) [with ψ(t) there replaced by φ(t) now] when dal rule Q the integrand f (x) is infinitely differentiable on (0, 1) and possibly has algebraic singularities at x = 0 and/or x = 1. As always, Euler–Maclaurin expansions concerning the trapezoidal rule b approximations of finite-range integrals a u(x) dx are the main analytical tool we use in our study. For the sake of easy reference, we reproduce here the relevant Euler–Maclaurin expansion due to the author (see, Sidi [11, Corollary 2.2]) as Theorem 3.1. This theorem is a special case of another very general theorem from [11], and is expressed in terms of the asymptotic expansions of u(x) as x → a+ and x → b− and is easy to write down and use. Theorem 3.1. Let u ∈ C ∞ (a, b), and assume that u(x) has the asymptotic expansions u (x) ∼ u (x) ∼

∞ i=0 ∞

ci (x − a)γi

as x → a+,

di (b − x)δi

as x → b−,

i=0

where the γi and δi are distinct complex numbers that satisfy −1 < γ0 γ1 γ2 · · · ; limi→∞ γi = +∞, −1 < δ0 δ1 δ2 · · · ; limi→∞ δi = +∞. Assume furthermore that, for each positive integer k, u(k) (x) has asymptotic expansions as x → a+ and x → b− that are obtained by differentiating those of u(x) term by term k times. Let also h = (b − a)/n for n = 1, 2, . . . . Then h

n−1 i=1

b

u(a + ih) ∼

u(x) dx +

a

+

∞ i=0 γi ∈{2,4,6,... } ∞ i=0 δi ∈{2,4,6,... }

where ζ (z) is the Riemann Zeta function.

ci ζ (−γi ) hγi +1

di ζ (−δi ) hδi +1

as h → 0,

404

Sidi

It is clear from Theorem 3.1 that even positive powers of (x − a) and (b − x), if present in the asymptotic expansions of u(x) as

x → a+ and x → b−, do not contribute to the asymptotic expansion of h n−1 i=1 u(a + ih) as h → 0. In addition, if γp is the first of the γi that is different from 2, 4, 6, . . . , and if δq is the first of the δi that is different from 2, 4, 6, . . . , then h

n−1

b

u(a + ih) −

u(x) dx = O hσ +1

as h → 0;

σ = min{γp , δq }.

a

i=1

Here is our main result: Theorem 3.2. Let f ∈ C ∞ (0, 1), and assume that f (x) has the asymptotic expansions f (x) ∼

∞

ci x γi

as x → 0+;

i=0

f (x) ∼

∞

di (1 − x)δi

as x → 1 − .

i=0

Here γi and δi are distinct complex numbers that satisfy −1 < γ0 γ1 γ2 · · · ; limi→∞ γi = +∞, −1 < δ0 δ1 δ2 · · · ; limi→∞ δi = +∞. Assume furthermore that, for each positive integer k, f (k) (x) has asymptotic expansions as x → 0+ and x → 1− that are obtained by differentiating those of f (x) term by term k times. Let us now make the transformation of var1 iable x = φ(t) in I [f ] = 0 f (x) dx, with φ ∈ Tr,s , the result of this being 1 I [f ] = 0 f(t) dt, where f(t) = f φ(t) φ (t). Finally, let us approximate the

n [f ] = h n−1 f(ih), where h = transformed I [f ] via the trapezoidal rule Q i=1 1/n, n = 1, 2, . . . . Then the following hold: (i)

In the worst case, n [f ] − I [f ] = O hω as h → 0; Q ω = min{(γ0 + 1)r, (δ0 + 1)s}.

(ii)

If γ0 and δ0 are real, and if r = (2k + 1)/(γ0 + 1) and s = (2l + 1)/ (δ0 + 1), where k and l are positive integers, then, at worst, n [f ] − I [f ] = O hω as h → 0, Q


405

where now ω = min{(γ0 + 2)r, (γ1 + 1)r, (δ0 + 2)s, (δ1 + 1)s}. Remark. If f (x) = x μ (1 − x)ν g(x), g(x) being infinitely differentiable on [0, 1], then f (x) satisfies the conditions of the theorem. In such a case, if f (x) has full Taylor series at x = 0 and x = 1, we have γs = μ + s and δs = ν + s, s = 0, 1, . . . . Note that this f (x) has an algebraic branch singularity at x = 0 if μ is not a positive integer. Similarly, it has an algebraic branch singularity at x = 1 if ν is not a positive integer. Proof. To keep the presentation simple, we do the proof with φ(t) as in subsection 2.2. It is clear from Theorem 3.1 that we need to the analyze asymptotic expansions of the transformed integrand f(t) = f φ(t) φ (t) as t → 0 and t → 1. Because φ(t) → 0 as t → 0+ and φ(t) → 1 as t → 1−, we first have the genuine asymptotic expansions f(t) ∼

∞

(0)

Ki (t) = ci [φ(t)]γi φ (t), (0)

as t → 0+;

Ki (t)

i=0

and f(t) ∼

∞

(1)

Ki (t) = di [1 − φ(t)]δi φ (t). (1)

as t → 1−;

Ki (t)

i=0

Invoking the relevant asymptotic expansions of φ(t) in subsection 2.2, and (0) re-expanding, we have that Ki (t) contributes the sum ∞ γi ∞ (0) Rkr (t) Rkr−1 (t) as t → 0+ Ki (t) = k=1

=

∞

k=1

R(γi +k)r−1 (t)

as t → 0+,

(3.1)

k=1 (1)

whereas Ki (t) contributes the sum (1) Ki (t)

= =

∞

k=1 ∞

δi Rks (1 − t)

∞

Rks−1 (1 − t) as t → 1−

k=1

R(δi +k)s−1 (1 − t) as t → 1−.

k=1

(3.2)

406

Sidi

To see this, we proceed as follows: As z → 0+, and omitting the argument z in Rp (z) (as z → 0+), and taking p > 0 so that Rp (z) = O(zp ) = o(1) as z → 0+, there holds σ ∞ ∞ σ ∞ ∞ Rp−1 1 + Rkp Rkp−1 = Rp 1 + Rkp Rkp k=1

k=1

= Rpσ 1 +

k=1 ∞

k=1

σ ∞ Rp−1 1 + Rkp Rkp

k=1 ∞

= Rσp 1 +

Rp−1 1 +

Rkp

k=1

k=1 ∞

Rkp

k=1

∞ ∞ Rkp 1 + Rkp = Rσp Rp−1 1 + = Rσp+p−1 1 +

k=1 ∞

k=1

Rkp

k=1

=

∞

R(σ +k)p−1 .

k=1

We note that the most dominant terms in the asymptotic expansions (0) (1) of f(t) as t → 0+ and as t → 1− come from K0 (t) and from K0 (t), respectively, and they are α t (γ0 +1)r−1 and β (1 − t)(δ0 +1)s−1 , respectively, for some α and β . Thus, by Theorem 3.1, the most dominant terms in n [f ] − I [f ] as h → 0 are αh(γ0 +1)r coming from the endthe expansion of Q point t = 0, and βh(δ0 +1)s coming from the endpoint t = 1, for some α and β. This proves part (i) of the theorem. To prove part (ii), we need a more refined study of the terms from (0) (1) K0 (t) and from K0 (t). The first term in the summation of (3.1), namely, R(γ0 +1)r−1 (t), contains only even positive powers of t in its asymptotic expansion as t → 0+ provided r is chosen such that (γ0 + 1)r = 2k + 1, where k is a positive integer. With this r, R(γ0 +1)r−1 (t) contributes nothing n [f ] − I [f ] by Theorem 3.1. The to the Euler–Maclaurin expansion of Q (0) most dominant terms that can contribute are (i) R(γ0 +2)r−1 (t) of K0 (t) (0) and (ii) R(γi +1)r−1 (t) of Ki (t), i 1, for which γi = γ1 . These terms contribute αh(γ0 +2)r and βi h(γi +1)r , for some α and βi . Similarly, the first term in the summation of (3.2), namely, R(δ0 +1)r−1 (1 − t), contains only even positive powers of (1 − t) in its asymptotic expansion as t → 1− provided s is chosen such that (δ0 + 1)s = 2l+1, where l is a positive integer. With this s, R(δ0 +1)s−1 (1 − t) contributes


407

n [f ] − I [f ] by Theorem nothing to the Euler–Maclaurin expansion of Q 3.1. The most dominant terms that can contribute are (i) R(δ0 +2)s−1 (1 − t) (1) (1) of K0 (t) and (ii) R(δi +1)s−1 (1 − t) of Ki (t), i 1, for which δi = δ1 . These terms contribute αh(δ0 +2)s and βi h(δi +1)s , for some α and βi . Combining the above, we complete the proof of the result in part (ii).

Remark. Note that the result in part (ii) of Theorem 3.2 is made possible by our definition of the class Tr,s transformations, where we have excluded the powers t r+2i+1 , i = 0, 1, . . . , r/2 − 1, and (1 − t)s+2i+1 , i = 0, 1, . . . , s/2 − 1, from the asymptotic expansions of φ(t) as t → 0+ and t → 1−. Corollary 3.3. In case f (x) = x μ (1 − x)ν g(x), g(x) being infinitely differentiable on [0, 1], the following hold: (i)

In the worst case, n [f ] − I [f ] = O hω as h → 0; Q ω = min{(μ + 1)r, (ν + 1)s}.

(ii)

If μ and ν are real, and if r = (2k + 1)/(μ + 1) and s = (2l + 1)/(ν + 1), where k and l are positive integers, then we have the optimal result n [f ] − I [f ] = O hω as h → 0; Q ω = min{(μ + 2)r, (ν + 2)s}.

Corollary 3.4. When μ = ν = c, let φ ∈ Tr,r in Corollary 3.3. Then the following hold: (i)

In the worst case, n [f ] − I [f ] = O hω Q

(ii)

as h → 0;

ω = (c + 1)r.

If c is real, and if r = (2k + 1)/(c + 1), where k is a positive integer, then we have the optimal result n [f ] − I [f ] = O hω Q

as h → 0;

ω = (c + 2)r.

In case c = 0 (that is, f ∈ C ∞ [0.1]) in Corollary 3.4, we have an error of O(hr ) with arbitrary r, whereas the error is O(h2r ) when r is a positive

408

Sidi

odd integer, and this result is the same as that given in [9, Theorem 3.5] pertaining to class Sr−1 variable transformations. When μ = ν in part (ii) of Corollary 3.3, we choose the integers k and l such that (μ + 2)r ≈ (ν + 2)s, that is, 2k + 1 ν + 2 μ + 1 ≈ · . 2l + 1 ν + 1 μ + 2 (Thus, by choosing k first, we can determine l, and vice versa.) This guarantees that the singularities of the transformed integrand f(t) = f (φ(t))φ (t) at the endpoints are of approximately the same strength. 3.1. An Improvement of the Numerical Quadrature Approximation to I [f ] In case f (x) = x μ (1 − x)v g(x), with μ, v real and > −1, g ∈ C ∞ [0, 1], and g(0) and g(1) available, we can exploit the result of Corollary 3.3 to improve the accuracy of the approximation to I [f ] as follows: Let v(x) be the linear function interpolating g(x) at x = 0 and x = 1; that is, v(x) = g(0) + [g(1) − g(0)]x. Let also g0 (x) = g(x) − v(x). Then f (x) = p(x) + f0 (x);

p(x) = x μ (1 − x)v v(x), f0 (x) = x μ (1 − x)v g0 (x),

and I [f ] = I [p] + I [f0 ]. We now compute I [p] exactly; we have I [p] = g(0)B(μ + 1, ν + 1) + [g(1) − g(0)]B(μ + 2, ν + 1), where B(x, y) is the Beta function (see, for example, Abramowitz and Stegun [1, Formula 6.2.1]) that is given as in

1

B(x, y) = 0

ξ x−1 (1 − ξ )y−1 dξ =

Γ (x)Γ (y) , Γ (x + y)

Γ (z) being the Gamma function. Next, because g0 (0) = g0 (1) = 0, we have that f0 (x) is now of the form f0 (x) = x μ+1 (1 − x)ν+1 g (x), g (x) being a regular function on [0, 1]. In view of this, we apply the trapezoidal rule to 1 the integral I [f0 ] = 0 f0 (t) dt, where f0 (t) = f0 φ(t) φ (t), where φ ∈ Tr,s with r = (2k + 1)/(μ + 2) and s = (2l + 1)/(ν + 2), k and l being positive integers. Then, by Corollary 3.3, we have the optimal result n [f0 ] − I [f ] = O hω as h → 0; ω = min{(μ + 3)r, (ν + 3)s}. I [p] + Q


409

In case μ = ν = c, and φ ∈ Tr,r , with r = (2k + 1)/(c + 2), where k is a positive integer, this result becomes, n [f0 ] − I [f ] = O hω as h → 0; ω = (c + 3)r. I [p] + Q If, furthermore, c = 0 [that is, f ∈ C ∞ [0, 1] and g(x) = f (x)], and 2r is a positive odd integer at least 3, then n [f0 ] − I [f ] = O h3r as h → 0. I [p] + Q 4. NUMERICAL EXAMPLES In this section, we provide two examples, considered already in [16], to illustrate the validity of the results of the preceding section. The computations for these examples were done in quadruple-precision arithmetic (approximately 35 decimal digits). Example 4.1.

Consider the integral 0

1

x μ dx =

1 , 1+μ

μ > −1.

In this case, we have f (x) = x μ

and f (x) =

∞ s=0

(−1)s

μ (1 − x)s . s

Of these, the first is a single-term series representing f (x) asymptotically as x → 0+ with γ0 = μ, while the second is a (convergent) series representing f (x) asymptotically as x → 1− with δs = s, s = 0, 1, . . . . (Note that, in the notation of Corollary 3.3, ν = 0 now.) Thus, if we choose r and s arbitrarily, we will obtain, by part (i) of Theorem 3.2 and Corollary 3.3, n [f ] − I [f ] = O hω as h → 0; ω = min{(μ + 1)r, s}. Q In case r = (2k + 1)/(μ + 1) and s = 2l + 1, with k, l positive integers, we will obtain, by part (ii) of Theorem 3.2, n [f ] − I [f ] = O hω as h → 0, ω = min{(μ + 2)r, 2s}. Q This is so because the asymptotic expansion of f (x) as x → 0+ consists of only the term x μ . In our computations, we have taken μ = 0.1.

410

Sidi

n [f ] for n = 2k , k = In Table I, we give the relative errors in the Q r,s 1, . . . , 10, obtained with the T -transformation. In column j of this table, we have chosen r = (j + 1.9)/(μ + 1) and s = (j + 1.9)/(ν + 1) when j is odd, while r = (j + 1)/(μ + 1) and s = (j + 1)/(ν + 1) when j is even. The superior convergence of the columns with j an even integer is clearly demonstrated. [Note that the r (the s), hence the clusterings of the effective abscissas xi = ψ(i/n) near x = 0 (near x = 1) with j = 2k − 1 and j = 2k are approximately the same for each k.] In Table II, we give the numbers 1 |Q2k [f ] − I [f ]| ρr,s,k = , · log 2k+1 [f ] − I [f ]| log 2 |Q for the same values of r and s and for k = 1, 2, . . . , 9. It is seen that, with increasing k, the ρr,s,k are tending to min{(μ + 1)r, s} when j an odd integer, and to min{(μ + 2)r, 2s} when j is an even integer, completely in accordance with Theorem 3.2 and Corollary 3.3. (With the floating-point arithmetic we are using, this convergence seems to be less visible for relatively large r and s in the columns with even j .)

Example 4.2. Consider the integral 1 0

f (x) = 0,

f (x) =

d μ+1 (1 − x)ν+1 w(x) , x dx

μ, ν > 0,

w ∈ C ∞ [0, 1].

In this case, we have f (x) = x μ (1 − x)ν g(x), where

g(x) = (μ + 1)(1 − x) − (ν + 1)x w(x) + x(1 − x)w (x).

If case w(0) and w(1) are both nonzero, we have that g(0) and g(1) are both nonzero as well, and this implies that γ0 = μ and δ0 = ν. Thus, if we choose r and s arbitrarily, we will obtain, by part (i) of Theorem 3.2 and Corollary 3.3, n [f ] − I [f ] = O hω as h → 0; ω = min{(μ + 1)r, (ν + 1)s}. Q In case r = (2k + 1)/(μ + 1) and s = (2l + 1)/(ν + 1), with k, l positive integers, we will obtain, by part (ii) of Theorem 3.2 and Corollary 3.3, n [f ] − I [f ] = O hω as h → 0, ω = min{(μ + 2)r, (ν + 2)s}. Q In our computations, we have taken μ = 0.1 and ν = 0.4 and w(x) = 1/(1 + x).

1.18D−01 4.94D−04 1.77D−04 2.44D−05 3.29D−06 4.42D−07 5.92D−08 7.93D−09 1.06D−09 1.42D−10

j =1

1.57D−01 1.26D−03 3.41D−06 6.33D−08 1.14D−09 2.03D−11 3.64D−13 6.57D−15 1.19D−16 2.18D−18

j =2

8.88D−01 9.11D−02 5.45D−04 1.04D−07 3.26D−09 1.09D−10 3.67D−12 1.23D−13 4.12D−15 1.38D−16

j =3 9.26D−01 1.01D−01 6.88D−04 1.27D−08 2.26D−14 2.83D−17 3.56D−20 4.52D−23 5.81D−26 7.54D−29

j =4 1.65D+00 3.62D−01 1.32D−02 7.06D−06 8.55D−12 6.59D−14 5.53D−16 4.63D−18 3.88D−20 3.25D−22

j =5 1.69D+00 3.78D−01 1.46D−02 8.83D−06 1.21D−12 1.62D−22 1.43D−26 1.28D−30 9.63D−34 2.89D−34

j =6 2.40D+00 7.09D−01 6.21D−02 2.11D−04 1.23D−09 6.99D−17 1.47D−19 3.08D−22 6.45D−25 1.35D−27

j =7 2.44D+00 7.27D−01 6.57D−02 2.40D−04 1.62D−09 2.96D−19 1.29D−32 0.00D+00 3.85D−34 3.85D−34

j =8 3.14D+00 1.07D+00 1.54D−01 1.62D−03 1.04D−07 1.26D−15 6.04D−23 3.17D−26 1.66D−29 7.90D−33

j =9 3.17D+00 1.09D+00 1.59D−01 1.75D−03 1.24D−07 1.80D−15 1.87D−32 1.93D−34 5.78D−34 1.54D−33

j = 10

n [f ] for the integral of Example 4.1 Obtained with n = 2k , k = 1(1)10, and with the T r,s -transformation Errors in the Rules Q

In column j , we have chosen r = (j + 1.9)/(μ + 1) and s = j + 1.9 when j is odd, while r = (j + 1)/(μ + 1) and s = j + 1 when j is even.

2 4 8 16 32 64 128 256 512 1024

n

Table I.

Symmetric and Nonsymmetric Periodizing Variable Transformations 411

j =1

7.904 1.482 2.855 2.892 2.898 2.900 2.900 2.900 2.900 2.9

n

1 2 3 4 5 6 7 8 9 ∞

3.284 7.386 12.357 4.994 4.896 4.899 4.900 4.900 4.900 4.9

j =3 3.197 7.198 15.726 19.098 9.642 9.637 9.620 9.603 9.589 9.545· · ·

j =4

The Numbers ρr,s,k = log1 2 · log

6.959 8.530 5.753 5.799 5.807 5.801 5.792 5.784 5.776 5.727· · ·

j =2

Table II.

2.189 4.777 10.868 19.656 7.020 6.897 6.899 6.900 6.900 6.9

j =5

k [f ]−I [f ]| |Q 2 k+1 [f ]−I [f ]| |Q 2

2.158 4.692 10.694 22.799 32.800 13.471 13.447 10.372 1.737 13.363· · ·

j =6 1.760 3.513 8.199 17.388 24.072 8.891 8.899 8.900 8.900 8.9

j =7 1.746 3.468 8.099 17.175 32.349 44.384 * * 0.000 17.181· · ·

j =8 1.552 2.800 6.570 13.924 26.296 24.317 10.898 10.899 11.036 10.9

j =9

n [f ] as in Table I, for k = 1(1)9 , with r, s, f (x), and Q

1.544 2.772 6.506 13.786 26.037 56.420 6.600 −1.585 −1.415 21

j = 10

412 Sidi


413

In Table III, we give the absolute errors (recall that I [f ] = 0) in the n [f ] for n = 2k , k = 1, . . . , 10, obtained with the T r,s -transformation. In Q column j of this table, we have chosen r = (j + 1.9)/(μ + 1) and s = (j + 1.9)/(ν + 1) when j is odd, while r = (j + 1)/(μ + 1) and s = (j + 1)/(ν + 1) when j is even. The superior convergence of the columns with j an even integer is again clearly demonstrated. [Note that the r (the s), hence the clusterings of the effective abscissas xi = ψ(i/n) near x = 0 (near x = 1) with j = 2k − 1 and j = 2k are approximately the same for each k.] In Table IV, we give the numbers ρr,s,k defined in the preceding example for the same values of r and s and for k = 1, 2, . . . , 9. It is seen that, with increasing k, the ρr,s,k are tending to min{(μ + 1)r, (ν + 1)s} when j an odd integer, and to min{(μ + 2)r, (ν + 2)s} when j is an even integer, completely in accordance with Theorem 3.2 and Corollary 3.3. (As in the preceding example, with the floating-point arithmetic we are using, this convergence seems to be less visible for relatively large r and s in the columns with even j .) 5. CONCLUDING REMARKS In this work, we presented a class of variable transformations, which we denoted Tr,s , whose members φ(t) have asymptotic expansions as t → 0+ and t → 1− of the forms given in (2.4). We also noted that what gives these transformations their exceptional effectiveness is the fact that their asymptotic expansions include the powers t r+2i , i = 0, 1, . . . , r/2, and (1 − t)s+2i , i = 0, 1, . . . , s/2, but exclude the powers t r+2i+1 , i = 0, 1, . . . , r/2 − 1, and (1 − t)s+2i+1 , i = 0, 1, . . . , s/2 − 1. To see what happens if we do include the powers t r+2i+1 , i = 0, 1, . . . , r/2 − 1, and (1 − t)s+2i+1 , i = 0, 1, . . . , s/2 − 1, let us modify the definition of class Tr,s by replacing the condition in (2.4) in Definition 2.1 by the following analogous condition: φ(t) =

2 r/2

αi t r+i + O t 2r

as t → 0+, (5.1)

i=0

φ(t) =

2 s/2

βi (1 − t)s+i + O (1 − t)2s

as t → 1−,

i=0

which can be realized by taking, for example, φ(t) = t r /[t r + (1 − t)s ], which is one of the variable transformations proposed in [7]. For convenience, let us denote by Tr,s the resulting modification of class Tr,s . With class Tr,s variable transformations φ(t), the quality of the trapezoidal rule on

9.10D−02 2.50D−04 3.58D−05 5.69D−06 7.82D−07 1.05D−07 1.41D−08 1.89D−09 2.53D−10 3.39D−11

j =1

9.24D−02 5.79D−05 5.30D−06 9.04D−08 1.38D−09 1.65D−11 3.75D−14 7.04D−15 3.52D−16 1.29D−17

j =2

9.23D−02 2.44D−02 1.62D−04 2.35D−08 7.79D−10 2.61D−11 8.74D−13 2.93D−14 9.80D−16 3.28D−17

j =3 9.09D−02 2.51D−02 2.01D−04 1.96D−10 1.98D−13 4.64D−16 1.14D−18 2.87D−21 7.39D−24 1.92D−26

j =4 3.94D−02 1.50D−02 1.06D−04 4.38D−07 1.88D−12 1.57D−14 1.32D−16 1.10D−18 9.23D−21 7.73D−23

j =5 3.53D−02 1.34D−02 7.88D−05 6.90D−07 5.67D−15 1.23D−20 2.98D−24 7.24D−28 1.76D−31 1.21D−34

j =6 6.63D−02 3.40D−02 1.18D−02 9.45D−05 1.48D−11 1.68D−17 3.51D−20 7.34D−23 1.54D−25 3.22D−28

j =7 7.29D−02 3.72D−02 1.29D−02 1.12D−04 2.63D−11 8.62D−23 8.38D−30 2.03D−34 1.51D−35 1.60D−35

j =8 2.19D−01 1.10D−01 4.36D−02 1.22D−03 8.07D−09 5.50D−18 1.44D−23 7.54D−27 3.95D−30 2.07D−33

j =9 2.28D−01 1.14D−01 4.56D−02 1.34D−03 9.68D−09 8.44D−18 5.63D−36 5.60D−35 4.58D−35 5.77D−35

j = 10

n [f ] for the Integral of Example 4.2, Obtained with n = 2k , k = 1(1)10, and with the T r,s -transformation Errors in the Rules Q

In column j , we have chosen r = (j + 1.9)/(μ + 1) and s = (j + 1.9)/(ν + 1) when j is odd, while r = (j + 1)/(μ + 1) and s = (j + 1)/(ν + 1) when j is even.

2 4 8 16 32 64 128 256 512 1024

n

Table III.

414 Sidi

j =1

8.508 2.803 2.655 2.864 2.894 2.899 2.900 2.900 2.900

2.9

k

1 2 3 4 5 6 7 8 9

∞

5.142· · ·

10.639 3.450 5.874 6.033 6.389 8.778 2.414 4.321 4.776

4.9

1.920 7.236 12.748 4.915 4.901 4.900 4.900 4.900 4.900

j =3

8.571· · ·

1.856 6.966 19.970 9.948 8.738 8.673 8.628 8.602 8.587

j =4

The Numbers ρp,q,k = log1 2 · log

j =2

Table IV.

6.9

1.391 7.140 7.924 17.831 6.901 6.900 6.900 6.900 6.900

j =5

k [f ]−I [f ]| |Q 2 k+1 [f ]−I [f ]| |Q 2

12

1.402 7.405 6.835 26.860 18.814 12.012 12.007 12.003 10.510

j =6

8.9

0.965 1.531 6.960 22.609 19.749 8.900 8.900 8.900 8.900

j =7

15.428· · ·

0.971 1.531 6.850 22.015 38.152 23.295 15.331 3.752 −0.088

j =8

10.9

0.998 1.333 5.160 17.203 30.452 18.541 10.900 10.900 10.896

j =9

n [f ] as in Table III, for k = 1(1)9 , with p, q, f (x), and Q

18.857· · ·

0.999 1.322 5.089 17.080 30.095 60.377 −3.314 0.290 −0.331

j = 10

Symmetric and Nonsymmetric Periodizing Variable Transformations 415

416

Sidi

1 the transformed integrals 0 f (φ(t))φ (t) dt drops drastically. In particular, Corollary 3.3 and Corollary 3.4 assume the following forms: Proposition 5.1. In case f (x) = x μ (1 − x)ν g(x), g(x) being infinitely differentiable on [0, 1], the following hold: (i)

(ii)


as h → 0;

ω = min{(μ + 1)r, (ν + 1)s}.

If μ and ν are real, and if r = (2k + 1)/(μ + 1) and s = (2l + 1)/(ν + 1), where k and l are positive integers, then we have the optimal result n [f ] − I [f ] = O hω as h → 0; Q ω = min{(μ + 1)r + 1, (ν + 1)s + 1}.

Proposition 5.2. When μ = ν = c, let φ ∈ Tr,r in Corollary 5.1. Then the following hold: (i)


(ii)

as h → 0;

ω = (c + 1)r.

If c is real, and if r = (2k + 1)/(c + 1), where k is a positive integer, then we have the optimal result n [f ] − I [f ] = O hω as h → 0; ω = (c + 1)r + 1. Q

The proofs of these propositions are the same as those of Corollaries 3.3 and 3.4 and are left to the reader. As can be seen by comparing parts (ii) of Corollaries 3.3 and 3.4 with n [f ] drops with optiparts (ii) of Propositions 5.1 and 5.2, the quality of Q mal values of r and s when we chose φ ∈ Tr,s since r > 1. That is, higher accuracy is achieved with class Tr,s transformations than with class Tr,s transformations when r and s assume their optimal values. For example, if we let c = 0 so that f ∈ C ∞ [0, 1], and take r = s = 2k + 1 with k a posin [f ] is O(hr+1 ) in Proposition 5.2, whereas it is tive integer, the error in Q 2r O(h ) in Corollary 3.4. REFERENCES 1. Abramowitz, M., and Stegun, I. A. (1964). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Number 55 in Nat. Bur. Standards Appl. Math. Series. US Government Printing Office, Washington, DC.


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