Magic efficiency of approximation of smooth functions by weighted means of two $N$-point Padé approximants
We consider the approximation of smooth functions by two weighted $N$-point Pad´e approximants. We present numerical examples and the inequalities between the Stietjes function and its $N$-point Padé approximant. Квартиры посуточно - онлайн-бронирование
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| author | Gilewicz, J. Jedynak, R. Гілевич, Я. Я. Єдинак, Р. |
| author_facet | Gilewicz, J. Jedynak, R. Гілевич, Я. Я. Єдинак, Р. |
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| description | We consider the approximation of smooth functions by two weighted $N$-point Pad´e approximants. We present numerical
examples and the inequalities between the Stietjes function and its $N$-point Padé approximant.
Квартиры посуточно - онлайн-бронирование
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UDC 517.5
R. Jedynak, J. Gilewicz (K. Pulaski Univ. Technology and Humanities, Poland)
MAGIC EFFICIENCY OF APPROXIMATION OF SMOOTH FUNCTIONS
BY WEIGHTED MEANS OF TWO \bfitN -POINT PADÉ APPROXIMANTS
ЧАРIВНА ЕФЕКТИВНIСТЬ НАБЛИЖЕННЯ ГЛАДКИХ ФУНКЦIЙ
ЗВАЖЕНИМИ СЕРЕДНIМИ ДВОХ \bfitN -ТОЧКОВИХ ПАДЕ АПРОКСИМАЦIЙ
We consider the approximation of smooth functions by two weighted N -point Padé approximants. We present numerical
examples and the inequalities between the Stietjes function and its N -point Padé approximant.
Статтю присвячено наближенню гладких функцiй двома N -точковими наближеннями Паде з вагами. Наведено
числовi приклади, нерiвностi мiж функцiєю Стiльтьєса та її N -точковим наближенням Паде.
We present the new developments of this intuitively discovered method of approximation of real
functions f characterized by the two-sided estimate (TSE) property by weighted means of two N -
point Padé approximants (NPA). This method was first presented in [13] and is briefly explained in
the Introduction. In Section 2 we recall the inequalities between the Stieltjes functions and NPAs
proved in [6, 7]. In the review article [6] these inequalities where extended outside the interval
[x1, xN ] of definition of the NPA. Unfortunately, in general these extended inequalities do not hold
for the entire interval ]xN ,\infty [, which is illustrated by an example. In Section 3 we prove a surprising
result: the weights used in the weighted approximation are convex. In Section 4 we discuss some
questions related to the rescaling of the reference functions used in our method, the problem of the
transformation of the function to be approximated, and the problem of the TSE property. Section 5
contains some illustrative examples.
1. Introduction. Let f be a function we wish to approximate on the interval [x1, xN ] knowing
p1 > 1, p2, . . . , pN coefficients of Taylor series expansion of f at the points x1, x2, . . . , xN . We
start by computing two neighboring NPAs of f, namely f1 = [m/n] and f2 = [m - 1/n] of f. The
property TSE says that these two NPA f1 and f2 (the second being computed with a reduced amount
of information by removing one coefficient from the expansion of f at one point xi, in general at x1)
bound f in each interval [xi, xi+1] on opposite sides. The TSE property holds for Stieltjes functions
[7], but also for many other functions of practical interest (it holds for a great number of convex
functions, but not all). In such a case, further steps become relatively simple. We select a known
function s having the TSE property rescaling s such that its values s(xi) are as close as possible to
the values f(xi). We then compute the approximants s1 = [m/n] and s2 = [m - 1/n] using the
values at the points xi and determine for all x the weight function \alpha from the equation
s(x) = \alpha (x)s1(x) + (1 - \alpha (x))s2(x). (1)
Applying this weight to calculate the weighted mean \alpha f1 + (1 - \alpha )f2 we obtain a significantly
improved approximation of f. This intuitive method of weighted means approximation was presented
in [13] and gives more accurate results than all previously proposed methods: see the examples of
the non-Stieltjes functions e - x, e - x/x and another one, playing an important role in tribology
[2, 12, 15, 18]. The similar idea, the use of weight functions, which is a unit factorization to the
c\bigcirc R. JEDYNAK, J. GILEWICZ , 2018
1192 ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 9
MAGIC EFFICIENCY OF APPROXIMATION OF SMOOTH FUNCTIONS BY WEIGHTED MEANS . . . 1193
Taylor series, was proposed by O. M. Lytvyn and V. L. Rvachev for approximations by partial sums
of the Taylor series [17], and then it was used for continuous fractions. The issue of continuous
fractions is broadly discussed in our previous paper [8].
2. Inequalities between a Stieltjes function and its \bfitN -point Padé approximant. Let f be an
analytic function at N different real points
- R < x1 < x2 < . . . < xN < \infty
having the Taylor expansion
pj - 1\sum
k=0
ck(xj)(x - xj)
k +O
\Bigl(
(x - xj)
pj
\Bigr)
, j = 1, . . . , N. (2)
Then the NPA to f, if it exists, is a rational function Pm/Qn denoted as follows [9, 14]:
[m/n]p1p2...pNx1x2...xN
(x) =
a0 + a1x+ . . .+ amxm
1 + b1x+ . . .+ bnxn
, m+ n+ 1 = p = p1 + p2 + . . .+ pN , (3)
and satisfying the following relations:
f(x) - [m/n](x) = O
\Bigl(
(x - xj)
pj
\Bigr)
, j = 1, 2, . . . , N,
where each pj represents the number of coefficients ck(xj) of the expansion (2) actually used for
the computation of the NPA given by (3). In the following we deal only with subdiagonal [n - 1/n]
and diagonal [n/n] NPAs. Let us introduce a nondecreasing step-wise function L
L(x) =
N\sum
j=1
pjH(x - xj),
where H is the Heaviside function. The value L(x) denotes the total number of given coefficients
of the power series expansions of f at all points xj \leq x:
L(xk) = p1 + p2 + . . .+ pk, L(xN ) = p =
N\sum
j=1
pj .
Theorem 1 (M. Barnsley). Let s be a Stieltjes function defined as follows:
s(z) =
1/R\int
0
d\mu (t)
1 + tz
, z \in \BbbC \setminus ] - \infty , - R], d\mu \geq 0,
then the subdiagonal [k - 1/k] and diagonal [k/k] NPA to s obey the following inequality:
( - 1)L(x)[m/n](x) \leq ( - 1)L(x)s(x), x \in ] - R,\infty [.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 9
1194 R. JEDYNAK, J. GILEWICZ
Notice that the parity of L controls the position of the NPA with respect to s and so, playing
with this parity, we can obtain two sided estimates of s. In particular, for x < x1, all NPAs are
less than s. Suppose, but it is without importance, that p = L(xN ) = 2k + 1. Then our NPA is
[k/k]p1...pNx1...xN . Removing one piece of information (one coefficient) from the development of s at x1
we obtain [k - 1/k]p1 - 1...pN
x1...xN which, starting from x1, bounds s on the opposite side with respect
to [k/k]. Immediately we can imagine a nice improvement of the approximation by computing the
mean value of both these NPAs. However, the result is not so nice: [k/k] is in fact better that this
mean because
\bigm| \bigm| s(x) - [k/k]
\bigm| \bigm| \ll \bigm| \bigm| s(x) - [k - 1/k]
\bigm| \bigm| , which shows that the convergence of the Padé
approximant to Stieltjes functions is very fast. This observation has suggested our weighted means
approximation method.
The following theorem was proved in [6].
Theorem 2. In the two first inequalities, the second NPA is computed with one piece of infor-
mation removed at x1. In the two last inequalities, the second NPA is computed with one piece of
information removed at an arbitrary point xr with r \leq i :
k \geq 1, p = 2k + 1, x \in ] - R, x1[ :
0 < s(x) - [k/k]p1...pNx1...xN
<
x1 - x
x1 +R
\Bigl\{
s(x) - [k - 1/k]p1 - 1...pN
x1...xN
\Bigr\}
,
k \geq 0, p = 2k + 2, x \in ] - R, x1[ :
0 < s(x) - [k/k + 1]p1...pNx1...xN
<
x1 - x
x1 +R
\Bigl\{
s(x) - [k/k]p1 - 1...pN
x1...xN
\Bigr\}
,
k \geq 1, p = 2k + 1, 1 \leq i \leq N - 1, x \in ]xi, xi+1[ :
0 < ( - 1)L(x)
\Bigl\{
s(x) - [k/k]p1...pNx1...xN
\Bigr\}
<
xi+1 - x
xi+1 - xi
( - 1)L(x) - 1
\Bigl\{
s(x) - [k - 1/k]p1...pr - 1...pN
x1...xr......xN
\Bigr\}
,
k \geq 0, p = 2k + 2, 1 \leq i \leq N - 1, x \in ]xi, xi+1[ :
0 < ( - 1)L(x)
\Bigl\{
s(x) - [k/k + 1]p1...pNx1...xN
\Bigr\}
<
xi+1 - x
xi+1 - xi
( - 1)L(x) - 1
\Bigl\{
f(x) - [k/k]p1...pr - 1...pN
x1...xr......xN
\Bigr\}
.
The two last inequalities presented in the original theorem [6] (Theorem 4.3, inequalities (74)
and (75)) for the interval ]xN ,\infty [ are not valid in general. They depend on each particular case: on
the Stieltjes function s and on the chosen NPAs. The classical inequalities between the ordinary PAs,
i.e., the 1-point Padé approximant to the Stieltjes functions [5, p. 264–265] in ]xN ,\infty [ \equiv ]0,\infty [ do
not imply general inequalities for errors in this interval.
Example. We consider three classical Padé Padé approximants to the Stieltjes function
s(x) =
\mathrm{l}\mathrm{n}(1 + x)
x
= 1 - 1
2
x+
1
3
x2 - . . . :
[0/0](x) = 1, [0/1](x) =
2
2 + x
and [1/1](x) =
6 + x
6 + 4x
.
In this case N = 1, x1 = xN = 0 and s(x) - [0/1](x) < [0/0](x) - s(x), that is,
\mathrm{l}\mathrm{n}(1 + x)
x
- 2
2 + x
< 1 - \mathrm{l}\mathrm{n}(1 + x)
x
(4)
holds for all x > 0, as shown in Fig. 1.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 9
MAGIC EFFICIENCY OF APPROXIMATION OF SMOOTH FUNCTIONS BY WEIGHTED MEANS . . . 1195
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 2 4 6 8 10
x
s(x)−[0/1](x)
[0/0](x)−s(x)
Fig. 1. In this case inequality (4) holds on all interval ]xN ,\infty [ = ]0,\infty [.
0
0.02
0.04
0.06
0.08
0.1
0.12
0 2 4 6 8 10
x
s(x)−[0/1](x)
[1/1](x)−s(x)
Fig. 2. In this case inequality (5) holds only on the interval ]x1, 5 + \varepsilon [ = ]0, 5 + \varepsilon [ \not = ]x1,\infty [.
On the contrary, the inequality [1/1](x) - s(x) < s(x) - [0/1](x), that is,
6 + x
6 + 4x
- \mathrm{l}\mathrm{n}(1 + x)
x
<
\mathrm{l}\mathrm{n}(1 + x)
x
- 2
2 + x
(5)
is satisfied only for x \in ]0, 5 + \varepsilon [ and is false for x = 6, as shown in Fig. 2.
3. Convexity of the weight function. We prove a new surprising result: the weight \alpha (x) =
=
s(x) - s2(x)
s1(x) - s2(x)
defined by s and two oscillating NPAs is a smooth convex function. Then, we
can use \alpha in the formulae (1) continuously. We prove this property in the particular case of the
Stieltjes function s(x) =
\mathrm{l}\mathrm{n}(1 + x)
x
and two NPAs, s1(x) = [2/1]211012(x) and s2(x) = [1/1]111012(x).
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 9
1196 R. JEDYNAK, J. GILEWICZ
The power series expansions of s at x = 0, 1, 2 are
s(x) =
\mathrm{l}\mathrm{n}(1 + x)
x
= 1 - 1
2
x+
1
3
x2 + . . . =
= \mathrm{l}\mathrm{n} 2 +
\biggl(
1
2
- \mathrm{l}\mathrm{n} 2
\biggr)
(x - 1) +
\biggl(
\mathrm{l}\mathrm{n} 2 - 5
8
(x - 1)2
\biggr)
+ . . . =
=
\mathrm{l}\mathrm{n} 3
2
+
\biggl(
1
6
- \mathrm{l}\mathrm{n} 3
4
\biggr)
(x - 2) +
\biggl(
\mathrm{l}\mathrm{n} 3
8
- 1
9
\biggr)
(x - 2)2 + . . . ,
s1(x) = [2/1]211012(x) =
=
(4 - 8 \mathrm{l}\mathrm{n} 2 + 2 \mathrm{l}\mathrm{n} 3) + ( - 6 + 12 \mathrm{l}\mathrm{n} 2 - 2 \mathrm{l}\mathrm{n} 3)x+ (2 - 4 \mathrm{l}\mathrm{n} 2 + \mathrm{l}\mathrm{n} 2\times \mathrm{l}\mathrm{n} 3)x2
(4 - 8 \mathrm{l}\mathrm{n} 2 + 2 \mathrm{l}\mathrm{n} 3) + ( - 4 + 8 \mathrm{l}\mathrm{n} 2 - \mathrm{l}\mathrm{n} 3)x
=
=
1.+ .184866x - .017006x2
1.+ .684866x
,
s2(x) = [1/1]111012(x) =
2(2 \mathrm{l}\mathrm{n} 2 - \mathrm{l}\mathrm{n} 3) + ( - 2 \mathrm{l}\mathrm{n} 2 + 2 \mathrm{l}\mathrm{n} 3 - \mathrm{l}\mathrm{n} 2\times \mathrm{l}\mathrm{n} 3)x
2(2 \mathrm{l}\mathrm{n} 2 - \mathrm{l}\mathrm{n} 3) + (2 - 4 \mathrm{l}\mathrm{n} 2 + \mathrm{l}\mathrm{n} 3)x
=
1.+ .085911x
1.+ .566639x
,
\alpha (x) =
\biggl(
x
\biggl(
\mathrm{l}\mathrm{n}
\biggl(
256
3
\biggr)
- 4
\biggr)
+ 4 - 8 \mathrm{l}\mathrm{n}(2) + \mathrm{l}\mathrm{n}(9)
\biggr)
x2(x2 - 3x+ 2)
\biggl(
4 - (\mathrm{l}\mathrm{n}(81) - 16) \mathrm{l}\mathrm{n}2(2) + \mathrm{l}\mathrm{n}2(3) \mathrm{l}\mathrm{n}(2) - \mathrm{l}\mathrm{n}(3) \mathrm{l}\mathrm{n}(4) - \mathrm{l}\mathrm{n}
\biggl(
65536
9
\biggr) \biggr) \times
\times
\biggl( \biggl(
\mathrm{l}\mathrm{n}
\biggl(
16
9
\biggr)
- x
\biggl(
\mathrm{l}\mathrm{n}
\biggl(
16
3
\biggr)
- 2
\biggr) \biggr)
\mathrm{l}\mathrm{n}(x+ 1) - x
\biggl(
x
\biggl(
\mathrm{l}\mathrm{n}
\biggl(
9
4
\biggr)
- \mathrm{l}\mathrm{n}(2) \mathrm{l}\mathrm{n}(3)
\biggr)
+ \mathrm{l}\mathrm{n}
\biggl(
16
9
\biggr) \biggr) \biggr)
x2(x2 - 3x+ 2)
\biggl(
4 - (\mathrm{l}\mathrm{n}(81) - 16) \mathrm{l}\mathrm{n}2(2) + \mathrm{l}\mathrm{n}2(3) \mathrm{l}\mathrm{n}(2) - \mathrm{l}\mathrm{n}(3) \mathrm{l}\mathrm{n}(4) - \mathrm{l}\mathrm{n}
\biggl(
65536
9
\biggr) \biggr)
We can observe that the weight \alpha (x) has the indeterminate form 0/0 at the points xi = 0, 1, 2.
They can be quickly evaluated with the help of L’Hospital’s rule. Fig. 3 shows plots of derivative of
numerator s\prime (x) - s\prime 2(x) and derivative of denominator s\prime 1(x) - s\prime 2(x) of \alpha (x). It is clear that for
those mentioned before points exist limits because denominator is not equal to 0. For example
\alpha (0) = \mathrm{l}\mathrm{i}\mathrm{m}
x\rightarrow 0
s(x) - s2(x)
s1(x) - s2(x)
=
0
0
= \mathrm{l}\mathrm{i}\mathrm{m}
x\rightarrow 0
s\prime (x) - s\prime 2(x)
s\prime 1(x) - s\prime 2(x)
= 1.
The weight \alpha is convex: the second derivative is positive, as shown in Fig. 4. We remark also
that if the numerical values are rounded to four digits, the resulting \alpha is incorrect. This is due to the
very fast convergence of NPAs to Stieltjes functions, i.e.,
\bigm| \bigm| s(x) - s1(x)
\bigm| \bigm| \ll \bigm| \bigm| s(x) - s2(x)
\bigm| \bigm| and then
a high numerical precision is necessary to obtain a correct result.
4. The TSE property, rescaling, modification of approximated function. In the next section
we show the practical application of the proposed method of approximation on a few examples.
They cover the inverse Langevin function with its transformed function and a statistical integral that is
popular in tribology. Before we apply this method, we have to consider some additional requirements.
They concern the problem of selecting the appropriate reference function s, its rescaling, and choice
of the NPA.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 9
MAGIC EFFICIENCY OF APPROXIMATION OF SMOOTH FUNCTIONS BY WEIGHTED MEANS . . . 1197
−0.02
−0.015
−0.01
−0.005
0
0.005
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x
s’(x)−s2’(x)
s1’(x)−s2’(x)
Fig. 3. Plots of derivative of numerator s\prime (x) - s\prime 2(x) and derivative of denominator s\prime 1(x) - s\prime 2(x) of \alpha (x).
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.5 1 1.5 2
x
α(x)
α(x)’’
Fig. 4. Convexity of the weight \alpha .
As discussed in our previous paper, we choose, for a reference function, functions that possess
the TSE property, i.e., the existence of the two sided estimates and the same kind of smoothness as
the approximated function f. Many real functions have these properties, such as Stieltjes functions,
and any function (a+ bx)h(x) where h is a Stieltjes function. Other examples used in our research
include the function (a+ bx) \mathrm{c}\mathrm{t}\mathrm{g} h(x), which is not Stieltjes.
To simplify the problem of rescaling the reference function s to be as close as possible to f, we
use a simple condition at one point x\ast of the interval [x1, xN ]:
s(x\ast ) = f(x\ast ).
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 9
1198 R. JEDYNAK, J. GILEWICZ
In our examples we use the inverse Langevin function and its transformation. Because the well-
known approximation formulas for this function are based on [3/1] and [3/2] PA: we choose the
NPAs f1[3/1] and f2[2/1] for our computation of the inverse Langevin function. For computation
of transformed function we use the NPAs f1[2/1] and f2[1/1].
In our further studies we use \alpha s, which means the average value of the function \alpha (x) on [x1, xN ].
It is computed from the following formula:
\alpha s =
1
xN - x1
xN\int
x1
\alpha (x) dx.
The mean value simplifies the computation of new approximation formulas and gives quite reasonable
results in comparison to using the continuous function \alpha (x) in certain cases. In the case of using the
continuous weight \alpha (x) we obtain the approximation formula
m = \alpha (x)f1 + (1 - \alpha (x))f2,
and in the case of using the mean value \alpha s we get
ms = \alpha sf1 + (1 - \alpha s)f2. (6)
Our examples show that these approximations can work equally well.
5. Examples. In this section we analyse different attempts to approximate the inverse Langevin
function and the Gaussian distribution of asperity heights in a tribology problem. The Langevin
function is defined by
\scrL (x) = \mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}x - 1/x = 1 +
2
e2x - 1
- 1
x
=
(x - 1)e2x + 1 + x
x(e2x - 1)
but its inverse \scrL - 1 has no analytical form. However, their power series expansions can be calculated
arbitrarily far [11]. For instance, its development at x = 0 is
\scrL - 1 (x) = 3x+
9x3
5
+
297x5
175
+
1539x7
875
+
126117x9
67375
+
43733439x11
21896875
+
231321177x13
109484375
+
+
20495009043x15
9306171875
+
1073585186448381x17
476522530859375
+
4387445039583x19
1944989921875
+O
\bigl(
x21
\bigr)
.
This series can be divided by x and multiplied by (1 - x). In fact it is not difficult to observe that
\scrL - 1 has an asymptote at x = 1. \scrL - 1 is defined on [0, 1] and plays an important role in the theoretical
physics analysis of polymers [1, 20, 21]. Their values in a neighbourhood of x = 1 are of a little
practical interest. For instance, the case x = 1 corresponds to the unreal case of a polymer with all
molecules aligned. A simple approximation of \scrL - 1 is needed in many theoretical calculations. The
following attempts at approximations will be presented in the following:
1) simple PA,
2) NPA,
3) different optimization methods,
4) our general method of weighted means approximation of two neighbouring NPAs.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 9
MAGIC EFFICIENCY OF APPROXIMATION OF SMOOTH FUNCTIONS BY WEIGHTED MEANS . . . 1199
0
20
40
60
80
100
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
exact
[3/2]
[1/4]
[5/0]
[1/2]
Fig. 5. Comparison of different PAs of the inverse Langevin function for x \in [0, 1].
The well known and the most popular PA [3/2] is given by Cohen [3] but it is usually used in the
rounded form
\scrL - 1 (x) = x
3 - x2
1 - x2
, \epsilon max \approx 4.9\%.
Thanks to the process of rounding off, the formula has the required singularity at x = 1. This
approximant has been widely accepted and developed by numerous researchers.
The proposals made by Darabi and Itskov [4]
\scrL - 1 (x) = x
3 - 3x+ x2
1 - x
, \epsilon max \approx 2.6\%
and Jedynak [10]
\scrL - 1 (x) = x
3.0 - 2.6x+ 0.7x2
(1 - x)(1 + 0.1x)
, \epsilon max \approx 1.5\%
are more exact than the PA. They are derived by the use of NPA and are presented in rounded forms.
\epsilon max shows the maximal relative error for the appropriate formula.
Fig. 5 compares the different PAs of the inverse Langevin function. Fig. 6 presents the relative
errors of these approximants.
Quite different approaches from those mentioned above can be found in the third attempt. These
methods use mathematical software to minimize the maximal error of the approximation formula by
modifying the coefficients of previous approximations in each particular case to obtain a better result.
A representative example of this method is the solution obtained by Marchi and Arruda.
In our study we would like to concentrate on the Padé approach therefore we do not discuss
further optimization methods in this paper.
In our further research we want to find the answer to the question whether the inverse Langevin
function has the TSE property. We also wish to find a function simply related to the original having
the TSE property. We initially considered 3 candidates: R1(x) =
\scrL - 1(x)
x
, R2(x) = (1 - x)\scrL - 1(x)
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1200 R. JEDYNAK, J. GILEWICZ
0
10
20
30
40
50
60
70
80
90
100
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Pe
rc
en
t r
el
at
iv
e
er
ro
r [
%
]
x
[3/2]
[1/4]
[5/0]
[1/2]
Fig. 6. Graphs of relative error for x \in [0, 1].
and R3(x) = (1 - x)
\scrL - 1(x)
x
. Our detailed inspection showed that only the first transformed function
has the partial TSE property.
In the next subsection we show in detail our intensive research on R1 function and next on the
inverse Langevin function. This section also documents extended study on the Gaussian integral,
which was introduced in our previous paper [13].
5.1. The case of \bfitf (\bfitx ) = \bfitR \bfone (\bfitx ). In this example we consider the following information at
four points x = 0, 0.3, 0.6, 0.99 : f(x1), f \prime (x1), f(x2), f(x3), f(x4) to compute the NPAs
f1 = [2/2]2111x1x2x3x4
and f2 = [1/2]1111x1x2x3x4
. Our reference function after rescaling process is
s(x) = 3x \mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}x = 3x
e2x + 1
e2x - 1
.
Fig. 7 presents R1 and s and their first derivatives.
We obtain the following NPAs:
f1(x) =
- 1.35932x2 + 0.166147x+ 3
- 1.05775x2 + 0.0553938x+ 1
,
f2(x) =
2.99999 - 1.6867x
- 0.405847x2 - 0.595011x+ 1
and the continuous weight function \alpha (x) shown in Fig. 8:
\alpha (x) =
0.157155
\bigl(
x2 + 0.0531582x+ 16.0798
\bigr)
x4 - 1.89x3 + 1.071x2 - 0.1782x
\times
\times
\bigl(
(x2 - 2.23917x - 1.7268)x \mathrm{c}\mathrm{o}\mathrm{t}\mathrm{h}(x) + 2.16865x+ 1.7268
\bigr)
x4 - 1.89x3 + 1.071x2 - 0.1782x
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MAGIC EFFICIENCY OF APPROXIMATION OF SMOOTH FUNCTIONS BY WEIGHTED MEANS . . . 1201
0
5
10
15
20
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
x
R1(x)
s(x)
0
20
40
60
80
100
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
x
R1’(x)
s’(x)
(a) (b)
Fig. 7. The functions R1 and s (a) and their first derivatives for x \in [0, 1] (b).
0.988
0.99
0.992
0.994
0.996
0.998
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
Fig. 8. Plot of \alpha for x \in [0, 1].
The error functions of the NPAs f1, f2, m and ms are depicted in Fig. 9 (a). From this figure
we can conclude that R1 has partial TSE property in the interval [0.3, 0.6]. Fig. 9 (b) shows the
plots from the previous picture without f2. It documents a slight superiority of the continuous \alpha and
simplified approximation with the weighted mean \alpha s over f1 in this interval. We observe that in the
interval [0.3, 0.95], the new method gives the best results (Table 1).
Table 1 presents the numerical results. Recall that m in the third column corresponds to the
continuous approximation used on the interval [0, 1] and the fourth column corresponds to the ap-
proximation ms (6) on the interval [0, 1] computed with the mean weight \alpha s = 0.99509. We can
observe that only at the beginning of the interval [0, 0.3] are our new proposals a bit worse than the
‘good’ NPA f1. It is surprising that the approximation with the mean weight \alpha s gives the best results
on the interval [0.3, 0.5].
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1202 R. JEDYNAK, J. GILEWICZ
−0.0005
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
0.3 0.35 0.4 0.45 0.5 0.55 0.6
x
f−f1
f−f2
f−ms
f−m
−0.0004
−0.00035
−0.0003
−0.00025
−0.0002
−0.00015
−0.0001
−5×10−5
0
5×10−5
0.3 0.35 0.4 0.45 0.5 0.55 0.6
x
f−f1
f−ms
f−m
(a) (b)
Fig. 9. Plot of errors of f1, f2, m, and ms (a) and without f2 (b).
0
5
10
15
20
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
x
L−1(x)
s(x)
0
20
40
60
80
100
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
x
L−1’(x)
s’(x)
(a) (b)
Fig. 10. Plot of \scrL - 1 and s functions (a) and their first derivatives for x \in [0, 1] (b).
5.2. The case of - \bfitf (\bfitx ) = \bfscrL - \bfone (\bfitx ). In this case we use the following information at four
points x = 0, 0.2, 0.6, 0.9 : f(x1), f \prime (x1), f(x2), f(x3), f(x4) to compute the NPAs f1 =
= [3/1]2111x1x2x3x4
and f2 = [2/1]1111x1x2x3x4
. The same points were used by Darabi and Itskov [4] to
compute their [3/1] approximation formula for the inverse Langevin function.
We choose the following reference function for this case:
s(x) = 3
\mathrm{l}\mathrm{n}(x+ 1)
1 - x
.
Fig. 10 presents the functions \scrL - 1 and s and their first derivatives.
We obtain the following NPAs:
f1(x) =
x(3 - 2.86755x+ 0.934859x2)
1 - 0.993467x
,
f2(x) =
x(2.89653 - 2.25965x)
1 - 1.02483x
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MAGIC EFFICIENCY OF APPROXIMATION OF SMOOTH FUNCTIONS BY WEIGHTED MEANS . . . 1203
Table 1. Errors of approximations for \bfitf (\bfitx ) = \bfitR \bfone (\bfitx )
x f - f1 f - m f - ms
0. 0 0 0
0.05 –0.0000219185 –0.0000228992 –0.0000399383
0.1 –0.000056229 –0.000059273 –0.0000820745
0.15 –0.0000706737 –0.0000755992 –0.0000966069
0.2 –0.0000562392 –0.0000617667 –0.0000766606
0.25 –0.0000236783 –0.000027706 –0.000034875
0.3 0 0 0
0.35 –0.0000231763 –0.0000167292 –0.0000116906
0.4 –0.000131737 –0.000117268 –0.000110236
0.45 –0.000342822 –0.000320593 –0.000314725
0.5 –0.000604877 –0.000578399 –0.000575948
0.55 –0.000695581 –0.000673674 –0.000674612
0.6 0 0 0
0.65 0.00297636 0.00292337 0.00293611
0.7 0.0116858 0.0115257 0.011576
0.75 0.0339223 0.0335597 0.0336958
0.8 0.0870261 0.0862857 0.0866021
0.85 0.209815 0.20834 0.209035
0.9 0.501434 0.498365 0.499928
0.95 1.34894 1.34133 1.34545
and the continuous weight function \alpha (x) which is shown in Fig. 11:
\alpha (x) =
1.57378(x - 1.00402)(x(x2 - 3.92326x+ 2.92326) + (3.00879x - 2.99161) \mathrm{l}\mathrm{n}(x+ 1))
(x - 1)x(x3 - 1.7x2 + 0.84x - 0.108)
(7)
The error functions of NPAs f1, f2, m and ms are depicted in Fig. 12. They clearly show that
our function has the TSE property in the whole interval. We inspected also percent relative error of
our continuous weighted mean approximation and compared to two well-known formulas: Jedynak
[3/2] and Darabi & Itskov [3/1]. Fig. 13 shows plots of errors for those mentioned before formulas.
The maximum relative error of our new proposition in the interval [0, 0.9] is equal to 0.35%. Our
previous approximation formula [3/2] gives 1.5 % in this interval, while Darabi & Itskov [3/1] 2.6%.
We are very surprised this excellent accuracy. We remind the fact that our ’good’ NPA is the same as
Darabi & Itskov [3/1]. The final approximation formula of our previous NPA [3/2] approximation
formula and Darabi & Itskov [3/1] one slightly differ from NPAs [3/2] and [3/1] because they were
a little rounded to capture the asymptotic property of approximation formula at the point x = 1. In
the literature we can find two other proposals of rational approximation R3,1 of the inverse Langevin
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 9
1204 R. JEDYNAK, J. GILEWICZ
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
x
Fig. 11. Plot of \alpha for x \in [0, 1].
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
x
f−f1
f−m
f−ms
f−f2
Fig. 12. Plot of errors of f1, f2, m and ms. Information at the following points x = 0, 0.2, 0.6, 0.9 is used for those
approximations.
function by Marchi and Arruda [19] and Kröger [16]. In both cases, the maximum relative error
of approximation formula is about 0.95% in that mentioned before interval. We cite these facts to
emphasize high precision of our new numerical method of approximation.
It is seen that our solution approximates very accurately the inverse Langevin function in the
interval [0, 0.9]. Outside of this interval we observe a decrease in accuracy. We suppose that this
decrease results from the location of the last node x = 0.9. We decided to do an experiment with
different values of this point to confirm this assumption. The obtained results for x = 0.95 are
illustrated in Fig. 14 and for x = 0.99 in Fig. 15. They clearly show that our solution is very
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MAGIC EFFICIENCY OF APPROXIMATION OF SMOOTH FUNCTIONS BY WEIGHTED MEANS . . . 1205
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
3
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Pe
rc
en
t r
el
at
iv
e
er
ro
r [
%
]
x
continuous weighted mean
Jedynak [3/2]
Darabi and Itskov [3/1]
Fig. 13. Plot of percent relative error of new proposition (weighted mean), Jedynak [3/2] and Darabi & Itskov [3/1].
Fig. 14. Plot of errors of f1, f2, m and ms. Information at the points x = 0, 0.2, 0.6, 0.95 is used for those
approximations.
sensitive to the location of the mentioned node. Both figures document that m and ms give a more
exact approximation than ’good’ NPA (f1) in a bigger interval than in the first case with x = 0.9.
Using the mean weights ms = 1.10592f1 - 0.10592f2 we get the results which are presented in
Table 2.
We notice that in the interval [0, 0.9] the best results are given by the approximation with contin-
uous \alpha . Our simplified approximation ms with the mean weight \alpha s gives slightly worse results than
f1. This situation changes radically when we consider higher values of the last node (it is x = 0.95
or x = 0.99). This effect is documented by Figs. 14 and 15.
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1206 R. JEDYNAK, J. GILEWICZ
Table 2. Errors of approximations for function \bfitf (\bfitx ) = \bfscrL - \bfone (\bfitx )
x f - f1 f - m f - ms
0. 0 0 0
0.05 –0.000194303 –0.0000646495 –0.000588925
0.1 –0.000473798 –0.000155885 –0.00097594
0.15 –0.000484451 –0.000157643 –0.000841124
0.2 0 0 0
0.25 0.0010798 0.000343309 0.00159598
0.3 0.00272264 0.000854084 0.00384982
0.35 0.00474827 0.00146613 0.00649942
0.4 0.00680342 0.00205991 0.00908647
0.45 0.00832601 0.00245746 0.0109124
0.5 0.00849974 0.00242318 0.010981
0.55 0.00620616 0.00168379 0.00793335
0.6 0 0 0
0.65 –0.0118168 –0.00263578 –0.0149539
0.7 –0.030816 –0.00551996 –0.0390682
0.75 –0.0572276 –0.00624079 –0.0731967
0.8 –0.086495 0.000631018 –0.112846
0.85 –0.098469 0.0203343 –0.133422
0.9 0 0 0
0.95 1.07753 –1.77336 1.92977
5.3. Approximation of the Gaussian distribution from a tribology problem. We demonstrated
in [12] that the simple PA gives more accurate results than all previously proposed formulas. In a
subsequent paper [13], we showed that the accuracy can be further improved by using the method
of weighted means. We used discrete values of \alpha and their means. In contrast to that method, we
now apply continuous values of the function \alpha and their mean in the interval [x1, xN ]. The reference
function s(x) is drawn from our previous paper [13]. Below we give the explicit formulas for the
NPAs. They are calculated by the use of the information for four points: 0.5; 1; 1.5 and 2. The
NPAs are denoted by g1 = [2/3] and g2 = [1/3]
g(x) =
1\surd
2\pi
e
x2
2
\infty \int
x
(t - x)
5
2 e -
t2
2 dt,
s(x) =
\mathrm{l}\mathrm{n}
\biggl(
x
2
+ 1
\biggr)
x
2
( - .279408x+ .616634).
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MAGIC EFFICIENCY OF APPROXIMATION OF SMOOTH FUNCTIONS BY WEIGHTED MEANS . . . 1207
−0.001
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
0.01
0 0.1 0.2 0.3 0.4 0.5 0.6
x
f−f
1
f−m
f−m
s
−1
−0.5
0
0.5
1
1.5
2
2.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
x
f−f
1
f−m
f−m
s
Fig. 15. Plot of errors of f1, f2, m and ms. Information at the following points x = 0, 0.2, 0.6, 0.99 is used for those
approximations.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 0.5 1 1.5 2
x
g(x)
s(x)
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
0 0.5 1 1.5 2
x
g’(x)
s’(x)
g’’(x)
s’’(x)
(a) (b)
Fig. 16. Functions g(x) and s (a) and their first and second derivatives for x \in [0, 2] (b).
Fig. 16 presents the functions g(x) and s and their first and second derivatives.
We obtain the following NPAs:
g1(x) =
0.0131008x2 - 0.133611x+ 0.617026
0.287364x3 + 0.917832x2 + 1.53293x+ 1
,
g2(x) =
0.621943 - 0.0611792x
0.500488x3 + 1.00966x2 + 1.71168x+ 1
and the continuous weight function \alpha (x) which is shown in Fig. 17.
Using the mean weights ms = 0.989638g1 + 0.010362g2 we get the results which are presented
in Table 3. We adopted some values from our previous paper [13] to compare with those computed
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1208 R. JEDYNAK, J. GILEWICZ
0.97
0.98
0.99
1
1.01
1.02
1.03
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
y
Fig. 17. Plot of \alpha for x \in [0, 2].
Table 3. Errors of approximation for \bfitg (\bfitx )
x g - g1 g - mi g - md g - m g - ms
0. –0.00039 –0.00043 –0.00048 –0.00029 –0.00044
0.25 –0.0000126 –0.000015 –0.000018 –0.00000969 –0.0000158
0.5 0 0 0 0 0
0.75 .000000464 .00000038 .00000027 .00000038 .000000348
1. 0 0 0 0 0
1.25 .000000763 .00000065 .00000066 .000000649 .000000699
1.5 0 0 0 0 0
1.75 .00000124 .0000011 .0000011 .00000108 .00000117
2. 0 0 0 0 0
by the use of the new method. The third and fourth columns are computed from discrete values but
the last two columns from the new continuous method.
Note that only in the interval [0, 0.5[ is the NPA g1 a little better than the results coming from the
‘old method’ m1 and md. The new method with the continuous \alpha gives a more exact approximation
at this interval. This fact is clearly demonstrated in Fig. 18 (a). Fig. 18 (b) shows advantage of all
our approximation methods compared to the ’good’ NPA g1 in the interval [0.5, 2]. Our new method
with continuous \alpha gives a more exact approximation at interval [1, 2].
6. Conclusions. The presented method, introduced in a previous paper, develops our cutting-
edge idea of constructing the weighted mean approximation of a given function. We show that the
weight \alpha (x) defined by s and two oscillating NPAs is a smooth convex function. Then, we can use
it in formula (1) continuously. This approach improves our ‘old’ method which relies on calculating
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MAGIC EFFICIENCY OF APPROXIMATION OF SMOOTH FUNCTIONS BY WEIGHTED MEANS . . . 1209
−0.0005
−0.00045
−0.0004
−0.00035
−0.0003
−0.00025
−0.0002
−0.00015
−0.0001
−5×10−5
0
0 0.1 0.2 0.3 0.4 0.5
y
g−g1
g−m
g−ms
g−md
g−mi
−1×10−6
−5×10−7
0
5×10−7
1×10−6
1.5×10−6
0.6 0.8 1 1.2 1.4 1.6 1.8 2
x
g−g1
g−m
g−ms
g−md
g−mi
(a) (b)
Fig. 18. Errors of approximations with reference to [12, 13]. A slight superiority of the continuous \alpha (presented by curve
g - m) over f1 (presented by curve g - f1 ) is detectable in these graphs.
the discrete values of the weights. We observe the fact that the accuracy of the approximation is
sensitive to the proximity of both the reference and the approximated functions.
The examples presented in the paper clearly demonstrate that the idea is very interesting and
promises good results in the numerical approximation of functions.
The weighted means approximation for the inverse Langevin function compared with the approx-
imations in Jedynak [10] and Darabi and Itskov [4] gives us satisfaction. Also the approximation by
continuous weighted means of two NPAs of the integral of the Gaussian distribution used in tribology
[18] is better than the discrete one presented in our previous research.
Because our method is substantially based on numerical investigations, many questions concern-
ing this method remain open:
1) characterize the functions having the TSE property,
2) improve the choice and the rescaling of the auxiliary functions defining the weight,
3) analyse the cases where the second NPA is calculated by removing one coefficient from the
expansion of f at the points xi with i \not = 1,
4) what are the limits of the use of this method for functions without the TSE property?
References
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Received 10.04.17
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|
| id | umjimathkievua-article-1628 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:09:29Z |
| publishDate | 2018 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/58/2b2bbb08acb6deb75e93d512cb3e6e58.pdf |
| spelling | umjimathkievua-article-16282019-12-05T09:21:25Z Magic efficiency of approximation of smooth functions by weighted means of two $N$-point Padé approximants Чарiвна ефективнiсть наближення гладких функцiй зваженими середнiми двох $N$ -точкових Паде апроксимацiй Gilewicz, J. Jedynak, R. Гілевич, Я. Я. Єдинак, Р. We consider the approximation of smooth functions by two weighted $N$-point Pad´e approximants. We present numerical examples and the inequalities between the Stietjes function and its $N$-point Padé approximant. Квартиры посуточно - онлайн-бронирование Статтю присвячено наближенню гладких функцiй двома $N$-точковими наближеннями Паде з вагами. Наведено числовi приклади, нерiвностi мiж функцiєю Стiльтьєса та її $N$-точковим наближенням Паде. Квартиры посуточно - онлайн-бронирование Institute of Mathematics, NAS of Ukraine 2018-09-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1628 Ukrains’kyi Matematychnyi Zhurnal; Vol. 70 No. 9 (2018); 1192-1210 Український математичний журнал; Том 70 № 9 (2018); 1192-1210 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1628/610 Copyright (c) 2018 Gilewicz J.; Jedynak R. |
| spellingShingle | Gilewicz, J. Jedynak, R. Гілевич, Я. Я. Єдинак, Р. Magic efficiency of approximation of smooth functions by weighted means of two $N$-point Padé approximants |
| title | Magic efficiency of approximation of smooth functions by weighted means of two $N$-point Padé approximants |
| title_alt | Чарiвна ефективнiсть наближення гладких функцiй
зваженими середнiми двох $N$ -точкових Паде апроксимацiй |
| title_full | Magic efficiency of approximation of smooth functions by weighted means of two $N$-point Padé approximants |
| title_fullStr | Magic efficiency of approximation of smooth functions by weighted means of two $N$-point Padé approximants |
| title_full_unstemmed | Magic efficiency of approximation of smooth functions by weighted means of two $N$-point Padé approximants |
| title_short | Magic efficiency of approximation of smooth functions by weighted means of two $N$-point Padé approximants |
| title_sort | magic efficiency of approximation of smooth functions by weighted means of two $n$-point padé approximants |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1628 |
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