# Exponentially convergent trapezoidal rules to approximate fractional powers
of operators

Lidia Aceto  Lidia Aceto
Università del Piemonte Orientale
Dipartimento di Scienze e Innovazione Tecnologica, viale Teresa Michel 11,
15121 Alessandria
Italy<EMAIL_ADDRESS>and  Paolo Novati  Paolo Novati
Università di Trieste
Dipartimento di Matematica e Geoscienze, via Valerio 12/1, 34127 Trieste
Italy<EMAIL_ADDRESS>
###### Abstract.

In this paper we are interested in the approximation of fractional powers of
self-adjoint positive operators. Starting from the integral representation of
the operators, we apply the trapezoidal rule combined with a single-
exponential and a double-exponential transform of the integrand function. For
the first approach our aim is only to review some theoretical aspects in order
to refine the choice of the parameters that allow a faster convergence. As for
the double exponential transform, in this work we show how to improve the
existing error estimates for the scalar case and also extend the analysis to
operators. We report some numerical experiments to show the reliability of the
estimates obtained.

###### Key words and phrases:

Matrix functions, Single-exponential transform, Double-exponential transform,
Trapezoidal rule, Fractional Laplacian

This work was partially supported by GNCS-INdAM and FRA-University of Trieste.
The authors are members of the INdAM research group GNCS

##  1\. Introduction

In this work we are interested in the numerical approximation of
${\mathcal{L}}^{-\alpha}$, $\alpha\in(0,1).$ Here ${\mathcal{L}}$ is a self-
adjoint positive operator acting in an Hilbert space ${\mathcal{H}}$ in which
the eigenfunctions of ${\mathcal{L}}$ form an orthonormal basis of
${\mathcal{H}},$ so that ${\mathcal{L}}^{-\alpha}$ can be written through the
spectral decomposition of ${\mathcal{L}}$. In other words, for a given
$g\in{\mathcal{H}}$, we have

(1) ${\mathcal{L}}^{-\alpha}g=\sum_{j=1}^{+\infty}\mu_{j}^{-\alpha}\langle
g,\varphi_{j}\rangle\varphi_{j}$

where $\mu_{j}$ and $\varphi_{j}$ are the eigenvalues and the eigenfunctions
of ${\mathcal{L}},$ respectively, and $\langle\cdot,\cdot\rangle$ denotes the
${\mathcal{H}}$-inner product. Throughout the paper we also assume
$\sigma(\mathcal{L})\subseteq[1,+\infty)$, where $\sigma(\mathcal{L})$ denotes
the spectrum of ${\mathcal{L}}$.

Applications of (1) include the numerical solution of fractional equations
involving the anomalous diffusion, in which ${\mathcal{L}}$ is related to the
Laplacian operator, and this is the main reason for which in recent years a
lot of attention has been placed on the efficient approximation of fractional
powers. Among the approaches recently introduced we quote here the methods
based on the best uniform rational approximations of functions closely related
to $\lambda^{-\alpha}$ that have been studied in [6, 7, 8, 9]. Another class
of methods relies on quadrature rules arising from the Dunford-Taylor integral
representation of $\lambda^{-\alpha}$ [1, 2, 3, 4, 5, 17, 18]. Very recently,
time stepping methods for a parabolic reformulation of fractional diffusion
equations, proposed in [19], have been interpreted by Hofreither in [10] as
rational approximations of $\lambda^{-\alpha}.$

In this work, starting from the integral representation

(2)
$\mathcal{L}^{-\alpha}=\frac{2\sin(\alpha\pi)}{\pi}\int_{0}^{+\infty}t^{2\alpha-1}({\mathcal{I}}+t^{2}\mathcal{L})^{-1}dt,\qquad\alpha\in(0,1),$

where ${\mathcal{I}}$ is the identity operator in ${\mathcal{H}}$, we consider
the trapezoidal rule applied to the single and the double-exponential
transform of the integrand function. The former approach has been extensively
studied in [5], where the authors also provide reliable error estimates. The
rate of convergence has been shown to be of type

(3) $\exp(-c\sqrt{n}),$

where $n$ is closely related to the number of nodes. Our aim here is just to
review some theoretical aspects in order to refine the choice of the
parameters that allow faster convergence, even if, still of type (3). As for
the double-exponential transform, widely investigated in [12, 13, 14, 15, 16]
for general scalar functions, in this work we show how to improve the existing
error estimates for the function $\lambda^{-\alpha}$. We also extend the
analysis to operators, showing that it is possible to reach a convergence rate
of type

$\exp\left(-c\sqrt{\frac{n}{\ln n}}\right).$

While theoretically disadvantageous with respect to the single-exponential
approach, we show that the double-exponential approach is actually faster at
least for $\alpha\in[1/2,1)$.

The paper is organized as follows. In Section 2 we make a short background
concerning the trapezoidal rule with particular attention to functions that
decay exponentially at infinity. In Section 3 we review the existing
convergence analysis for the trapezoidal rule combined with a single-
exponential transform and we refine the choice of the parameters that allow a
faster convergence. Section 4 is devoted to the trapezoidal rule combined with
a double-exponential transform. Here the convergence analysis is derived for
the approximation of the scalar function $\lambda^{-\alpha}$ and is then
extended to the case of the operator ${\mathcal{L}}^{-\alpha}.$ Some
concluding remarks are finally reported in Section 5.

##  2\. A general convergence result for the trapezoidal rule

Given a generic continuous function $f:\mathbb{R}\rightarrow\mathbb{R}$, in
this section we make a short background concerning the trapezoidal
approximation

$I(f)=\int_{-\infty}^{+\infty}f(x)dx\approx
h\sum_{\ell=-\infty}^{+\infty}f(\ell h),$

where $h$ is a suitable positive value. Given $M$ and $N$ positive integers,
we denote the truncated trapezoidal rule by

$T_{M,N,h}(f)=h\sum_{\ell=-M}^{N}f(\ell h).$

For the error we have

$\mathcal{E}_{M,N,h}(f):=\left|I(f)-T_{M,N,h}(f)\right|\leq\mathcal{E}_{D}+\mathcal{E}_{T_{L}}+\mathcal{E}_{T_{R}},$

where

$\displaystyle\mathcal{E}_{D}$
$\displaystyle=\left|\int_{-\infty}^{+\infty}f(x)dx-h\sum_{\ell=-\infty}^{+\infty}f(\ell
h)\right|,$ $\displaystyle\mathcal{E}_{T_{L}}$
$\displaystyle=h\sum_{\ell=-\infty}^{-M-1}\left|f(\ell
h)\right|,\qquad\mathcal{E}_{T_{R}}=h\sum_{\ell=N+1}^{+\infty}\left|f(\ell
h)\right|.$

The quantities $\mathcal{E}_{D}$ and
$\mathcal{E}_{T}:=\mathcal{E}_{T_{L}}+\mathcal{E}_{T_{R}}$ are referred to as
the discretization error and the truncation error, respectively.

######  Definition 1.

[11, Definition 2.12] For $d>0,$ let $\mathcal{D}_{d}$ be the infinite strip
domain of width $2d$ given by

$\mathcal{D}_{d}=\\{\zeta\in\mathbb{C}:\,|\operatorname{Im}(\zeta)|<d\\}.$

Let $B(\mathcal{D}_{d})$ be the set of functions analytic in $\mathcal{D}_{d}$
that satisfy

$\int_{-d}^{d}|f(x+i\eta)|d\eta=\mathcal{O}(|x|^{a}),\quad
x\rightarrow\pm\infty,\quad 0\leq a<1,$

and

$\mathcal{N}(f,d)=\lim_{\eta\rightarrow
d^{-}}\left\\{\int_{-\infty}^{+\infty}|f(x+i\eta)|dx+\int_{-\infty}^{+\infty}|f(x-i\eta)|dx\right\\}<+\infty.$

The next theorem gives an estimate for the discretization error of the
trapezoidal rule when applied to functions in $B(\mathcal{D}_{d})$.

######  Theorem 1.

[11, Theorem 2.20] Assume $f\in B(\mathcal{D}_{d})$. Then

(4) $\mathcal{E}_{D}\leq\frac{\mathcal{N}(f,d)}{2\sinh(\pi d/h)}e^{-\pi d/h}.$

######  Theorem 2.

Assume $f\in B(\mathcal{D}_{d})$ and that there are positive constants
$\beta,\gamma$ and $C$ such that

(5) $|f(x)|\leq C\left\\{\begin{array}[]{cc}\exp(-\beta|x|),&x<0,\\\
\exp(-\gamma|x|),&x\geq 0.\end{array}\right.$

Then,

(6) $\mathcal{E}_{M,N,h}(f)\leq\frac{N(f,d)}{2\sinh(\pi d/h)}e^{-\pi
d/h}+\frac{C}{\beta}e^{-\beta Mh}+\frac{C}{\gamma}e^{-\gamma Nh}.$

###### Proof.

By (5) we immediately have

$\mathcal{E}_{T_{L}}\leq\frac{C}{\beta}e^{-\beta
Mh},\quad\mathcal{E}_{T_{R}}\leq\frac{C}{\gamma}e^{-\gamma Nh}.$

Using Theorem 1 we obtain (6). ∎

The above result states that for functions that decay exponentially for
$x\rightarrow\pm\infty$ it may be possible to have exponential convergence
after a proper selection of $h$. When working with the more general situation

(7) $I(g):=\int_{a}^{b}g(t)dt,$

one can consider a suitable conformal map

$\psi:(-\infty,+\infty)\rightarrow(a,b),$

and, through the change of variable $t=\psi(x)$, transform (7) to

$I(g_{\psi}):=\int_{-\infty}^{+\infty}g_{\psi}(x)dx,\qquad
g_{\psi}(x)=g(\psi(x))\psi^{\prime}(x).$

A suitable choice of the mapping $\psi$ may allow to work with a function
$g_{\psi}$ that fulfills the hypothesis of Theorem 2 so that $I(g)$ can be
evaluated with an error that decays exponentially.

Since the aim of the paper is the computation of $\mathcal{L}^{-\alpha}$ with
$\sigma(\mathcal{L})\subseteq[1,+\infty),$ for $\lambda\geq 1$ we consider now
the integral representation (2)

(8)
$\lambda^{-\alpha}=\frac{2\sin(\alpha\pi)}{\pi}\int_{0}^{+\infty}t^{2\alpha-1}(1+t^{2}\lambda)^{-1}dt,\qquad\alpha\in(0,1).$

Defining

(9) $g_{\lambda}(t):=t^{2\alpha-1}(1+t^{2}\lambda)^{-1},$

and a change of variable $t=\psi(x)$,
$\psi:(-\infty,+\infty)\rightarrow(0,+\infty)$, let

(10) $g_{\lambda,\psi}(x)=g_{\lambda}(\psi(x))\psi^{\prime}(x).$

Let moreover

$\mathcal{Q}_{M,N,h}^{\alpha}(g_{\lambda,\psi})=\frac{2\sin(\alpha\pi)}{\pi}h\sum_{\ell=-M}^{N}g_{\lambda,\psi}(\ell
h)$

be the truncated trapezoidal rule for the computation of $\lambda^{-\alpha}$,
that is, for the computation of

$\frac{2\sin(\alpha\pi)}{\pi}\int_{0}^{+\infty}g_{\lambda}(t)dt=\frac{2\sin(\alpha\pi)}{\pi}\int_{-\infty}^{+\infty}g_{\lambda,\psi}(x)dx.$

We denote the error by

$\displaystyle E_{M,N,h}(\lambda)$
$\displaystyle=\left|\lambda^{-\alpha}-\mathcal{Q}_{M,N,h}^{\alpha}(g_{\lambda,\psi})\right|$
(11)
$\displaystyle=\frac{2\sin(\alpha\pi)}{\pi}\mathcal{E}_{M,N,h}(g_{\lambda,\psi}),$

and for operator argument

(12)
$E_{M,N,h}(\mathcal{L})=\left\|{\mathcal{L}}^{-\alpha}-\mathcal{Q}_{M,N,h}^{\alpha}(g_{{\mathcal{L}},\psi})\right\|_{\mathcal{H}\rightarrow\mathcal{H}}.$

The remainder of the paper is devoted to the analysis of two special choices
for $\psi$, the single-exponential (SE) and the double-exponential (DE)
transforms.

##  3\. The Single-Exponential transform

The SE transform is defined by

(13) $\psi_{SE}(x)=\exp(x),$

so that from (9) and (10) we get

$g_{\lambda,\psi_{SE}}(x)=e^{2\alpha x}(1+e^{2x}\lambda)^{-1}.$

Since the poles of this function are given by

$x_{k}=-\frac{1}{2}\log\lambda-i(2k+1)\frac{\pi}{2},\qquad k\in\mathbb{Z},$

we have that $g_{\lambda,\psi_{SE}}$ is analytic in

$\mathcal{D}_{\psi_{SE}}=\left\\{\zeta\in\mathbb{C}:\,|\operatorname{Im}(\zeta)|<\frac{\pi}{2}\right\\},$

that is, the strip domain with $d=\pi/2,$ independently of $\alpha$ and
$\lambda.$ Now, in order to prove that $g_{\lambda,\psi_{SE}}$ belongs to
$B(\mathcal{D}_{\psi_{SE}})$ (see Definition 1), following the analysis given
in [5] we first note that for $\eta\in\mathbb{R},$ $|\eta|<\pi/2$ and
$\lambda\geq 1,$

$\left|(1+e^{2(x+i\eta)}\lambda)^{-1}\right|\leq\left\\{\begin{array}[]{cl}1,&x<0,\\\
e^{-2x},&x\geq 0.\end{array}\right.$

Therefore,

(14)
$\left|g_{\lambda,\psi_{SE}}(x+i\eta)\right|\leq\left\\{\begin{array}[]{ll}e^{2\alpha
x},&\mbox{for }x<0,\\\ e^{-2(1-\alpha)x},&\mbox{for }x\geq
0.\end{array}\right.$

This implies that

$\displaystyle\mathcal{N}\left(g_{\lambda,\psi_{SE}},\pi/2\right)$
$\displaystyle=\lim_{\eta\rightarrow(\pi/2)^{-}}\left\\{\int_{-\infty}^{+\infty}\left|g_{\lambda,\psi_{SE}}(x+i\eta)\right|dx+\int_{-\infty}^{+\infty}\left|g_{\lambda,\psi_{SE}}(x-i\eta)\right|dx\right\\}$
$\displaystyle\leq\frac{1}{\alpha(1-\alpha)},$

and also that

$\int_{-\pi/2}^{\pi/2}|g_{\lambda,\psi_{SE}}(x+i\eta)|d\eta=\mathcal{O}(1),\quad
x\rightarrow+\infty.$

Therefore, by (1) we can conclude that $g_{\lambda,\psi_{SE}}$ belongs to
$B(\mathcal{D}_{\psi_{SE}}).$ By (4), for the discretization error we have

$\left|\int_{-\infty}^{+\infty}g_{\lambda,\psi_{SE}}(x)dx-h\sum_{\ell=-\infty}^{+\infty}g_{\lambda,\psi_{SE}}(\ell
h)\right|\leq\frac{1}{\alpha(1-\alpha)}\frac{e^{-\pi d/h}}{2\sinh(\pi
d/h)},\quad d=\pi/2.$

Since

$\frac{e^{-t}}{2\sinh(t)}=e^{-2t}\left(1+\mathcal{O}(e^{-2t})\right)\quad\text{as
}t\rightarrow+\infty,$

by (2), for $h\leq 2\pi d$ we obtain

$\displaystyle\mathcal{E}_{M,N,h}(g_{\lambda,\psi_{SE}})$
$\displaystyle=\left|\int_{-\infty}^{+\infty}g_{\lambda,\psi_{SE}}(x)dx-h\sum_{\ell=-M}^{N}g_{\lambda,\psi_{SE}}(\ell
h)\right|$ (15) $\displaystyle\leq\frac{1}{\alpha(1-\alpha)}e^{-2\pi
d/h}+\frac{1}{2\alpha}e^{-2\alpha
Mh}+\frac{1}{2(1-\alpha)}e^{-2(1-\alpha)Nh}.$

After choosing $h$, the contribute of the three exponentials can be equalized
by taking $M$ and $N$ such that

$\pi d/h\approx\alpha Mh\approx(1-\alpha)Nh,$

that is

$M=\left\lceil\frac{\pi d}{\alpha h^{2}}\right\rceil,\quad
N=\left\lceil\frac{\pi d}{(1-\alpha)h^{2}}\right\rceil,$

where $\left\lceil\cdot\right\rceil$ is the ceiling function. Denoting by
$n=M+N+1$ the total number of inversions we have that

$n\approx\frac{\pi d}{h^{2}}\frac{1}{\alpha(1-\alpha)},$

and therefore by (15) we obtain

$\mathcal{E}_{M,N,h}(g_{\lambda,\psi_{SE}})\leq\frac{3}{2}\frac{1}{\alpha(1-\alpha)}\exp\left(-2\sqrt{\pi
d\alpha(1-\alpha)}\sqrt{n}\right).$

By (12), since $\mathcal{L}$ is assumed to be self-adjoint and
$\sigma(\mathcal{L})\subseteq[1,+\infty)$, we have that

$E_{M,N,h}(\mathcal{L})\leq\frac{2\sin(\alpha\pi)}{\pi}\max_{\lambda\geq
1}\mathcal{E}_{M,N,h}(g_{\lambda,\psi_{SE}}),$

and then, finally, taking $d=\pi/2$ we obtain

(16)
$E_{M,N,h}(\mathcal{L})\leq\frac{\sin(\alpha\pi)}{\pi}\frac{3}{\alpha(1-\alpha)}\exp\left(-\pi\sqrt{2\alpha(1-\alpha)}\sqrt{n}\right),\quad
n=M+N+1.$

The analysis just presented is almost identical to the one given in [5].
Nevertheless in that paper the authors define $d=\pi/4$, while here we have
shown that one can take $d=\pi/2$. This choice produces a remarkable speedup,
as shown in Figure 1, where for different values of $\alpha$ we have
considered the error in the computation of $\mathcal{L}^{-\alpha}$ for the
artificial operator

(17)
$\mathcal{L}=\left[\operatorname{diag}(1,2,\dots,100)\right]^{8},\quad\sigma(\mathcal{L})\subseteq[1,10^{16}]\text{.}$

Figure 1. Error for the trapezoidal rule applied with the single-exponential
transform with $d=\pi/4$ and $d=\pi/2$, and error estimate given by (16).

##  4\. Double-Exponential transformation

The DE transform we use here is given by

(18) $\psi_{DE}(x)=\exp\left(\frac{\pi}{2}\sinh(x)\right).$

We consider in (8) the change of variable

$\tau t^{2}=\left(\psi_{DE}(x)\right)^{2}=\exp(\pi\sinh(x)),\qquad\tau>0.$

The function involved in this case is

$\displaystyle g_{\lambda,\psi_{DE}}(x)$
$\displaystyle=\frac{\pi}{2}\tau^{1-\alpha}\frac{\exp(\alpha\pi\sinh(x))}{\tau+\lambda\exp(\pi\sinh(x))}\cosh(x)$
(19)
$\displaystyle=\frac{\pi}{2}\lambda^{-\alpha}\frac{\left(\lambda/\tau\exp(\pi\sinh(x))\right)^{\alpha}}{1+\lambda/\tau\exp(\pi\sinh(x))}\cosh(x),$

and we employ the trapezoidal rule to compute

$\lambda^{-\alpha}=\frac{2\sin(\alpha\pi)}{\pi}\int_{-\infty}^{+\infty}g_{\lambda,\psi_{DE}}(x)dx.$

The parameter $\tau$ needs to be selected in some way and the analysis is
provided in Section 4.5. Its introduction is motivated by the fact that, when
moving from $\lambda$ to $\mathcal{L}$, the method (the choice of $M$, $N$ and
$h$) and the error estimates have to be derived by working uniformly in the
interval $[1,+\infty)$ containing $\sigma(\mathcal{L})$. As in the SE case,
the function $g_{\lambda,\psi_{DE}}(x)$ exhibits a fast decay for
$x\rightarrow\pm\infty$, but the definition of the strip of analyticity is now
much more difficult to handle since everything now depends on $\lambda$ and
$\tau$.

###  4.1. Asymptotic behavior of the integrand function

In order to apply Theorem 2 we need to study
$|g_{\lambda,\psi_{DE}}(x+i\eta)|$. From (19) we have

$\left|g_{\lambda,\psi_{DE}}(x+i\eta)\right|=\frac{\pi}{2}\lambda^{-\alpha}\left|\frac{\left(\lambda/\tau\exp(\pi\sinh(x+i\eta))\right)^{\alpha}}{1+\lambda/\tau\exp(\pi\sinh(x+i\eta))}\right||\cosh(x+i\eta)|.$

After simple manipulations based on standard relations we find

$\left|\cosh(x+i\eta)\right|=\sqrt{\cosh^{2}x-\sin^{2}\eta},$

and therefore

$\left|\cosh(x+i\eta)\right|\leq\cosh x.$

Moreover

$\left|\left(\lambda/\tau\exp(\pi\sinh(x+i\eta))\right)^{\alpha}\right|=\left(\frac{\lambda}{\tau}\right)^{\alpha}\left|\exp(\alpha\pi\sinh
x\cos\eta))\right|.$

In addition, we can bound the denominator using the results given in [13, p.
388], that is,

$\left|\frac{1}{1+\lambda/\tau\exp(\pi\sinh(x+i\eta))}\right|\leq\frac{1}{1+\lambda/\tau\exp(\pi\sinh
x\cos\eta)\cos(\pi/2\sin\eta)}.$

From the above relations we find

$\left|g_{\lambda,\psi_{DE}}(x+i\eta)\right|\leq\frac{\pi}{2}\lambda^{-\alpha}\frac{\cosh
x}{\cos(\pi/2\sin\eta)}G_{\alpha}(x,\eta),$

where

$G_{\alpha}(x,\eta)=\frac{\left(\lambda/\tau\exp(\pi\sinh
x\cos\eta)\right)^{\alpha}}{1+\lambda/\tau\exp(\pi\sinh x\cos\eta)}.$

Let $x^{\ast}$ be such that

$\pi\sinh x^{\ast}\cos\eta=\ln(\tau/\lambda);$

we have

(20)
$G_{\alpha}(x,\eta)\leq\left\\{\begin{array}[]{ll}(\lambda/\tau)^{\alpha}\exp(\alpha\pi\cos\eta\sinh
x),&x\leq x^{\ast},\\\
(\lambda/\tau)^{\alpha-1}\exp(-(1-\alpha)\pi\cos\eta\sinh
x),&x>x^{\ast}.\end{array}\right.$

###  4.2. Error estimate for the scalar case

The bound (20) implies that

$\displaystyle\mathcal{N}\left(g_{\lambda,\psi_{DE}},d\right)$
$\displaystyle=\lim_{\eta\rightarrow
d^{-}}\left\\{\int_{-\infty}^{+\infty}\left|g_{\lambda,\psi_{DE}}(x+i\eta)\right|dx+\int_{-\infty}^{+\infty}\left|g_{\lambda,\psi_{DE}}(x-i\eta)\right|dx\right\\}$
$\displaystyle\leq\lim_{\eta\rightarrow
d^{-}}\pi\lambda^{-\alpha}\left\\{\frac{1}{\cos(\pi/2\sin\eta)}\int_{-\infty}^{+\infty}G_{\alpha}(x,\eta)\cosh
xdx\right\\}$ $\displaystyle\leq\lim_{\eta\rightarrow
d^{-}}\frac{\pi\lambda^{-\alpha}}{\cos(\pi/2\sin\eta)}\left\\{(\lambda/\tau)^{\alpha}\int_{-\infty}^{x^{\ast}}\exp(\alpha\pi\cos\eta\sinh
x)\cosh x\,dx\right.+$
$\displaystyle+\left.(\lambda/\tau)^{\alpha-1}\int_{x^{\ast}}^{+\infty}\exp(-(1-\alpha)\pi\cos\eta\sinh
x)\cosh x\,dx\right\\}$
$\displaystyle\leq\frac{1}{\alpha(1-\alpha)}\frac{2}{\cos d\cos(\pi/2\sin
d)}\lambda^{-\alpha}.$

In addition, assuming $d=d(\lambda,\tau)<\pi/2$, it can be observed that

$\displaystyle\int_{-d(\lambda,\tau)}^{d(\lambda,\tau)}|g_{\lambda,\psi_{DE}}(x+i\eta)|d\eta$
$\displaystyle\leq\frac{\pi}{2}\lambda^{-\alpha}\int_{-d(\lambda,\tau)}^{d(\lambda,\tau)}\frac{G_{\alpha}(x,\eta)\cosh
x}{\cos(\pi/2\sin\eta)}d\eta$ $\displaystyle=\mathcal{O}(1)\quad\text{for
}x\rightarrow\pm\infty.$

Using (1), for the discretization error we have

$\left|\int_{-\infty}^{+\infty}g_{\lambda,\psi_{DE}}(x)dx-h\sum_{\ell=-\infty}^{+\infty}g_{\lambda,\psi_{DE}}(\ell
h)\right|\leq\xi(d)\frac{1}{\alpha(1-\alpha)}\lambda^{-\alpha}\frac{e^{-\pi
d/h}}{2\sinh(\pi d/h)},$

where

(22) $\xi(d)=\frac{2}{{\cos d\cos(\pi/2\sin d)}}.$

The remaining task is to estimate the truncation error. Using (20) we obtain

$\displaystyle
h\sum\nolimits_{\ell=-\infty}^{-M-1}\left|g_{\lambda,\psi_{DE}}(\ell
h)\right|$
$\displaystyle\leq\frac{\pi}{2}\tau^{-\alpha}h\sum\nolimits_{\ell=-\infty}^{-M-1}\exp(\alpha\pi\sinh(\ell
h))\cosh(\ell h)$
$\displaystyle\leq\frac{\pi}{2}\tau^{-\alpha}\int_{-\infty}^{-Mh}\exp(\alpha\pi\sinh
x)\cosh(x)dx$
$\displaystyle\leq\frac{\tau^{-\alpha}}{2\alpha}\exp\left(-\alpha\pi\sinh(Mh)\right)$
$\displaystyle\leq\frac{\tau^{-\alpha}}{2\alpha}\exp\left(\frac{\alpha\pi}{2}\right)\exp\left(-\frac{\alpha\pi}{2}\exp(Mh)\right).$

Similarly,

$\displaystyle
h\sum\nolimits_{\ell=N+1}^{+\infty}\left|g_{\lambda,\psi_{DE}}(\ell h)\right|$
$\displaystyle\leq\frac{\pi}{2}\lambda^{-1}\tau^{1-\alpha}h\sum\nolimits_{\ell=N+1}^{+\infty}\exp(-(1-\alpha)\pi\sinh(\ell
h))\cosh(\ell h)$
$\displaystyle\leq\frac{\pi}{2}\lambda^{-1}\tau^{1-\alpha}h\int_{Nh}^{+\infty}\exp(-(1-\alpha)\pi\sinh
x)\cosh(x)dx$
$\displaystyle\leq\frac{\lambda^{-1}\tau^{1-\alpha}}{2(1-\alpha)}\exp\left(\frac{(1-\alpha)\pi}{2}\right)\exp\left(-\frac{(1-\alpha)\pi}{2}\exp(Nh)\right).$

The above results are summarized as follows.

######  Proposition 1.

Using the double-exponential transform, for the quadrature error it holds

(23) $\displaystyle\mathcal{E}_{M,N,h}(g_{\lambda,\psi_{DE}})$
$\displaystyle\leq\frac{1}{\alpha(1-\alpha)}\xi(d)\lambda^{-\alpha}\frac{e^{-\pi
d/h}}{2\sinh(\pi d/h)}+$ (24)
$\displaystyle\frac{\tau^{-\alpha}}{2\alpha}\exp\left(\frac{\alpha\pi}{2}\right)\exp\left(-\frac{\alpha\pi}{2}\exp(Mh)\right)+$
(25)
$\displaystyle\frac{\lambda^{-1}\tau^{1-\alpha}}{2(1-\alpha)}\exp\left(\frac{(1-\alpha)\pi}{2}\right)\exp\left(-\frac{(1-\alpha)\pi}{2}\exp(Nh)\right),$

where $\xi(d)$ is defined by (22).

Defining

(26)
$h=\ln\left(\frac{4dn}{\mu}\right)\frac{1}{n},\quad\text{for\quad}n\geq\frac{\mu
e}{4d},\quad\mu=\min(\alpha,1-\alpha)$

as in [13, Theorem 2.14], we first observe that (see (23))

(27) $\frac{\exp\left(-\frac{\pi d}{h}\right)}{2\sinh\left(\frac{\pi
d}{h}\right)}\leq\frac{1}{1-e^{-\frac{\pi}{2}\mu e}}\exp\left(\frac{-2\pi
dn}{\ln\left(\frac{4dn}{\mu}\right)}\right).$

Setting $M=N=n$, the choice of $h$ as in (26) leads to a truncation error that
decays faster than the discretization one, because for an arbitrary constant
$c$ (see (24)-(25))

$\exp\left(-c\exp\left(nh\right)\right)=\exp\left(-\frac{4cdn}{\mu}\right).$

As consequence the idea is to assume the discretization error as estimator for
the global quadrature error, that is, using (23) and (27),

(28)
$\mathcal{E}_{n,h}(g_{\lambda,\psi_{DE}})=\mathcal{E}_{M,N,h}(g_{\lambda,\psi_{DE}})\approx
K_{\alpha}\xi(d)\lambda^{-\alpha}\exp\left(\frac{-2\pi
dn}{\ln\left(\frac{4dn}{\mu}\right)}\right),$

where

(29) $K_{\alpha}=\frac{1}{\alpha(1-\alpha)}\frac{1}{1-e^{-\frac{\pi}{2}\mu
e}}.$

Formula (28) is very similar to the one given in [13, Theorem 2.14], that
reads

(30)
$\mathcal{\hat{E}}_{M,N,h}(g_{\lambda,\psi_{DE}})\approx\frac{\tau^{-\alpha}}{\mu}\alpha(1-\alpha)\left(K_{\alpha}\xi(d)+e^{\frac{\pi}{2}\nu}\right)\exp\left(\frac{-2\pi
dn}{\ln\left(\frac{4dn}{\mu}\right)}\right)$

where $\nu=\max\left(\alpha,1-\alpha\right)$ and $M=n$, $N=n-\chi$ (or
viceversa depending on $\alpha$), where $\chi>0$ is defined in order to
equalize the contribute of the truncation errors [13, Theorem 2.11]. The
important difference is given by the factor $\lambda^{-\alpha}$ that replaces
$\tau^{-\alpha}$, and this is crucial to correctly handle the case of
$\lambda\rightarrow+\infty$. In this situation the error of the trapezoidal
rule goes 0 because $g_{\lambda,\psi_{DE}}(x)\rightarrow 0$ as
$\lambda\rightarrow+\infty$ (see (19)). Anyway, as we shall see, $d\rightarrow
0$ as $\lambda\rightarrow+\infty$, so that the exponential term itself is not
able to reproduce this situation. An example is given in Figure 2 in which we
consider $\lambda=10^{12}$ and $\tau=100.$

Figure 2. Error for the trapezoidal rule applied with the double-exponential
transform (error DE), estimates (28) and (30) vs the number of inversions, for
the computation of $\lambda^{-\alpha}$ with $\lambda=10^{12}$ and $\tau=100$.

###  4.3. The poles of the integrand function

All the analysis presented so far is based on the assumption that the
integrand function

$g_{\lambda,\psi_{DE}}(x)=\frac{\pi}{2}\tau^{1-\alpha}\frac{\exp\left(\alpha\pi\sinh
x\right)}{\tau+\lambda\exp\left(\pi\sinh x\right)}\cosh x$

is analytic in the strip $\mathcal{D}_{d}$, for a certain $d=d(\lambda,\tau)$.
Therefore we have to study the poles of this function, that is, we have to
study the equation

$\tau+\lambda\exp\left(\pi\sinh x\right)=0.$

We have

$\displaystyle\exp\left(\pi\sinh x\right)$
$\displaystyle=\frac{\tau}{\lambda}e^{i\pi},$ $\displaystyle\sinh x$
$\displaystyle=\frac{1}{\pi}\ln\frac{\tau}{\lambda}+i(2k+1),\quad
k\in\mathbb{Z}.$

By solving the above equation for each $k$, we obtain the complete set of
poles. Assuming to work with the principal value of the logarithm and taking
$k=0$, we obtain the poles closest to the real axis $x_{0}$ and its conjugate
$\overline{x_{0}}$, where

$\displaystyle x_{0}$
$\displaystyle=\sinh^{-1}\left(\frac{1}{\pi}\ln\frac{\tau}{\lambda}+i\right)$
(31)
$\displaystyle=\ln\left(\frac{1}{\pi}\ln\frac{\tau}{\lambda}+i+\sqrt{\left(\frac{1}{\pi}\ln\frac{\tau}{\lambda}\right)^{2}+2i\frac{1}{\pi}\ln\frac{\tau}{\lambda}}\right).$

In order to apply the bound on the strip we have to define

(32) $d=d(\lambda,\tau)=r\operatorname{Im}x_{0},\quad 0<r<1.$

The introduction of the factor $r$ is necessary to avoid
${\xi}{(d)\rightarrow+\infty}$ as $\operatorname{Im}x_{0}\rightarrow\pi/2$,
which verifies for $\lambda\rightarrow\tau$ (see (22)).

####  4.3.1. Asymptotic behaviors

Setting

$s=\frac{1}{\pi}\ln\frac{\lambda}{\tau},$

we have

$\frac{1}{\pi}\ln\frac{\tau}{\lambda}=-s,$

and therefore we can write (31) as

$x_{0}=\ln\left(s\left(-1+\frac{i}{s}+\sqrt{1-\frac{2i}{s}}\right)\right).$

Assuming $\lambda\gg\tau$, that is, $s\gg 1$, and using

(33) $\sqrt{1-x}\approx
1-\frac{1}{2}x-\frac{1}{8}x^{2}-\frac{1}{16}x^{3},\quad x\approx 0,$

we obtain

$\sqrt{1-\frac{2i}{s}}\approx
1-\frac{i}{s}+\frac{1}{2s^{2}}+\frac{i}{2s^{3}}.$

Using also $\ln(1+x)\approx x$,

$\displaystyle x_{0}$
$\displaystyle\approx\ln\left(s\left(-1+\frac{i}{s}+1-\frac{i}{s}+\frac{1}{2s^{2}}+\frac{i}{2s^{3}}\right)\right)$
$\displaystyle=\ln\left(s\left(\frac{1}{2s^{2}}+\frac{i}{2s^{3}}\right)\right)$
$\displaystyle=\ln\left(\frac{1}{2s}\right)+\ln\left(1+\frac{i}{s}\right)$
$\displaystyle\approx\ln\left(\frac{1}{2s}\right)+\frac{i}{s}.$

Therefore, for $\lambda\gg\tau$,

(34)
$\operatorname{Im}x_{0}\approx\frac{1}{s}=\frac{\pi}{\ln\frac{\lambda}{\tau}}.$

Assume now $\lambda=1$ and $\tau\gg 1$. By (31) we have

$x_{0}=\ln\left(\frac{1}{\pi}\ln\tau+i+\sqrt{\left(\frac{1}{\pi}\ln\tau\right)^{2}+2i\frac{1}{\pi}\ln\tau}\right).$

Setting

$s=\frac{1}{\pi}\ln\tau,$

we have

$\displaystyle x_{0}$
$\displaystyle=\ln\left(s\left(1+\frac{i}{s}+\sqrt{1+\frac{2i}{s}}\right)\right)$
$\displaystyle\approx\ln\left(s\left(1+\frac{i}{s}+1+\frac{i}{s}\right)\right)$
$\displaystyle=\ln\left(2s\left(1+\frac{i}{s}\right)\right)$
$\displaystyle\approx\ln\left(2s\right)+\frac{i}{s},$

that finally leads to

(35) $\operatorname{Im}x_{0}\approx\frac{1}{s}=\frac{\pi}{\ln\tau}.$

###  4.4. The minimax problem

Let us define the function

$\varphi(\lambda,\tau)=\xi(d)\lambda^{-\alpha}\exp\left(\frac{-2\pi
dn}{\ln\left(\frac{4dn}{\mu}\right)}\right),\quad d=d(\lambda,\tau),$

representing the $(\lambda,\tau)$-dependent factor of the error estimate given
by (28), that is,

$\mathcal{E}_{n,h}(g_{\lambda,\psi_{DE}})\approx
K_{\alpha}\varphi(\lambda,\tau),$

where $K_{\alpha}$ is defined by (29). Since our aim is to work with a self-
adjoint operator with spectrum contained in $[1,+\infty)$ the problem consists
in defining properly the parameter $\tau.$ This can be done by solving

(36) $\min_{\tau\geq 1}\max_{\lambda\geq 1}\varphi(\lambda,\tau).$

As for the true error, experimentally one observes that $\tau$ must be taken
much greater than 1, independently of $\alpha$. Therefore, from now on the
analysis will be based on the assumption $\tau\gg 1$. Regarding the function
$\varphi(\lambda,\tau)$, by taking $d=d(\lambda,\tau)$ as in (32) and $n$
sufficiently large, again, one experimentally observes that with respect to
$\lambda$ the function initially decreases, reaches a local minimum (for
$\lambda=\tau$ in which $d=r\pi/2$), then a local maximum (much greater than
$\tau$), and finally goes to 0 for $\lambda\rightarrow+\infty$ (see Figure 3).
In this view, denoting by $\overline{\lambda}$ the local maximum, for $n$
sufficiently large the problem (36) reduces to the solution of

(37) $\varphi(1,\tau)=\varphi(\overline{\lambda},\tau).$

####  4.4.1. Evaluating the local maximum

Since $0<d\leq r\pi/2$, $0<r<1$, we have

$0<C\leq\cos d\cos\left(\frac{\pi}{2}\sin d\right)<1,$

where $C$ is a constant depending on $r$. Therefore by (22),

$2<\xi(d)\leq\frac{2}{C},$

so that we neglect the contribute of this function in what follows.

Since the maximum is seen to be much larger than $\tau$, we consider the
approximation (34). Therefore we have to solve

$\frac{d}{d\lambda}\lambda^{-\alpha}\exp\left(-\frac{2\pi
r\frac{\pi}{\ln\frac{\lambda}{\tau}}n}{\ln\left(\frac{4}{\mu}nr\frac{\pi}{\ln\frac{\lambda}{\tau}}\right)}\right)=0,$

that, after some manipulation leads to

$\frac{d}{d\lambda}\lambda^{-\alpha}\exp\left(-\frac{c_{1}n}{\ln\frac{\lambda}{\tau}\,q(\lambda)}\right)=0,$

where

(38) $c_{1}=2\pi^{2}r,\quad
q(\lambda)=\ln\left(c_{2}n\right)-\ln\left(\ln\frac{\lambda}{\tau}\right),\quad
c_{2}=\frac{4}{\mu}\pi r.$

We find the equation

$-\alpha\lambda^{-1}-\frac{d}{d\lambda}\left(\frac{c_{1}n}{\ln\frac{\lambda}{\tau}\,q(\lambda)}\right)=0,$

and since

$\displaystyle\frac{d}{d\lambda}\left(\frac{c_{1}n}{\ln\frac{\lambda}{\tau}\,q(\lambda)}\right)$
$\displaystyle=\frac{c_{1}n}{\lambda}\frac{1-q(\lambda)}{\left(\ln\frac{\lambda}{\tau}\right)^{2}q(\lambda)^{2}},$

we finally have to solve

(39)
$\alpha+c_{1}n\frac{1-q(\lambda)}{\left(\ln\frac{\lambda}{\tau}\right)^{2}q(\lambda)^{2}}=0.$

For large $n$ we have

$\displaystyle q(\lambda)$ $\displaystyle\approx\ln\left(c_{2}n\right),$
$\displaystyle\frac{q(\lambda)-1}{q(\lambda)^{2}}$
$\displaystyle\approx\frac{1}{q(\lambda)}\approx\frac{1}{\ln\left(c_{2}n\right)},$

so that the solution of (39) can be approximated by

(40)
$\lambda^{\ast}=\tau\exp\left(\sqrt{\frac{c_{1}n}{\alpha\ln\left(c_{2}n\right)}}\right).$

For any given $\tau\geq 1,$ it can be observed experimentally that
$\lambda^{\ast}$ is a very good approximation of the local maximum (see Figure
3).

We also remark that the assumption on $n$ stated in (26), that leads to the
error estimate (28), is automatically fulfilled for $\lambda=\lambda^{\ast}$,
at least for $\alpha$ not too small. Indeed, using (32) and (34) we first
observe that

(41)
$d(\lambda^{\ast},\tau)\approx\frac{r\pi}{\ln\frac{\lambda^{\ast}}{\tau}}=r\pi\sqrt{\frac{\alpha\ln\left(c_{2}n\right)}{c_{1}n}}.$

Then by (38), using $\mu\leq 1/2$ and assuming for instance $0.9<r<1,$ after
some simple computation we find

$\displaystyle\frac{\mu e}{4d(\lambda^{\ast},\tau)}$
$\displaystyle\approx\frac{\mu e}{4\pi
r}\sqrt{\frac{c_{1}n}{\alpha\ln\left(c_{2}n\right)}}$
$\displaystyle\leq\frac{1}{3}\sqrt{\frac{n}{\alpha}}.$

Therefore the condition (26) holds true for $n\geq 1/(9\alpha).$

####  4.4.2. The error at the local maximum

By (41) clearly $d(\lambda^{\ast},\tau)\rightarrow 0$ for
$n\rightarrow+\infty$, and therefore from (22) we deduce that
$\xi(d(\lambda^{\ast},\tau))\rightarrow 2$ for $n\rightarrow+\infty$. As
consequence

$\varphi(\lambda^{\ast},\tau)\approx
2\left(\lambda^{\ast}\right)^{-\alpha}\exp\left(\frac{-2\pi
d(\lambda^{\ast},\tau)n}{\ln\left(\frac{4d(\lambda^{\ast},\tau)n}{\mu}\right)}\right).$

By defining

(42) $s_{n}=\sqrt{\frac{c_{1}n}{\ln\left(c_{2}n\right)}},$

from (40) and (41) we have

$\displaystyle\lambda^{\ast}$
$\displaystyle=\tau\exp\left(\frac{s_{n}}{\sqrt{\alpha}}\right),$
$\displaystyle d(\lambda^{\ast},\tau)$
$\displaystyle\approx\frac{\sqrt{\alpha}r\pi}{s_{n}},$

and hence, after some computation

$\displaystyle\left(\lambda^{\ast}\right)^{-\alpha}\exp\left(\frac{-2\pi
d(\lambda^{\ast},\tau)n}{\ln\left(\frac{4d(\lambda^{\ast},\tau)n}{\mu}\right)}\right)$
$\displaystyle\approx\tau^{-\alpha}\exp\left(-\sqrt{\alpha}s_{n}\right)\exp\left(\frac{-2\pi\frac{\sqrt{\alpha}r\pi}{s_{n}}n}{\ln\left(\frac{4\frac{\sqrt{\alpha}r\pi}{s_{n}}n}{\mu}\right)}\right)$
$\displaystyle=\tau^{-\alpha}\exp\left(-\sqrt{\alpha}\left(s_{n}+\frac{c_{1}n}{s_{n}\ln\left(c_{2}n\frac{\sqrt{\alpha}}{s_{n}}\right)}\right)\right).$

By (42) we have

$\displaystyle
s_{n}+\frac{c_{1}n}{s_{n}\ln\left(c_{2}n\frac{\sqrt{\alpha}}{s_{n}}\right)}$
$\displaystyle=\sqrt{\frac{c_{1}n}{\ln\left(c_{2}n\right)}}+\frac{c_{1}n}{\sqrt{\frac{c_{1}n}{\ln\left(c_{2}n\right)}}\ln\left(c_{2}n\frac{\sqrt{\alpha}}{s_{n}}\right)}$
$\displaystyle=\sqrt{\frac{c_{1}n}{\ln\left(c_{2}n\right)}}\left(1+\frac{\ln\left(c_{2}n\right)}{\ln\left(c_{2}n\frac{\sqrt{\alpha}}{s_{n}}\right)}\right)$
$\displaystyle\approx 3\sqrt{\frac{c_{1}n}{\ln\left(c_{2}n\right)}},$

because

$\frac{\ln\left(c_{2}n\right)}{\ln\left(c_{2}n\frac{\sqrt{\alpha}}{s_{n}}\right)}\rightarrow
2\text{\quad for\quad}n\rightarrow+\infty.$

Joining the above approximations we finally obtain

$\displaystyle\varphi(\lambda^{\ast},\tau)$ $\displaystyle\approx
2\tau^{-\alpha}\exp\left(-3\sqrt{\alpha}\sqrt{\frac{c_{1}n}{\ln\left(c_{2}n\right)}}\right)$
(43) $\displaystyle=2\tau^{-\alpha}\exp\left(-3\sqrt{\alpha}s_{n}\right).$

####  4.4.3. Error at $\lambda=1$

By (35), that is,

$d(1,\tau)\approx r\frac{\pi}{\ln\tau},\quad\tau\gg 1,$

we have again $\xi(d(1,\tau))\approx 2$ and therefore

$\varphi(1,\tau)\approx 2\exp\left(\frac{-2\pi
d(1,\tau)n}{\ln\left(\frac{4d(1,\tau)n}{\mu}\right)}\right).$

Using (38) we find

$\displaystyle\varphi(1,\tau)$ $\displaystyle\approx 2\exp\left(-\frac{2\pi
r\frac{\pi}{\ln\tau}n}{\ln\left(\frac{4}{\mu}nr\frac{\pi}{\ln\tau}\right)}\right)$
$\displaystyle=2\exp\left(-\frac{c_{1}n}{\ln\tau\left(\ln\left(c_{2}n\right)-\ln\left(\ln\tau\right)\right)}\right)$
$\displaystyle\approx
2\exp\left(-\frac{c_{1}n}{\ln\tau\ln\left(c_{2}n\right)}\right)$ (44)
$\displaystyle=2\exp\left(-\frac{s_{n}^{2}}{\ln\tau}\right).$

###  4.5. Approximating the optimal value for $\tau$

We need to solve (37). Using the approximations (44) and (43) we impose

$\displaystyle\exp\left(-\frac{s_{n}^{2}}{\ln\tau}\right)$
$\displaystyle=\tau^{-\alpha}\exp\left(-3\sqrt{\alpha}s_{n}\right)$
$\displaystyle=\exp\left(-3\sqrt{\alpha}s_{n}-\alpha\ln\tau\right),$

that is,

$-\frac{s_{n}^{2}}{\ln\tau}=-3\sqrt{\alpha}s_{n}-\alpha\ln\tau.$

Solving the above equation we find

$\displaystyle\ln\tau$
$\displaystyle=\frac{\left(-3+\sqrt{13}\right)\sqrt{\alpha}s_{n}}{2\alpha}$
$\displaystyle\approx 0.3\frac{s_{n}}{\sqrt{\alpha}},$

so that

(45) $\tau^{\ast}=\exp\left(0.3\frac{s_{n}}{\sqrt{\alpha}}\right)$

represents an approximate solution of (37).

Figure 3. Plot of the function $\varphi(\lambda,\tau^{\ast})$ for $n=40$ and
$\alpha=1/2$. The asterisk represents the approximation of the local maximum
given by (40), that is, the point
$\left(\lambda^{\ast},\varphi(\lambda^{\ast},\tau^{\ast})\right)$. The diamond
represents the approximation of $\varphi(\lambda^{\ast},\tau^{\ast})$ stated
in (43). Finally the circle is the approximation of $\varphi(1,\tau^{\ast})$
given in (44).

In Figure 3 we plot the function $\varphi(\lambda,\tau^{\ast})$ for
$\lambda\in[1,10^{20}]$, in an example in which $n=40$, $\alpha=1/2$, and
$\tau^{\ast}\cong 84.4$ defined by (45). Moreover we show the results of the
approximations (40), (43) and (44), for $\tau=\tau^{\ast}$. Clearly the ideal
situation would be to have $\tau^{\ast}$ such that
$\varphi(1,\tau^{\ast})=\varphi(\overline{\lambda},\tau^{\ast})$, but
notwithstanding all the approximations used, the results are fairly good and
allow to have a simple expression for $\tau^{\ast}$.

By using (45) in (43) we obtain

(46) $\varphi(\lambda^{\ast},\tau^{\ast})\approx
2\exp\left(-3.3\sqrt{\alpha}\sqrt{\frac{c_{1}n}{\ln\left(c_{2}n\right)}}\right).$

Remembering that

$E_{n,h}(\mathcal{L})\leq
2\frac{\sin\left(\alpha\pi\right)}{\pi}\max_{\lambda\geq
1}\mathcal{E}_{n,h}(g_{\mathcal{\lambda},\psi_{DE}}),$

using (46) we finally obtain the error estimate

(47)
$E_{n,h}(\mathcal{L})\approx\overline{K}_{\alpha}\exp\left(-3.3\sqrt{\alpha}\sqrt{\frac{c_{1}n}{\ln\left(c_{2}n\right)}}\right),$

where

$\displaystyle\overline{K}_{\alpha}$
$\displaystyle=4\frac{\sin\left(\alpha\pi\right)}{\pi}K_{\alpha}$
$\displaystyle=4\frac{\sin\left(\alpha\pi\right)}{\pi}\frac{1}{\alpha(1-\alpha)}\frac{1}{1-e^{-\frac{\pi}{2}\mu
e}}.$

In Figure 4 we show the behavior of the method for the computation of
$\mathcal{L}^{-\alpha}$, with $\mathcal{L}$ defined in (17), together with the
estimate (47). For comparison, in the same pictures we also plot the error of
the SE approach. As mentioned in the introduction, the DE approach appears to
be faster for $1/2\leq\alpha<1.$

Figure 4. Error for the trapezoidal rule applied with the double-exponential
transform (error DE), with the single-exponential transform (error SE) and
error estimate given by (47).

##  5\. Conclusions

In this work we have analyzed the behavior of the trapezoidal rule for the
computation of $\mathcal{L}^{-\alpha}$, in connection with the single and the
double-exponential transformations. All the analysis has been based on the
assumption of $\mathcal{L}$ unbounded, so that the results can be applied even
to discrete operators, with spectrum arbitrarily large, without the need to
know its amplitude, that is, the largest eigenvalue. In particular we have
revised the analysis for the single-exponential transform and we have
introduced new error estimates for the scalar and the operator case for the
double-exponential transform. The sharp estimate obtained for the scalar case
has been fundamental for the proper selection of the parameter $\tau$ that is
necessary to obtain good results also for the operator case.

## References

  * [1] L. Aceto, D. Bertaccini, F. Durastante, P. Novati. Rational Krylov methods for functions of matrices with applications to fractional partial differential equations. J. Comput. Phys. 396, 470-482 (2019)
  * [2] L. Aceto, P. Novati. Padé-type approximations to the resolvent of fractional powers of operators. J. Sci. Comput. 83 (2020): 13, DOI:10.1007/s10915-020-01198-w
  * [3] L. Aceto, P. Novati. Rational approximations to fractional powers of self-adjoint positive operators. Numer. Math. 143, 1-16 (2019)
  * [4] L. Aceto, P. Novati. Fast and accurate approximations to fractional powers of operators. IMA J. Numer. Anal. (2021), drab002, https://doi.org/10.1093/imanum/drab002
  * [5] A. Bonito, J.E. Pasciak. Numerical approximation of fractional powers of elliptic operators. Math. Comp. 84, 2083-2110 (2015)
  * [6] S. Harizanov, R. Lazarov, S. Margenov, P. Marinov, Y. Vutov. Optimal solvers for linear systems with fractional powers of sparse SPD matrices. Numer. Linear Algebra Appl. 25(5), e2167 (2018)
  * [7] S. Harizanov, R. Lazarov, P. Marinov, S. Margenov, J.E. Pasciak. Analysis of numerical methods for spectral fractional elliptic equations based on the best uniform rational approximation. arXiv:1905.08155v2 (2019)
  * [8] S. Harizanov, R. Lazarov, P. Marinov, S. Margenov, J.E. Pasciak. Comparison analysis of two numerical methods for fractional diffusion problems based on the best rational approximations of $t^{\gamma}$ on $[0,1].$ In: Apel T., Langer U., Meyer A., Steinbach O. (eds) Advanced Finite Element Methods with Applications. FEM 2017. Lecture Notes in Computational Science and Engineering, vol 128. Springer, Cham, 2019
  * [9] S. Harizanov, S. Margenov. Positive approximations of the inverse of fractional powers of SPD M-Matrices. In: Feichtinger G., Kovacevic R., Tragler G. (eds) Control Systems and Mathematical Methods in Economics. Lecture Notes in Economics and Mathematical Systems, vol 687. Springer, Cham, 2018
  * [10] C. Hofreither. A unified view of some numerical methods for fractional diffusion. Comput. Math. Appl. 80(2), 351-366 (2020)
  * [11] J. Lund, K.L. Bowers. Sinc Methods for Quadrature and Differential Equations. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR1171217 (93i:65004)
  * [12] T. Okayama, K. Tanaka, T. Matsuo, M. Sugihara. DE-Sinc methods have almost the same convergence property as SE-Sinc methods even for a family of functions fitting the SE-Sinc methods Part I: definite integration and function approximation Numer. Math. 125, 511-543 (2013)
  * [13] T. Okayama, T. Matsuo, M. Sugihara. Error estimates with explicit constants for Sinc approximation, Sinc quadrature and Sinc indefinite integration. Numer. Math. 124, 361-394 (2013)
  * [14] M. Mori. Discovery of the double exponential transformation and its developments. Publ. RIMS, Kyoto Univ. 41, 897-935 (2005)
  * [15] K. Tanaka, M. Sugihara, K. Murota, M. Mori. Function classes for double exponential integration formulas. Numer. Math. 111, 631-655 (2009)
  * [16] L.N. Trefethen, J.A.C. Weideman, The exponentially convergent trapezoidal rule. SIAM Review 56(3), 385-458 (2014)
  * [17] P.N. Vabishchevich. Approximation of a fractional power of an elliptic operator. CoRR abs/1905.10838 (2019)
  * [18] P.N. Vabishchevich. Numerical solution of time-dependent problems with fractional power elliptic operator. Comput. Meth. in Appl. Math. 18(1), 111-128 (2018)
  * [19] P.N. Vabishchevich. Numerically solving an equation for fractional powers of elliptic operators. J. Comput. Phys. 282, 289-302 (2015)


* [98] L.-y. Dong, S.-j. Ji, C.-j. Zhang, Q. Zhang, D. W. Chiu, L.-q. Qiu, and D. Li, “An unsupervised topic-sentiment joint probabilistic model for detecting deceptive reviews,” _Expert Systems with Applications_ , vol. 114, pp. 210–223, 2018.
  * [99] G. Xu, M. Hu, C. Ma, and M. Daneshmand, “Gscpm: Cpm-based group spamming detection in online product reviews,” in _ICC 2019 - 2019 IEEE International Conference on Communications (ICC)_ , 2019, pp. 1–6.
  * [100] C. Zhu, W. Zhao, Q. Li, P. Li, and Q. Da, “Network embedding-based anomalous density searching for multi-group collaborative fraudsters detection in social media,” _Computers, Materials and Continua_ , vol. 60, no. 1, pp. 317–333, 2019. [Online]. Available: http://dx.doi.org/10.32604/cmc.2019.05677
  * [101] R. Wen, J. Wang, C. Wu, and J. Xiong, “Asa: Adversary situation awareness via heterogeneous graph convolutional networks,” in _Companion Proceedings of the Web Conference 2020_ , ser. WWW ’20. New York, NY, USA: Association for Computing Machinery, 2020, p. 674–678. [Online]. Available: https://doi.org/10.1145/3366424.3391266
  * [102] R. Narayan, J. K. Rout, and S. K. Jena, “Review spam detection using semi-supervised technique,” in _Progress in Intelligent Computing Techniques: Theory, Practice, and Applications_. Springer, 2018, pp. 281–286.
  * [103] R. Hassan and M. R. Islam, “Detection of fake online reviews using semi-supervised and supervised learning,” in _2019 International Conference on Electrical, Computer and Communication Engineering (ECCE)_ , 2019, pp. 1–5.
  * [104] A. Pathak and R. K. Srihari, “Breaking! presenting fake news corpus for automated fact checking,” in _Proceedings of the 57th annual meeting of the association for computational linguistics: student research workshop_ , 2019, pp. 357–362.
  * [105] H. Rashkin, E. Choi, J. Y. Jang, S. Volkova, and Y. Choi, “Truth of varying shades: Analyzing language in fake news and political fact-checking,” in _Proceedings of the 2017 conference on empirical methods in natural language processing_ , 2017, pp. 2931–2937.
  * [106] R. Vijjali, P. Potluri, S. Kumar, and S. Teki, “Two stage transformer model for covid-19 fake news detection and fact checking,” _arXiv preprint arXiv:2011.13253_ , 2020.
  * [107] S. Wang, W. Mao, P. Wei, and D. D. Zeng, “Knowledge structure driven prototype learning and verification for fact checking,” _Knowledge-Based Systems_ , vol. 238, p. 107910, 2022. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S0950705121010650
  * [108] A. Porshnev, I. Redkin, and A. Shevchenko, “Machine learning in prediction of stock market indicators based on historical data and data from twitter sentiment analysis,” in _2013 IEEE 13th International Conference on Data Mining Workshops_. IEEE, 2013, pp. 440–444.
  * [109] S. Li, Y. Wang, J. Xue, N. Zhao, and T. Zhu, “The impact of covid-19 epidemic declaration on psychological consequences: a study on active weibo users,” _International journal of environmental research and public health_ , vol. 17, no. 6, p. 2032, 2020.
  * [110] J. Marciano, “Fake online reviews cost $152 billion a year. Here’s how e-commerce sites can stop them,” https://www.weforum.org/agenda/2021/08/fake-online-reviews-are-a-152-billion-problem-heres-how-to-silence-them/, Aug 10, 2021, [Online; accessed 04-Aug-2022].
  * [111] S. Dickens, “Introducing New Machine Learning Techniques to Help Stop Scams,” https://www.facebook.com/notes/10157814548136886/, March 26, 2018, [Online; accessed 04-Aug-2022].
  * [112] J. D. Day and H. Zimmermann, “The osi reference model,” _Proceedings of the IEEE_ , vol. 71, no. 12, pp. 1334–1340, 1983.
  * [113] P. Gupta, S. Gandhi, and B. R. Chakravarthi, “Leveraging transfer learning techniques-bert, roberta, albert and distilbert for fake review detection,” in _Forum for Information Retrieval Evaluation_ , 2021, pp. 75–82.
  * [114] M. Matassoni, R. Gretter, D. Falavigna, and D. Giuliani, “Non-native children speech recognition through transfer learning,” in _2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 2018, pp. 6229–6233.
  * [115] P. G. Shivakumar and P. Georgiou, “Transfer learning from adult to children for speech recognition: Evaluation, analysis and recommendations,” _Computer speech & language_, vol. 63, p. 101077, 2020.
  * [116] S. Shen, M. Sadoughi, M. Li, Z. Wang, and C. Hu, “Deep convolutional neural networks with ensemble learning and transfer learning for capacity estimation of lithium-ion batteries,” _Applied Energy_ , vol. 260, p. 114296, 2020.
  * [117] Y. Yao, B. Viswanath, J. Cryan, H. Zheng, and B. Y. Zhao, “Automated crowdturfing attacks and defenses in online review systems,” in _Proceedings of the 2017 ACM SIGSAC conference on computer and communications security_ , 2017, pp. 1143–1158.
  * [118] M. A. Al-Garadi, K. D. Varathan, S. D. Ravana, E. Ahmed, and V. Chang, “Identifying the influential spreaders in multilayer interactions of online social networks,” _Journal of Intelligent & Fuzzy Systems_, vol. 31, no. 5, pp. 2721–2735, 2016.
  * [119] B. Oselio, A. Kulesza, and A. O. Hero, “Multi-layer graph analysis for dynamic social networks,” _IEEE Journal of Selected Topics in Signal Processing_ , vol. 8, no. 4, pp. 514–523, 2014.
  * [120] S. Sivasankari and G. Vadivu, “Tracing the fake news propagation path using social network analysis,” _Soft Computing_ , pp. 1–9, 2021.
  * [121] W. L. Hamilton, R. Ying, and J. Leskovec, “Inductive representation learning on large graphs,” in _Proceedings of the 31st International Conference on Neural Information Processing Systems_ , ser. NIPS’17. Red Hook, NY, USA: Curran Associates Inc., 2017, p. 1025–1035.