dependence in order to keep formulas more compact.

The form of approximants (3.123) resembles that of nested radicals [180, 181],
because of which we call them nested approximants.

At large $x$, approximant (3.123) behaves as

$f_{k}^{*}(x)\simeq B_{k}^{*}x^{n_{1}S_{k}}\qquad(x\rightarrow\infty)\;,$
(3.124)

with the amplitude

$B_{k}^{*}=\left(B_{1}B_{2}^{n_{2}}B_{3}^{n_{2}n_{3}}\ldots
B_{k}^{n_{2}n_{3}\ldots n_{k}}\right)^{n_{1}}$ (3.125)

and the notation

$S_{k}\equiv\alpha_{1}+(\alpha_{2}-\alpha_{1})n_{2}+(\alpha_{3}-\alpha_{2})n_{2}n_{3}+\ldots+(\alpha_{k}-\alpha_{k-1})n_{2}n_{3}\ldots
n_{k}\;.$ (3.126)

If the behavior of the sought function at large variable is known, being

$f(x)\simeq Bx^{\beta}\qquad(x\rightarrow\infty)\;,$ (3.127)

then we have two conditions defining

$B_{k}^{*}=B\;,\qquad n_{1}(k)=\frac{\beta}{S_{k}}\;.$ (3.128)

In the case, when the powers $\alpha_{j}=j$ in expansion (3.117) are integers,
approximant (3.123) simplifies to the form

$f_{k}^{*}(x)=\left(1+B_{1}x\left(1+B_{2}x\ldots\left(1+B_{k}x\right)^{n_{k}}\right)^{n_{k-1}}\ldots\right)^{n_{1}}$
(3.129)

and sum (3.126) becomes

$S_{k}=1+n_{2}+n_{2}n_{3}+\ldots+n_{2}n_{3}\ldots n_{k}\;.$ (3.130)

Generally, we can try to define the control parameters $B_{j}$ from the
accuracy-through-order procedure at small variable by equating the expansion
of approximant (3.123) with expansion (3.117). For example, for $B_{j}$ of low
orders this gives

$B_{1}=\frac{a_{1}}{n_{1}}=\frac{a_{1}}{\beta}\;S_{k}\;,\qquad
B_{2}=\frac{a_{1}^{2}+n_{1}(2a_{2}-a_{1}^{2})}{2a_{1}n_{1}n_{2}}=\frac{a_{1}^{2}S_{k}+\beta(2a_{2}-a_{1}^{2})}{2a_{1}\beta
n_{2}}\;.$

However the powers $n_{j}$ cannot be uniquely defined from such a procedure,
either at small or large variables. To define the powers $n_{j}$, some
additional conditions are needed. Also, to avoid complex numbers the
coefficients $B_{j}$ should be positive for keeping the function
$f_{k}^{*}(x)$ real valued. But $B_{j}$ can be negative, if the described
problem has critical points.

Notice that setting in Eq. (3.129) all powers $n_{j}=\pm 1$, we get Padé
approximants $P_{M/N}(x)$ or continued fractions, while setting the internal
powers $n_{j}=\pm 1$, with the external power $n_{1}$ given by condition
(3.128), we come to the modified Padé approximants
$[P_{M/N}(x)]^{\beta/(M-N)}$. However, such a choice is not uniquely defined,
since it is possible to construct several Padé approximants satisfying the
condition $M+N=k$. In addition, dealing with Padé approximants, we confront
the known difficulty of unphysical poles.

The other way could be by setting all powers, including $n_{1}$, equal,
$n_{j}=q$, and defining $q$ by the expression $q=\beta/S_{k}$ following from
condition (3.128). Then

$S_{k}=\sum_{j=0}^{k-1}q^{j}=\frac{1-q^{k}}{1-q}\qquad(k\geq 1)\;$

But in this way, to define $q$ uniquely, it is necessary to neglect the term
$q^{k}$, which disturbs condition (3.128) (see Ref. [182]).

One more possibility could be as follows. We can take the same number $q$ only
for all internal powers $n_{j}$, with $j=2,3,\ldots,k$. But the external power
$n_{1}(k)$ should be defined, as earlier, by condition (3.128), so that
$n_{1}(k)S_{k}=\beta$. This leads to the self-similar nested approximant

$f_{k}^{*}(x)=\left(1+B_{1}x\left(1+B_{2}x\ldots\left(1+B_{k}x\right)^{q}\right)^{q}\ldots\right)^{m_{k}}\;.,$

in which

$m_{k}\equiv n_{1}(k)=\frac{\beta}{S_{k}}=\frac{1-q}{1-q^{k}}\;\beta\;.$

This way does not impose constraints on the value of $\beta$. The power $q$
can be defined from additional conditions that would guarantee good numerical
convergence of the sequence $\\{f_{k}^{*}(x)\\}$.

To make the internal expressions nonnegative, we may set $q=2$, which gives

$m_{k}=\frac{\beta}{2^{k}-1}\qquad(q=2)\;.$

###  3.10 Self-Similar Exponential Approximants

An important particular case of the nested approximants can be derived by
setting the powers $s_{j}$ of the fractal transform (3.80) as
$s_{j}\rightarrow\infty$. Consequently, $n_{j}\rightarrow-\infty$,
$A_{j}\rightarrow 0$, and $B_{j}\rightarrow 0$. Then expression (3.123)
reduces to a self-similar exponential approximant [173, 183]

$f_{k}^{*}(x)=\exp\left(C_{1}x^{\alpha_{1}}\exp\left(C_{2}x^{\alpha_{2}-\alpha_{1}}\ldots\exp\left(C_{k}x^{\alpha_{k}-\alpha_{k-1}}\right)\right)\right)\;,$
(3.131)

in which

$C_{j}\equiv\frac{a_{j}}{a_{j-1}}\;t_{j}\qquad(j=1,2,\ldots,k)\;,$ (3.132)

where $a_{0}=1$. For integer $\alpha_{j}=j$, this leads to the expression

$f_{k}^{*}(x)=\exp\left(C_{1}x\exp\left(C_{2}x\ldots\exp\left(C_{k}x\right)\right)\right)\;.$
(3.133)

It is useful to stress that the obtained self-similar exponential approximants
are rather different from the iterated Euler [184] exponentials (see also
[185, 186]). The latter have the same radius of convergence as the initial
series (3.103). Hence, if the series (3.103) are divergent, the Euler
exponentials do not allow for an effective extrapolation. This is because the
coefficients in the Euler exponentials are defined by the accuracy-through-
order procedure. While the parameters of the self-similar exponentials are
given by definition (3.132).

The control functions $t_{j}$ should be found from additional conditions. For
example by minimizing a cost functional [12, 187]. The other way could be as
follows [173, 183]. Let us set in function (3.133) the control parameters
$t_{j}=1$ for $j=0,1,2,\ldots,k-1$, denoting the result as $F_{k}(x,t_{k})$.
The control parameters can be defined by the minimal difference optimization
condition

$F_{k}(x,t_{k})-F_{k-1}(x,t_{k})=0\;,$ (3.134)

which yields the equation for the control function $t_{k}=t_{k}(x)$,

$t_{k}=\exp\left(\frac{a_{k}}{a_{k-1}}\;t_{k}x\right)\;.$ (3.135)

This defines the exponential approximant

$f_{k}^{*}(x)=F_{k}(x,t_{k}(x))\;.$

As is mentioned above, it is possible to define the parameters $C_{n}$ by
minimizing some cost functionals [12, 187], which gives other forms of the
control parameters, for instance

$C_{n}=\frac{a_{n}(1+a_{1}^{2})}{na_{n-1}(1+a_{n}^{2})}\qquad(n=1,2,\ldots,k)\;.$

###  3.11 Self-Similar Additive Approximants

The small-variable expansion (3.103) can be reorganized using the asymptotic
relation

$1+\sum_{n=1}^{k}a_{n}x^{n}\simeq\sum_{j=1}^{k}d_{j}(1+\lambda
x)^{j}\qquad(x\rightarrow 0)\;,$ (3.136)

where $d_{j}$ are defined through $a_{n}$, with the condition

$\sum_{j=1}^{k}d_{j}=1\;.$

Accomplishing the self-similar renormalization of the linear terms, as in Sec.
3.7, we get the approximant

$f_{k}^{*}(x)=\sum_{j=1}^{k}d_{j}(1+\lambda_{j}x)^{n_{j}}\;.$ (3.137)

Here $d_{j}$ are known from relation (3.136), while $2k$ parameters
$\lambda_{j}$ and $n_{j}$ are control parameters. Rewriting the above
approximant in a bit different way, as

$f_{k}^{*}(x)=\sum_{j=1}^{k}D_{j}(1+\lambda x)^{n_{j}}\;,$ (3.138)

we have $2k+1$ control parameters $D_{j}$, $\lambda$, and $n_{j}$. But only
$2k$ of them are independent, since there is the additional condition

$\sum_{j=1}^{k}D_{j}=1\;.$

Let the large-variable expansion be

$f(x)\simeq\sum_{j=1}^{k}b_{j}x^{\beta_{j}}\qquad(x\rightarrow\infty)\;,$
(3.139)

where the powers are arranged in the descending order,

$\beta_{j}>\beta_{j+1}\qquad(j=1,2,\ldots,k-1)\;.$

Requiring that the approximant (3.138) would reproduce the terms of the large-
variable expansion (3.139), we set

$n_{j}=\beta_{j}\qquad(j=1,2,\ldots,k)\;.$ (3.140)

This gives

$f_{k}^{*}(x)=\sum_{j=1}^{k}D_{j}(1+\lambda x)^{\beta_{j}}\;.$ (3.141)

However, strictly speaking, expanding expression (3.141) at a large variable
$x$, in addition to the terms with the powers $\beta_{j}$, we also will get
the terms with the powers $\beta_{j}-1$. If such terms pertain to the set
$\\{\beta_{j}\\}$, then there is no problem. However, if they are not in that
set, then we need to add to expression (3.141) compensating terms canceling
the terms of wrong powers arising at $x\rightarrow\infty$. Thus the general
form of the additive approximant is

$f_{k}^{*}(x)=\sum_{j=1}^{k}D_{j}(1+\lambda
x)^{\beta_{j}}+\sum_{j}C_{j}(1+\lambda x)^{\beta_{j}-1}\;.$ (3.142)

The parameters $D_{j}$ and $\lambda$ are defined by the accuracy-through-order
procedure either at small or at large variable, or at both, while the role of
$C_{j}$ is to cancel the terms yielding wrong powers at the large variable
expansion. Employing the accuracy-through-order procedure, we confront the
problem of multiple solutions. Among them there are real as well as complex-
valued solutions. The latter come in complex conjugate pairs, so that their
sum is real. It is admissible to either select only real solutions or to take
the sum of all solutions. In both these cases, the results are of comparable
accuracy [188].

###  3.12 Self-Similar Factor Approximants

Self-similar renormalization of the expansion

$f_{k}(x)=1+\sum_{n=1}^{k}a_{n}x^{n}$ (3.143)

can also be done in the following way [189, 190, 191, 192]. The above sum can
be rewritten as

$f_{k}(x)=\prod_{j=1}^{k}(1+b_{j}x)\;,$ (3.144)

with the coefficients $b_{j}$ expressed through $a_{n}$. Applying the
procedure of the self-similar renormalization to each of the factors results
in the transformation

$1+b_{j}x\longrightarrow(1+A_{j}x)^{n_{j}}\;.$

Then we get the self-similar factor approximant

$f_{k}^{*}(x)=\prod_{j=1}^{N_{k}}(1+A_{j}x)^{n_{j}}\;,$ (3.145)

where the number of factors $N_{k}$ is prescribed by the number of terms in
the initial expansion (3.143). This number $N_{k}$ will be specified below.

When the order $k$ of the expansion (3.143) is even, then we can take
$N_{k}=k/2$ and define all control parameters $A_{j}$ and $n_{j}$ from the
accuracy-through-order procedure at small variables,

$f_{k}^{*}(x)\simeq f_{k}(x)\qquad(x\rightarrow 0)\;,$ (3.146)

resulting in the equations

$\sum_{j=1}^{k/2}n_{j}A_{j}^{n}=D_{n}\qquad(n=1,2,\ldots,k)\;,$ (3.147)

where

$D_{n}\equiv\frac{(-1)^{n-1}}{(n-1)!}\;\lim_{x\rightarrow
0}\;\frac{d^{n}}{dx^{n}}\;\ln\left(1+\sum_{m=1}^{n}a_{m}x^{m}\right)\;.$
(3.148)

Thus we have $k/2$ parameters $A_{j}$ and $k/2$ parameters $n_{j}$ that are
uniquely defined by $k$ equations (3.147).

When $k$ is odd, we take $N_{k}=(k+1)/2$, obtaining the equations

$\sum_{j=1}^{(k+1)/2}n_{j}A_{j}^{n}=D_{n}\qquad(n=1,2,\ldots,k)\;.$ (3.149)

Now we have $(k+1)/2$ parameters $A_{j}$ and $(k+1)/2$ parameters $n_{j}$,
hence altogether $k+1$ parameters, but only $k$ equations (3.149). Thus all
parameters $A_{j}$ and $n_{j}$ can be expressed through one of the parameters,
say through $A_{1}$. The latter is to be defined by an additional condition.

The accuracy-through-order equations with respect to the control parameters
$A_{j}$ are polynomial, the first equation being of first order, the second,
of second order, and so on, with the last equation being of order $k$. Each
polynomial equation of order $n$ possesses $n$ solutions. So that the total
number of solutions is $1\times 2\times 3\times\ldots\times k=k!$. At the same
time, from the form of the factor approximants it is evident that each of them
is invariant with respect to the $k!$ permutations

$A_{i}\longrightarrow A_{j}\;,\qquad n_{i}\longrightarrow n_{j}\;.$

Therefore, the multiplicity of solutions for $A_{j}$ is trivial, being related
to the enumeration of the parameters. Up to this enumeration, the solutions
are unique. If the parameters $A_{j}$ are given, then we get the linear
algebraic equations with respect to $n_{j}$ that are uniquely defined [37].

In the large-variable limit, approximant (3.145) behaves as

$f_{k}^{*}(x)\simeq B_{k}x^{\beta_{k}}\qquad(x\rightarrow\infty)\;,$ (3.150)

with the amplitude and power given by the expressions

$B_{k}=\prod_{j=1}^{N_{k}}A_{j}^{n_{j}}\;,\qquad\beta_{k}=\sum_{j=1}^{N_{k}}n_{j}\;,$
(3.151)

and with the number of factors

$\displaystyle N_{k}=\left\\{\begin{array}[]{ll}k/2\;,{}&~{}k=2,4,\ldots\\\
(k+1)/2\;,{}&~{}k=3,5,\ldots\end{array}\;.\right.$ (3.154)

Note that the values of the parameters $A_{j}=A_{j}(k)$ and $n_{j}=n_{j}(k)$
depend on the considered approximation order $k$. If the large-variable
behavior of the sought function is known, being

$f(x)\simeq Bx^{\beta}\qquad(x\rightarrow\infty)\;,$ (3.155)

then we have two additional equations

$B_{k}=B\;,\qquad\beta_{k}=\beta\;.$ (3.156)

As is mentioned above, in the case of an odd approximation order $k$, one of
the parameters $A_{j}$ remains undefined. It would be possible to fix
$A_{1}=A_{1}(k)$ in the odd order $k$ equal to $A_{1}(k-1)$ from the previous
even order $k-1$ or to define $A_{1}$ in the odd order approximant by a
variational procedure [193]. Another way could be based on scaling arguments
[192]. One more possibility could be based on the assumption of convergence.

Assume that the sequence $\\{f_{k}^{*}(x)\\}$ converges. This implies that the
sequence of the large-variable amplitudes $B_{k}$ also converges, so that
there exists a limit

$\lim_{k\rightarrow\infty}B_{k}=B^{*}\;.$ (3.157)

For an even approximation order $k=2p$, we have

$B_{2p}=\prod_{j=1}^{p}A_{j}^{n_{j}}\;,$ (3.158)

where $A_{j}=A_{j}(2p)$ and $n_{j}=n_{j}(2p)$. And for the next, odd,
approximation order $k=2p+1$, the amplitude is

$B_{2p+1}=\prod_{j=1}^{p+1}A_{j}^{n_{j}}\;,$ (3.159)

where $A_{j}=A_{j}(2p+1)$ and $n_{j}=n_{j}(2p+1)$. The latter expression can
be represented as

$B_{2p+1}=\left(\prod_{j=1}^{p}A_{j}^{n_{j}}\right)A_{p+1}^{n_{p+1}}\;.$
(3.160)

Taking here the limit $p\rightarrow\infty$ gives

$B^{*}=B^{*}\lim_{p\rightarrow\infty}A_{p+1}^{n_{p+1}}\;,$ (3.161)

from where

$\lim_{p\rightarrow\infty}A_{p+1}^{n_{p+1}}=1\;.$ (3.162)

Generally, there are two possibilities: either $A_{p+1}$ tends to a finite
value and $n_{p+1}$ tends to zero, or $A_{p+1}$ tends to one, while $n_{p+1}$
tends to a finite value. Aiming at using the limiting behavior as an
approximation at finite orders, we cannot take $n_{p+1}$ as zero, since then
the term $(1+A_{p+1}x)^{n_{p+1}}$ disappears at al. In order that the term
$(1+A_{p+1}x)^{n_{p+1}}$ in the general form of approximant (3.145) be
nontrivial and unambiguously defined, the parameters $A_{p+1}$ and $n_{p+1}$
have to be finite and nonzero. Therefore we have to accept that limit (3.162)
yields

$\lim_{p\rightarrow\infty}A_{p+1}=1\;,$ (3.163)

which does not exclude that $n_{p+1}$ simultaneously tends to zero. Moreover,
if $n_{p+1}$ tends to zero, then a finite value of $A_{p+1}$ is not of
importance, hence can be set to one. Thus condition (3.163) is sufficient for
the validity of limit (3.162). This suggests that for large approximation
orders one of the parameters $A_{j}$ in the odd approximant can be set to one.
Keeping in mind this asymptotic equality, we can apply it, as an
approximation, for finite orders $k$. By re-enumerating the parameters, it is
admissible to set to one the term $A_{1}$. Since setting $A_{1}=1$ for finite
odd approximation orders is an approximation, we have to accept that the most
reliable are the even factor approximants that do not involve such an
additional condition. But considering odd factor approximants gives us
information how close we are to the saturation limit of large $k$, where even
and odd approximants should coincide.

Obtaining the sequence of the factor approximants $f_{k}^{*}(x)$, it is
possible to estimate their accuracy by the error bar

$\pm\frac{1}{2}\;\left|f_{k}^{*}(x)-f_{k-1}^{*}(x)\right|\;.$ (3.164)

It may happen that the accuracy-through-order equations result in complex-
valued parameters $A_{j}$ and $n_{j}$. But this is not a problem, since such
solutions come in complex conjugated pairs, so that the total approximant is
always real.

Factor approximants can provide good approximations for various functions,
rational, irrational, as well as transcendental [189, 190, 191, 192].
Moreover, there exists a large class of functions that can be reproduced
exactly by self-similar approximants of finite order. This class is composed
of the functions of the form

$R_{k}(x)=\prod_{j=1}^{M}P_{m_{j}}^{\alpha_{j}}(x)\qquad\left(k=\sum_{j=1}^{M}m_{j}\right)\;,$

where $P_{m}(x)$ is a polynomial and $\alpha_{j}$ are complex-valued numbers.
Transcendental functions arise as limits of the above form. For instance,
applying factor approximants to the series

$e^{x}\simeq\sum_{n=0}^{k}\frac{x^{n}}{n!}\;,$

we exactly reconstruct the exponential for any order $k\geq 2$ because of the
limit

$\lim_{a\rightarrow 0}(1+ax)^{1/a}=e^{x}\;.$

Note that the number of factors in a factor approximant, defined in Eq.
(3.154), corresponds to the case when only a small-variable expansion (3.143)
is available. In the case, when the large-variable asymptotic behavior (3.155)
is also given, the number of the factors depends on what information is
additionally used. Thus, if both the amplitude $B$ as well as the power
$\beta$ are used as known values, then

$\displaystyle
N_{k}=\left\\{\begin{array}[]{ll}(k+2)/2\;,{}&~{}k=2,4,\ldots\\\
(k+3)/2\;,{}&~{}k=3,5,\ldots\end{array}\qquad(B_{k}=B\;,~{}\beta_{k}=\beta)\;,\right.$
(3.167)

with the additional condition $A_{1}=1$ for odd approximants.

And if only the power $\beta$ is employed, while the amplitude $B$ is not
known, then the number of factors is

$\displaystyle
N_{k}=\left\\{\begin{array}[]{ll}(k+2)/2\;,{}&~{}k=2,4,\ldots\\\
(k+1)/2\;,{}&~{}k=3,5,\ldots\end{array}\qquad(\beta_{k}=\beta)\;,\right.$
(3.170)

with the additional condition $A_{1}=1$ for even approximants.

It is also useful to mention that, defining the parameters $A_{j}$ and powers
$n_{j}$, one can meet complex values. However, the related parameters and
powers appear in complex conjugate pairs, which makes the overall approximant
real, since

$\left|(1+Ax)^{m}\right|=|1+Ax|^{{\rm Re}\;m}\exp\left\\{-({\rm Im}\;m){\rm
arg}(1+Ax)\right\\}\;.$

###  3.13 Self-Similar Combined Approximants

It is possible to combine different self-similar approximants in the following
way [194]. Suppose we have expansion (3.143) of order $k$. We may consider a
subexpansion of order $n<k$,

$f_{n}(x)=1+\sum_{j=1}^{n}a_{j}x^{j}\qquad(n<k)\;,$ (3.171)

for which we can construct a self-similar approximant $f_{n}^{*}(x)$. Then we
introduce the function

$C_{k/n}(x)\equiv\frac{f_{k}(x)}{f_{n}^{*}(x)}\;,$ (3.172)

which is expanded in powers of $x$, giving

$C_{k/n}(x)\simeq 1+\sum_{j=n+1}^{k}b_{j}x^{j}\qquad(x\rightarrow 0)\;,$
(3.173)

with the coefficients $b_{j}$ expressed through $a_{j}$. On the basis of the
latter expansion, we construct another self-similar approximant
$C_{k/n}^{*}(x)$. The result is the combined approximant

$f_{k}^{*}(x)=f_{n}^{*}(x)C^{*}_{k/n}(x)\;.$ (3.174)

Such combined approximants, composed of two (or more) factors, can be used if
one of the factors catches better the behavior at small variables, while the
other is more accurate at large variables.

On the basis of expansion (3.173), it is also admissible to construct a Padé
approximant $P_{M/N}(x)$, with $M+N=k-n$, defining the Padé approximant
parameters by the standard condition

$P_{M/N}(x)\simeq C_{k/n}(x)\qquad(x\rightarrow 0)\;.$

In that way, we obtain the combined expression

$f_{k}^{*}(x)=f_{n}^{*}(x)P_{M/N}(x)$ (3.175)

that is a self-similarly corrected Padé approximant. Such combined
approximants are useful when one tries to approximate an irrational function
that cannot be well approximated by Padé approximants [195] and when the
latter are not applicable because of the problems typical of Padé
approximants, such as the appearance of spurious poles, incompatibility of
small-variable and large-variable behaviors, and like that [196].

###  3.14 Acceleration of Convergence

Obtaining a sequence $\\{f_{k}^{*}(x)\\}$ of self-similar approximants, it is
possible to accelerate its convergence by employing additional tricks. One
method for this purpose is by using splines based on several last approximants
[37].

For example, we can treat the last approximants $f_{k-2}^{*}(x)$,
$f_{k-1}^{*}(x)$, and $f_{k}^{*}(x)$ as the terms of a new sequence. Then we
define a polynomial spline, say a quadratic one,

$g(x,t)=a(x)+b(x)t+c(x)t^{2}\;,$ (3.176)

whose coefficients are prescribed by the conditions

$g(x,0)=f_{k-2}^{*}(x)\;,\qquad g(x,1)=f_{k-1}^{*}(x)\;,\qquad
g(x,2)=f_{k}^{*}(x)\;,$ (3.177)

which give

$a(x)=f_{k-2}^{*}(x)\;,\qquad
b(x)=-\;\frac{1}{2}\;\left[f_{k}^{*}(x)-4f_{k-1}^{*}(x)+3f_{k-2}^{*}(x)\right]\;,$
$c(x)=\frac{1}{2}\;\left[f_{k}^{*}(x)-2f_{k-1}^{*}(x)+f_{k-2}^{*}(x)\right]\;.$

Considering spline (3.176) as an expansion in powers of $t$, we construct a
self-similar approximant

$g^{*}(x,t)=a(x)[1+A(x)t]^{n(x)}\;,$ (3.178)

in which the control functions $A(x)$ and $n(x)$ are defined by the accuracy-
through-order procedure at small $t$, resulting in

$A(x)=\frac{b^{2}(x)-a(x)c(x)}{a(x)b(x)}\;,\qquad
n(x)=\frac{b^{2}(x)}{b^{2}(x)-a(x)c(x)}\;.$

The final answer is given by the expression

$g^{*}(x)=\frac{1}{2}\;\left[g^{*}(x,2)+g^{*}(x,3)\right]\;,$ (3.179)

with the error bar

$\pm\;\frac{1}{2}\left|g^{*}(x,2)-g^{*}(x,3)\right]\;.$ (3.180)

Note that if $f_{k}^{*}(x)\propto x^{\beta}$, then also $g^{*}(x)\propto
x^{\beta}$.

It is possible to employ some other tricks improving convergence. For example,
one can treat the sequence of self-similar approximants $\\{f_{k}^{*}(x)\\}$
similarly to a set of expansions $f_{k}(x)$, defining a new approximation
cascade, whose trajectory is bijective to the sequence $\\{f_{k}^{*}(x)\\}$.
Then, following the standard scheme, one can construct a twice renormalized
self-similar approximant [196].

It is also possible to introduce more control functions, for instance by using
power transforms [197]

$P_{k}(x,m)\equiv\left[f_{k}(x)\right]^{m}\;,$

with the power $m$ playing the role of a control parameter. Then the
transformed expression is re-expanded in powers of $x$ resulting in

$P_{k}(x,m)\simeq\sum_{n=0}^{k}b_{n}(m)x^{n}\qquad(x\rightarrow 0)\;,$

with $b_{n}(m)$ expressed through $a_{n}$. On the basis of the latter
expansion, we construct a self-similar approximant $P_{k}^{*}(x,m)$.
Accomplishing the inverse power transformation, we get

$F^{*}_{k}(x,m)=\left[P^{*}_{k}(x,m)\right]^{1/m}\;.$

Defining control functions $m_{k}=m_{k}(x)$ by some optimization conditions,
we come to the self-similar approximant

$f^{*}_{k}(x,m)=F^{*}_{k}(x,m_{k}(x))\;.$

This way, however, is quite cumbersome. The method of using splines looks much
easier, being rather accurate.

###  3.15 Electron Gas

Self-similar approximation theory has been used for solving hundreds of
problems, as can be inferred from the cited references. Here we illustrate its
application just for a few examples.

Let us consider the correlation energy of one-dimensional uniform electron gas
as a function of the Wigner-Seitz radius $r_{s}$ defined by the relation

$r_{s}\equiv\frac{1}{2\rho}\qquad\left(\rho\equiv\frac{N}{L}\right)\;,$

in which $\rho$ is mean density. The asymptotic behavior of the correlation
energy at small $r_{s}$, or high density, is given [198] by the expansion

$E_{cor}(r_{s})\simeq-\;\frac{\pi^{2}}{360}+0.00845r_{s}\qquad(r_{s}\rightarrow
0)\;.$ (3.181)

At large $r_{s}$, hence low density, the behavior is also known [198] being

$E_{cor}(r_{s})\simeq\frac{b_{1}}{r_{s}}+\frac{b_{2}}{r_{s}^{3/2}}\qquad(r_{s}\rightarrow\infty)\;,$
(3.182)

with the coefficients

$b_{1}=-\left(\ln\sqrt{2\pi}\;-\;\frac{3}{4}\right)=-0.168939\;,\qquad
b_{2}=0.359933\;.$

When the small, as well as large, variable behaviors are known, it is
reasonable to resort to self-similar root approximants. The root approximant,
satisfying both the small and large $r_{s}$ expansions reads as [199]

$E_{3}^{*}(r_{s})=-\;\frac{\pi^{2}}{360}\left(\left((1+A_{1}r_{s})^{3/2}+A_{2}r_{s}^{2}\right)^{5/4}+A_{3}r^{3}\right)^{-1/3}\;,$
(3.183)

where

$A_{1}=0.493150\;,\qquad A_{2}=0.056122\;,\qquad A_{3}=0.004274\;.$

This correlation energy has been calculated by diffusion Monte Carlo
simulations [198] in the interval of the Wigner-Seitz radius $r_{s}\in[0,20]$.
Comparing the numerical results [198] with the root approximant (3.183), we
find that the maximal error of the latter is $8\%$ for $r_{s}$ in that
interval.

###  3.16 Harmonium Atom

An $N$-electron harmonium atom is described by the Hamiltonian

$\hat{H}=\frac{1}{2}\sum_{i=1}^{N}\left(-\nabla^{2}_{i}+\omega^{2}r_{i}^{2}\right)+\frac{1}{2}\sum_{i\neq
j}^{N}\frac{1}{r_{ij}}\;,$ (3.184)

in which dimensionless units are used and

$r_{i}\equiv|{\bf r}_{i}|\;,\qquad r_{ij}\equiv|{\bf r}_{i}-{\bf r}_{j}|\;.$

This Hamiltonian provides a realistic modeling of many finite quantum systems,
such as trapped ions, quantum dots, atomic nuclei, and metallic grains [58].
Here we consider the ground-state energy of a two-electron harmonium $(N=2)$.

When the trapping is weak, so that $\omega$ is small, the ground-state energy
reads as

$E(\omega)\simeq
c_{0}\omega^{3/2}+c_{1}\omega+c_{2}\omega^{4/3}\qquad(\omega\rightarrow 0)\;,$
(3.185)

with the coefficients

$c_{0}=\frac{3}{2^{4/3}}=1.19055\;,\qquad
c_{1}=\frac{1}{2}\;(3+\sqrt{3})=2.36603\;,\qquad c_{2}=\frac{7}{36\cdot
2^{2/3}}=0.122492\;.$

And for a hard trapping by a rigid potential with a large $\omega$, the energy
behaves as

$E(\omega)\simeq
b_{0}\omega+b_{1}\omega^{1/2}+b_{2}+b_{3}\omega^{-1/2}\qquad(\omega\rightarrow\infty)\;,$
(3.186)

where

$b_{0}=3\;,\qquad b_{1}=\sqrt{\frac{2}{\pi}}=0.797885\;,\qquad
b_{2}=-\;\frac{2}{\pi}\;\left(1-\;\frac{\pi}{2}+\ln 2\right)=-0.077891\;,$
$b_{3}=\left(\frac{2}{\pi}\right)^{3/2}\;\left[2-2G-\;\frac{3}{2}\;\pi+(3+\pi)\ln
2+\frac{3}{2}\;(\ln 2)^{2}-\;\frac{\pi^{2}}{24}\right]=0.0112528\;,$

and $G$ is the Catalan constant

$G\equiv\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n+1)^{2}}=0.91596559\;.$

Having both the small-variable and large-variable expansions, it is again
reasonable to employ self-similar root approximants. Such a root approximant,
satisfying both asymptotic expansions, at small as well as at large $\omega$,
is [199]

$E_{6}^{*}(\omega)=c_{0}\omega^{2/3}\left(\left(\left(\left(\left(\left(1+A_{1}\omega^{1/3}\right)^{1/2}+A_{2}\omega^{2/3}\right)^{3/4}+A_{3}\omega\right)^{5/6}+\right.\right.\right.$
$\left.\left.\left.+A_{4}\omega^{4/3}\right)^{7/8}+A_{5}\omega^{5/3}\right)^{9/10}+A_{6}\omega^{2}\right)^{1/6}\;,$
(3.187)

where

$A_{1}=48.4532\;,\qquad A_{2}=564.108\;,\qquad A_{3}=1088.39\;,$
$A_{4}=1221.08\;,\qquad A_{5}=796.791\;,\qquad A_{6}=256\;.$

The accuracy of this approximant, as compared with numerical calculations
[200], is very good, having an error of only $0.9\%$. Note that Padé
approximants cannot be used in the case of harmonium, since the small-variable
and large-variable asymptotic expansions are incompatible.

###  3.17 Schwinger Model

The Schwinger model [201, 202] is a lattice gauge theory in $1+1$ dimensions
representing Euclidean quantum electrodynamics with a Dirac fermion field. It
possesses many properties in common with QCD, such as confinement, chiral
symmetry breaking, and charge shielding. For this reason, it has become a
standard test bed for the study of numerical techniques in field theory. Here
we consider the ground state of the model, corresponding to a vector boson of
mass $M(x)$ as a function of the variable $x=m/g$, where $m$ is the electron
mass and $g$ is the coupling parameter having the dimension of mass, so that
$x$ is dimensionless. The energy is given by the relation $E=M-2m$.

The small-$x$ expansion for the ground-state energy [203, 204, 205, 206] is

$E(x)\simeq 0.5642-0.219x+0.1907x^{2}\qquad(x\rightarrow 0)\;.$ (3.188)

While the large-$x$ expansion [206, 207, 208, 209] reads as

$E(x)\simeq
0.6418x^{-1/3}-\;\frac{1}{\pi}\;x^{-1}-0.25208x^{-5/3}\qquad(x\rightarrow\infty)\;.$
(3.189)

Using again self-similar root approximants, we find the form agreeing with
both asymptotic expansions,

$E_{5}^{*}(x)=A\left(\left(\left(\left((1+A_{1}x)^{4/3}+A_{2}x^{2}\right)^{7/6}+A_{3}x^{3}\right)^{10/9}\right.\right.+$
$+\left.\left.A_{4}x^{4}\right)^{13/12}+A_{5}x^{5}\right)^{-1/15}\;,$ (3.190)

where

$A=0.5642\;,\qquad A_{1}=3.109547\;,\qquad A_{2}=3.640565\;,$
$A_{3}=4.028571\;,\qquad A_{4}=1.070477\;,\qquad A_{5}=0.144711\;.$

The root approximant (3.190) provides a very good accuracy, as compared with
the calculations using a density-matrix renormalization group approach [210]
and fast moving frame estimates [211]. Table 2 shows that the root approximant
$E_{5}^{*}(x)$ gives the results coinciding with those of these numerical
techniques, in the frame of their accuracy.

Table 2:  Ground-state energy of the Schwinger model, for varying
dimensionless parameter $x=m/g$, obtained by using a density matrix
renormalization group, $E_{DMRG}$; fast moving frame estimates, $E_{FMFE}$;
and the self-similar root approximants, $E_{5}^{*}$.
$x$ | $E_{DMRG}$ | $E_{FMFE}$ | $E^{*}_{5}$
---|---|---|---
0.125 | 0.540 | 0.528 | 0.540
0.25 | 0.519 | 0.511 | 0.519
0.5 | 0.487 | 0.489 | 0.487
1 | 0.444 | 0.455 | 0.444
2 | 0.398 | 0.394 | 0.392
4 | 0.340 | 0.339 | 0.337
8 | 0.287 | 0.285 | 0.284
16 | 0.238 | 0.235 | 0.235

###  3.18 Large-Variable Prediction: Partition Function

It is often of great interest to learn the behavior of a function at large
variables, while only a small-variable expansion is available. This problem
can be solved by resorting to self-similar factor approximants.

To illustrate how it is possible to predict the large-variable behavior, let
us consider the partition function of the so-called zero-dimensional field
theory

$Z(g)=\frac{1}{\sqrt{\pi}}\int_{-\infty}^{\infty}e^{-\varphi^{2}-g\varphi^{4}}\;d\varphi\;,$
(3.191)

whose structure is mathematically typical of many problems in statistical
physics and field theory.

The weak-coupling expansion yields

$Z_{k}(g)=1+\sum_{n=1}^{k}a_{n}g^{n}\qquad(g\rightarrow 0)\;,$ (3.192)

with the coefficients

$a_{n}=\frac{(-1)^{n}}{\sqrt{\pi}\;n!}\;\Gamma\left(2n+\frac{1}{2}\right)$

factorially diverging for large orders $n$. We wish to find out what would be
the behavior of the partition function in the strong-coupling limit
$g\rightarrow\infty$. No information from this limit is supposed to be used,
just the weak-coupling expansion. The accuracy of the predicted result can be
defined from the exact asymptotic limit

$Z(g)\simeq 1.022765g^{-1/4}\qquad(g\rightarrow\infty)\;.$ (3.193)

Constructing the factor approximants $Z^{*}_{k}(g)$, we find the strong-
coupling limit

$Z_{k}^{*}(g)\simeq B_{k}g^{-\beta_{k}}\qquad(g\rightarrow\infty)\;.$ (3.194)

The amplitude $B$ and power $\beta$ are predicted within the errors of the
order of $20\%$, starting from the seventh approximation [192].

If we are interested solely in the value of the power $\beta$, then it is
possible to employ another procedure. We define

$\beta_{k}(g)\equiv\frac{d\ln Z_{k}(g)}{d\ln g}$ (3.195)

and expand it in powers of $g$,

$\beta_{k}(g)\simeq a_{1}g\left(1+\sum_{n=1}^{k}b_{n}g^{n}\right)\;.$ (3.196)

On the basis of this expansion, we construct the corresponding factor
approximants

$\beta_{k}^{*}(g)\simeq a_{1}g\prod_{j=1}^{N_{k}}(1+A_{j}g)^{n_{j}}\;,$
(3.197)

imposing an additional constraint requiring the finiteness of the limit
$g\rightarrow\infty$,

$1+\sum_{j=1}^{N_{k}}n_{j}=0\;.$ (3.198)

Then the sought answer is given by the limit

$\beta_{k}^{*}\equiv\lim_{g\rightarrow\infty}\beta_{k}^{*}(g)=a_{1}\prod_{j=1}^{N_{k}}A_{j}^{n_{j}}\;.$
(3.199)

Applying this method to the considered problem gives the exponent with an
error close to, although slightly larger, than the direct method of
constructing $Z^{*}_{k}(g)$.

###  3.19 Large-Variable Prediction: Eigenvalue Problem

As another example, illustrating the use of factor approximants for predicting
the large-variable behavior, we take the anharmonic oscillator with the
Hamiltonian

$\hat{H}=-\;\frac{1}{2}\;\frac{d^{2}}{dx^{2}}+\frac{1}{2}\;x^{2}+gx^{4}\;.$
(3.200)

The weak-coupling expansion for the ground-state energy reads as [26, 212]

$E_{k}(g)\simeq\frac{1}{2}\;\left(1+\sum_{n=1}^{k}a_{n}g^{n}\right)\;,$
(3.201)

with the first ten coefficients

$a_{1}=1.5\;,\qquad a_{2}=-5.25\;,\qquad a_{3}=41.625\;,\qquad
a_{4}=-482.578125\;,$ $a_{5}=7161.9609375\;,\qquad
a_{6}=-127965.6269532\;,\qquad a_{7}=2659467.4541\;,$
$a_{8}=-62896429.3856\;,\qquad a_{9}=1667083206.526\;,\qquad
a_{10}=-48957881405.6\;.$

And the strong-coupling limit is known to be

$E(g)\simeq 0.667986g^{1/3}\qquad(g\rightarrow\infty)\;.$ (3.202)

Using factor approximants, we get

$E^{*}_{k}(g)\simeq B_{k}g^{\beta_{k}}\qquad(g\rightarrow\infty)\;.$ (3.203)

Starting from the seventh order, the amplitude and power are predicted [192]
within the errors of about $10\%$.

###  3.20 Exact Solutions: Initial-Value Problem

As has been mentioned above, in some cases self-similar factor approximants
provide exact answers. This is illustrated below by solving nonlinear
differential equations [213]. An equation is termed singular, if it does not
allow for finding a solution by using perturbation theory in powers of
$\varepsilon$ [214, 215, 216].

(i) Linear singular problem

Consider the singular differential equation

$\varepsilon\;\frac{d^{2}f}{dt^{2}}+2\;\frac{df}{dt}+\frac{f}{\varepsilon}=0$
(3.204)

for a function $f(t)$ with $t\geq 0$ and with the initial conditions

$f(0)=1\;,\qquad\left.\frac{df}{dt}\right|_{t=0}=-\;\frac{1}{\varepsilon}\;.$

Looking for the solution at small time $t$, we get the expansion

$f(t)\simeq
1-\;\frac{t}{\varepsilon}+\frac{t^{2}}{2\varepsilon^{2}}+\ldots\qquad(t\rightarrow
0)\;.$ (3.205)

Constructing factor approximants, we find in any order, starting from $k=2$,
the form

$f_{k}^{*}(t)=e^{-t/\varepsilon}\qquad(k\geq 2)\;,$ (3.206)

which is the exact solution of Eq. (3.204).

(ii) Nonlinear singular problem

Let us consider the singular equation

$(\varepsilon f+t)\;\frac{df}{dt}+f-1=0\;,$ (3.207)

with the initial condition $f(0)=2$.

Using again factor approximants, we get [213], in any order starting from the
fourth order, the exact solution

$f^{*}_{k}(t)=\sqrt{4+\frac{2t}{\varepsilon}+\frac{t^{2}}{\varepsilon^{2}}}\;-\;\frac{t}{\varepsilon}\qquad(k\geq
4)\;.$ (3.208)

(iii) Singular logistic equation

Logistic equations are widespread in various applications [217]. Here we
consider a singular logistic equation

$\varepsilon\;\frac{df}{dt}=f(1-f)\;,$ (3.209)

with an initial condition $f(0)=f_{0}$.

Factor approximants for any order after the second, give [213] the exact
solution

$f_{k}^{*}(t)=\frac{f_{0}}{f_{0}-(f_{0}-1)e^{-t/\varepsilon}}\qquad(k\geq
3)\;.$ (3.210)

###  3.21 Exact Solutions: Boundary-Value Problem

Factor approximants also allow for finding exact solutions for some boundary-
value problems [213], as is shown below.

(i) Kink soliton equation

The $\varphi^{4}$ model, with particle mass $1/\varepsilon$, is described by
the nonlinear Schrödinger equation

$\frac{\varepsilon}{2}\;\frac{d^{2}f}{dx^{2}}+f-f^{3}=0\;.$ (3.211)

As boundary conditions, we take

$f(-\infty)=-1\;,\qquad f(\infty)=1\;.$ (3.212)

Considering expansions near the boundaries and employing factor approximants,
we find [213], in any order starting from the fourth, the exact kink soliton
solution

$f_{k}^{*}(x)=\tanh\left(\frac{x}{\sqrt{\varepsilon}}\right)\qquad(k\geq
4)\;.$ (3.213)

Such solitons are called topological [218].

(ii) Bell soliton equation

A nonlinear Schrödinger equation, with mass $-1/\varepsilon$ is

$\frac{\varepsilon}{2}\;\frac{d^{2}f}{dx^{2}}-f+f^{3}=0\;.$ (3.214)

We assume the boundary conditions

$f(-\infty)=0\;,\qquad f(\infty)=0\;.$ (3.215)

Again, we derive expansions near the boundaries and use factor approximants,
which results [213], in any order after the second, in the exact bell soliton
solution

$f_{k}^{*}(x)=\sqrt{2}\;{\rm
sech}\left(\sqrt{\frac{2}{\varepsilon}}\;x\right)\qquad(k\geq 3)\;.$ (3.216)

This is an example of a nontopological soliton [218].

###  3.22 Vortex-Line Equation

The radial nonlinear Schrödinger equation

$\frac{d^{2}f}{dr^{2}}+\frac{1}{r}\;\frac{df}{dr}\;-\;\frac{f}{r^{2}}+f-f^{3}=0\;,$
(3.217)

where $r>0$ is a dimensionless radial variable, can be met in different
branches of physics. Complimented by the boundary conditions

$f(0)=0\;,\qquad f(\infty)=1\;,$ (3.218)

it describes vortex lines.

At short radius $r$, we can find the expansion

$f_{k}(r)\simeq cr\left(1+\sum_{n=1}^{k}a_{n}r^{2n}\right)\qquad(r\rightarrow
0)\;,$ (3.219)

with the coefficients

$a_{1}=-\;\frac{1}{8}\;,\qquad a_{2}=\frac{1+8c^{2}}{192}\;,\qquad
a_{3}=-\;\frac{1+80c^{2}}{9216}\;,\qquad
a_{4}=\frac{1+656c^{2}+1152c^{4}}{737280}\;.$

The coefficient $c$ is defined from the boundary conditions.

On the basis of expansion (3.219), we construct factor approximants

$f_{k}^{*}(r)\simeq c_{k}r\prod_{j=1}^{N_{k}}(1+A_{j}r^{2})^{n_{j}}\;.$
(3.220)

The boundary condition $f_{k}^{*}(0)=0$ is automatically valid. And the
boundary condition $f_{k}^{*}(\infty)=1$ imposes the constraints

$c_{k}\prod_{j=1}^{N_{k}}A_{j}^{n_{j}}=1\;,\qquad
1+2\sum_{j=1}^{N_{k}}n_{j}=0\;,$ (3.221)

defining the coefficient $c_{k}$.

To characterize the accuracy of differential equations, it is possible to
proceed as follows [219]. Denoting the equation in the symbolic form as

$\hat{E}[f(r)]=0\;,$ (3.222)

one defines the solution defect of $f_{k}^{*}(r)$ as

$D[f_{k}^{*}(r)]\equiv|\hat{E}[f_{k}^{*}(r)]|\;.$ (3.223)

The maximal solution defect is given by the maximal value

$D[f_{k}^{*}]\equiv\sup_{r}D[f_{k}^{*}(r)]\;.$ (3.224)

The other characteristic of accuracy is the solution error

$\Delta[f_{k}^{*}(r)]\equiv|f_{k}^{*}(r)-f(r)|$ (3.225)

defining the deviation of the approximant from the exact solution at each
point $r$. Respectively, the maximal solution error is

$\Delta[f_{k}^{*}]\equiv\sup_{r}\Delta[f_{k}^{*}(r)]\;.$ (3.226)

The advantage of using as the characteristics of accuracy the solution defects
is that they can be defined without knowing the exact solution. For the factor
approximants (3.220), the maximal solution defects (3.224), are found to be

$D[f_{2}^{*}]=0.12\;,\qquad D[f_{3}^{*}]=0.017\;,$
$D[f_{4}^{*}]=0.015\;,\qquad D[f_{5}^{*}]=0.002\;,\qquad
D[f_{6}^{*}]=0.0018\;.$

These results demonstrate a high accuracy of the factor approximants and their
good numerical convergence.

If we wish to employ self-similar root approximants from Sec. 3.8 for
approximating the solutions to Eq. (3.217), we need to find out the asymptotic
expansion at large $r$. The latter is

$f(r)\simeq
1-\;\frac{1}{2}\;r^{-2}-\;\frac{9}{8}\;r^{-4}-\;\frac{161}{16}\;r^{-6}\qquad(r\rightarrow\infty)\;.$
(3.227)

The first several self-similar root approximants, that we denote as
$R_{k}^{*}(r)$ to distinguish them from the factor approximants, are

$R_{2}^{*}(r)=\frac{r}{2}\left(1+\frac{1}{4}\;r^{2}\right)^{-1/2}\;,\qquad
R_{3}^{*}(r)=\frac{r}{\sqrt{2}}\left(1+\frac{1}{2}\;r^{2}+\frac{1}{4}\;r^{4}\right)^{-1/4}\;,$
$R_{4}^{*}(r)=\frac{r}{4^{1/3}}\left(1+\frac{3}{4}\;r^{2}+\frac{3}{16}\;r^{4}+\frac{1}{16}\;r^{6}\right)^{-1/6}\;,$
$R_{5}^{*}(r)=\frac{r}{136^{1/8}}\left(1+r^{2}+\frac{9}{68}\;r^{4}+\frac{1}{34}\;r^{6}+\frac{1}{136}\;r^{8}\right)^{-1/8}\;.$

Their maximal solution defects are close to $0.1$, which is worse than that of
the factor approximants.

###  3.23 Critical Phenomena: Spin Systems

Self-similar approximants allow for a straightforward description of critical
phenomena. As and example, let us consider the three-dimensional spin - $1/2$
cubic Ising model with the Hamiltonian

$\hat{H}=-J\sum_{\langle ij\rangle}\sigma_{i}^{z}\sigma_{j}^{z}\;,$ (3.228)

where $\sigma_{j}=\pm 1$, the summation is over the nearest neighbors and
$J>0$. Thermodynamic characteristics of the model can be calculated by
perturbation theory with respect to the parameter

$v=\tanh\left(\frac{J}{T}\right)\;,$ (3.229)

in which $T$ is temperature. Series in powers of $v$ correspond to the weak-
coupling or high-temperature asymptotic expansions.

One often considers the second derivative of the susceptibility

$f(v)\equiv\left.\frac{d^{2}\chi}{dB^{2}}\right|_{B=0}\;.$ (3.230)

The related weak-coupling expansion [220] reads as

$f_{k}(v)=-2\left(1+\sum_{n=1}^{k}a_{n}v^{n}\right)\;,$ (3.231)

where the several first coefficients are

$a_{1}=24\;,\qquad a_{2}=318\;,\qquad a_{3}=3240\;,\qquad a_{4}=28158\;.$

The fourth-order factor approximant is

$f_{4}^{*}(v)=-2(1+A_{1}v)^{n_{1}}(1+A_{2}v)^{n_{2}}\;,$ (3.232)

with the parameters

$A_{1}=-4.602\;,\quad n_{1}=-4.30\;,\qquad A_{2}=7.373\;,\quad
n_{1}=0.571\;,\qquad.$

The critical point, corresponds to the closest to zero pole of the considered
function, with the related power defining the critical exponent, which gives

$v_{c}=\frac{1}{|A_{1}|}=0.217\;,\qquad|n_{1}|=4.3\;.$ (3.233)

These results are in good agreement with the values found by Monte Carlo
simulations [220, 221] yielding $v_{c}=0.218092$ and $|n_{1}|=4.37$. The
accuracy of the best Padé approximant of the same order is much worse, leading
to $v_{c}=0.153$ and to the critical exponent equal to one.

Critical points and related critical exponents can also be found by employing
self-similar root approximants [171, 222].

###  3.24 Critical Phenomena: Bose Gas

Calculation of the critical temperature shift of the Bose-Einstein
condensation for a weakly interacting Bose gas is discussed in Secs. 2.12 and
2.13, where optimized perturbation theory is employed. Recall that the aim is
to find the coefficient $c_{1}$ in the temperature shift

$\frac{\Delta T_{c}}{T_{0}}\simeq c_{1}\gamma\qquad(\gamma\rightarrow 0)$
(3.234)

for an asymptotically weak gas parameter $\gamma\equiv\rho^{1/3}a_{s}$.
Resorting to loop expansion in $O(N)$ field theory $\varphi^{4}$ yields [53,
54, 55] the expansion

$c_{1}(x)\simeq\sum_{n=1}^{5}a_{n}x^{n}\qquad(x\rightarrow 0)$ (3.235)

in powers of the variable (2.175) and with the coefficients listed in Table 3.
The required answer is given by the limit

$c_{1}=\lim_{x\rightarrow\infty}c_{1}(x)\;.$

Employing for this purpose self-similar factor approximants, we construct the
second-order approximant

$f_{2}^{*}(x)=a_{1}x(1+x)^{n_{1}}(1+A_{2}x)^{n_{2}}\;,$ (3.236)

with the condition

$n_{1}+n_{2}+1=0\;,$

the third-order approximant

$f_{3}^{*}(x)=a_{1}x(1+A_{1}x)^{n_{1}}(1+A_{2}x)^{n_{2}}\;,$ (3.237)

with the same condition $n_{1}+n_{2}+1=0$, and the fourth-order approximant

$f_{4}^{*}(x)=a_{1}x(1+x)^{n_{1}}(1+A_{2}x)^{n_{2}}(1+A_{3}x)^{n_{3}}\;,$
(3.238)

with the constraint

$n_{1}+n_{2}+n_{3}+1=0.$

Then, using the convergence acceleration scheme of Sec. 3.14, we find [37, 56]
the values of $c_{1}$ for a different number of field components $N$. Our
results are presented in Table 4 together with the results of the known Monte
Carlo simulations. As is seen, our results coincide with the Monte Carlo ones,
within the accuracy of the latter.

Table 3: Coefficients $a_{n}$ of the asymptotic expansion for $c_{1}(x)$ for a different number of field components $N$. $N$ | 0 | 1 | 2 | 3 | 4
---|---|---|---|---|---
$a_{1}$ | 0.111643 | 0.111643 | 0.111643 | 0.111643 | 0.111643
$a_{2}$ |  $-$0.0264412 |  $-$0.0198309 |  $-$0.0165258 |  $-$0.0145427 |  $-$0.0132206
$a_{3}$ | 0.0086215 | 0.00480687 | 0.00330574 | 0.00253504 | 0.0020754
$a_{4}$ |  $-$0.0034786 |  $-$0.00143209 |  $-$0.000807353 |  $-$0.000536123 |  $-$0.000392939
$a_{5}$ | 0.00164029 | 0.00049561 | 0.000227835 | 0.000130398 | 0.0000852025

Table 4: Coefficient $c_{1}$ of the critical temperature shift for a different number of components $N$, found by using self-similar factor approximants, and compared with the available Monte Carlo simulations. $N$ | $c_{1}$ | Monte Carlo
---|---|---
0 | 0.77$\pm$ 0.03 |
1 | 1.06$\pm$ 0.05 | 1.09$\pm$ 0.09 [52]
2 | 1.29$\pm$ 0.07 | 1.29$\pm$ 0.05 [41]
|  | 1.32$\pm$ 0.02 [39]
3 | 1.46$\pm$ 0.08 |
4 | 1.60$\pm$ 0.09 | 1.60$\pm$ 0.10 [52]

###  3.25 Critical Exponents

Critical exponents describe the behavior of thermodynamic characteristics in
the vicinity of second order phase transitions (see, e.g., Ref. [223] and also
Sec. 4). Here we show how such exponents can be calculated by using self-
similar factor approximants. Asymptotic series for critical exponents can be
derived by employing the so-called epsilon expansion (see Sec. 4). One usually
derives the series in powers of $\varepsilon=d-4$ for the critical exponents
$\eta$, $\nu^{-1}$, and $\omega$. The other exponents can be found from the
scaling relations

$\alpha=2-\nu
d\;,\qquad\beta=\frac{\nu}{2}\;(d-2+\eta)\;,\qquad\gamma=\nu(2-\eta)\;,\qquad\delta=\frac{d+2-\eta}{d-2+\eta}\;,$
(3.239)

where $d$ is space dimensionality. For the three-dimensional space $(d=3)$, we
have

$\alpha=2-3\nu\;,\qquad\beta=\frac{\nu}{2}\;(1+\eta)\;,$
$\gamma=\nu(2-\eta)\;,\qquad\delta=\frac{5-\eta}{1+\eta}\qquad(d=3)\;.$
(3.240)

Different $N$ correspond to different physical systems. Thus $N=0$ corresponds
to dilute polymer solutions, $N=1$, to the Ising model, $N=2$, to magnetic
$XY$ models and to superfluids, and $N=3$ characterizes the Heisenberg model.
But, in general, it is possible to consider other $N$.

For $N=-2$, the critical exponents for any $d$ are defined exactly:

$\alpha=\frac{1}{2}\;,\qquad\beta=\frac{1}{4}\;,\qquad\gamma=1\;,\qquad\delta=5\;,$
$\eta=0\;,\qquad\nu=\frac{1}{2}\qquad(N=-2)\;.$ (3.241)

In the limit of $N\rightarrow\infty$, the exponents also are known exactly:

$\alpha=\frac{d-4}{d-2}\;,\qquad\beta=\frac{1}{2}\;,\qquad\gamma=\frac{2}{d-2}\;,\qquad\delta=\frac{d+4}{d-2}\;,$
$\eta=0\;,\qquad\nu=\frac{1}{d-2}\;,\qquad\omega=4-d\qquad(N\rightarrow\infty)\;,$
(3.242)

which for the three dimensional space reduces to

$\alpha=-1\;,\qquad\beta=\frac{1}{2}\;,\qquad\gamma=2\;,\qquad\delta=5\;,$
$\eta=0\;,\qquad\nu=1\;,\qquad\omega=1\qquad(d=3,~{}N\rightarrow\infty)\;.$
(3.243)

The $\varepsilon$\- expansions for the critical exponents $\eta$, $\nu^{-1}$,
and $\omega$, in three dimensions for arbitrary $N$ can be found in the book
[224]. Applying to these expansions self-similar factor approximants for small
$\varepsilon$ and then setting $\varepsilon=1$, we obtain [225, 226] the
results summarized in Table 5. For the components $N=-2$ and
$N\rightarrow\infty$, factor approximants give the exact values. The results
for all $N$ are in good agreement with the known experimental exponents, as
well as with Monte Carlo simulations, Borel-Padé summation, and other methods,
as can be inferred from reviews [33, 224, 227, 228] (see also [229, 230, 231,
232]).

Table 5: Critical exponents for the $N$-component $\varphi^{4}$ field theory, obtained by the summation of $\varepsilon$-expansions using self-similar factor approximants. $N$ | $\alpha$ | $\beta$ | $\gamma$ | $\delta$ | $\eta$ | $\nu$ | $\omega$
---|---|---|---|---|---|---|---
-2 | 0.5 | 0.25 | 1 | 5 | 0 | 0.5 | 0.79838
-1 | 0.36612 | 0.27742 | 1.0791 | 4.8897 | 0.01874 | 0.54463 | 0.79380
0 | 0.23466 | 0.30268 | 1.1600 | 4.8323 | 0.02875 | 0.58845 | 0.79048
1 | 0.10645 | 0.32619 | 1.2412 | 4.8050 | 0.03359 | 0.63118 | 0.78755
2 | -0.01650 | 0.34799 | 1.3205 | 4.7947 | 0.03542 | 0.67217 | 0.78763
3 | -0.13202 | 0.36797 | 1.3961 | 4.7940 | 0.03556 | 0.71068 | 0.78904
4 | -0.23835 | 0.38603 | 1.4663 | 4.7985 | 0.03476 | 0.74612 | 0.79133
5 | -0.33436 | 0.40208 | 1.5302 | 4.8057 | 0.03347 | 0.77812 | 0.79419
6 | -0.41963 | 0.41616 | 1.5873 | 4.8142 | 0.03197 | 0.80654 | 0.79747
7 | -0.49436 | 0.42836 | 1.6376 | 4.8231 | 0.03038 | 0.83145 | 0.80108
8 | -0.55920 | 0.43882 | 1.6816 | 4.8320 | 0.02881 | 0.85307 | 0.80503
9 | -0.61506 | 0.44774 | 1.7196 | 4.8406 | 0.02729 | 0.87169 | 0.80935
10 | -0.66297 | 0.45530 | 1.7524 | 4.8489 | 0.02584 | 0.88766 | 0.81408
50 | -0.98353 | 0.50113 | 1.9813 | 4.9537 | 0.00779 | 0.99451 | 0.93176
100 | -0.93643 | 0.49001 | 1.9564 | 4.9926 | 0.00123 | 0.97881 | 0.97201
1000 | -0.99528 | 0.49933 | 1.9966 | 4.9986 | 0.00023 | 0.99842 | 0.99807
10000 | -0.99952 | 0.49993 | 1.9997 | 4.9999 | 0.00002 | 0.99984 | 0.99979
$\infty$ | -1 | 0.5 | 2 | 5 | 0 | 1 | 1

###  3.26 Time Series

There exist various time series consisting of sets of data $f_{n}$ at
different moments of time $t_{n}$. One of the most important tasks concerning
time series is the possibility of forecasting their future behavior being
based on the knowledge of their past data.

Let a time series be characterized by a data set

$\mathbb{D}_{k}=\\{f_{n},\;t_{n}:~{}n=0,1,2,\ldots,k\\}\;,$ (3.244)

in which all $t_{n}$ are distinct and ordered either in ascending or
descending order. It is straightforward to define the interpolating polynomial

$f_{k}(t)=\sum_{n=0}^{k}c_{n}t^{n}$ (3.245)

that passes through all point of the data set,

$f_{k}(t_{n})=f_{n}\qquad(n=0,1,2,\ldots,k)\;.$ (3.246)

The latter condition is nothing but a system of linear algebraic equations
with respect to the coefficients $c_{m}$,

$\sum_{m=0}^{k}c_{m}t_{n}^{m}=f_{n}\;.$ (3.247)

The solution to this system of equations involves the Vandermonde determinant

$\displaystyle
V_{k}=\left|\begin{array}[]{ccccc}1{}&~{}t_{0}{}&~{}t_{0}^{2}{}&~{}\ldots{}&~{}t_{0}^{k}\\\
1{}&~{}t_{1}{}&~{}t_{1}^{2}{}&~{}\ldots{}&~{}t_{1}^{k}\\\
\ldots{}&~{}\ldots{}&~{}\ldots{}&~{}\ldots{}&~{}\ldots\\\
1{}&~{}t_{k}{}&~{}t_{k}^{2}{}&~{}\ldots{}&~{}t_{k}^{k}\end{array}\right|=\prod_{0\leq
i<j\leq k}(t_{j}-t_{i})\;.$ (3.252)

For all $t_{n}$ distinct, the determinant is nonzero, hence the solution is
unique.

The other way of defining the interpolating polynomial is by composing a
function

$f_{k}(t)=\sum_{n=0}^{k}f_{n}\lambda_{n}^{k}(t)\;,$ (3.253)

which is an expansion over the Lagrange basis polynomials

$\lambda_{n}^{k}(t)\equiv\prod_{j(\neq
n)}^{k}\frac{t-t_{j}}{t_{n}-t_{j}}\qquad(n\leq k)\;,$

having the property

$\lambda_{n}^{k}(t_{m})=\delta_{mn}\;.$

On the basis of the interpolating polynomial $f_{k}(t)$, it is possible to
construct a self-similar approximant $f_{k}^{*}(t)$ that allows for the
extrapolation of the data set for times $t$ larger than the maximal of
$t_{n}$. Thus we obtain a forecast $f_{k}^{*}(t)$ for future time. Such an
extrapolation can be done using exponential approximants [173, 187, 233, 234,
235, 236, 237, 238] or factor approximants.

###  3.27 Probabilistic Scenarios

Quite regularly, accomplishing measurements for the same process may result in
different data. This can be due, e.g., to the existence of poorly controlled
noise, because the measurements have been accomplished by different devices or
with differing methodics, or just realized in slightly different times. Then
the results composing different data sets have to be characterized in a
probabilistic picture. The idea of a probabilistic approach comes from the
problem of pattern selection in the case of multiple solutions of nonlinear
equations [187, 237, 239, 240, 241].

Suppose we have several data sets enumerated by an index $\alpha=1,2,\ldots$,

$\mathbb{D}_{k\alpha}=\\{f_{n\alpha},\;t_{n\alpha}:~{}n=0,1,2,\ldots,k\\}\;.$
(3.254)

For each data set, we can construct, as is explained in the previous section,
a self-similar approximant $f_{k\alpha}^{*}(t)$ forecasting for times outside
the data set. Following Sec. 3.3, we interpret the sequence of
$f_{k\alpha}^{*}(t)$ as the points corresponding to a trajectory of a
dynamical system for which we can define the map multipliers

$m_{k\alpha}^{*}(t)=\frac{\delta f_{k\alpha}^{*}(t)}{\delta
f^{*}(t)}=\frac{\partial f_{k\alpha}^{*}(t)/\partial t}{\partial
f^{*}(t)/\partial t}\;,$ (3.255)

where $f^{*}(t)$ is a first nontrivial self-similar approximant. Then we
introduce the average multiplier $M_{k}(t)$ by the relation

$\frac{1}{M_{k}(t)}\equiv\sum_{\alpha}\frac{1}{|m_{k\alpha}^{*}(t)|}\;.$
(3.256)

The probability of a scenario $\alpha$ at time $t$ is

$p_{\alpha}(t)=\left|\frac{M_{k}(t)}{m_{k\alpha}^{*}(t)}\right|\;.$ (3.257)

By construction, it is normalized,

$\sum_{\alpha}p_{\alpha}(t)=1\;.$ (3.258)

The optimal scenario corresponds to the maximal probability

$p_{opt}(t)\equiv\sup_{\alpha}p_{\alpha}(t)\;.$ (3.259)

###  3.28 Discrete Scaling

One calls a function $f(x)$ scale invariant, when it satisfies the scaling
relation

$f(\lambda x)=\mu f(x)\;.$ (3.260)

The evident solution to this equation is the power-law behavior

$f(x)=Cx^{\alpha}\;,$ (3.261)

with the condition

$\lambda^{\alpha}=\mu e^{i2\pi n}\qquad(n=0,1,2,\ldots)\;.$ (3.262)

Therefore the power $\alpha$ writes as

$\alpha=\frac{\ln\mu}{\ln\lambda}+i\;\frac{2\pi n}{\ln\lambda}\;.$ (3.263)

In the case of continuous scaling, the scale $\lambda$ can be taken as
$\lambda=1$, which requires to set $n=0$. Then

$\alpha=\frac{\ln\mu}{\ln\lambda}\qquad(n=0)\;.$ (3.264)

This defines the power $\alpha$ as a real number, provided $\lambda$ and $\mu$
are real.

But for discrete scaling, $\lambda$ can pertain to a set not including
$\lambda=1$. In such a case, the power $\alpha$ becomes complex-valued [242].
The processes with complex-valued exponents, resulting in log-periodic
oscillations, occur, for instance, in financial time series [243, 244, 245,
246, 247, 248, 249, 250] and materials rupture [251].

###  3.29 Complex Renormalization

Under discrete scaling, when characteristic exponents become complex-valued,
there appears the necessity of considering renormalization equations on a
complex plane. Below, we give an example of such a procedure.

Let us study a system characterized by a function $I(t)$. This can be a market
price, financial index, or the intensity of acoustic radiation from a loaded
material as a function of time [252, 253]. The function is considered on the
time interval $0<t<t_{c}$, with $t_{c}$ being a critical point, where $I(t)$
experiences a sharp increase or a drastic fall. It is convenient to introduce
the dimensionless variable

$x\equiv\frac{t_{c}-t}{t_{c}}=x(t)\;,$ (3.265)

with the boundary conditions

$x(0)=1\;,\qquad x(t_{c})=0\;.$ (3.266)

Also, we define the dimensionless function

$f(x)\equiv\frac{I(t_{c})-I(t)}{I(t_{c})-I(0)}$ (3.267)

satisfying the boundary conditions

$f(0)=0\qquad(t=t_{c}\;,~{}x(t_{c})=0)\;,$ $f(1)=1\qquad(t=0\;,~{}x(0)=1)\;.$
(3.268)

The function $I(t)$ is expressed through function (3.267) as

$I(t)=I(t_{c})-[I(t_{c})-I(0)]f(x(t))\;.$ (3.269)

Since we keep in mind a discrete scaling, requiring to work on the complex
plane, we analytically continue the real function $f(x)$ to a complex function
$F(x)$, such that

$f(x)={\rm Re}F(x)\;.$ (3.270)

And $F(x)$ is assumed to obey the same boundary conditions as $f(x)$,

$F(0)=0\;,\qquad F(1)=1\;.$ (3.271)

The renormalization-group equation for $F(x)$ reads [243] as

$\frac{d\ln F}{d\ln x}=G\;,$ (3.272)

where $G=G(x)$ is a Gell-Mann law function. This equation has to be
complemented either by an equation for the complex-conjugate function
$F^{*}(x)$ or by an equation for the function

$W=|F|^{2}=W(x)\;,$ (3.273)

with the boundary conditions

$W(0)=0\;,\qquad W(1)=1\;.$ (3.274)

From Eq. (3.272), we get

$\frac{d\ln W}{d\ln x}=2{\rm Re}G\;.$ (3.275)

The function $W$ equals the squared amplitude of $F$ describing the main
tendency of the variance of $F$, while $F$ itself also contains a phase
producing oscillations around the main tendency. Therefore it is reasonable to
treat $F$ as a fast function, as compared to the slow function $W$. Then it is
admissible to employ the idea of the averaging techniques and solve the
equation for the fast function $F$, keeping the slow function as an integral
of motion. This gives

$F=x^{G}\;.$ (3.276)

Generally, $G$ is a complex function, hence the previous expression can be
represented as

$F=x^{{\rm Re}G}\left\\{\cos\left[({\rm Im}G)\ln x\right]+i\sin\left[({\rm
Im}G)\ln x\right]\right\\}\;.$ (3.277)

Note that the necessary and sufficient condition for classifying the function
$F$ as fast, while $W$ as slow is $\rm{Re}G\ll\rm{Im}G$.

We need to find out the dependence of $G$ on $x$. If we assume that the
function $W(x)$ can be expanded in powers of some $x^{s}$ then from equation
(3.275) it follows that $G(x)$ is expandable in the same powers, so that its
$k$-th order expansion reads as

$G(x)\simeq G_{k}(x)=\sum_{n=0}^{k}a_{n}x^{sn}\qquad(x\rightarrow 0)\;,$
(3.278)

under the relation

$s=2{\rm Re}a_{0}\equiv 2\alpha\;.$ (3.279)

The extrapolation of the small variable asymptotic expansion (3.278) to the
finite values of $x$ can be done by means of the self-similar exponential
approximants (see Sec. 3.10). This yields

$G_{k}^{*}(x)=a_{0}\exp\left(C_{1}x^{2\alpha}\exp\left(C_{2}x^{2\alpha}\ldots\exp\left(C_{k}x^{2\alpha}\right)\right)\right)\;,$
(3.280)

where $C_{j}$ are control parameters. To first order, we have

$G_{1}^{*}(x)=a_{0}\exp\left(C_{1}x^{2\alpha}\right)\;.$ (3.281)

Taking into account that the parameters, in general, are complex, we separate
their real and imaginary parts,

$a_{0}=\alpha+i\omega\;,\qquad C_{1}=\beta+i\gamma\;.$ (3.282)

Then we get

$G_{1}^{*}(x)=(\alpha+i\omega)\exp\left\\{(\beta+i\gamma)x^{2\alpha}\right\\}=$
$=\exp\left(\beta x^{2\alpha}\right)\left\\{\alpha\cos\left(\gamma
x^{2\alpha}\right)-\omega\sin\left(\gamma
x^{2\alpha}\right)+i\left[\alpha\sin\left(\gamma
x^{2\alpha}\right)+\omega\cos\left(\gamma
x^{2\alpha}\right)\right]\right\\}\;.$ (3.283)

Using the approximant (3.283), we obtain the sought function

$f(x)={\rm Re}F(x)=\exp[({\rm Re}G)\ln x]\cos[({\rm Im}G)\ln x]\;,$ (3.284)

in which

${\rm Re}G=\left[\alpha\cos\left(\gamma
x^{2\alpha}\right)-\omega\sin\left(\gamma
x^{2\alpha}\right)\right]\exp\left(\beta x^{2\alpha}\right)\;,$ ${\rm
Im}G=\left[\alpha\sin\left(\gamma x^{2\alpha}\right)+\omega\cos\left(\gamma
x^{2\alpha}\right)\right]\exp\left(\beta x^{2\alpha}\right)\;.$ (3.285)

The four parameters, $\alpha$, $\beta$, $\gamma$, and $\omega$ should be found
from additional conditions, or from empirical data.

Note that close to the critical point $t_{c}$, where

$x\rightarrow 0\;,\qquad{\rm Re}G\rightarrow\alpha\;,\qquad{\rm
Im}G\rightarrow\omega\;,$

function (3.284) reduces to

$f(x)\simeq x^{\alpha}\cos(\omega\ln x)\qquad(x\rightarrow 0)\;,$

which is the simple case of a power law behavior decorated by log-periodic
oscillations.

Taking higher orders of approximant (3.280) makes the function $f(x)$ more
complicated and increases the number of parameters. In the $k$-th order, the
number of control parameters is $2(1+k)$.

###  3.30 Quark-Gluon Plasma

In Quantum Chromodynamics (QCD), perturbation theory with respect to the
coupling parameter is valid at asymptotically large temperatures. One assumes
that the temperature is so high that it is admissible to set the chemical
potential zero and to neglect quark masses. The weak-coupling free energy
expansion can be represented either in powers of the quantum chromodynamics
coupling $\alpha_{s}$ or in powers of the coupling parameter $g$, using the
relation

$\alpha_{s}=\frac{g^{2}}{4\pi}\;.$ (3.286)

The perturbative expansion for the pressure of high-temperature $SU(N_{c})$
gauge theory, with massless fermions, in four dimensions, with $N_{c}=3$
colours and with $n_{f}$ quark flavors in the fundamental representation, has
been calculated to fourth order [254] and to fifth order [255, 256]. In terms
of $\alpha_{s}$, the expansion of pressure reads as

$P(\alpha_{s})\simeq\frac{8\pi^{2}}{45}\;T^{4}\;\left[\sum_{n=0}^{5}a_{n}\left(\frac{\alpha_{s}}{\pi}\right)^{n/2}+a_{4}^{\prime}\left(\frac{\alpha_{s}}{\pi}\right)^{2}\ln\;\frac{\alpha_{s}}{\pi}\right]\;,$
(3.287)

with the coefficients

$a_{0}=1+\frac{21}{32}\;n_{f}\;,\qquad a_{1}=0\;,\qquad
a_{2}=-\;\frac{15}{4}\;\left(1+\frac{15}{12}\;n_{f}\right)\;,\qquad
a_{3}=30\left(1+\frac{1}{6}\;n_{f}\right)^{3/2}\;,$
$a_{4}=273.2+15.97\;n_{f}-0.413\;n_{f}^{2}+\frac{135}{8}\;\left(1+\frac{1}{6}\;n_{f}\right)\ln\left(1+\frac{1}{6}\;n_{f}\right)-$
$-\;\frac{165}{8}\left(1+\frac{5}{12}\;n_{f}\right)\left(1-\;\frac{2}{32}\;n_{f}\right)\ln\;\frac{\mu}{2\pi
T}\;,$
$a_{5}=\left(1+\frac{1}{6}\;n_{f}\right)^{1/2}\left[\frac{495}{2}\left(1+\frac{1}{6}\;n_{f}\right)\left(1-\;\frac{2}{33}\;n_{f}\right)\ln\;\frac{\mu}{2\pi
T}-799.2-21.96\;n_{f}-1.926\;n_{f}^{2}\right]\;,$
$a_{4}^{\prime}=\frac{135}{2}\;\left(1+\frac{1}{6}\;n_{f}\right)\;.$

Here, $\mu$ is the renormalization scale parameter in the $\overline{\rm MS}$
scheme and the dimensional regularization is used.

The weak-coupling expansion for the pressure in terms of $g$ is

$P(g)\simeq\frac{8\pi^{2}}{45}\;T^{4}\left(\sum_{n=0}^{5}c_{n}g^{n}+c_{4}^{\prime}g^{4}\ln
g\right)\;,$ (3.288)

with the coefficients

$c_{0}=1+\frac{21}{32}\;n_{f}\;,\qquad c_{1}=0\;,\qquad
c_{2}=-\;\frac{15}{(4\pi)^{2}}\;\left(1+\frac{5}{12}\;n_{f}\right)=-0.09499\left(1+\frac{5}{12}\;n_{f}\right)\;,$
$c_{3}=\frac{240}{(4\pi)^{3}}\;\left(1+\frac{1}{6}\;n_{f}\right)^{3/2}=0.12094\;\left(1+\frac{1}{6}\;n_{f}\right)^{3/2}$
$c_{4}=0.04331\left(1+\frac{1}{6}\;n_{f}\right)\ln\left(1+\frac{1}{6}\;n_{f}\right)+0.01733-0.00763\;n_{f}-0.00088\;n_{f}^{2}-$
$-0.01323\left(1+\frac{5}{12}\;n_{f}\right)\left(1-\;\frac{2}{33}\;n_{f}\right)\ln\;\frac{\mu}{T}\;,$
$c_{5}=0.02527\left(1+\frac{1}{6}\;n_{f}\right)^{3/2}\left(1-\;\frac{2}{33}\;n_{f}\right)\ln\;\frac{\mu}{T}-$
$-\left(0.12806+0.00717\;n_{f}-0.00027\;n_{f}^{2}\right)\left(1+\frac{1}{6}\;n_{f}\right)^{1/2}\;,$
$c_{4}^{\prime}=0.08662\left(1+\frac{1}{6}\;n_{f}\right)\;.$

It is convenient to introduce the reduced pressure

$p_{k}\equiv\frac{P_{k}}{P_{0}}=p_{k}(g,\mu,T)\;,$ (3.289)

that is the ratio of a $k$-th order pressure $P_{k}$, to the Stefan-Boltzmann
pressure

$P_{0}=\frac{8\pi^{2}}{45}\;T^{4}\left(1+\frac{21}{32}\;n_{f}\right)\;.$
(3.290)

The expansion for the reduced pressure reads as

$p_{k}=1+\sum_{n=0}^{5}\overline{c}_{n}g^{n}+\overline{c}_{4}^{\prime}g^{4}\ln
g\;,$ (3.291)

where

$\overline{c}_{n}\equiv\frac{c_{n}}{c_{0}}\;,\qquad\overline{c}_{4}^{\prime}\equiv\frac{c_{4}^{\prime}}{c_{0}}\;.$

The parameter $g$ is the running coupling, whose dependence $g(\mu)$ on the
renormalization scale $\mu$ is given by the renormalization group equation

$\mu\;\frac{\partial g}{\partial\mu}=\beta(g)\;.$ (3.292)

The weak-coupling expansion for the Gell-Mann-Low function $\beta(g)$ can be
written as

$\beta_{k}(g)=-\sum_{n=0}^{k}b_{n}g^{2n+3}\;.$ (3.293)

The coefficients for $N_{c}=3$ have been calculated in four-loop order [257,
258] and, recently, in five-loop order [259, 260]. These coefficients are

$b_{0}=\frac{1}{(4\pi)^{2}}\;\left(11-\;\frac{2}{3}\;n_{f}\right)\;,\qquad
b_{1}=\frac{2}{(4\pi)^{4}}\;\left(51-\;\frac{19}{3}\;n_{f}\right)\;,$
$b_{2}=\frac{1}{2(4\pi)^{6}}\;\left(2857-\;\frac{5033}{9}\;n_{f}+\frac{325}{27}\;n_{f}^{2}\right)\;,$
$b_{3}=\frac{1}{(4\pi)^{8}}\;\left[\frac{149753}{6}+3564\zeta(3)-\left(\frac{1078361}{162}+\frac{6508}{27}\;\zeta(3)\right)n_{f}+\right.$
$+\left.\left(\frac{50065}{162}+\frac{6472}{81}\;\zeta(3)\right)n_{f}^{2}+\frac{1093}{729}\;n_{f}^{3}\right]\;,$
$b_{4}=\frac{1}{(4\pi)^{10}}\;\left[\frac{8157455}{16}+\frac{621885}{2}\;\zeta(3)-\;\frac{88209}{2}\;\zeta(4)-288090\zeta(5)-\right.$
$-\;\left(\frac{336460813}{1944}+\frac{4811164}{81}\;\zeta(3)-\;\frac{33935}{6}\;\zeta(4)-\;\frac{1358995}{27}\;\zeta(5)\right)n_{f}+$
$+\left(\frac{25960913}{1944}+\frac{698531}{81}\;\zeta(3)-\;\frac{10526}{9}\;\zeta(4)-\;\frac{381760}{81}\;\zeta(5)\right)n_{f}^{2}-$
$-\left.\left(\frac{630559}{5832}+\frac{48722}{243}\;\zeta(3)-\;\frac{1618}{27}\;\zeta(4)-\;\frac{460}{9}\;\zeta(5)\right)n_{f}^{3}+\left(\frac{1205}{2916}-\;\frac{152}{81}\;\zeta(3)\right)n_{f}^{4}\right]\;.$

As an initial condition for Eq. (3.290), we can take the value of the running
coupling $\alpha_{s}(m_{Z})=0.119$ for the $Z^{0}$ boson mass $m_{Z}$. In
terms of the coupling $g$, this initial condition is

$g(m_{Z})=1.222856\qquad(m_{Z}=91187\;{\rm MeV})\;.$ (3.294)

The equation of state for quantum chromodynamics, employing exponential self-
similar approximants (see Sec. 3.10), has been derived in Ref. [261]. The
procedure has been as follows. The renormalization scale $\mu$ is treated as a
control function defined by the minimal difference condition $p_{4}-p_{3}=0$,
which gives $\mu=\mu(g,T)$. By using the expansions $\beta_{k}(g)$ and
$p_{k}(g,T)$, exponential self-similar approximants $\beta_{k}^{*}(g)$ and
$p_{k}^{*}(g,T)$ are constructed. The renormalization group equation (3.289),
with the initial condition (3.291), yields $g=g(\mu)$. From $\mu=\mu(g,T)$ and
$g=g(\mu)$, we find $\mu(T)$ and $g(T)$. Then the self-similar approximant
$p_{k}^{*}(g,\mu,T)$ defines the pressure as a function of temperature
$p_{k}^{*}(T)$. When diminishing temperature, the pressure varies from the
Stefan-Boltzmann limit to a sharp drop to zero at temperature $T_{c}\approx
150$ Mev. This temperature $T_{c}$ can be associated with the confinement-
deconfinement phase transition. These results agree with other models of
confinement [262, 263, 264], as well as with lattice data [265]. Hard-thermal-
loop perturbation theory, that is a variant of optimized perturbation theory,
also yields a similar behavior of the QCD pressure [266].

###  3.31 Gell-Mann-Low Function: Quantum Chromodynamics

The overall behavior of a Gell-Mann-Low function, as a function of a scaling
variable provides important information on the related field theory. Of
special interest is the large variable asymptotic form of the function. In
Secs. 3.18 and 3.19, it has been shown that the large-variable behavior of a
function can be predicted by extrapolating its small-variable expansion by
means of self-similar factor approximants. The accuracy of such a prediction
is about $10\%$. Using these approximants, we also can estimate the large-
variable behavior of Gell-Mann-Low functions.

The Gell-Mann-Low function for QCD can be defined as

$\beta(a_{s})=\mu^{2}\;\frac{da_{s}}{d\mu^{2}}\qquad\left(a_{s}\equiv\frac{\alpha_{s}}{\pi}\right)\;.$
(3.295)

In the space-time four dimensions, within the minimal subtraction scheme
$(\overline{\rm MS})$, a weak-coupling expansion of the beta function in
powers of $a_{s}$, in five-loop order [260, 267], reads as

$\beta(a_{s})\simeq-\sum_{n=0}^{k}\beta_{n}a_{s}^{n+2}\qquad(a_{s}\rightarrow
0)\;,$ (3.296)

with the coefficients $\beta_{n}$ expressed through the coefficients $b_{n}$
from the previous section by the relation

$\beta_{n}=(2\pi)^{2n+2}b_{n}\;.$ (3.297)

In numerical form, these coefficients are

$\beta_{0}=2.75-0.166667\;n_{f}\;,\qquad\beta_{1}=6.375-0.791667\;n_{f}\;,$
$\beta_{2}=22.3203-4.36892\;n_{f}+0.0940394\;n_{f}^{2}\;,$
$\beta_{3}=114.23-27.1339\;n_{f}+1.58238\;n_{f}^{2}+0.0058567\;n_{f}^{3}\;,$
$\beta_{4}=524.56-181.8\;n_{f}+17.16\;n_{f}^{2}-0.22586\;n_{f}^{3}-0.0017993\;n_{f}^{4}\;.$

Applying self-similar factor approximants for summing series (3.296), we can
find the strong-coupling limit

$\beta(a_{s})\simeq-Ba_{s}^{\gamma}\qquad(a_{s}\rightarrow\infty)$ (3.298)

that is connected with the limit of the function $\beta(g)$, defined by Eq.
(3.292), as

$\beta(g)=\frac{(2\pi)^{2}}{g}\;\beta(a_{s})\simeq-\;\frac{B}{(2\pi)^{2\gamma-2}}\;g^{2\gamma-1}\qquad(g\rightarrow\infty)\;.$

It turns out that for the factor approximants with the $n_{f}$ from zero to
five the limit $g$ tending to infinity does not exist, since the approximants
become complex-valued. For the physically reasonable number of flavors
$n_{f}=6$, there exist the real-valued approximants:

$\beta_{2}^{*}(a_{s})=-1.75a_{s}^{2}(1+1.1554a_{s})^{0.598}\;,$
$\beta_{3}^{*}(a_{s})=-1.75a_{s}^{2}(1+a_{s})^{0.912}(1+32.53a_{s})^{0.0005}\;,$
(3.299)

whose strong-coupling asymptotics are

$\beta_{2}^{*}(a_{s})\simeq-2.277a_{s}^{2.598}\;,$
$\beta_{3}^{*}(a_{s})\simeq-1.753a_{s}^{2.913}\qquad(a_{s}\rightarrow\infty)\;.$
(3.300)

Similarly, applying self-similar factor approximants for the beta function
defined in equation (3.292), with expansion (3.293), we obtain

$\beta_{2}^{*}(g)=-0.044g^{3}(1+0.039g^{2})^{0.598}\;,$
$\beta_{3}^{*}(g)=-0.044g^{3}(1+g^{2})^{0.000277}(1+0.028g^{2})^{0.833}\;.$

The strong-coupling limit for these approximants gives

$\beta_{2}^{*}(g)\simeq-0.0064g^{4.195}\;,\qquad\beta_{3}^{*}(g)\simeq-0.0022g^{4.667}\qquad(g\rightarrow\infty)\;.$

It seems that there are no reliable estimates of the strong-coupling behavior
of the QCD beta function, which it would be possible to compare with the above
results. There exists an estimate [268, 269] of $\gamma=-13$ for $n_{f}=0$ and
the negative $g$ tending to minus infinity. The change of the sign of $g$ is
motivated by the fact that the used Borel-Leroy summation can be applied to
alternating series, while the series (3.246) are not alternating, so that,
strictly speaking, the Borel summation is not applicable. Changing the sign of
$g$ produces a kind of alternating series. However, the expansions for
$g\rightarrow\infty$ and $g\rightarrow-\infty$ can be drastically different
[270].

###  3.32 Gell-Mann-Low Function: Quantum Electrodynamics

The Gell-Mann-Low function in Quantum Electrodynamics (QED) is known in the
five-loop approximation [267, 271]. The weak-coupling expansion of the beta-
function depends on the used renormalization scheme.

(i) Minimal subtraction scheme

In the modified minimal subtraction scheme $(\overline{\rm MS})$, the Gell-
Mann-Low function is defined by the renormalization group equation

$\beta_{MS}(\overline{\alpha})=\mu^{2}\;\frac{\partial}{\partial\mu^{2}}\left(\frac{\overline{\alpha}}{\pi}\right)\;,$
(3.301)

in which $\overline{\alpha}$ is the renormalized $(\overline{\rm MS})$ scheme
QED coupling parameter and $\mu$ is a scale parameter. In the five-loop
approximation for the beta-function of an electron, neglecting the
contributions of leptons with higher masses, that is, of muons and tau-
leptons, the expansion of the beta-function with respect to
$\overline{\alpha}$ [271] has the form

$\beta_{MS}(\overline{\alpha})\simeq\sum_{n=0}^{k}b_{n}\left(\frac{\overline{\alpha}}{\pi}\right)^{n+2}\;,$
(3.302)

with the coefficients

$b_{0}=\frac{1}{3}\;,\qquad b_{1}=\frac{1}{4}\;,\qquad
b_{2}=-\;\frac{31}{288}\;,\qquad
b_{3}=-\;\frac{2785}{31104}\;-\;\frac{13}{36}\;\zeta(3)\;,$
$b_{4}=-\;\frac{195067}{497664}\;-\;\frac{25}{96}\;\zeta(3)-\;\frac{13}{96}\;\zeta(4)+\frac{215}{96}\;\zeta(5)\;.$

In numerical form, the coefficients are

$b_{0}=0.3333\;,\qquad b_{1}=0.25\;,\qquad b_{2}=-0.1076\;,\qquad
b_{3}=-0.5236\;,\qquad b_{4}=1.471\;.$

(ii) On-shell scheme

In the on-shell scheme (OS), one sets $\mu=m$, with $m$ being the electron
mass. The beta-function, is defined by the equation

$\beta_{OS}(\alpha)=m^{2}\;\frac{\partial}{\partial
m^{2}}\left(\frac{\alpha}{\pi}\right)\;,$ (3.303)

where $\alpha=e^{2}/4\pi$ is the QED coupling parameter. The weak-coupling
expansion reads as

$\beta_{OS}(\alpha)\simeq\sum_{n=0}^{k}b_{n}\left(\frac{\alpha}{\pi}\right)^{n+2}\qquad(\alpha\rightarrow
0)\;,$ (3.304)

with the coefficients

$b_{0}=0.3333\;,\qquad b_{1}=0.25\;,\qquad b_{2}=-0.4201\;,\qquad
b_{3}=-0.5712\;,\qquad b_{4}=-0.3462\;.$

The latter are denoted here by the same notation $b_{n}$, which should not
lead to confusion.

(iii) Momentum subtraction scheme

In the frame of the momentum subtraction scheme (MOM), the beta-function
expansion can be written as in Eq. (3.302), but with different coefficients:

$b_{0}=0.3333\;,\qquad b_{1}=0.25\;,\qquad b_{2}=0.04999\;,\qquad
b_{3}=-0.6010\;,\qquad b_{4}=1.434\;.$

Constructing self-similar factor approximants using the beta-function
expansions for each of the scheme, we take those of them that are defined for
all coupling parameters $\alpha\in[0,\infty]$. The highest orders of these
approximants give the strong-coupling limits

$\beta_{MS}^{*}(\overline{\alpha})\simeq
0.476\;\overline{\alpha}^{2.096}\;,\qquad\beta_{OS}^{*}(\alpha)\simeq
0.433\;\alpha^{2.189}\;,$ $\beta_{MOM}^{*}(\overline{\alpha})\simeq
0.493\;\overline{\alpha}^{2.309}\;.$ (3.305)

As is seen, the strong-coupling asymptotic behavior of the beta function is
practically the same for all schemes. This suggests that self-similar factor
approximants provide scheme independent expressions. The Borel-Leroy summation
[269, 272] in the MOM scheme gives the strong-coupling limit with the beta
function proportional to $\overline{\alpha}$ in first order.

###  3.33 Gell-Mann-Low Function: Quartic Field Theory

The Gell-Mann-Low function for the $O(N)$-symmetric $\varphi^{4}$ field
theory, within the minimal subtraction scheme, is known in the six-loop
approximation [273]. Defining the beta-function as

$\beta(g)=\mu\;\frac{\partial g}{\partial\mu}\;,$ (3.306)

where $g=\lambda/(4\pi)^{2}$, for its weak-coupling expansion one has

$\beta(g)\simeq\sum_{n=0}^{k}b_{n}g^{n+2}\;.$ (3.307)

The six coefficients $b_{n}$ for a different number of components are given in
Table 6.

Schnetz [274] derived the coefficients for $N=1$ to seven-loop order:

$b_{0}=3\;,\qquad b_{1}=-5.66667\;,\qquad b_{2}=32.5497\;,\qquad
b_{3}=-271.606\;,$ $b_{4}=2848.57\;,\qquad b_{5}=-34776.1\;,\qquad
b_{6}=474651\;.$

The general six-loop expansion of the beta function for the $N$-component
field theory [273] has the coefficients

$b_{0}=\frac{N+8}{3}\;,\qquad b_{1}=-\;\frac{3N+14}{3}\;,$
$b_{2}=\frac{1}{216}\left[96(5N+22)\zeta(3)+33N^{2}+922N+2960\right]\;,$
$b_{3}=-\;\frac{1}{3888}\left[1920\left(2N^{2}+55N+186\right)\zeta(5)-288(N+8)(5N+22)\zeta(4)+\right.$
$+\left.96\left(63N^{2}+764N+2332\right)\zeta(3)-\left(5N^{3}-6320N^{2}-80456N-196648\right)\right]\;,$
$b_{4}=\frac{1}{62208}\left[112896\left(14N^{2}+189N+526\right)\zeta(7)-768\left(6N^{3}+59N^{2}-446N-3264\right)\zeta^{2}(3)-\right.$
$-9600(N+8)\left(2N^{2}+55N+186\right)\zeta(6)+256\left(305N^{3}+7466N^{2}+66986N+165084\right)\zeta(5)-$
$-288\left(63N^{3}+1388N^{2}+9532N+21120\right)\zeta(4)-$
$-16\left(9N^{4}-1248N^{3}-67640N^{2}-552280N-1314336\right)\zeta(3)+$
$+\left.13N^{4}+12578N^{3}+808496N^{2}+6646336N+13177344\right]\;,$
$b_{5}=-\;\frac{1}{41990400}\left[204800\left(1819N^{3}+97823N^{2}+901051N+2150774\right)\zeta(9)+\right.$
$+14745600\left(N^{3}+65N^{2}+619N+1502\right)\zeta^{3}(3)+$
$+995328\left(42N^{3}+2623N^{2}+25074N+59984\right)\zeta(3,5)-$
$-20736\left(28882N^{3}+820483N^{2}+6403754N+14174864\right)\zeta(8)-$
$-5529600\left(8N^{3}-635N^{2}-9150N-25944\right)\zeta(3)\zeta(5)+$
$+11520\left(440N^{4}+126695N^{3}+2181660N^{2}+14313152N+29762136\right)\zeta(7)+$
$+207360(N+8)\left(6N^{3}+59N^{2}-446N-3264\right)\zeta(3)\zeta(4)-$
$-23040\left(188N^{4}+132N^{3}-93363N^{2}-862604N-2207484\right)\zeta^{2}(3)-$
$-28800\left(595N^{4}+20286N^{3}+277914N^{2}+1580792N+2998152\right)\zeta(6)+$
$+5760\left(4698N^{4}+131827N^{3}+2250906N^{2}+14657556N+29409080\right)\zeta(5)+$
$+2160\left(9N^{5}-1176N^{4}-88964N^{3}-1283840N^{2}-6794096N-12473568\right)\zeta(4)-$
$-720\left(33N^{5}+2970N^{4}-477740N^{3}-10084168N^{2}-61017200N-117867424\right)\zeta(3)-$
$-\left.45\left(29N^{5}+22644N^{4}-3225892N^{3}-88418816N^{2}-536820560N-897712992\right)\right]\;.$

Here $\zeta(3,5)$ implies the double zeta function

$\zeta(3,5)=\sum_{1\leq n<m}\frac{1}{n^{3}m^{5}}=0.037707673\;.$

Table 6: Coefficients of weak-coupling expansion for the Gell-Man-Low function of the $N$-component $\varphi^{4}$ field theory, in the six-loop approximation. $N$ | 0 | 1 | 2 | 3 | 4
---|---|---|---|---|---
$b_{0}$ | 2.667 | 3.000 | 3.333 | 3.667 | 4.000
$b_{1}$ |  $-$4.667 |  $-$5.667 |  $-$6.667 |  $-$7.667 |  $-$8.667
$b_{2}$ | 25.46 | 32.55 | 39.95 | 47.65 | 55.66
$b_{3}$ |  $-$200.9 |  $-$271.6 |  $-$350.5 |  $-$437.6 |  $-$533.0
$b_{4}$ | 2004 | 2849 | 3845 | 4999 | 6318
$b_{5}$ |  $-$23315 |  $-$34776 |  $-$48999 |  $-$66243 |  $-$86768

Employing self-similar factor approximants for the six-loop expansions, we
find the beta-functions for a different number of components. We present below
the approximants of even orders that are more reliable then those of odd
orders, although all of them are close to each other. Thus for $N=0$, we have

$\beta_{2}^{*}(g)=\frac{2.667g^{2}}{(1+9.161g)^{0.191}}\qquad(N=0)\;,$
$\beta_{4}^{*}(g)=\frac{2.667g^{2}}{(1+6.083g)^{0.194}(1+15.544g)^{0.0366}}\;,$
(3.308)

which yields the strong-coupling behavior

$\beta_{2}^{*}(g)\simeq 1.747g^{1.809}\;,\qquad\beta_{4}^{*}(g)\simeq
1.699g^{1.769}\qquad(N=0,~{}g\rightarrow\infty)\;.$ (3.309)

For $N=2$, we get

$\beta_{2}^{*}(g)=\frac{3.333g^{2}}{(1+9.984g)^{0.2003}}\qquad(N=2)\;,$
$\beta_{4}^{*}(g)=\frac{3.333g^{2}}{(1+5.812g)^{0.206}(1+16.234g)^{0.0493}}\;,$
(3.310)

with the strong-coupling limit

$\beta_{2}^{*}(g)\simeq 2.102g^{1.7997}\;,\qquad\beta_{4}^{*}(g)\simeq
2.020g^{1.7443}\qquad(N=2,~{}g\rightarrow\infty)\;.$ (3.311)

In the case of $N=3$, we find

$\beta_{2}^{*}(g)=\frac{3.667g^{2}}{(1+10.339g)^{0.202}}\qquad(N=3)\;,$
$\beta_{4}^{*}(g)=\frac{3.667g^{2}}{(1+5.544g)^{0.213}(1+16.537g)^{0.055}}\;,$
(3.312)

giving the limit

$\beta_{2}^{*}(g)\simeq 2.286g^{1.798}\;,\qquad\beta_{4}^{*}(g)\simeq
2.183g^{1.732}\qquad(N=3,~{}g\rightarrow\infty)\;.$ (3.313)

And the case of $N=4$ results in

$\beta_{2}^{*}(g)=\frac{4g^{2}}{(1+10.677g)^{0.203}}\qquad(N=4)\;,$
$\beta_{4}^{*}(g)=\frac{4g^{2}}{(1+5.274g)^{0.219}(1+16.866g)^{0.05988}}\;,$
(3.314)

with the strong-coupling limit

$\beta_{2}^{*}(g)\simeq 2.474g^{1.797}\;,\qquad\beta_{4}^{*}(g)\simeq
2.345g^{1.721}\qquad(N=4,~{}g\rightarrow\infty)\;.$ (3.315)

Using the seven-loop expansion for $N=1$ yields the beta functions

$\beta_{2}^{*}(g)=\frac{3g^{2}}{(1+9.599g)^{0.197}}\qquad(N=1)\;,$
$\beta_{4}^{*}(g)=\frac{3g^{2}}{(1+6.014g)^{0.2004}(1+15.9204g)^{0.0479}}\;,$
$\beta_{6}^{*}(g)=\frac{3g^{2}}{(1+5.357g)^{0.187}(1+13.72g)^{0.0597}(1+22.096)^{0.00316}}\;,$
(3.316)

whose strong-coupling behavior is

$\beta_{2}^{*}(g)\simeq 1.922g^{1.803}\;,\qquad\beta_{4}^{*}(g)\simeq
1.859g^{1.757}\;,$ $\beta_{6}^{*}(g)\simeq
1.857g^{1.750}\qquad(N=1,~{}g\rightarrow\infty)\;.$ (3.317)

As is seen, all powers of the strong-coupling asymptotic behavior
$\beta(g)\propto g^{\gamma}$ are close to each other, being between
$\gamma=1.7$ and $\gamma=1.8$. In literature, it is possible to find the
estimates for the single-component field theory with $N=1$. For example,
Kazakov [275] gives the estimate of $\gamma=2$. Borel summation, with
conformal mapping [276] gives $\gamma=1.9$. The use of optimized perturbation
theory [277] results in $\gamma=1.5$. Borel-Leroy summation [269, 278, 279]
leads to $\gamma=1$. So, our result is between that of optimized perturbation
theory [277] and that of Borel summation [275, 276].

###  3.34 Other Applications

Self-similar approximation theory has been applied to hundreds of various
problems, as can be inferred from the literature cited in this review. In
addition to the cited publications, it is possible to mention here several
more applications. For example, self-similar approximants were used for
describing systems with Bose-Einstein condensate [280, 281, 282, 283, 284],
barrier crossing processes [285, 286], stochastic processes [287],
seismological phenomena [288, 289, 290], and time series [291, 292]. Self-
similar approximants have been widely used for a large class of problems in
the theory of structured media [293, 294].

##  4 Basics of Renormalization Group

Throughout the previous chapters, it has been mentioned several times that the
ideas of optimized perturbation theory and especially of self-similar
approximation theory share many properties with the method of renormalization
group. Here we recall the basics of the latter approach in order to better
explain the similarities in the foundations of the different theories.

###  4.1 Renormalization-Group Classes

It is necessary to distinguish two main classes among the available variants
of renormalization group [295]. One class incorporates the methods where the
introduction of a renormalization group is based on the existence of some
properties of internal symmetry of a studied physical system formulated in the
language of natural variables of this system. Here, as an example, one may
point to the derivation of the quantum-field renormalization group, as is
described in the book [296]. In the models of condensed matter physics, one
also sometimes tries to discover internal symmetries allowing one, at least
approximately, involving several simplifications, to formulate renormalization
group equations [297, 298, 299, 300].

The other class derives renormalization group equations from the construction
of a set of models differing from each other by the scale of some variable. It
is this class of renormalization group that is used in considering critical
phenomena [301, 302, 303, 304]. This class, actually, corresponds to
semigroups, since the inverse operation is not defined.

Thus, the first class reflects internal symmetries of the considered systems
and is formed by groups. While the second class describes scaling
transformations and represents semigroups. Below we briefly sketch the main
ideas of both these classes.

###  4.2 Critical Phenomena

Under critical phenomena one understands the peculiarities of a system
behavior in the vicinity of a continuous phase transition, that is, near a
second order phase transition [223, 305, 306]. At the critical point, two
thermodynamic phases become indistinguishable. The point, where several phases
are indistinguishable, is termed multicritical [307]. A multicritical point on
a critical line usually separates second- and first-order phase transitions.
The critical behavior near a multicritical point is more complicated than in
the vicinity of a critical point. Below we keep in mind critical points.

The vicinity of a system to a critical temperature $T_{c}$ is described by the
relative temperature deviation

$\tau\equiv\frac{T-T_{c}}{T_{c}}\;.$ (4.1)

The asymptotic behavior of a statistical system near a critical point is
characterized by critical exponents. Thus the asymptotic behavior of specific
heat is

$\displaystyle
C_{V}\propto\left\\{\begin{array}[]{ll}(-\tau)^{\alpha^{\prime}}\;,{}&~{}T<T_{c}\\\
\tau^{\alpha}\;,{}&~{}T>T_{c}\end{array}\right..$ (4.4)

The order parameter, such as magnetization, behaves as

$\displaystyle
M\propto\left\\{\begin{array}[]{ll}(-\tau)^{\beta^{\prime}}\;,{}&~{}T<T_{c}\\\
\tau^{\beta}\;,{}&~{}T>T_{c}\end{array}\right..$ (4.7)

For the dependence of a magnetic field $B$ on magnetization, at the critical
temperature, one has

$B\propto M^{\delta}\qquad(T=T_{c})\;.$ (4.8)

Susceptibility diverges as

$\displaystyle\chi\propto\left\\{\begin{array}[]{ll}(-\tau)^{-\gamma^{\prime}}\;,{}&~{}T<T_{c}\\\
\tau^{-\gamma}\;,{}&~{}T>T_{c}\end{array}\right..$ (4.11)

Correlation length also diverges as

$\displaystyle\xi\propto\left\\{\begin{array}[]{ll}(-\tau)^{\nu^{\prime}}\;,{}&~{}T<T_{c}\\\
\tau^{-\nu}\;,{}&~{}T>T_{c}\end{array}\right..$ (4.14)

The spatial dependence of correlation function at the critical temperature,

$C(r)\propto r^{-(d-2+\eta)}\qquad(T=T_{c})\;,$ (4.15)

where $d$ is the space dimensionality, is characterized by the exponent
$\eta$. The derivative of the Gell-Mann-Low function at a fixed-point coupling
defines the exponent

$\omega\equiv\lim_{g\rightarrow g^{*}}\;\frac{d\beta(g)}{dg}$ (4.16)

specifying the correction to the critical scaling behavior.

Assuming that the critical exponents are the same above as well as below the
critical temperature, one gets the symmetry relations

$\alpha=\alpha^{\prime}\;,\qquad\beta=\beta^{\prime}\;,\qquad\gamma=\gamma^{\prime}\;,\qquad\nu=\nu^{\prime}\;.$
(4.17)

And one also assumes the validity of the scaling relations

$\alpha+2\beta+\gamma=2\;,\qquad\alpha+\beta(1+\delta)=2\;,\qquad\gamma+\beta(1-\delta)=0\;,$
$\gamma+\nu(\eta-2)=0\;,\qquad\alpha+\nu d=2$ (4.18)

motivated by thermodynamic equalities.

In the case of a multicritical point, the symmetry relations do not hold.
Approaching a multicritical point from the side of a second-order transition,
it is possible to introduce the critical exponents as above. But at the point
itself, the exponents change by a jump.

###  4.3 Universality Classes

When all critical exponents for two systems coincide, one tells that these
systems are equivalent in the sense of their critical behavior. And the
collection of the critical exponents is termed the set of universal
quantities.

Historically, the assumption on the universality of critical behavior has been
based on the results of mean-field theory, where all critical exponents are
the same for all physical systems [308]. A particular case of mean-field
theory is the Landau method [309] of expanding the free energy of an
$N$-particle system

$f\equiv\frac{F}{N}=-\;\frac{T}{N}\;\ln{\rm Tr}e^{-\hat{H}/T}$ (4.19)

in powers of an order parameter, say magnetization,

$f\simeq f_{0}+f_{2}M^{2}+f_{4}M^{4}\qquad(M\ll 1\;,~{}T<T_{c})\;.$ (4.20)

If $(f_{0}-f)/f_{2}\propto\tau$, then

$M\propto(-\tau)^{\beta}\qquad\left(\beta=\frac{1}{2}\right)\;,$

irrespectively of the system nature.

The Landau expansion is applicable in the temperature region, where the order
parameter is already small, while the fluctuations, increasing in the vicinity
of the critical temperature, are not yet too large, so that the Ginzburg
criterion be valid:

$Gi\propto|\tau|^{(d-4)/2}\ll 1\;.$

This criterion shows that in the space of dimensionality $d>4$ critical
exponents can be defined by mean-field theory. While for dimensionality $d<4$
the Landau expansion is not appropriate. The dimensionality $d=4$ is marginal,
requiring a separate investigation.

Experiments have demonstrated that critical exponents of the majority of
systems are not of the mean-field type, being quite different for different
systems. At the same time, there are many systems demonstrating very similar
critical behavior, with the critical exponents being practically the same,
within the accuracy of experiments. Thus the notion of universality classes
has arisen.

A universality class is a family of systems, equivalent in the sense of their
critical behavior, hence possessing the same set of critical exponents.

Among the characteristics of physical systems there are the main that define a
class of universality of a system. The systems having at least one of such
characteristics different, pertain to different universality classes. These
main characteristics are:

  1. (I)

The system dimensionality.

  2. (II)

The number of the order parameter components.

  3. (III)

The order parameter symmetry.

  4. (IV)

The range of particle interactions.

  5. (V)

The interaction potential anisotropy.

Particle interactions can be of the following types.

Long-range potentials. For a discrete system, such potentials

$J_{ij}\equiv J({\bf r}_{ij})\qquad({\bf r}_{ij}\equiv{\bf r}_{i}-{\bf
r}_{j})\;,$

in thermodynamic limit, satisfy the limiting property

$\lim_{N\rightarrow\infty}J_{ij}=0\;,\qquad\lim_{N\rightarrow\infty}\;\frac{1}{N}\sum_{i\neq
j}^{N}J_{ij}=const\neq 0\;.$ (4.21)

The equivalent property for a continuous system is

$\lim_{V\rightarrow\infty}\Phi({\bf
r})=0\;,\qquad\lim_{V\rightarrow\infty}\;\int_{V}\Phi({\bf r})\;d{\bf
r}=const\neq 0\;.$ (4.22)

The models with such long-range potentials allow for asymptotically exact
solutions in thermodynamic limit [310] and enjoy the critical exponents of
mean-field type.

Short-range potentials. These are the potentials with a finite interaction
radius. For instance, the exponential potentials

$J_{ij}\propto\exp\left(-\;\frac{|{\bf r}_{ij}|}{b}\right)\;,$ (4.23)

or a sufficiently quickly decaying power-law potentials

$J_{ij}\propto\frac{1}{|{\bf r}_{ij}|^{d+n}}\qquad(n>2)\;.$ (4.24)

Short-range potentials can be well classified into different universality
classes.

Mid-range potentials. These potentials are intermediate between long-range and
short-range potentials, e.g.,

$J_{ij}\propto\frac{1}{|{\bf r}_{ij}|^{d+n}}\qquad(0<n<2)\;.$ (4.25)

Such potentials do not allow for classifying systems into universality
classes, since critical exponents depend on the power $n$.

Anisotropic potentials. This is a type of potentials depending on the angles
and requiring a special consideration. Thus the dipolar potential

$J_{ij}\propto\frac{1}{|{\bf
r}_{ij}|^{d}}\;\left(\delta_{\alpha\beta}-\;\frac{r_{ij}^{\alpha}r_{ij}^{\beta}}{|{\bf
r}_{ij}|^{2}}\right)\;,$ (4.26)

although it looks like a long-range potential, but has the critical exponents
coinciding, up to logarithmic corrections, with those of short-range systems.
This is because of the property

$\frac{1}{N}\sum_{i\neq j}J_{ij}=0\;,$

valid for sufficiently large systems. Dipolar potentials also lead to some
complications in describing the systems with such forces. Thus the Fourier
transform of a dipolar potential is ill-defined, so that to overcome this
problem one needs to take into account screening effects [311, 312, 313].

The reason why the universality classes do exist is as follows. In the
vicinity of critical points, large-scale fluctuations become predominant and
the correlation length tends to infinity. This makes small-scale peculiarities
less important, while the system behavior is governed by long-range
fluctuations. But let us recall that the concept of universality classes is
not applicable to mid-range interactions and to some anisotropic models [314].

In practice, the universality can be employed by describing critical phenomena
resorting to the simplest system in the given universality class.

###  4.4 Kadanoff Transformations

Let a statistical system be characterized by a Hamiltonian $\hat{H}$ being a
functional of a set of variables $s$. It is convenient to define the
dimensionless Hamiltonian

$H[s]\equiv\frac{\hat{H}}{T}\;,$ (4.27)

where $T$ is temperature. Thermodynamic characteristics are described by the
free energy

$f=-\;\frac{T}{N}\;\ln Z\;,$ (4.28)

in which $Z$ is the partition function

$Z={\rm Tr}_{s}\exp(-H[s])\;.$ (4.29)

The Kadanoff idea is to define a set of transformations moving from small to
large scales and allowing for a step-by-step simplification of the Hamiltonian
[315]. At the first step, one splits the set of variables $s$ into two
subsets, $s_{1}$ and $s_{1}^{\prime}$, and integrates out (or sums out) the
variables $s_{1}^{\prime}$ according to the rule

$\exp(-H_{1}[s_{1}])\equiv{\rm
Tr}_{s_{1}^{\prime}}\exp(-H[s_{1},s_{1}^{\prime}])$ (4.30)

defining the effective Hamiltonian

$H_{1}[s_{1}]=-\ln{\rm Tr}_{s_{1}^{\prime}}\exp(-H[s_{1},s_{1}^{\prime}])\;.$
(4.31)

Hence the partition function $Z$ becomes

$Z={\rm Tr}_{s_{1}}\exp(-H_{1}[s_{1}])\;.$ (4.32)

Splitting again the set of variables $s_{1}$ into two subsets, $s_{2}$ and
$s_{2}^{\prime}$, and summing out the variables $s_{2}^{\prime}$,

$\exp(-H_{2}[s_{2}])\equiv{\rm
Tr}_{s_{2}^{\prime}}\exp(-H_{1}[s_{2},s_{2}^{\prime}])\;,$ (4.33)

one introduces the effective Hamiltonian

$H_{2}[s_{2}]=-\ln{\rm
Tr}_{s_{2}^{\prime}}\exp(-H_{1}[s_{2},s_{2}^{\prime}])\;.$ (4.34)

This yields the partition function

$Z={\rm Tr}_{s_{2}}\exp(-H_{2}[s_{2}])\;.$ (4.35)

Repeating this procedure $n$ times results in the recurrence relation

$H_{n}[s_{n}]=-\ln{\rm
Tr}_{s_{n}^{\prime}}\exp(-H_{n-1}[s_{n},s_{n}^{\prime}])$ (4.36)

and in the partition function

$Z={\rm Tr}_{s_{n}}\exp(-H_{n}[s_{n}])\;,$ (4.37)

where $H_{0}[s_{0}]\equiv H[s]$.

Introducing the renormalization transformation

$\hat{R}(s_{n}^{\prime})H_{n-1}[s_{n},s_{n}^{\prime}]\equiv-\ln{\rm
Tr}_{s_{n}^{\prime}}\exp(-H_{n-1}[s_{n},s_{n}^{\prime}])$ (4.38)

makes it straightforward to rewrite the recurrence relation (4.36) as

$H_{n}[s_{n}]=\hat{R}(s_{n}^{\prime})H_{n-1}[s_{n},s_{n}^{\prime}]\qquad(n\geq
1)\;.$ (4.39)

Defining the double transformation

$\hat{R}(s_{m}^{\prime},s_{n}^{\prime})H_{k}\equiv-\ln{\rm
Tr}_{s_{m}^{\prime}}{\rm Tr}_{s_{n}^{\prime}}\exp(-H_{k})$ (4.40)

and the identity transformation

$\hat{R}(0)\equiv 1$ (4.41)

gives us the property

$\hat{R}(s_{m}^{\prime},s_{n}^{\prime})=\hat{R}(s_{m}^{\prime})\hat{R}(s_{n}^{\prime})=\hat{R}(s_{n}^{\prime})\hat{R}(s_{m}^{\prime})\;.$
(4.42)

This tells us that the family of the renormalization operators
$\hat{R}(s_{j}^{\prime})$ forms a commutative semigroup. This is a semigroup,
since the inverse operation is not defined.

The free energy (4.28) acquires the form

$f=\frac{T}{N}\;\hat{R}(s_{n})H_{n}[s_{n}]=\frac{T}{N}\;\hat{R}(s_{n})\hat{R}(s_{n}^{\prime})\hat{R}(s_{n-1}^{\prime})\ldots\hat{R}(s_{1}^{\prime})H[s]=$
$=-\;\frac{T}{N}\;\ln{\rm Tr}_{s_{n}}\exp(-H_{n}[s_{n}])\;.$ (4.43)

The above scheme lies in the core of renormalization semigroups used in
statistical physics. Practical realizations can be different, but the
principal idea is as described above.

###  4.5 Momentum Scaling

A widely employed approach of introducing renormalization operations is based
on different variants of scaling in the momentum space [301, 303, 316, 317,
318]. The typical procedure can be characterized as follows.

Let the Hamiltonian $H[\varphi]$ be a functional of a field depending on
momentum,

$\varphi=\varphi({\bf k})\qquad(0\leq k\leq\Lambda\;,~{}k\equiv|{\bf k}|)\;,$
(4.44)

with $\Lambda$ being an ultraviolet cutoff. The partition function, or
generating functional, is given by the functional integral

$Z=\int\exp(-H[\varphi])\;{\cal D}\varphi\;.$ (4.45)

One separates the momenta into two regions, one is the region of low energy,
where

$0\leq k\leq\frac{\Lambda}{b}\;,$ (4.46)

and the other is the region of high energy, where

$\frac{\Lambda}{b}<k\leq\Lambda\;,$ (4.47)

with $b\geq 1$ being an arbitrary real number. Then the field splits into two
parts:

$\displaystyle\varphi(k)=\left\\{\begin{array}[]{ll}\varphi_{b}({\bf
k})\;,{}&~{}k\leq\Lambda/b\\\ \varphi_{b}^{\prime}({\bf
k})\;,{}&~{}k>\Lambda/b\end{array}\right..$ (4.50)

Hence partition function (4.45) takes the form

$Z=\int\exp(-H[\varphi_{b},\varphi_{b}^{\prime}])\;{\cal D}\varphi_{b}{\cal
D}\varphi_{b}^{\prime}\;.$ (4.51)

One defines the effective Hamiltonian $H_{b}[\varphi_{b}]$ by integrating out
high-energy momenta in the equality

$\exp(-H_{b}[\varphi_{b}])\equiv\int\exp(-H[\varphi_{b},\varphi_{b}^{\prime}])\;{\cal
D}\varphi_{b}^{\prime}\;,$ (4.52)

which gives

$H_{b}[\varphi_{b}]=-\ln\int\exp(-H[\varphi_{b},\varphi_{b}^{\prime}])\;{\cal
D}\varphi_{b}^{\prime}\;.$ (4.53)

So that the partition function reads as

$Z=\int\exp(-H_{b}[\varphi_{b}])\;{\cal D}\varphi_{b}\;.$ (4.54)

The domain of the momenta in the field $\varphi_{b}(k)$ can be extended to the
whole momentum space by the change

${\bf k}\rightarrow b{\bf k}\;,\qquad\varphi_{b}({\bf
k})\rightarrow\zeta\varphi({\bf k})\;,$ (4.55)

in which $\zeta$ is a scale factor to be specified later. Then partition
function (4.54) transforms into

$Z=\int\exp(-H_{b}[\varphi])J(\zeta)\;{\cal D}\varphi\;,$ (4.56)

where $J(\zeta)$ is the Jacobian of transformation (4.55) and the effective
Hamiltonian is

$H_{b}[\varphi]\equiv-\ln\int\exp(-H[\zeta\varphi,\varphi_{b}^{\prime}])\;{\cal
D}\varphi_{b}^{\prime}\;.$ (4.57)

The procedure of obtaining the effective Hamiltonian (4.57) can be
symbolically denoted as the action of a renormalization transformation
$\hat{R}(b)$,

$\hat{R}(b)H[\varphi]=H_{b}[\varphi]\;.$ (4.58)

The unitary element is defined by the equality

$H_{1}[\varphi]=H[\varphi]\;,\qquad\hat{R}(1)=1\;.$ (4.59)

The transformation $\hat{R}(b)$ enjoys the property

$\hat{R}(b_{1}b_{2})=\hat{R}(b_{1})\hat{R}(b_{2})=\hat{R}(b_{2})\hat{R}(b_{1})\;.$
(4.60)

Thus the family of transformations $\hat{R}(b)$ forms a commutative
renormalization semigroup.

###  4.6 Fixed Points

The renormalization transformations $\hat{R}(b)$ generate a continuous
sequence of effective Hamiltonians (4.58) starting from the initial
Hamiltonian $H[\varphi]$. One says that the Hamiltonian moves along a
trajectory parameterized by the scaling factor $b\in[1,\infty)$. However, in
that motion the form of the Hamiltonian varies. Thus, if the initial
Hamiltonian contains a set of coupling parameters $g$, in the effective
renormalized Hamiltonian, not merely the coupling parameters become
renormalized, but also there appear new terms with additional coupling
parameters. This can be formulated as the motion in the space of coupling
parameters:

$g\rightarrow\overline{g}(b,g)=\\{\overline{g}_{1}(b,g),\overline{g}_{2}(b,g),\ldots\\}$
(4.61)

with the initial condition

$\overline{g}(1,g)=g\;.$ (4.62)

During this motion, one confronts with the problem of proliferation of
coupling parameters. Even if initially the Hamiltonian $H[\varphi]$ had a
finite number of such parameters, with increasing $b$, their number tends to
infinity.

But the hope is that in the vicinity of a critical point the Hamiltonian
becomes invariant with respect to the renormalization transformations, since
at a critical point the correlation length diverges, making irrelevant the
fine structure of the Hamiltonian. The Hamiltonian invariance implies the
existence of a fixed point $H^{*}[\varphi]$, where

$\hat{R}(b)H^{*}[\varphi]=H^{*}[\varphi]\;.$ (4.63)

In the space of coupling parameters, this is equivalent to the equation

$\overline{g}(b,g^{*})=g^{*}\;.$ (4.64)

As far as at the fixed point the Hamiltonian is invariant, conversely,
assuming the Hamiltonian invariance, one could approximately find the fixed
point.

###  4.7 Wilson Method

To be more specific, let us illustrate how the Wilson method works by
considering the scalar $\varphi^{4}$ field theory. We shall follow the scheme
of Refs. [301, 303, 319].

The Hamiltonian of the scalar $\varphi^{4}$ field theory can be represented in
the form

$H[\varphi]=\frac{1}{2}\int_{0}^{\Lambda}\left(m^{2}+k^{2}\right)|\varphi({\bf
k})|^{2}\;d{\bf k}+$ $+g\int_{0}^{\Lambda}\varphi({\bf k}_{1})\varphi({\bf
k}_{2})\varphi({\bf k}_{3})\varphi({\bf k}_{4})\delta_{d}({\bf k}_{1}+{\bf
k}_{2}+{\bf k}_{3}+{\bf k}_{4})\;d{\bf k}_{1}d{\bf k}_{2}d{\bf k}_{3}d{\bf
k}_{4}\;,$ (4.65)

where the momentum is a vector in a $d$-dimensional space,

${\bf k}=\\{k_{\alpha}:~{}\alpha=1,2,\ldots,d\\}\;,$

and the notations are used:

$\int_{0}^{\Lambda}d{\bf
k}\equiv\left(\frac{a}{2\pi}\right)^{d}\prod_{\alpha=1}^{d}\int_{0}^{\Lambda}dk_{\alpha}\;,\qquad\delta_{d}({\bf
k})\equiv\left(\frac{2\pi}{a}\right)^{d}\prod_{\alpha=1}^{d}\delta(k_{\alpha})\;.$

Here $1/a^{d}\sim N/V$ plays the role of an effective density.

Integrating out the high-energy momenta, as is explained in Sec. 4.5, the
integrals are calculated involving perturbation theory with respect to the
coupling parameter $g$. And, assuming that we are close to a fixed point,
hence the Hamiltonian has to be invariant under the renormalization
transformation, the terms spoiling the Hamiltonian structure are neglected.
Then the term $m^{2}+k^{2}$ transforms as

$m^{2}+k^{2}\rightarrow m^{2}(b)+f(b)k^{2}\;,$ (4.66)

where

$m^{2}(b)=\frac{\zeta^{2}}{b^{d}}\;\left[m^{2}+12g\int_{\Lambda/b}^{\Lambda}\frac{d{\bf
k}}{m^{2}+k^{2}}\right]+{\cal O}(g^{2})\;,\qquad
f(b)=\frac{\zeta^{2}}{b^{d+2}}+{\cal O}(g^{2})\;.$ (4.67)

The coupling parameter experiences the transformation

$g\rightarrow g(b)\;,$ (4.68)

in which

$g(b)=\frac{\zeta^{4}}{b^{3d}}\;\left[g-36g^{2}\int_{\Lambda/b}^{\Lambda}\frac{d{\bf
k}}{(m^{2}+k^{2})^{2}}\right]+{\cal O}(g^{3})\;.$ (4.69)

The scaling parameter, introduced in scaling (4.55), is arbitrary and can be
chosen such that the factor at $k^{2}$ be unchanged, that is, setting it to
one, which means

$f(b)=1\;,\qquad\zeta^{2}=b^{d+2}\;.$ (4.70)

Thus we get the recurrence relations

$m^{2}(b)=b^{2}\;\left[m^{2}+12g\int_{\Lambda/b}^{\Lambda}\frac{d{\bf
k}}{m^{2}+k^{2}}\right]+{\cal O}(g^{2})\;,$
$g(b)=b^{4-d}g\left[1-36g\int_{\Lambda/b}^{\Lambda}\frac{d{\bf
k}}{(m^{2}+k^{2})^{2}}\right]+{\cal O}(g^{2})\;.$ (4.71)

The fixed point in the parameter space is defined by the equations

$(m^{*})^{2}=b^{2}\;\left[(m^{*})^{2}+12g^{*}\int_{\Lambda/b}^{\Lambda}\frac{d{\bf
k}}{(m^{*})^{2}+k^{2}}\right]\;,$
$g^{*}=b^{4-d}g^{*}\left[1-36g^{*}\int_{\Lambda/b}^{\Lambda}\frac{d{\bf
k}}{((m^{*})^{2}+k^{2})^{2}}\right]\;.$ (4.72)

These equations have a trivial fixed point, where both $m^{*}$ and $g^{*}$ are
zero. But we wish to find a nontrivial fixed point. By noticing that
$(m^{*})^{2}$ is of order $g^{*}$, the integrals in Eqs (4.72) can be
simplified:

$\int_{\Lambda/b}^{\Lambda}\frac{d{\bf
k}}{(m^{*})^{2}+k^{2}}\simeq\int_{\Lambda/b}^{\Lambda}\frac{d{\bf
k}}{k^{2}}=\frac{C_{d}}{d-2}\left[\Lambda^{d-2}-\left(\frac{\Lambda}{b}\right)^{d-2}\right]\;,$
$\int_{\Lambda/b}^{\Lambda}\frac{d{\bf
k}}{((m^{*})^{2}+k^{2})^{2}}\simeq\int_{\Lambda/b}^{\Lambda}\frac{d{\bf
k}}{k^{4}}=\frac{C_{d}}{d-4}\left[\Lambda^{d-4}-\left(\frac{\Lambda}{b}\right)^{d-4}\right]\;,$

where

$C_{d}=\left(\frac{a}{2\pi}\right)^{d}\int
d^{d}\Omega=\frac{a^{d}}{2^{d-1}\pi^{d/2}\Gamma(d/2)}\;.$ (4.73)

Thus we come to the fixed point characterized by the approximate values

$(m^{*})^{2}=-\;\frac{12g^{*}b^{2}C_{d}}{(b^{2}-1)(d-2)}\;\left[\Lambda^{d-2}-\left(\frac{\Lambda}{b}\right)^{d-2}\right]\;,$
$g^{*}=\frac{(1-b^{d-4})(d-4)}{36C_{d}}\;\left[\Lambda^{d-4}-\left(\frac{\Lambda}{b}\right)^{d-4}\right]^{-1}\;.$
(4.74)

This answer, however, is not yet satisfactory, since it contains an arbitrary
scaling factor $b$ that makes the obtained values unphysical.

###  4.8 Fractional Dimension

We may notice that the dependence on $b$ disappears for dimension $d=4$,
although then the fixed point reduces to trivial zero. But it would be
possible to get rid of $b$ dependence supposing that the deviation from the
dimension $d=4$ is small, however not zero. This suggests to introduce the
variable

$\varepsilon\equiv 4-d$ (4.75)

assumed to be small. Then it is admissible to expand the quantities containing
$\varepsilon$ in powers of $\varepsilon$ employing the expansion

$x^{\varepsilon}=\exp(\varepsilon\ln x)\simeq 1+\varepsilon\ln x\;.$

Such an $\varepsilon$\- expansion for the fixed point (4.74), to first order
in $\varepsilon$, gives

$(m^{*})^{2}=-\;\frac{\Lambda^{2}}{6}\;\varepsilon\;,\qquad
g^{*}=\frac{\Lambda}{36C_{d}}\;\varepsilon\;.$ (4.76)

At the end, one has to remember that the real space is not $d=4$, but it is
$d=3$, or in the limiting cases it can be $d=1$ or $d=2$. Respectively, one
should set $\varepsilon=1$ or even $\varepsilon=2$ or $\varepsilon=3$. For
instance, quantity (4.73) becomes

$C_{1}=\frac{a}{\pi}\qquad(d=1\;,~{}\varepsilon=3)\;,$
$C_{2}=\frac{a^{2}}{2\pi}\qquad(d=2\;,~{}\varepsilon=2)\;,$
$C_{3}=\frac{a^{3}}{2\pi^{2}}\qquad(d=3\;,~{}\varepsilon=1)\;.$

And for the fixed point (4.76), we find

$\displaystyle\left(\frac{m^{*}}{\Lambda}\right)^{2}=\left\\{\begin{array}[]{ll}-0.5\;,{}&~{}d=1\\\
-0.333\;,{}&~{}d=2\\\ -0.167\;,{}&~{}d=3\end{array}\right.\;,\qquad
g^{*}\;\frac{a^{d}}{\Lambda}=\left\\{\begin{array}[]{ll}0.262\;,{}&~{}d=1\\\
0.349\;,{}&~{}d=2\\\ 0.548\;,{}&~{}d=3\end{array}\right.\;.$ (4.83)

The described $\varepsilon$ \- expansion results in asymptotic series.
Substituting there finite values of $\varepsilon$, strictly speaking, is not
correct. Before such a substitution, the series have to be extrapolated to
finite values of $\varepsilon$, e.g., as is done in Sec. 3.25.

###  4.9 Differential Formulation

It is possible to derive differential equations for the renormalized
parameters [303, 320] by considering infinitesimally small variations of the
scaling factor

$b=1+\delta x\qquad(\delta x\rightarrow 0)\;.$ (4.84)

Then the integrals in the recurrence relations (4.71) can be represented as

$\int_{\Lambda/b}^{\Lambda}\frac{d{\bf
k}}{m^{2}+k^{2}}\simeq\frac{C_{d}\Lambda^{d}}{m^{2}+\Lambda^{2}}\;\delta
x\;,\qquad\int_{\Lambda/b}^{\Lambda}\frac{d{\bf
k}}{(m^{2}+k^{2})^{2}}\simeq\frac{C_{d}\Lambda^{d}}{(m^{2}+\Lambda^{2})^{2}}\;\delta
x\;.$

Hence for the quantities

$m^{2}(b)=m^{2}(1+\delta x)\;,\qquad g(b)=g(1+\delta x)$ (4.85)

we obtain the relations

$\frac{m^{2}(1+\delta x)-m^{2}}{\delta
x}=2m^{2}+\frac{12C_{d}\Lambda^{d}}{m^{2}+\Lambda^{2}}\;g\;,$
$\frac{g(1+\delta x)-g}{\delta
x}=(4-d)g-\;\frac{36C_{d}\Lambda^{d}}{(m^{2}+\Lambda^{2})^{2}}\;g^{2}\;.$
(4.86)

Thus we come to the differential equations

$\frac{dm^{2}(x)}{dx}=2m^{2}(x)+\frac{12C_{d}\Lambda^{d}}{m^{2}(x)+\Lambda^{2}}\;g(x)\;,$
$\frac{dg(x)}{dx}=(4-d)g(x)-\;\frac{36C_{d}\Lambda^{d}}{(m^{2}(x)+\Lambda^{2})^{2}}\;g^{2}(x)\;,$
(4.87)

with the initial conditions

$m(0)=m\;,\qquad g(0)=g\;.$ (4.88)

The fixed points of the dynamical system are defined by the zero derivatives

$\left.\frac{dm^{2}(x)}{dx}\right|_{m^{*},g^{*}}=0\;,\qquad\left.\frac{g(x)}{dx}\right|_{m^{*},g^{*}}=0\;,$
(4.89)

which leads to the equations

$6g^{*}C_{d}\Lambda^{d}+(m^{*})^{2}\left((m^{*})^{2}+\Lambda^{2}\right)=0\;,$
$36g^{*}C_{d}\Lambda^{d}-(4-d)\left((m^{*})^{2}+\Lambda^{2}\right)^{2}=0\;.$
(4.90)

Taking into account that the ultraviolet cutoff is sufficiently large, such
that $\Lambda\gg|m|$, we find

$(m^{*})^{2}=-\;\frac{4-d}{6}\;\Lambda^{2}\;,\qquad
g^{*}=\frac{4-d}{36C_{d}}\;\Lambda^{4-d}\;.$ (4.91)

Let us stress that in this way, we have not involved the
$\varepsilon$-expansion. Nevertheless, if we set $4-d=\varepsilon$, we return
to the fixed point (4.76). That is, the differential formulation is equivalent
to the $\varepsilon$-expansion, resulting in the same form of fixed points.

###  4.10 Decimation Procedure

Renormalization group in real space can be accomplished employing the Kadanoff
transformations described in Sec. 4.4. Summing out a part of variables leads
to the coarsening of the real-space scale [315, 321]. Such a procedure is
often called decimation. This term is borrowed from the language of the
ancient Rome, where the decimation meant the killing of one in every ten of
soldiers from a mutinous or retreated Roman legion.

Let us consider the two-dimensional Ising model with the Hamiltonian

$\hat{H}=N\;\frac{U}{2}\;-\;\frac{1}{2}\sum_{\langle
ij\rangle}J\sigma_{i}\sigma_{j}\;,$ (4.92)

where the spin variables take the values $\sigma_{j}=\pm 1$ and the summation
is over the nearest neighbors. We keep here a constant term to show that it
also needs renormalization [319]. The set of all spin variables is denoted as

$s=\\{\sigma_{j}=\pm 1\;:~{}j=1,2,\ldots,N\\}\;.$ (4.93)

Introducing the dimensionless parameters

$u\equiv\frac{U}{2T}\;,\qquad g\equiv\frac{J}{2T}\;,$ (4.94)

gives us the dimensionless Hamiltonian (4.27) in the form

$H[s]=Nu-g\sum_{\langle ij\rangle}\sigma_{i}\sigma_{j}\;.$ (4.95)

Considering a square lattice with the side $a$, we sum out the nearest
neighbors, which coarsens the lattice side into $\sqrt{2}a$. The effective
Hamiltonian (4.31) becomes

$H_{1}[s_{1}]=Nu_{2}-g_{2}\sum_{\langle
ij\rangle}\sigma_{i}\sigma_{j}-g_{2}^{\prime}\sum_{(ij)}\sigma_{i}\sigma_{j}-g_{2}^{\prime\prime}\sum_{[ijkl]}\sigma_{i}\sigma_{j}\sigma_{k}\sigma_{l}\;,$
(4.96)

where the initial parameters $u$ and $g$ are renormalized into

$u_{2}=2u-\ln 2-\;\frac{1}{8}\;[\ln\cosh(4g)+4\ln\cosh(2g)]\;,\qquad
g_{2}=\frac{1}{4}\;\ln\cosh(4g)\;,$ (4.97)

and there arise two additional terms, with the couplings

$g_{2}^{\prime}=\frac{1}{2}\;g_{2}\;,\qquad
g_{2}^{\prime\prime}=\frac{1}{8}\;[\ln\cosh(4g)-4\ln\cosh(2g)]\;.$ (4.98)

In these additional terms, the notation $(ij)$ implies the next to nearest
neighbors, and $[ijkl]$ denotes a placket, that is a square of sites
containing no sites inside.

As is seen, the number of the coupling parameters has been doubled,

$\\{u,g\\}\rightarrow\\{u_{2},g_{2},g_{2}^{\prime},g_{2}^{\prime\prime}\\}\;.$

This is an example of the coupling-parameter proliferation, discussed in Sec.
4.4.

Assuming that near a fixed point the Hamiltonian is invariant under the
renormalization transformation, one neglects the appearance of the additional
terms. It is straightforward to continue the procedure, as is explained in
Sec. 4.4. If we limit ourselves by one transformation, we get the fixed point
equations

$u^{*}=\ln 2+\frac{1}{2}\;[g^{*}+\ln\cosh(2g^{*})]\;,\qquad
g^{*}=\frac{1}{4}\;\ln\cosh(4g^{*})\;.$ (4.99)

However, these equations have only a trivial solution $u^{*}=\ln 2$ and
$g^{*}=0$ yielding incompatible values for the critical temperature. Thus the
first equality gives $T_{c}=U/2\ln 2$, while the second equality means that
$T_{c}\rightarrow\infty$. This signifies that just one step of the
renormalization procedure is not sufficient for defining a nontrivial fixed
point. But the procedure has to be continued.

###  4.11 Exact Semigroup

There exist rather rare cases, when the proliferation of parameters does not
occur. In such exceptional cases, the renormalization procedure can be
realized exactly. As an illustration, let us consider the one-dimensional
Ising model with the Hamiltonian

$H[s]=\sum_{j=1}^{N}(u-g\sigma_{j}\sigma_{j+1})\;,$ (4.100)

in which $\sigma_{j}=\pm 1$ and the periodic boundary condition

$\sigma_{N+1}=\sigma_{1}$ (4.101)

is imposed.

Following the scheme of Sec. 4.4, we split the set of spin variables into two
subsets, $s_{1}$ and $s_{1}^{\prime}$, the first subset comprising the spins
with odd indices, while the second, containing even spins,

$s_{1}=\left\\{\sigma_{2j+1}:~{}j=0,1,2,\ldots,\frac{N}{2}\right\\}\;,\qquad
s_{1}^{\prime}=\left\\{\sigma_{2j}:~{}j=1,2,\ldots,\frac{N}{2}\right\\}\;.$

Summing out the even spins at the first renormalization step, we pass from a
chain with the interspin distance $a$ to that with the distance $2a$. For the
effective Hamiltonian (4.31), we get

$H_{1}[s_{1}]=\sum_{j=0}^{N/2}(u_{2}-g_{2}\sigma_{2j+1}\sigma_{2j+3})\;,$
(4.102)

with the renormalized parameters

$u_{2}=2u-\ln 2-2\ln\cosh g+\ln\cosh g_{2}\;,$ $g_{2}={\rm
arctanh}\left(\tanh^{2}g\right)\;.$ (4.103)

To simplify the recurrence relations, it is useful to introduce the notations

$c\equiv e^{-u}\cosh g\;,\qquad t\equiv\tanh g\;,$ $c_{n}\equiv
e^{-u_{n}}\cosh g_{n}\;,\qquad t_{n}\equiv\tanh g_{n}\;.$ (4.104)

Then relations (4.103) take the form

$c_{2}=2u^{2}\;,\qquad t_{2}=t^{2}\;.$ (4.105)

At the second renormalization step, we pass to the chain with the interspin
distance $4a$. The related twice renormalized parameters read as

$c_{4}=2c_{2}^{2}=2^{3}c^{4}\;,\qquad t_{4}=t_{2}^{2}=t^{4}\;.$ (4.106)

After the third renormalization, we get a chain with the interspin distance
$8a$, and with the parameters

$c_{8}=2^{7}c^{8}\;,\qquad t_{8}=t^{8}\;.$ (4.107)

Accomplishing $k$ renormalizations results in the parameters

$c_{n}=2^{n-1}c^{n}\;,\qquad t_{n}=t^{n}\qquad(n=2^{k})\;,$ (4.108)

which can be rewritten as

$u_{n}=nu-(n-1)\ln 2-n\ln\cosh g+\ln\cosh g_{n}\;,$ $g_{n}={\rm
arctanh}\left(\tanh^{n}g\right)\qquad(n=2^{k})\;.$ (4.109)

At all renormalization steps, no additional parameters arise.

Fixed points are given by the equations

$c^{*}=2^{n+1}(c^{*})^{n}\;,\qquad t^{*}=(t^{*})^{n}\;,$ (4.110)

having the solutions $c^{*}=0,1/2$ and $t^{*}=0,1$. The sole solution,
yielding compatible values in terms of the parameters $u$ and $g$, is

$u^{*}=\infty\;,\qquad g^{*}=\infty\;.$ (4.111)

This defines the critical point $T_{c}=0$, which is the exact answer for the
one-dimensional Ising model.

The $k$ times renormalized effective Hamiltonian is

$H_{k}[s_{k}]=H_{1,n+1}+H_{n+1,2n+1}+H_{2n+1,3n+1}+\ldots$ (4.112)

containing $N/n$ terms

$H_{i,j}\equiv u_{n}-g_{n}\sigma_{i}\sigma_{j}\qquad(n=2^{k})\;.$

Increasing the number $k$ of renormalizations, we get the parameters

$u_{n}\simeq n[u-\ln(2\cosh g)]\;,\qquad g_{n}\simeq
0\qquad(n\rightarrow\infty)\;.$ (4.113)

Then the effective Hamiltonian (4.112) tends to

$H_{k}[s_{k}]\simeq u_{n}\qquad(n=2^{k}\rightarrow\infty)\;.$ (4.114)

The free energy (4.43) becomes

$f\simeq\frac{T}{N}\;u_{n}\qquad(n\rightarrow\infty)\;.$ (4.115)

Since the maximal number of the described renormalizations is connected with
the number of spins by the relations

$N=2^{k}\;,\qquad\max\;k=\frac{\ln N}{\ln 2}\;,$

we finally obtain the free energy

$f\simeq T[u-\ln(2\cosh g)]\;.$ (4.116)

This result is asymptotically exact in thermodynamic limit.

Recall that the situation when there is no proliferation of parameters is
quite exceptional. In general, the process of renormalization generates
additional terms with additional coupling parameters. To overcome this
problem, it is possible to use the idea of optimized perturbation theory and
to introduce control functions canceling the appearing additional terms [322].

###  4.12 Field-Theory Group

Statistical renormalization groups, considered above, are actually semigroups,
since they do not include inverse transformations. They are based on some kind
of scaling transformations resulting in the consecutive coarse graining of the
system and constructing a sequence of effective models. Thus in the Wilson
method, it is the scaling of momentum variables, while in the decimation
procedure, it is the real-space coarsening of a lattice.

Field-theory renormalization groups are groups, containing inverse
transformations. It is based on discovering the natural symmetries of the
system. This approach can be used in field theory as well as in statistical
physics. As an application of the latter kind, the field-theory approach is
illustrated below for the Ising model [323].

The spin variables in the Ising model can be scaled by the transformation
$\sigma_{j}\rightarrow\nu\sigma_{j}$, which results in the Hamiltonian

$\hat{H}_{\nu}=-\;\frac{\nu^{2}}{2}\sum_{\langle
ij\rangle}J\sigma_{i}\sigma_{j}-\nu\sum_{j}B_{j}\sigma_{j}\;,$ (4.117)

where $B_{j}$ is an external magnetic field. With the dimensionless parameters

$g\equiv\frac{J}{2T}\;,\qquad h_{j}\equiv\frac{B_{j}}{T}\;,$ (4.118)

we have the dimensionless Hamiltonian

$H_{\nu}[s]=-\nu^{2}g\sum_{\langle
ij\rangle}\sigma_{i}\sigma_{j}-\nu\sum_{j}h_{j}\sigma_{j}\;.$ (4.119)

The partition function reads as

$Z(\nu,g,h)=\sum_{s}\exp(-H_{\nu}[s])\;,$ (4.120)

where $h$ is the set of the fields $h_{j}$.

We wish to find such a transformation of the parameters

$\nu\rightarrow z_{1}\nu\;,\qquad g\rightarrow z_{2}g\;,\qquad
h_{j}\rightarrow z_{3}h_{j}$ (4.121)

that would leave invariant the partition function,

$Z(z_{1}\nu,z_{2}g,z_{3}h)=Z(\nu,g,h)\;.$ (4.122)

From Eq. (4.120), it follows that the condition of invariance is satisfied if

$z_{1}^{2}z_{2}=1\;,\qquad z_{1}z_{3}=1\;.$

This tells us that of the three quantities $z_{1}$, $z_{2}$, and $z_{3}$ only
one is independent. Denoting one of them, say $z_{2}\equiv z$, we have

$z_{1}=z^{-1/2}\;,\qquad z_{2}=z\;,\qquad z_{3}=z^{1/2}\;.$ (4.123)

The property of invariance (4.122) takes the form

$Z(z^{-1/2}\nu,zg,z^{1/2}h)=Z(\nu,g,h)\;.$ (4.124)

Because of invariance (4.124), the correlation function

$C_{ij}(\nu,g)\equiv\left.\frac{\partial^{2}\ln Z(\nu,g,h)}{\partial
h_{i}\partial h_{j}}\right|_{h=0}$ (4.125)

enjoys the property

$C_{ij}(z^{-1/2}\nu,zg)=z^{-1}C_{ij}(\nu,g)\;.$ (4.126)

The Fourier transform

$C({\bf k},\nu,g)\equiv\frac{1}{N}\sum_{ij}C_{ij}(\nu,g)e^{-i{\bf k}\cdot{\bf
r}_{ij}}\;,$ (4.127)

in which ${\bf r}_{ij}\equiv{\bf r}_{i}-{\bf r}_{j}$, possesses the same
property

$C({\bf k},z^{-1/2}\nu,zg)=z^{-1}C({\bf k},\nu,g)\;.$ (4.128)

Keeping in mind a spatially isotropic situation, we can define the function

$K(k^{2},\nu,g)\equiv gC({\bf k},\nu,g)$ (4.129)

that is invariant under the transformation

$K(k^{2},z^{-1/2}\nu,zg)=K(k^{2},\nu,g)\;.$ (4.130)

The latter means that this function actually depends on two combinations of
the variables,

$K(k^{2},\nu,g)=\overline{K}(k^{2},\nu^{2}g)\;.$ (4.131)

In the vicinity of a critical point, where the correlation length diverges, it
is possible [323] to introduce the dimensionless function

$D\left(\frac{k^{2}}{\lambda^{2}}\;,g\right)\equiv\frac{\overline{K}(k^{2},g)}{\overline{K}(\lambda^{2},g)}$
(4.132)

having the boundary condition

$D(1,g)=1\;.$ (4.133)

Representing the correlation functions as series in powers of the coupling
parameter $g$, it is possible [323] to explicitly demonstrate that function
(4.132) is invariant under the transformation

$\lambda\rightarrow\mu\;,\qquad g\rightarrow zg\;,\qquad D\rightarrow zD\;,$
(4.134)

satisfying the relation

$D\left(\frac{k^{2}}{\lambda^{2}}\;,g\right)=zD\left(\frac{k^{2}}{\mu^{2}}\;,zg\right)\;,$
(4.135)

with the property

$D\left(\frac{\mu^{2}}{\lambda^{2}}\;,g\right)=z\;.$ (4.136)

From here, with the use of the dimensionless variables,

$x\equiv\frac{k^{2}}{\lambda^{2}}\;,\qquad
t\equiv\frac{\mu^{2}}{\lambda^{2}}\;,$ (4.137)

we come to the renormalization group equation

$D(x,g)=D(t,g)D\left(\frac{x}{t}\;,gD(t,g)\right)$ (4.138)

for which condition (4.133) is valid.

###  4.13 Invariant Charge

The renormalization group equations of type (4.138) are often met in quantum
field theory. The other form of such equations can be given by introducing the
invariant charge

$G(x,g)\equiv gD(x,g)$ (4.139)

with the boundary condition

$G(1,g)=g\;.$ (4.140)

Scaling the variable $x$ as $x\rightarrow tx$, we get the renormalization
group equation

$gG(tx,g)=G(t,g)G(x,G(t,g))\;.$ (4.141)

Differentiating with respect to $x$ and setting $x\rightarrow 1$ and
$t\rightarrow x$, we obtain the renormalization group equation in differential
form

$\frac{\partial G(x,g)}{\partial\ln x}=\beta(G(x,g),g)\;,$ (4.142)

where the Gell-Mann-Low function is

$\beta(G(x,g),g)=\frac{G(x,g)}{g}\;\left.\left[\frac{\partial}{\partial
y}\;G(y,G(x,g))\right]\right|_{y=1}\;.$ (4.143)

Integrating this equation yields

$\int_{g}^{G}\frac{df}{\beta(f,g)}=\ln x\;.$ (4.144)

Fixed points correspond to the condition

$G(x,g^{*})=g^{*}\;,$ (4.145)

hence to the zero of the Gell-Mann-Low function,

$\beta(g^{*},g^{*})=0\;.$ (4.146)

Thus the field-theory group is a Lie group of continuous transformations,
possessing the inverse transformation and allowing for the presentation in the
form of a differential equation.

##  5 Discussion

The aim of this review has been twofold: First, it presents two approaches in
approximation theory, developed by the author, Optimized Perturbation Theory
and Self-Similar Approximation Theory. Secondly, it emphasizes that these
approaches are based on the ideas having several similarities with
renormalization group techniques.

The main idea of Optimized Perturbation Theory is to reorganize a divergent
sequence into a convergent sequence by introducing control functions. Such a
reorganization of the approximants for the sought quantity is analogous to the
renormalization of these approximants. The optimization conditions for control
functions follow from the Cauchy criterion, which for sequences plays the role
of a fixed-point condition. The derivation of the optimization conditions from
the Cauchy criterion makes the theory logical and self-consistent. This makes
the theory principally different from ad hoc prescriptions used by some
authors. The developed theory allows for the derivation of different
optimization conditions, justifying them on the basis of the Cauchy criterion.
Since the latter plays the role of a fixed-point condition, different
particular forms of optimization conditions, such as the minimal-difference
and minimal-derivative conditions, have the same meaning of fastest-
convergence conditions. Control functions can be incorporated either into an
initial approximation or into the perturbative series with respect to an
asymptotically small variable by means of a change of the variable. The theory
makes it possible to obtain good approximation accuracy with just a few
perturbative terms.

In Self-Similar-Approximation Theory, the passage between subsequent
approximations is treated as a motion with respect to the approximation order.
The sequence limit is equivalent to a fixed point. In the vicinity of a fixed
point, there exists a self-similar relation analogous to a renormalization
group equation. The transformation between subsequent approximants forms a
dynamical system in discrete time, called an approximation cascade, whose
trajectory is bijective to the sequence of the approximants. By embedding the
approximation cascade into a flow results in an approximation flow, where the
approximation order is considered as a continuous variable. Using a continuous
approximation order reminds us the use of a continuous dimensionality in the
Wilson renormalization group. Since both the equations of renormalization
group and self-similar approximation theory can be written either in a
functional form or as Lie differential equations, they allow for the analysis
of stability by considering the related map multipliers, as for dynamical
systems. Control functions can be introduced either into an initial
approximation or by a transformation of the treated sequence. A very
convenient is the fractal transform effectively rising the approximation
order. By employing the fractal transform it is possible to derive different
forms of approximants, such as self-similar root approximants, self-similar
nested approximants, self-similar exponential approximants, self-similar
additive approximants, self-similar factor approximants, and self-similar
combined approximants. The wide applicability of these approximants is
illustrated by several examples. The main features of the self-similar
approximants is their generality, simplicity, and good accuracy. In the cases,
where a problem enjoys an exact solution, these approximants reconstruct this
exact solution.

The main equation of self-similar approximation theory is based on the
assumption of the existence of the sequence limit, similarly to the assumption
of the existence of a critical point in statistical renormalization group. The
field-theory renormalization group equations are derived by employing the
natural symmetry of the considered system. The efficiency of both optimized
perturbation theory and self-similar approximation theory can be understood as
being due to their ability of discovering the hidden symmetry in asymptotic
sequences. By their formulation, they make it possible to discover the hidden
transformation law when moving from one to another terms of a perturbative
sequence. The discovered transformation law allows us to represent it as an
equation of motion analogous to renormalization group equations. The
difference is that in the renormalizaion group equations the motion is with
respect to some kind of a scaling parameter. While in self-similar
approximation theory, the motion is considered with respect to the
approximation order.

## Acknowledgement

I am grateful for many useful discussions and collaboration to S. Gluzman and
E.P. Yukalova.

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