text: 1 The so called “heavy-quark puzzle” had already begun to manifest itself in measurements of the suppression of high transverse momentum (high-p ) non-photonic electrons at the Relativistic Heavy-Ion Collider (RHIC). Measurements T by both the STAR [10] and PHENIX [11] detectors showed a slightly higher suppression than expected, based on a calculation that included both drag and radiative loss [8, 12, 13]. This trend has continued at the Large Hadron Collider (LHC) where the ALICE experiment has measured D and B meson suppression separately, and ﬁnds a larger than expected suppression in the semi-hard regime of heavy-quark momentum (we note that the case is not very clear for B-meson suppression which has so far only been presented as p integrated points)[14, 15]. T A considerable amount of theoretical work has been devoted to understand this larger than expected suppression of single electrons or heavy mesons arising from the fragmentation of a heavy-quark. However, most of these may be understood as falling in two categories: Calculations that have extended the base formalism of radiated energy loss for light ﬂavors to include mass dependent terms, as well as a drag term to include the prominent role played by drag in heavy ﬂavor energy loss [8, 16–20]. Calculations that have ignored the role of radiative loss and only focussed on drag loss [21–23]. In all calculations above, radiative loss is stimulated by transverse momentum diﬀusion experienced by the heavy quark or radiated gluon, which, in some cases, is quantiﬁed by the jet transport coeﬃcient qˆ [24, 25]. The drag loss is quantiﬁed using the drag coeﬃcient referred to as dE/dx (energy loss per unit distance) or eˆ [26]. To the best of our knowledge, no calculation of heavy ﬂavor energy loss has explored the possibility that the drag coeﬃcient eˆ (or the longitudinal diﬀusion coeﬃcient eˆ ) may lead to an additional source of radiative loss, beyond that 2 provided by qˆ. This possibility is immediately clear in the higher twist framework, where the drag (and longitudinal diﬀusion) coeﬃcient eˆ (eˆ ) has the boost invariant deﬁnition as the loss of light-cone momentum (ﬂuctuation in 2 light-cone momentum) per unit light-cone length, (assuming a parton moving in the negative light-cone direction) − − 2 d(cid:104)∆p (cid:105) d(cid:104)∆p (cid:105) eˆ = dL− , eˆ2 = dL− . (1) While such a transport coeﬃcient leads to little change in the oﬀ-shellness of a near on-shell massless quark, it has a considerable impact on the oﬀ-shellness of a near on-shell massive quark. No doubt, such a term will only have an eﬀect on the radiative loss of a patron where the momentum p is comparable to the mass M, thus for light ﬂavors, and for energetic heavy-quarks where, p (cid:29) M, this term will have a minimal eﬀect. This was explicitly explored for photon radiation from a light quark in Ref. [27]. We point out that such an eﬀect is by no means limited to the higher-twist scheme, but eﬀects several other formalisms that have considered the radiative loss from a heavy-quark in a quark gluon plasma. In this paper, we will explore the modiﬁcation to the calculation of radiative loss, due to the presence of this additional source within the higher twist formalism. To delineate the importance of these terms, we will use power- counting techniques borrowed from Soft-Collinear-Eﬀective-Theory (SCET) [28–31] to identify the regime where these mass dependent terms will cause detectable eﬀects on the gluon bremsstrahlung spectrum. As a ﬁrst attempt, we will consider only the case of single scattering and single emission. Arguments presented in subsequent sections will demonstrate this to be the leading contribution, given the short formation time of the radiated gluons. In this paper, the analytical expressions for the eˆ (qˆand eˆ ) induced gluon radiation spectrum will be derived. Numerical calculations 2 for the suppression of B and D mesons, as well as the suppression and azimuthal anisotropy of non-photonic leptons from the decay of these mesons at LHC and RHIC energies, and comparisons with experimental data will be carried out in a subsequent eﬀort. The article is organized as follows: in Sec. II, we will setup the basic formalism of Deep Inelastic Scattering (DIS) on a large nucleus, where the hard virtual photon strikes a heavy-quark, assumed to be produced in a rare high Q2 ﬂuctuation inside a proton. In Sec. III, we will carry out diagrammatic studies on the induced gluon radiation oﬀ the heavy quark in this system, within the higher twist formalism. We will present and discuss ﬁnal expressions for gluon bremsstrahlung from such a heavy quark, containing both qˆ, eˆ and eˆ in Sec. IV. We oﬀer concluding discussions and 2 an outlook in Sec. V. Involved expressions for a set of diagrams are contained in the appendix. I. DEEP INELASTIC SCATTERING AND THE SEMI-HARD HEAVY QUARK The set-up is based on the deep-inelastic scattering of a virtual photon oﬀ a heavy quark within a large nucleus with mass number A. We will study the case where the hard virtual photon scatters with the hard heavy quark converting it to slow moving heavy quark (this is deﬁned below). The propagation of the heavy-quark will be factorized from the hard scattering vertex which produces the outgoing slow moving heavy-quark. The nucleus has a momentum P = pA, where p is the average momentum of a nucleon in this nucleus. In the Breit frame, the exchanged virtual photon possesses no transverse momentum, ##LTLine## ##LTLine##

text: The

text: so

text: called

text: “heavy-quark

text: puzzle”

text: had

text: already

text: begun

text: to

text: manifest

text: itself

text: in

text: measurements

text: of

text: the

text: suppression

text: of

text: high

text: transverse

text: momentum

text: (high-p

text: non-photonic

text: electrons

text: at

text: the

text: Relativistic

text: Heavy-Ion

text: Collider

text: (RHIC).

text: Measurements

text: T

text: by

text: both

text: the

text: STAR

text: and

text: PHENIX

text: detectors

text: showed

text: a

text: slightly

text: higher

text: suppression

text: than

text: expected,

text: based

text: on

text: a

text: calculation

text: that

text: included

text: both

text: drag

text: and

text: radiative

text: loss

text: This

text: trend

text: has

text: continued

text: at

text: the

text: Large

text: Hadron

text: Collider

text: (LHC)

text: where

text: the

text: ALICE

text: experiment

text: has

text: measured

text: D

text: and

text: B

text: meson

text: suppression

text: separately,

text: and

text: ﬁnds

text: a

text: larger

text: than

text: expected

text: suppression

text: in

text: the

text: semi-hard

text: regime

text: of

text: heavy-quark

text: momentum

text: (we

text: note

text: that

text: the

text: case

text: is

text: not

text: very

text: clear

text: for

text: B-meson

text: suppression

text: which

text: has

text: so

text: far

text: only

text: been

text: presented

text: as

text: p

text: integrated

text: T

text: A

text: considerable

text: amount

text: of

text: theoretical

text: work

text: has

text: been

text: devoted

text: to

text: understand

text: this

text: larger

text: than

text: expected

text: suppression

text: of

text: single

text: electrons

text: or

text: heavy

text: mesons

text: arising

text: from

text: the

text: fragmentation

text: of

text: a

text: heavy-quark.

text: However,

text: most

text: of

text: these

text: may

text: be

text: understood

text: as

text: falling

text: in

text: two

text: categories:

text: Calculations

text: that

text: have

text: extended

text: the

text: base

text: formalism

text: of

text: radiated

text: energy

text: loss

text: for

text: light

text: ﬂavors

text: to

text: include

text: mass

text: dependent

text: terms,

text: as

text: well

text: as

text: a

text: drag

text: term

text: to

text: include

text: the

text: prominent

text: role

text: played

text: by

text: drag

text: in

text: heavy

text: ﬂavor

text: energy

text: loss

text: Calculations

text: that

text: have

text: ignored

text: the

text: role

text: of

text: radiative

text: loss

text: and

text: only

text: focussed

text: on

text: drag

text: loss

text: In

text: all

text: calculations

text: above,

text: radiative

text: loss

text: is

text: stimulated

text: by

text: transverse

text: momentum

text: diﬀusion

text: experienced

text: by

text: the

text: heavy

text: quark

text: or

text: radiated

text: gluon,

text: which,

text: in

text: some

text: cases,

text: is

text: quantiﬁed

text: by

text: the

text: jet

text: transport

text: coeﬃcient

text: qˆ

text: The

text: drag

text: loss

text: is

text: quantiﬁed

text: using

text: the

text: drag

text: coeﬃcient

text: referred

text: to

text: as

text: dE/dx

text: (energy

text: loss

text: per

text: unit

text: distance)

text: or

text: eˆ

text: To

text: the

text: best

text: of

text: our

text: knowledge,

text: no

text: calculation

text: of

text: heavy

text: ﬂavor

text: energy

text: loss

text: has

text: explored

text: the

text: possibility

text: that

text: the

text: drag

text: coeﬃcient

text: eˆ

text: (or

text: the

text: longitudinal

text: diﬀusion

text: coeﬃcient

text: eˆ

text: may

text: lead

text: to

text: an

text: additional

text: source

text: of

text: radiative

text: loss,

text: beyond

text: that

text: provided

text: by

text: qˆ.

text: This

text: possibility

text: is

text: immediately

text: clear

text: in

text: the

text: higher

text: twist

text: framework,

text: where

text: the

text: drag

text: (and

text: longitudinal

text: diﬀusion)

text: coeﬃcient

text: eˆ

text: (eˆ

text: has

text: the

text: boost

text: invariant

text: deﬁnition

text: as

text: the

text: loss

text: of

text: light-cone

text: momentum

text: (ﬂuctuation

text: in

text: light-cone

text: momentum)

text: per

text: unit

text: light-cone

text: length,

text: (assuming

text: a

text: parton

text: moving

text: in

text: the

text: negative

text: light-cone

text: direction)

text: eˆ

text: dL−

text: dL−

text: While

text: such

text: a

text: transport

text: coeﬃcient

text: leads

text: to

text: little

text: change

text: in

text: the

text: oﬀ-shellness

text: of

text: a

text: near

text: on-shell

text: massless

text: quark,

text: it

text: has

text: a

text: considerable

text: impact

text: on

text: the

text: oﬀ-shellness

text: of

text: a

text: near

text: on-shell

text: massive

text: quark.

text: No

text: doubt,

text: such

text: a

text: term

text: will

text: only

text: have

text: an

text: eﬀect

text: on

text: the

text: radiative

text: loss

text: of

text: a

text: patron

text: where

text: the

text: momentum

text: p

text: is

text: comparable

text: to

text: the

text: mass

text: M,

text: thus

text: for

text: light

text: ﬂavors,

text: and

text: for

text: energetic

text: heavy-quarks

text: where,

text: p

text: M,

text: this

text: term

text: will

text: have

text: a

text: minimal

text: eﬀect.

text: This

text: was

text: explicitly

text: explored

text: for

text: photon

text: radiation

text: from

text: a

text: light

text: quark

text: in

text: Ref.

text: We

text: point

text: out

text: that

text: such

text: an

text: eﬀect

text: is

text: by

text: no

text: means

text: limited

text: to

text: the

text: higher-twist

text: scheme,

text: but

text: eﬀects

text: several

text: other

text: formalisms

text: that

text: have

text: considered

text: the

text: radiative

text: loss

text: from

text: a

text: heavy-quark

text: in

text: a

text: quark

text: gluon

text: plasma.

text: In

text: this

text: paper,

text: we

text: will

text: explore

text: the

text: modiﬁcation

text: to

text: the

text: calculation

text: of

text: radiative

text: loss,

text: due

text: to

text: the

text: presence

text: of

text: this

text: additional

text: source

text: within

text: the

text: higher

text: twist

text: formalism.

text: To

text: delineate

text: the

text: importance

text: of

text: these

text: terms,

text: we

text: will

text: use

text: power-

text: counting

text: techniques

text: borrowed

text: from

text: Soft-Collinear-Eﬀective-Theory

text: (SCET)

text: to

text: identify

text: the

text: regime

text: where

text: these

text: mass

text: dependent

text: terms

text: will

text: cause

text: detectable

text: eﬀects

text: on

text: the

text: gluon

text: bremsstrahlung

text: spectrum.

text: As

text: a

text: ﬁrst

text: attempt,

text: we

text: will

text: consider

text: only

text: the

text: case

text: of

text: single

text: scattering

text: and

text: single

text: emission.

text: Arguments

text: presented

text: in

text: subsequent

text: sections

text: will

text: demonstrate

text: this

text: to

text: be

text: the

text: leading

text: contribution,

text: given

text: the

text: short

text: formation

text: time

text: of

text: the

text: radiated

text: gluons.

text: In

text: this

text: paper,

text: the

text: analytical

text: expressions

text: for

text: the

text: eˆ

text: (qˆand

text: eˆ

text: induced

text: gluon

text: radiation

text: spectrum

text: will

text: be

text: derived.

text: Numerical

text: calculations

text: for

text: the

text: suppression

text: of

text: B

text: and

text: D

text: mesons,

text: as

text: well

text: as

text: the

text: suppression

text: and

text: azimuthal

text: anisotropy

text: of

text: non-photonic

text: leptons

text: from

text: the

text: decay

text: of

text: these

text: mesons

text: at

text: LHC

text: and

text: RHIC

text: energies,

text: and

text: comparisons

text: with

text: experimental

text: data

text: will

text: be

text: carried

text: out

text: in

text: a

text: subsequent

text: eﬀort.

text: The

text: article

text: is

text: organized

text: as

text: follows:

text: in

text: Sec.

text: II,

text: we

text: will

text: setup

text: the

text: basic

text: formalism

text: of

text: Deep

text: Inelastic

text: Scattering

text: (DIS)

text: on

text: a

text: large

text: nucleus,

text: where

text: the

text: hard

text: virtual

text: photon

text: strikes

text: a

text: heavy-quark,

text: assumed

text: to

text: be

text: produced

text: in

text: a

text: rare

text: high

text: ﬂuctuation

text: inside

text: a

text: proton.

text: In

text: Sec.

text: III,

text: we

text: will

text: carry

text: out

text: diagrammatic

text: studies

text: on

text: the

text: induced

text: gluon

text: radiation

text: oﬀ

text: the

text: heavy

text: quark

text: in

text: this

text: system,

text: within

text: the

text: higher

text: twist

text: formalism.

text: We

text: will

text: present

text: and

text: discuss

text: ﬁnal

text: expressions

text: for

text: gluon

text: bremsstrahlung

text: from

text: such

text: a

text: heavy

text: quark,

text: containing

text: both

text: qˆ,

text: eˆ

text: and

text: eˆ

text: in

text: Sec.

text: IV.

text: We

text: oﬀer

text: concluding

text: discussions

text: and

text: an

text: outlook

text: in

text: Sec.

text: V.

text: Involved

text: expressions

text: for

text: a

text: set

text: of

text: diagrams

text: are

text: contained

text: in

text: the

text: appendix.

text: I.

text: DEEP

text: INELASTIC

text: SCATTERING

text: AND

text: THE

text: SEMI-HARD

text: HEAVY

text: QUARK

text: The

text: set-up

text: is

text: based

text: on

text: the

text: deep-inelastic

text: scattering

text: of

text: a

text: virtual

text: photon

text: oﬀ

text: a

text: heavy

text: quark

text: within

text: a

text: large

text: nucleus

text: with

text: mass

text: number

text: A.

text: We

text: will

text: study

text: the

text: case

text: where

text: the

text: hard

text: virtual

text: photon

text: scatters

text: with

text: the

text: hard

text: heavy

text: quark

text: converting

text: it

text: to

text: slow

text: moving

text: heavy

text: quark

text: (this

text: is

text: deﬁned

text: below).

text: The

text: propagation

text: of

text: the

text: heavy-quark

text: will

text: be

text: factorized

text: from

text: the

text: hard

text: scattering

text: vertex

text: which

text: produces

text: the

text: outgoing

text: slow

text: moving

text: heavy-quark.

text: The

text: nucleus

text: has

text: a

text: momentum

text: P

text: pA,

text: where

text: p

text: is

text: the

text: average

text: momentum

text: of

text: a

text: nucleon

text: in

text: this

text: nucleus.

text: In

text: the

text: Breit

text: frame,

text: the

text: exchanged

text: virtual

text: photon

text: possesses

text: no

text: transverse

text: momentum,

text: ##LTLine##

text: ##LTLine##


text: ∗ Observation of excitonic resonances in the second harmonic spectrum of MoS 2 Mads L. Trolle, Yao-Chung Tsao, Kjeld Pedersen and Thomas G. Pedersen † ‡ We measure the second-harmonic (SH) response of MoS in a broad fundamental photon energy 2 range between 0.85 and 1.7 eV, and present a continuous SH spectrum capturing signatures of both the fundamental A and B excitons in addition to the intense C resonance. Moreover, we interpret our results in terms of the ﬁrst exciton simulation of the SH properties of multi-layered MoS samples, 2 represented by a study of trilayer MoS . The good agreement between theory and experiments 2 allows us to establish a connection between the measured spectrum and the underlying electronic structure, in the process elucidating the non-linear excitation of electron-hole pairs. In recent years, two-dimensional transition metal dichalco- bulk MoS crystal. The ﬁrst approach gives access to 2 genides (TMDs), being semiconductors1–4 with band a relatively smooth surface suitable for optical experi- gaps in the 2-3 eV range4–7, have been applied as the ments, with little diﬀuse scattering. However, here ﬂake basis for a host of novel two-dimensional optoelectronic thickness and surface coverage are diﬃcult parameters to devices8–13. Accordingly, a great deal of attention has control within the area of a mm-sized laser beam. The been devoted to the linear optical properties of these thickness of the bulk crystal, on the other hand, is im- compounds1–3. Also, non-linear optical techniques have material due to absorption. Unfortunately, the rather been demonstrated as particularly powerful probes for rough surface of weakly van der Waals bound bulk TMD the microscopic structure of few-layered TMD crystals, crystals typically make systematic optical studies diﬃ- revealing e.g., their crystallographic orientation7,14–17, cult. Hence, due to large amounts of diﬀuse scattering, and stacking angle18. Moreover, second-harmonic (SH) we only measure reliable results for maximized SH sig- spectroscopy allows for insights into the electronic struc- nals at the C resonance. Thus, the ﬂake samples give ture of TMD ﬂakes14, exposing edge-localized eﬀects19, access to highly resolved SH spectra, whereas absolute or valley-coherent excitations7. values may be extracted from the bulk sample, which, in While experimentally determined linear response func- turn, yield very little spectral information. By combining tions of various TMDs, dominated by electron-hole pairs the measured SH response from these two sample types, bound by several hundred meVs5,6,20, are reproduced rea- we are able to calibrate the intensity measured from the sonably well at the Bethe-Salpeter equation (BSE) level exfoliated ﬂakes against the bulk crystal measurements, of complexity6,21–23, similar comparisons for non-linear thereby allowing for estimation of absolute parameters. cases are lacking. Although several proposed theoreti- In Figs. 1(b) and (c), we include microscope images of cal models, based on both independent-particle19,24 and the produced ﬂakes. Atomic force microscopy and pro- excitonic22,23 approaches, have been published, the SH ﬁler scans conﬁrm most ﬂakes to consist of 40 - 100 mono- spectrum of MoS has been investigated experimentally14 layers (MLs). Indeed, no measurable photo-luminescence 2 only in a narrow fundamental photon energy range be- (PL) was observed, regardless of pump wavelength. This tween 1.2 and 1.7 eV, probing the so-called C resonance suggests very little ML coverage, since only ML domains also known from linear optics. Theoretical models have posses the direct gap necessary for eﬃcient PL1–3. so far been benchmarked by their ability to reproduce At ﬁrst glance, dipole selection rules prevent genera- this single feature22–24. However, the parabolic disper- tion of even-ordered non-linearities in MoS due to in- 2 sion, and the split valence bands near the K-points of version symmetry, which is only broken in ﬁnite sam- the Brillouin zone, translates into bound electron-hole ples consisting of an odd number of 2H-stacked MoS 2 pairs in the BSE picture. These give rise to sharp peaks layers14–16. On the other hand, thick ﬂakes, such as those in the absorption spectrum due to the so-called A and B investigated here, are not expected to display properties excitons6,25 at photon energies near 1.9 eV, and the ques- markedly diﬀerent from bulk samples. From a simplistic tion remains how they might aﬀect the SH response. In point of view, the layer-selective vanishing SH response particular, the relative intensities and lineshapes of such may be understood from the fact that adjacent layers are features are of interest. mirror images of each other in the yz-plane, as depicted In the present letter, we report an experimental op- in Fig. 1(a). Hence, in the limit of weak inter-layer cou- tical SH spectrum generated from many-layered MoS , pling, adjacent layers contribute anti-symmetrically to 2 with fundamental photon energies varied in a broad range the SH response function χ(2). While this means that between 0.85 and 1.7 eV, capturing both the fundamen- the net polarization of two adjacent MoS layers can- 2 tal A and B excitons in addition to the aforementioned cels in the dipole approximation, the spatial variation C resonance. We perform our optical experiments on of the driving ﬁeld makes this cancellation only approx- (i) many-layered MoS ﬂakes exfoliated onto fused sil- imate. Indeed, previous measurements on bulk MoS 2 2 ica by the well-known scotch tape method8, and (ii) a crystals conﬁrm this, with non-vanishing (although very

label: paragraph

text: Observation

label: title

text: of

text: excitonic

text: resonances

text: in

text: the

text: second

text: harmonic

text: spectrum

text: of

text: MoS

text: Mads

label: author

text: L.

text: Trolle,

text: Yao-Chung

text: Tsao,

text: Kjeld

text: Pedersen

text: and

text: Thomas

text: G.

text: Pedersen

text: We

label: abstract

text: measure

text: the

text: second-harmonic

text: (SH)

text: response

text: of

text: MoS

text: in

text: a

text: broad

text: fundamental

text: photon

text: energy

text: range

text: between

text: and

text: eV,

text: and

text: present

text: a

text: continuous

text: SH

text: spectrum

text: capturing

text: signatures

text: of

text: both

text: the

text: fundamental

text: A

text: and

text: B

text: excitons

text: in

text: addition

text: to

text: the

text: intense

text: C

text: resonance.

text: Moreover,

text: we

text: interpret

text: our

text: results

text: in

text: terms

text: of

text: the

text: ﬁrst

text: exciton

text: simulation

text: of

text: the

text: SH

text: properties

text: of

text: multi-layered

text: MoS

text: samples,

text: represented

text: by

text: a

text: study

text: of

text: trilayer

text: MoS

text: The

text: good

text: agreement

text: between

text: theory

text: and

text: experiments

text: allows

text: us

text: to

text: establish

text: a

text: connection

text: between

text: the

text: measured

text: spectrum

text: and

text: the

text: underlying

text: electronic

text: structure,

text: in

text: the

text: process

text: elucidating

text: the

text: non-linear

text: excitation

text: of

text: electron-hole

text: pairs.

text: In

text: recent

text: years,

text: two-dimensional

text: transition

text: metal

text: dichalco-

text: bulk

text: MoS

text: crystal.

text: The

text: ﬁrst

text: approach

text: gives

text: access

text: to

text: genides

text: (TMDs),

text: being

text: with

text: band

text: a

text: relatively

text: smooth

text: surface

text: suitable

text: for

text: optical

text: experi-

text: gaps

text: in

text: the

text: eV

text: have

text: been

text: applied

text: as

text: the

text: ments,

text: with

text: little

text: diﬀuse

text: scattering.

text: However,

text: here

text: ﬂake

text: basis

text: for

text: a

text: host

text: of

text: novel

text: two-dimensional

text: optoelectronic

text: thickness

text: and

text: surface

text: coverage

text: are

text: diﬃcult

text: parameters

text: to

text: Accordingly,

text: a

text: great

text: deal

text: of

text: attention

text: has

text: control

text: within

text: the

text: area

text: of

text: a

text: mm-sized

text: laser

text: beam.

text: The

text: been

text: devoted

text: to

text: the

text: linear

text: optical

text: properties

text: of

text: these

text: thickness

text: of

text: the

text: bulk

text: crystal,

text: on

text: the

text: other

text: hand,

text: is

text: im-

text: Also,

text: non-linear

text: optical

text: techniques

text: have

text: material

text: due

text: to

text: absorption.

text: Unfortunately,

text: the

text: rather

text: been

text: demonstrated

text: as

text: particularly

text: powerful

text: probes

text: for

text: rough

text: surface

text: of

text: weakly

text: van

text: der

text: Waals

text: bound

text: bulk

text: TMD

text: the

text: microscopic

text: structure

text: of

text: few-layered

text: TMD

text: crystals,

text: crystals

text: typically

text: make

text: systematic

text: optical

text: studies

text: diﬃ-

text: revealing

text: e.g.,

text: their

text: crystallographic

text: cult.

text: Hence,

text: due

text: to

text: large

text: amounts

text: of

text: diﬀuse

text: scattering,

text: and

text: stacking

text: Moreover,

text: second-harmonic

text: (SH)

text: we

text: only

text: measure

text: reliable

text: results

text: for

text: maximized

text: SH

text: sig-

text: spectroscopy

text: allows

text: for

text: insights

text: into

text: the

text: electronic

text: struc-

text: nals

text: at

text: the

text: C

text: resonance.

text: Thus,

text: the

text: ﬂake

text: samples

text: give

text: ture

text: of

text: TMD

text: exposing

text: edge-localized

text: access

text: to

text: highly

text: resolved

text: SH

text: spectra,

text: whereas

text: absolute

text: or

text: valley-coherent

text: values

text: may

text: be

text: extracted

text: from

text: the

text: bulk

text: sample,

text: which,

text: in

text: While

text: experimentally

text: determined

text: linear

text: response

text: func-

text: turn,

text: yield

text: very

text: little

text: spectral

text: information.

text: By

text: combining

text: tions

text: of

text: various

text: TMDs,

text: dominated

text: by

text: electron-hole

text: pairs

text: the

text: measured

text: SH

text: response

text: from

text: these

text: two

text: sample

text: types,

text: bound

text: by

text: several

text: hundred

text: are

text: reproduced

text: rea-

text: we

text: are

text: able

text: to

text: calibrate

text: the

text: intensity

text: measured

text: from

text: the

text: sonably

text: well

text: at

text: the

text: Bethe-Salpeter

text: equation

text: (BSE)

text: level

text: exfoliated

text: ﬂakes

text: against

text: the

text: bulk

text: crystal

text: measurements,

text: of

text: similar

text: comparisons

text: for

text: non-linear

text: thereby

text: allowing

text: for

text: estimation

text: of

text: absolute

text: parameters.

text: cases

text: are

text: lacking.

text: Although

text: several

text: proposed

text: theoreti-

text: In

text: Figs.

text: and

text: (c),

text: we

text: include

text: microscope

text: images

text: of

text: cal

text: models,

text: based

text: on

text: both

text: and

text: the

text: produced

text: ﬂakes.

text: Atomic

text: force

text: microscopy

text: and

text: pro-

text: approaches,

text: have

text: been

text: published,

text: the

text: SH

text: ﬁler

text: scans

text: conﬁrm

text: most

text: ﬂakes

text: to

text: consist

text: of

text: mono-

text: spectrum

text: of

text: MoS

text: has

text: been

text: investigated

text: layers

text: (MLs).

text: Indeed,

text: no

text: measurable

text: photo-luminescence

text: only

text: in

text: a

text: narrow

text: fundamental

text: photon

text: energy

text: range

text: be-

text: (PL)

text: was

text: observed,

text: regardless

text: of

text: pump

text: wavelength.

text: This

text: tween

text: and

text: eV,

text: probing

text: the

text: so-called

text: C

text: resonance

text: suggests

text: very

text: little

text: ML

text: coverage,

text: since

text: only

text: ML

text: domains

text: also

text: known

text: from

text: linear

text: optics.

text: Theoretical

text: models

text: have

text: posses

text: the

text: direct

text: gap

text: necessary

text: for

text: eﬃcient

text: so

text: far

text: been

text: benchmarked

text: by

text: their

text: ability

text: to

text: reproduce

text: At

text: ﬁrst

text: glance,

text: dipole

text: selection

text: rules

text: prevent

text: genera-

text: this

text: single

text: However,

text: the

text: parabolic

text: disper-

text: tion

text: of

text: even-ordered

text: non-linearities

text: in

text: MoS

text: due

text: to

text: in-

text: sion,

text: and

text: the

text: split

text: valence

text: bands

text: near

text: the

text: K-points

text: of

text: version

text: symmetry,

text: which

text: is

text: only

text: broken

text: in

text: ﬁnite

text: sam-

text: the

text: Brillouin

text: zone,

text: translates

text: into

text: bound

text: electron-hole

text: ples

text: consisting

text: of

text: an

text: odd

text: number

text: of

text: MoS

text: pairs

text: in

text: the

text: BSE

text: picture.

text: These

text: give

text: rise

text: to

text: sharp

text: peaks

text: On

text: the

text: other

text: hand,

text: thick

text: ﬂakes,

text: such

text: as

text: those

text: in

text: the

text: absorption

text: spectrum

text: due

text: to

text: the

text: so-called

text: A

text: and

text: B

text: investigated

text: here,

text: are

text: not

text: expected

text: to

text: display

text: properties

text: at

text: photon

text: energies

text: near

text: eV,

text: and

text: the

text: ques-

text: markedly

text: diﬀerent

text: from

text: bulk

text: samples.

text: From

text: a

text: simplistic

text: tion

text: remains

text: how

text: they

text: might

text: aﬀect

text: the

text: SH

text: response.

text: In

text: point

text: of

text: view,

text: the

text: layer-selective

text: vanishing

text: SH

text: response

text: particular,

text: the

text: relative

text: intensities

text: and

text: lineshapes

text: of

text: such

text: may

text: be

text: understood

text: from

text: the

text: fact

text: that

text: adjacent

text: layers

text: are

text: features

text: are

text: of

text: interest.

text: mirror

text: images

text: of

text: each

text: other

text: in

text: the

text: yz-plane,

text: as

text: depicted

text: In

text: the

text: present

text: letter,

text: we

text: report

text: an

text: experimental

text: op-

text: in

text: Fig.

text: Hence,

text: in

text: the

text: limit

text: of

text: weak

text: inter-layer

text: cou-

text: tical

text: SH

text: spectrum

text: generated

text: from

text: many-layered

text: MoS

text: pling,

text: adjacent

text: layers

text: contribute

text: anti-symmetrically

text: to

text: with

text: fundamental

text: photon

text: energies

text: varied

text: in

text: a

text: broad

text: range

text: the

text: SH

text: response

text: function

text: While

text: this

text: means

text: that

text: between

text: and

text: eV,

text: capturing

text: both

text: the

text: fundamen-

text: the

text: net

text: polarization

text: of

text: two

text: adjacent

text: MoS

text: layers

text: can-

text: tal

text: A

text: and

text: B

text: excitons

text: in

text: addition

text: to

text: the

text: aforementioned

text: cels

text: in

text: the

text: dipole

text: approximation,

text: the

text: spatial

text: variation

text: C

text: resonance.

text: We

text: perform

text: our

text: optical

text: experiments

text: on

text: of

text: the

text: driving

text: ﬁeld

text: makes

text: this

text: cancellation

text: only

text: approx-

text: (i)

text: many-layered

text: MoS

text: ﬂakes

text: exfoliated

text: onto

text: fused

text: sil-

text: imate.

text: Indeed,

text: previous

text: measurements

text: on

text: bulk

text: MoS

text: ica

text: by

text: the

text: well-known

text: scotch

text: tape

text: and

text: (ii)

text: a

text: crystals

text: conﬁrm

text: this,

text: with

text: non-vanishing

text: (although

text: very


text: (cid:82) A RIGOROUS COMPUTATIONAL APPROACH TO LINEAR RESPONSE 11 ∈ ∞ R R ⊂ − where j C ( , +), supp(j) [ 1/2, 1/2] and R j(y)dy = 1. Lemma 4.1 (Properties of K ). (cid:15) ∈ ∈ { } (1) For f Ck, k 0, 1, 2 ≤ || || ≤ || || Var(K f) Var(f) K f f ; (cid:15) (cid:15) Ck Ck ∈ (2) for f C1 || − || ≤ || || K f f ∞ (cid:15) f ; ∈ (cid:15) C1 (3) for f C2 || − || ≤ || || K f f (cid:15) f . (cid:15) C1 C2 (cid:90) Proof. The ﬁrst assertion is a standard property of convolution. For (2), we have | − | | − − − − | ≤ || (cid:48)|| K f(x) f(x) = (cid:15) 1j((cid:15) 1(x y))(f(y) f(x))dy (cid:15) f ∞. (cid:15) T − Since the support of j is contained in [ (cid:15)/2, (cid:15)/2]. To prove (3), observe that ∂ − − − ∂ − − j((cid:15) 1(x y)) = j((cid:15) 1(x y)). ∂x (cid:90) ∂y Therefore, | (cid:48) − (cid:48) | | − − − ∂ − (cid:48) | (K f(x)) f (x) = (cid:15) 1j((cid:15) 1(x y)) f(y)dy f (x) . (cid:15) T ∂y Using integration by parts and the compactness of the support of j, we obtain: | (cid:48) − (cid:48) | ≤ || (cid:48)(cid:48)|| (K f(x)) f (x) (cid:15) f ∞. (cid:15) (cid:3) We now deﬁne a family of operators by setting L L := K . ε ε (cid:90) and deﬁne L L (cid:48) | | (4.1) ˆf := γ( f) γ := j(ξ) ξ dξ. By (4.1) one obtains (see [32]) || − L − L − L || ∀ ∈ lim (cid:15) 1( )f ˆf = 0 f C2([0, 1]); → (cid:15) C1 (cid:15) 0 i.e. assumption (4) of Proposition 2.1 is satisﬁed. For the other assumptions we refer to the following remark. (cid:90) L Remark 4.2. Let h be the ﬁxed point in C2([0, 1]) of . By Lemma 3.1 we have J(h) = 0. Therefore, 1 (cid:48) h dx = 0. 0 ##LTLine## ##LTLine## ##LTLine##

text: A

text: RIGOROUS

text: COMPUTATIONAL

text: APPROACH

text: TO

text: LINEAR

text: RESPONSE

text: R

text: R

text: where

text: j

text: C

text: supp(j)

text: and

text: R

text: j(y)dy

text: Lemma

text: (Properties

text: of

text: K

label: list

text: For

text: f

text: Ck,

text: k

text: Var(K

text: f)

text: Var(f)

text: K

text: f

text: f

text: Ck

text: Ck

text: for

text: f

text: K

text: f

text: f

text: f

text: for

text: f

text: K

text: f

text: f

text: f

text: Proof.

text: The

text: ﬁrst

text: assertion

text: is

text: a

text: standard

text: property

text: of

text: convolution.

text: For

text: we

text: have

text: K

text: f(x)

text: f(x)

text: y))(f(y)

text: f(x))dy

text: f

text: T

text: Since

text: the

text: support

text: of

text: j

text: is

text: contained

text: in

text: To

text: prove

text: observe

text: that

text: y))

text: y)).

text: ∂x

text: ∂y

text: Therefore,

text: (K

text: f(x))

text: f

text: (x)

text: y))

text: f(y)dy

text: f

text: (x)

text: T

text: ∂y

text: Using

text: integration

text: by

text: parts

text: and

text: the

text: compactness

text: of

text: the

text: support

text: of

text: j,

text: we

text: obtain:

text: (K

text: f(x))

text: f

text: (x)

text: f

text: We

text: now

text: deﬁne

text: a

text: family

text: of

text: operators

text: by

text: setting

text: L

text: L

text: K

text: ε

text: ε

text: and

text: deﬁne

text: L

text: L

text: ˆf

text: γ(

text: f)

text: γ

text: j(ξ)

text: ξ

text: dξ.

text: By

text: one

text: obtains

text: (see

text: L

text: L

text: L

text: lim

text: )f

text: ˆf

text: f

text: i.e.

text: assumption

text: of

text: Proposition

text: is

text: satisﬁed.

text: For

text: the

text: other

text: assumptions

text: we

text: refer

text: to

text: the

text: following

text: remark.

text: L

text: Remark

text: Let

text: h

text: be

text: the

text: ﬁxed

text: point

text: in

text: of

text: By

text: Lemma

text: we

text: have

text: J(h)

text: Therefore,

text: h

text: dx

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 14 WAEL BAHSOUN, STEFANO GALATOLO, ISAIA NISOLI, AND XIAOLONG NIU 1.01 0.15 1.005 0.1 1 0.05 0.995 0.2 0.4 0.6 0.8 1 0.99 -0.05 0.985 -0.1 0.98 -0.15 0 0.2 0.4 0.6 0.8 1 (a) A plot of of the invariant density (b) The computed linear response Figure 1. The computed response in the stochastic case, and a plot of the invariant density for the non-perturbed map. (cid:48) To compute the linear response we need to compute an approximation f to γh ; to η do so we use the discretization in Subsection . Let us choose l = 12 and η = 524288 both for the approximation of the density in C2 and the computation of the linear (cid:88) response; we approximate the linear response by 11 L hˆ = f . appr η η i=0 In ﬁgure 1 a plot of the invariant density and of the linear response of the map under stochastic perturbation are presented. We are going to estimate the error using the algorithm developed in the present paper (refer to Theorem 2.2 and the subsequent discussion): || (cid:88)− || ≤ (cid:88) (cid:88) (cid:88) hˆ hˆ ∞ appr ∞ ∗− ∗− ∗− || + L L || || l 1 L − l 1 L || || l 1 L − L || i ˆh ∞ + if i f ∞ + i(f ˆh) ∞. η η η η ∗ i=l i=0 i=0 i=0 In the following subsections we are going to estimate the diﬀerent summands sep- arately. Remark 4.5. To do our validated numerics we used SAGE [39] and the validated numerics packages shipped with it (the interval package is a binding to MPFI [37]), running either on locale computers or on a cloud based version called SageMath- Cloud, https://sagemathcloud.com/. The discretized operators are computed using a rigorous interval Newton method [40], while the estimates for the norm of the discretized operators are made using Scipy [26], with rigorous error bounds on the error made during matrix-vector products. ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: WAEL

text: BAHSOUN,

text: STEFANO

text: GALATOLO,

text: ISAIA

text: NISOLI,

text: AND

text: XIAOLONG

text: NIU

font: default

text: (a)

text: A

label: caption

text: plot

text: of

text: of

text: the

text: invariant

text: density

text: (b)

text: The

text: computed

text: linear

text: response

text: Figure

text: The

text: computed

text: response

text: in

text: the

text: stochastic

text: case,

text: and

text: a

text: plot

text: of

text: the

text: invariant

text: density

text: for

text: the

text: non-perturbed

text: map.

text: To

text: compute

text: the

text: linear

text: response

text: we

text: need

text: to

text: compute

text: an

text: approximation

text: f

text: to

text: γh

text: to

text: η

text: do

text: so

text: we

text: use

text: the

text: discretization

text: in

text: Subsection

text: Let

text: us

text: choose

text: l

text: and

text: η

text: both

text: for

text: the

text: approximation

text: of

text: the

text: density

text: in

text: and

text: the

text: computation

text: of

text: the

text: linear

text: response;

text: we

text: approximate

text: the

text: linear

text: response

text: by

text: L

text: hˆ

text: f

text: appr

text: η

text: η

text: In

text: ﬁgure

text: a

text: plot

text: of

text: the

text: invariant

text: density

text: and

text: of

text: the

text: linear

text: response

text: of

text: the

text: map

text: under

text: stochastic

text: perturbation

text: are

text: presented.

text: We

text: are

text: going

text: to

text: estimate

text: the

text: error

text: using

text: the

text: algorithm

text: developed

text: in

text: the

text: present

text: paper

text: (refer

text: to

text: Theorem

text: and

text: the

text: subsequent

text: discussion):

text: hˆ

text: hˆ

text: appr

text: L

text: L

text: l

text: L

text: l

text: L

text: l

text: L

text: L

text: i

text: ˆh

text: if

text: i

text: f

text: i(f

text: ˆh)

text: η

text: η

text: η

text: η

text: i=l

text: In

text: the

text: following

text: subsections

text: we

text: are

text: going

text: to

text: estimate

text: the

text: diﬀerent

text: summands

text: sep-

text: arately.

text: Remark

text: To

text: do

text: our

text: validated

text: numerics

text: we

text: used

text: SAGE

text: and

text: the

text: validated

text: numerics

text: packages

text: shipped

text: with

text: it

text: (the

text: interval

text: package

text: is

text: a

text: binding

text: to

text: MPFI

text: running

text: either

text: on

text: locale

text: computers

text: or

text: on

text: a

text: cloud

text: based

text: version

text: called

text: SageMath-

text: Cloud,

text: The

text: discretized

text: operators

text: are

text: computed

text: using

text: a

text: rigorous

text: interval

text: Newton

text: method

text: while

text: the

text: estimates

text: for

text: the

text: norm

text: of

text: the

text: discretized

text: operators

text: are

text: made

text: using

text: Scipy

text: with

text: rigorous

text: error

text: bounds

text: on

text: the

text: error

text: made

text: during

text: matrix-vector

text: products.

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: 16 WAEL BAHSOUN, STEFANO GALATOLO, ISAIA NISOLI, AND XIAOLONG NIU Therefore, the error on the response is || − || ≤ ≤ hˆ hˆ ∞ γ(0.0062 + 0.0033 + 0.0051) 0.0137γ. appr 5. Linear response for deterministic perturbations We now consider deterministic perturbations of a C3 expanding circle map6 T . L 0 We use a result of [23], where the structure of the operator ˆ for this case is studied. Let T (x) = T (x) + (cid:15)S(x) + o ((cid:15)); (cid:15) 0 C3 ∈ T where S(x) C3( ) and o ((cid:15)) is a term whose C3 norm goes to 0 faster than (cid:15) C3 as (cid:15) goes to 0. In [23(cid:18), Propo(cid:19)sition 3.1](cid:18)it is pr(cid:19)oved that,(cid:18)under these a(cid:19)ssumptions, the operator · (cid:48) (cid:48) · (cid:48)(cid:48) L −L f S − L f S L · · T ˆf(x) = (cid:48) (x) (cid:48) (x) + f S 0(cid:48) (x). 0 T 0 T 0 (T )2 0 0 0 is such that || − L − L − L || ∀ ∈ lim (cid:15) 1( )f ˆf = 0 f C3 → (cid:15) C1 (cid:15) 0 L where is the Perron-Frobenius operator associated to T . (cid:15) (cid:15) || − Remark 5.1. If we suppose that the perturbation is small in the C2 norm: T || ≤ L L 0 T K(cid:15) it follows that and , satisfy a uniform Lasota-Yorke inequality. (cid:15) C2 (cid:15) This implies that assumption (1) of Proposition 2.1 holds. Moreover, by [18], Sec- tion 6, assumption (2) and (3) of Proposition 2.1 also hold. Hence Proposition 2.1 holds and we have the linear response for these perturbations. − ||h h − || lim (cid:15) hˆ ∞ = 0, → (cid:15) 0 (cid:15) − L − L where hˆ := (Id ) 1 ˆh. ∈ In this example we study a family T , (cid:15) [0, 1) of C3-small deterministic per- (cid:15) turbations. We consider the family T : (cid:15) (cid:15) 1 T (x) = 2x + (cos(4πx) + cos(8πx)) mod 1. (cid:15) 16 4 For (cid:15) = 0 the dynamics is given by the map T (x) = 2x mod 1; 0 6After posting the ﬁrst version of our work on Arxiv, which did not include an example of a deterministic perturbation, Pollicott and Vytnova [36] studied the problem of approximating lin- ear response of given observables for deterministic perturbations. Their approach, which is based on the periodic point structure of the system, requires the maps to be analytic and the observ- ables to have a certain structure. Here we show that our approach also works for deterministic perturbations and it requires only little regularity and no information on the observable. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: WAEL

text: BAHSOUN,

text: STEFANO

text: GALATOLO,

text: ISAIA

text: NISOLI,

text: AND

text: XIAOLONG

text: NIU

text: Therefore,

text: the

text: error

text: on

text: the

text: response

text: is

text: hˆ

text: hˆ

text: appr

label: section

text: Linear

text: response

text: for

text: deterministic

text: perturbations

text: We

text: now

text: consider

text: deterministic

text: perturbations

text: of

text: a

text: expanding

text: circle

text: T

text: L

text: We

text: use

text: a

text: result

text: of

text: where

text: the

text: structure

text: of

text: the

text: operator

text: ˆ

text: for

text: this

text: case

text: is

text: studied.

text: Let

text: T

text: (x)

text: T

text: (x)

text: o

text: T

text: where

text: S(x)

text: and

text: o

text: is

text: a

text: term

text: whose

text: norm

text: goes

text: to

text: faster

text: than

text: as

text: goes

text: to

text: In

text: is

text: these

text: the

text: operator

text: L

text: −L

text: f

text: S

text: L

text: f

text: S

text: L

text: T

text: ˆf(x)

text: (x)

text: (x)

text: f

text: S

text: (x).

text: T

text: T

text: (T

text: is

text: such

text: that

text: L

text: L

text: L

text: lim

text: )f

text: ˆf

text: f

text: L

text: where

text: is

text: the

text: Perron-Frobenius

text: operator

text: associated

text: to

text: T

text: Remark

text: If

text: we

text: suppose

text: that

text: the

text: perturbation

text: is

text: small

text: in

text: the

text: norm:

text: T

text: L

text: L

text: T

text: it

text: follows

text: that

text: and

text: satisfy

text: a

text: uniform

text: Lasota-Yorke

text: inequality.

text: This

text: implies

text: that

text: assumption

text: of

text: Proposition

text: holds.

text: Moreover,

text: by

text: Sec-

text: tion

text: assumption

text: and

text: of

text: Proposition

text: also

text: hold.

text: Hence

text: Proposition

text: holds

text: and

text: we

text: have

text: the

text: linear

text: response

text: for

text: these

text: perturbations.

text: ||h

text: h

text: lim

text: hˆ

text: L

text: L

text: where

text: hˆ

text: (Id

text: ˆh.

text: In

text: this

text: example

text: we

text: study

text: a

text: family

text: T

text: of

text: deterministic

text: per-

text: turbations.

text: We

text: consider

text: the

text: family

text: T

text: T

text: (x)

text: mod

text: For

text: the

text: dynamics

text: is

text: given

text: by

text: the

text: map

text: T

text: (x)

text: mod

text: posting

text: the

text: ﬁrst

text: version

text: of

text: our

text: work

text: on

text: Arxiv,

text: which

text: did

text: not

text: include

text: an

text: example

text: of

text: a

text: deterministic

text: perturbation,

text: Pollicott

text: and

text: Vytnova

text: studied

text: the

text: problem

text: of

text: approximating

text: lin-

text: ear

text: response

text: of

text: given

text: observables

text: for

text: deterministic

text: perturbations.

text: Their

text: approach,

text: which

text: is

text: based

text: on

text: the

text: periodic

text: point

text: structure

text: of

text: the

text: system,

text: requires

text: the

text: maps

text: to

text: be

text: analytic

text: and

text: the

text: observ-

text: ables

text: to

text: have

text: a

text: certain

text: structure.

text: Here

text: we

text: show

text: that

text: our

text: approach

text: also

text: works

text: for

text: deterministic

text: perturbations

text: and

text: it

text: requires

text: only

text: little

text: regularity

text: and

text: no

text: information

text: on

text: the

text: observable.

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text: A RIGOROUS COMPUTATIONAL APPROACH TO LINEAR RESPONSE 31 ∀ ∈ ||L || ≤ || || (6.15) v V, n1(v) λ v δ w 2 w ∈ with λ < 1. Let us consider a starting measure: g V , let us denote g = L 2 0 i+1 n1gi. If the system is as above, putting together the Lasota-Yorke inequality, (6.14) (cid:26) and (6.15) ||L || ≤ || || || || (6.16) ||L || ≤n1||gLi s ||λn11 gi s||+ |B| gi w || || , (cid:26) n1gi w nδ1gi w + δ(C gi s + n1D gi w) ||L || ≤ || || || || ||L || n≤1gi s|| ||Aλn11 gi ||s +|| B gi w|| || . n1gi w λ2 gi w + δ(C gi s + n1D gi w) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) Compacting it in a vector notation, || || || || g (cid:22) Aλn1 B g (6.17) || i+1||s 1 || i||s g δC δn D + λ g i+1 w 1 2 i w (cid:22) ≤ where indicates the component-wise relation (both coordinates are less or (cid:22) (cid:18) (cid:19) equal). The relation can be used because the matrix is positive. The relation (6.17) and the assumptions allow to estimate explicitly the contraction rate, by M Aλn1 B approximating the matrix and its iterations. Let = 1 . (cid:18) (cid:19) (cid:18) (cid:19) δC δn D + λ || || || || 1 2 Consequently, we can bound g and g by a sequence i s i w || || || || g (cid:22) M g || i||s i || 0||s g g i w 0 w which can be computed explicitly. This gives an explicit estimate on the speed of || || || || convergence for the norms and at a given time7. w We need an asymptotic estimation as the one given in (2.2) and in particular an estimation for C and ρ. This can be done by the eigenvalues and eigenvectors of M 1 . Indeed, let the leading eigenvalue be denoted by ρM and a left positive eigen- vector (a, b), such that a + b = 1. For each pair of values (a, b) such that a + b = 1 we can deﬁne a norm || || || || || || g = a g + b g . (cid:18) (cid:19) (a,b) s w We have that || || ||L || ||L || ||L || ≤ · M · g g = a g + b g (a, b) || ||s (a,b) s w g w then ||L || ≤ || || kn1g ρkM g . (a,b) (a,b) By estimating of ρM and the coeﬃcients (a, b) we can have upper estimates for C 1 and ρ. 7Moreover, λn11, λ2 < 1 and the quantities δC, δn1D have a chance to be very small when δ is very small. This is nLot automatic because n1 depend on δ. HoLwever, in the case of piecewise expanding maps, with being an Ulam-type approximation of , as we consider in this paper, δ δn1D can be made suﬃciently small (see [20], Theorem 12 ). ##LTLine##

text: A

text: RIGOROUS

text: COMPUTATIONAL

text: APPROACH

text: TO

text: LINEAR

text: RESPONSE

label: equation

text: ||L

text: v

text: V,

text: λ

text: v

text: δ

text: w

text: w

text: with

text: λ

text: Let

text: us

text: consider

text: a

text: starting

text: measure:

text: g

text: V

text: let

text: us

text: denote

text: g

text: L

text: If

text: the

text: system

text: is

text: as

text: above,

text: putting

text: together

text: the

text: Lasota-Yorke

text: inequality,

text: and

text: ||L

text: ||L

text: s

text: gi

text: s||+

text: |B|

text: gi

text: w

text: w

text: w

text: δ(C

text: gi

text: s

text: gi

text: w)

text: ||L

text: ||L

text: s||

text: gi

text: ||s

text: B

text: gi

text: w||

text: w

text: λ2

text: gi

text: w

text: δ(C

text: gi

text: s

text: gi

text: w)

text: Compacting

text: it

text: in

text: a

text: vector

text: notation,

text: g

text: B

text: g

text: i||s

text: g

text: δC

text: δn

text: D

text: λ

text: g

text: w

text: i

text: w

text: where

text: indicates

text: the

text: component-wise

text: relation

text: (both

text: coordinates

text: are

text: less

text: or

text: equal).

text: The

text: relation

text: can

text: be

text: used

text: because

text: the

text: matrix

text: is

text: positive.

text: The

text: relation

text: and

text: the

text: assumptions

text: allow

text: to

text: estimate

text: explicitly

text: the

text: contraction

text: rate,

text: by

text: M

text: B

text: approximating

text: the

text: matrix

text: and

text: its

text: iterations.

text: Let

text: δC

text: δn

text: D

text: λ

text: Consequently,

text: we

text: can

text: bound

text: g

text: and

text: g

text: by

text: a

text: sequence

text: i

text: s

text: i

text: w

text: g

text: M

text: g

text: i||s

text: i

text: g

text: g

text: i

text: w

text: w

text: which

text: can

text: be

text: computed

text: explicitly.

text: This

text: gives

text: an

text: explicit

text: estimate

text: on

text: the

text: speed

text: of

text: convergence

text: for

text: the

text: norms

text: and

text: at

text: a

text: given

text: w

text: We

text: need

text: an

text: asymptotic

text: estimation

text: as

text: the

text: one

text: given

text: in

text: and

text: in

text: particular

text: an

text: estimation

text: for

text: C

text: and

text: ρ.

text: This

text: can

text: be

text: done

text: by

text: the

text: eigenvalues

text: and

text: eigenvectors

text: of

text: M

text: Indeed,

text: let

text: the

text: leading

text: eigenvalue

text: be

text: denoted

text: by

text: ρM

text: and

text: a

text: left

text: positive

text: eigen-

text: vector

text: (a,

text: b),

text: such

text: that

text: a

text: b

text: For

text: each

text: pair

text: of

text: values

text: (a,

text: b)

text: such

text: that

text: a

text: b

text: we

text: can

text: deﬁne

text: a

text: norm

text: g

text: a

text: g

text: b

text: g

text: (a,b)

text: s

text: w

text: We

text: have

text: that

text: ||L

text: ||L

text: ||L

text: M

text: g

text: g

text: a

text: g

text: b

text: g

text: (a,

text: b)

text: ||s

text: (a,b)

text: s

text: w

text: g

text: w

text: then

text: ||L

text: ρkM

text: g

text: (a,b)

text: (a,b)

text: By

text: estimating

text: of

text: ρM

text: and

text: the

text: coeﬃcients

text: (a,

text: b)

text: we

text: can

text: have

text: upper

text: estimates

text: for

text: C

text: and

text: ρ.

text: λ2

text: and

text: the

text: quantities

text: δC,

text: have

text: a

text: chance

text: to

text: be

text: very

text: small

text: when

text: δ

text: is

text: very

text: small.

text: This

text: is

text: nLot

text: automatic

text: because

text: depend

text: on

text: δ.

text: HoLwever,

text: in

text: the

text: case

text: of

text: piecewise

text: expanding

text: maps,

text: with

text: being

text: an

text: Ulam-type

text: approximation

text: of

text: as

text: we

text: consider

text: in

text: this

text: paper,

text: δ

text: can

text: be

text: made

text: suﬃciently

text: small

text: (see

text: Theorem

text: ##LTLine##


text: 32 WAEL BAHSOUN, STEFANO GALATOLO, ISAIA NISOLI, AND XIAOLONG NIU References [1] Bahsoun, W., Rigorous numerical approximation of escape rates, Nonlinearity, 19 (2006), no. 11, 2529–2542. [2] Bahsoun, W., Bose, C., Invariant Densities and Escape Rates: Rigorous and Computable ∞ Approximations in The L -norm. Nonlinear Analysis, 74 (2011), 4481–4495. [3] Bahsoun, W. and Bose, C. and Duan, Y., Rigorous Pointwise approximations for invariant densities of nonuniformly expanding maps, Ergodic Theory and Dynamical Systems, 35 (2015), 1028–1044. [4] Bahsoun, W., Galatolo, S., Nisoli, I. and Niu, X., Rigorous approximation of diﬀusion coeﬃ- cients for expanding maps. Journal of Statistical Physics V. 163 (2016) Issue 6, pp 1486–1503. [5] Bahsoun, W., Saussol, B., Linear response in the intermittent family: diﬀerentiation in a weighted C0-norm. Available on http://arxiv.org/abs/1512.01080. To appear in Discrete Con- tin. Dynam. Sys. A. [6] Baladi, V., The quest for the ultimate anisotropic Banach space arXiv:1607.00654 [7] Baladi, V., Positive transfer operators and decay of correlations, Advanced Series in Nonlinear Dynamics, 16. World Sci. Publ., NJ, 2000. [8] Baladi, V., On the susceptibility function of piecewise expanding interval maps, Comm. Math. Phy., (2007) 839-859. [9] Baladi, V., Linear response, or else. Available at http://arxiv.org/pdf/1408.2937v1.pdf [10] Baladi, V., Smania, D., Linear response formula for piecewise expanding unimodal maps. Nonlinearity, (2008) 677–711. [11] Baladi, V., Todd, M., Linear response for intermittent maps. To appear in Comm. Math. Phys. [12] Carminati, C., Tiozzo, G., Tuning and plateaux for the entropy of α-continued fractions. Nonlinearity, 26 (2013), 1049–1070. [13] Dellnitz, M., Junge, O., On the approximation of complicated dynamical behavior. SIAM J. Num. Anal. 36 (1999) 491–515. [14] Dolgopyat, D., On diﬀerentiability of SRB states for partially hyperbolic systems. Invent. Math. (2004), 389–449. [15] Froyland, G., Finite approximation of Sinai-Bowen-Ruelle measures for Anosov systems in two dimensions. Random Comput. Dynam. 3 (1995), no. 4, 251–263. [16] Froyland, G., Using Ulam’s method to calculate entropy and other dynamical invariants. Nonlinearity 12 (1999), no. 1, 79–101. [17] Froyland, G., Computer-assisted bounds for the rate of decay of correlations. Comm. Math. Phys. 189 (1997), no. 1, 237-257. [18] Galatolo, S. Statistical properties of dynamics. Introduction to the functional analytic ap- proach . Lecture notes http://www.math.sci.hokudai.ac.jp/ Hokkaido-Pisa/Galatolo.pdf [19] Galatolo, S., Monge, M., Nisoli, I., Poloni F. ”Eﬃcient Recipes for the computation of in- variant measures and other objects related to the statistical properties of dynamics”, Work in Progress [20] Galatolo, S., Nisoli, I., An elementary approach to rigorous approximation of invariant mea- sures. SIAM J. Appl. Dyn. Syst. 13 (2014), no. 2, 958–985. [21] Galatolo, S., Nisoli, I. and Saussol, S., An elementary way to rigorously estimate convergence to equilibrium and escape rates. Journal of Computational Dynamics V. 2, Issue 1, 2015 Pages 51-64 [22] Galatolo, S., Nisoli, I., Rigorous computation of invariant measures and fractal dimension for piecewise hyperbolic maps: 2D Lorenz like maps. To appear in Ergodic Theory and Dynamical Systems doi: 10.1017/etds.2014.145 [23] Galatolo S., Pollicott M, Controlling the statistical properties of expanding maps, preprint, http://arxiv.org/abs/1603.07762 [24] Gou¨ezel, S., Liverani, C., Banach spaces adapted to Anosov systems, Ergodic Theory Dynam. Systems 26 (2006), 189–217. [25] Higham, N., The accuracy of ﬂoating point summation, SIAM Journal on Scientiﬁc Comput- ing, 14 (4) (1993) [26] Jones E., Oliphant T., Peterson P. et Alia, SciPy: Open source scientiﬁc tools for Python,2001, http://www.scipy.org/

text: WAEL

text: BAHSOUN,

text: STEFANO

text: GALATOLO,

text: ISAIA

text: NISOLI,

text: AND

text: XIAOLONG

text: NIU

text: References

label: reference

text: Bahsoun,

text: W.,

text: Rigorous

text: numerical

text: approximation

text: of

text: escape

text: rates,

text: Nonlinearity,

text: no.

text: Bahsoun,

text: W.,

text: Bose,

text: C.,

text: Invariant

text: Densities

text: and

text: Escape

text: Rates:

text: Rigorous

text: and

text: Computable

text: Approximations

text: in

text: The

text: L

text: -norm.

text: Nonlinear

text: Analysis,

text: Bahsoun,

text: W.

text: and

text: Bose,

text: C.

text: and

text: Duan,

text: Y.,

text: Rigorous

text: Pointwise

text: approximations

text: for

text: invariant

text: densities

text: of

text: nonuniformly

text: expanding

text: maps,

text: Ergodic

text: Theory

text: and

text: Dynamical

text: Systems,

text: Bahsoun,

text: W.,

text: Galatolo,

text: S.,

text: Nisoli,

text: I.

text: and

text: Niu,

text: X.,

text: Rigorous

text: approximation

text: of

text: diﬀusion

text: coeﬃ-

text: cients

text: for

text: expanding

text: maps.

text: Journal

text: of

text: Statistical

text: Physics

text: V.

text: Issue

text: pp

text: Bahsoun,

text: W.,

text: Saussol,

text: B.,

text: Linear

text: response

text: in

text: the

text: intermittent

text: family:

text: diﬀerentiation

text: in

text: a

text: weighted

text: Available

text: on

text: To

text: appear

text: in

text: Discrete

text: Con-

text: tin.

text: Dynam.

text: Sys.

text: A.

text: Baladi,

text: V.,

text: The

text: quest

text: for

text: the

text: ultimate

text: anisotropic

text: Banach

text: space

text: Baladi,

text: V.,

text: Positive

text: transfer

text: operators

text: and

text: decay

text: of

text: correlations,

text: Advanced

text: Series

text: in

text: Nonlinear

text: Dynamics,

text: World

text: Sci.

text: Publ.,

text: NJ,

text: Baladi,

text: V.,

text: On

text: the

text: susceptibility

text: function

text: of

text: piecewise

text: expanding

text: interval

text: maps,

text: Comm.

text: Math.

text: Phy.,

text: Baladi,

text: V.,

text: Linear

text: response,

text: or

text: else.

text: Available

text: at

text: Baladi,

text: V.,

text: Smania,

text: D.,

text: Linear

text: response

text: formula

text: for

text: piecewise

text: expanding

text: unimodal

text: maps.

text: Nonlinearity,

text: Baladi,

text: V.,

text: Todd,

text: M.,

text: Linear

text: response

text: for

text: intermittent

text: maps.

text: To

text: appear

text: in

text: Comm.

text: Math.

text: Phys.

text: Carminati,

text: C.,

text: Tiozzo,

text: G.,

text: Tuning

text: and

text: plateaux

text: for

text: the

text: entropy

text: of

text: α-continued

text: fractions.

text: Nonlinearity,

text: Dellnitz,

text: M.,

text: Junge,

text: O.,

text: On

text: the

text: approximation

text: of

text: complicated

text: dynamical

text: behavior.

text: SIAM

text: J.

text: Num.

text: Anal.

text: Dolgopyat,

text: D.,

text: On

text: diﬀerentiability

text: of

text: SRB

text: states

text: for

text: partially

text: hyperbolic

text: systems.

text: Invent.

text: Math.

text: Froyland,

text: G.,

text: Finite

text: approximation

text: of

text: Sinai-Bowen-Ruelle

text: measures

text: for

text: Anosov

text: systems

text: in

text: two

text: dimensions.

text: Random

text: Comput.

text: Dynam.

text: no.

text: Froyland,

text: G.,

text: Using

text: Ulam’s

text: method

text: to

text: calculate

text: entropy

text: and

text: other

text: dynamical

text: invariants.

text: Nonlinearity

text: no.

text: Froyland,

text: G.,

text: Computer-assisted

text: bounds

text: for

text: the

text: rate

text: of

text: decay

text: of

text: correlations.

text: Comm.

text: Math.

text: Phys.

text: no.

text: Galatolo,

text: S.

text: Statistical

text: properties

text: of

text: dynamics.

text: Introduction

text: to

text: the

text: functional

text: analytic

text: ap-

text: proach

text: Lecture

text: notes

text: Hokkaido-Pisa/Galatolo.pdf

text: Galatolo,

text: S.,

text: Monge,

text: M.,

text: Nisoli,

text: I.,

text: Poloni

text: F.

text: ”Eﬃcient

text: Recipes

text: for

text: the

text: computation

text: of

text: in-

text: variant

text: measures

text: and

text: other

text: objects

text: related

text: to

text: the

text: statistical

text: properties

text: of

text: dynamics”,

text: Work

text: in

text: Progress

text: Galatolo,

text: S.,

text: Nisoli,

text: I.,

text: An

text: elementary

text: approach

text: to

text: rigorous

text: approximation

text: of

text: invariant

text: mea-

text: sures.

text: SIAM

text: J.

text: Appl.

text: Dyn.

text: Syst.

text: no.

text: Galatolo,

text: S.,

text: Nisoli,

text: I.

text: and

text: Saussol,

text: S.,

text: An

text: elementary

text: way

text: to

text: rigorously

text: estimate

text: convergence

text: to

text: equilibrium

text: and

text: escape

text: rates.

text: Journal

text: of

text: Computational

text: Dynamics

text: V.

text: Issue

text: Pages

text: Galatolo,

text: S.,

text: Nisoli,

text: I.,

text: Rigorous

text: computation

text: of

text: invariant

text: measures

text: and

text: fractal

text: dimension

text: for

text: piecewise

text: hyperbolic

text: maps:

text: Lorenz

text: like

text: maps.

text: To

text: appear

text: in

text: Ergodic

text: Theory

text: and

text: Dynamical

text: Systems

text: doi:

text: Galatolo

text: S.,

text: Pollicott

text: M,

text: Controlling

text: the

text: statistical

text: properties

text: of

text: expanding

text: maps,

text: preprint,

text: Gou¨ezel,

text: S.,

text: Liverani,

text: C.,

text: Banach

text: spaces

text: adapted

text: to

text: Anosov

text: systems,

text: Ergodic

text: Theory

text: Dynam.

text: Systems

text: Higham,

text: N.,

text: The

text: accuracy

text: of

text: ﬂoating

text: point

text: summation,

text: SIAM

text: Journal

text: on

text: Scientiﬁc

text: Comput-

text: ing,

text: Jones

text: E.,

text: Oliphant

text: T.,

text: Peterson

text: P.

text: et

text: Alia,

text: SciPy:

text: Open

text: source

text: scientiﬁc

text: tools

text: for


text: where ρ denotes the spectral radius and L is the Leja interpolation polynomial of m,c degree m on the symmetric interval [−c, c], see (4) and (5). Note that Ω is open in m,c C n×n and contains a neighborhood of 0 for m ≥ 2, since ξ = 0. For X ∈ Ω we deﬁne 2 m,c the function h (X) = log(e−XL (X)). (7) m+1,c m,c Here log denotes the principal logarithm [8, Thm. 1.31]. As h (X) commutes with m+1,c X we get Lm,c(X) = eX+hm+1,c(X) for X ∈ Ωm,c. By introducing a scaling factor s such C that s−1A ∈ Ω for A ∈ n×n we obtain m,c − L (s−1A)s = eA+shm+1,c(s 1A) =: eA+∆A, (8) m,c where ∆A = sh (s−1A) is the backward error resulting from the approximation m+1,c of eA by the Leja method L (s−1A)s. m,c On the set Ω the function h has a series expansion of the form m,c m+1,c (cid:88) ∞ h (X) = a Xk. (9) m+1,c k,c k=0 In order to bound the backward error by a speciﬁed tolerance tol we need to ensure (cid:107)∆A(cid:107) (cid:107)h (s−1A)(cid:107) m+1,c = ≤ tol (10) (cid:107)A(cid:107) (cid:107)s−1A(cid:107) for a given matrix norm. As a consequence of this bound, one can select the scaling factor s (always a positive integer) such that (10) is satisﬁed for a chosen degree of interpolation m. In contrast to [1] we have the endpoint c of the interpolation interval as an additional parameter to m and s for our analysis. In the following, we are going to introduce two diﬀerent ways of bounding the backward error. The ﬁrst one is closely related to the analysis presented in [1]. We study how (10) can be used to select the interpolation parameters when we perform a power-series expansion of the backward error. In the second approach we consider a contour integral formulation of h and estimate the m+1,c error along the contour. 3.1 Power-series expansion of the backward error In this section we investigate bounds on the backward error represented by h . m+1,c The analysis is based on a power-series expansion of h . m+1,c 5 ##LTLine## ##LTLine##

text: where

text: ρ

text: denotes

text: the

text: spectral

text: radius

text: and

text: L

text: is

text: the

text: Leja

text: interpolation

text: polynomial

text: of

text: m,c

text: degree

text: m

text: on

text: the

text: symmetric

text: interval

text: [−c,

text: c],

text: see

text: and

text: Note

text: that

text: Ω

text: is

text: open

text: in

text: m,c

text: C

text: n×n

text: and

text: contains

text: a

text: neighborhood

text: of

text: for

text: m

text: since

text: ξ

text: For

text: X

text: Ω

text: we

text: deﬁne

text: m,c

text: the

text: function

text: h

text: (X)

text: log(e−XL

text: (X)).

text: m,c

text: Here

text: log

text: denotes

text: the

text: principal

text: logarithm

text: Thm.

text: As

text: h

text: (X)

text: commutes

text: with

text: X

text: we

text: get

text: Lm,c(X)

text: for

text: X

text: Ωm,c.

text: By

text: introducing

text: a

text: scaling

text: factor

text: s

text: such

text: C

text: that

text: Ω

text: for

text: A

text: n×n

text: we

text: obtain

text: m,c

text: L

text: eA+∆A,

text: m,c

text: where

text: ∆A

text: sh

text: is

text: the

text: backward

text: error

text: resulting

text: from

text: the

text: approximation

text: of

text: eA

text: by

text: the

text: Leja

text: method

text: L

text: m,c

text: On

text: the

text: set

text: Ω

text: the

text: function

text: h

text: has

text: a

text: series

text: expansion

text: of

text: the

text: form

text: m,c

text: h

text: (X)

text: a

text: Xk.

text: k,c

text: In

text: order

text: to

text: bound

text: the

text: backward

text: error

text: by

text: a

text: speciﬁed

text: tolerance

text: tol

text: we

text: need

text: to

text: ensure

text: tol

text: for

text: a

text: given

text: matrix

text: norm.

text: As

text: a

text: consequence

text: of

text: this

text: bound,

text: one

text: can

text: select

text: the

text: scaling

text: factor

text: s

text: (always

text: a

text: positive

text: integer)

text: such

text: that

text: is

text: satisﬁed

text: for

text: a

text: chosen

text: degree

text: of

text: interpolation

text: m.

text: In

text: contrast

text: to

text: we

text: have

text: the

text: endpoint

text: c

text: of

text: the

text: interpolation

text: interval

text: as

text: an

text: additional

text: parameter

text: to

text: m

text: and

text: s

text: for

text: our

text: analysis.

text: In

text: the

text: following,

text: we

text: are

text: going

text: to

text: introduce

text: two

text: diﬀerent

text: ways

text: of

text: bounding

text: the

text: backward

text: error.

text: The

text: ﬁrst

text: one

text: is

text: closely

text: related

text: to

text: the

text: analysis

text: presented

text: in

text: We

text: study

text: how

text: can

text: be

text: used

text: to

text: select

text: the

text: interpolation

text: parameters

text: when

text: we

text: perform

text: a

text: power-series

text: expansion

text: of

text: the

text: backward

text: error.

text: In

text: the

text: second

text: approach

text: we

text: consider

text: a

text: contour

text: integral

text: formulation

text: of

text: h

text: and

text: estimate

text: the

text: error

text: along

text: the

text: contour.

text: Power-series

text: expansion

text: of

text: the

text: backward

text: error

text: In

text: this

text: section

text: we

text: investigate

text: bounds

text: on

text: the

text: backward

text: error

text: represented

text: by

text: h

text: The

text: analysis

text: is

text: based

text: on

text: a

text: power-series

text: expansion

text: of

text: h

text: ##LTLine##

text: ##LTLine##


text: We choose θ according to the third option, which favors normal matrices where all m,c eigenvalues lie in an interval. More precisely, we select θ = min{c: θ = c}. (15) m m,c This means that θ is the ﬁrst intersection point of the graph (c, θ ) with the m m,c diagonal, cf. Figure 1. The behavior of the curves c (cid:55)→ (c, θ ) is not unexpected. Let us consider the m,c approximation of eθ by L (θ) for θ ≥ 0 and s = 1. Then h˜ (θ)/θ is an overestimate m,c m+1,c of the relative backward error h (θ)/θ. The value θ represents the maximum m+1,c m,c value for which Lm,c(θm,c) is an acceptable approximation of eθm,c. The value θm,0 corresponds to interpolation at a set of conﬂuent points at c = 0, i.e. the truncated Taylor series approximation. If we slightly increase the interpolation interval [−c, c], about half of the interpolation points lie in [0, c]. Therefore, it is possible to have an acceptable interpolation up to θ ≥ θ . If we continue to increase c, the mutual m,c m,0 distance between interpolation points increases as well. When the interval gets too large, the number of interpolation points is too small to achieve the desired accuracy and the value θ starts to decrease. m,c Figure 1 shows that θ ≥ θ for all 0 ≤ c ≤ θ . Therefore, we can safely interpolate m,c m m with degree m for all intervals [−c, c] with 0 ≤ c ≤ θ . We will use this fact in Section . m We compute θ by a combination of two root ﬁnding algorithms based on Newton’s m method. The inner equation (13) for computing θ is solved by an exact Newton m,c iteration with an accuracy of 10−20. The outer equation θ = c is also solved by m,c Newton’s method. This time, however, the necessary derivative is approximated by numerical diﬀerentiation. We compute the result up to an accuracy of 10−18. The resulting values are truncated to 16 digits (double precision) and used henceforth as the θ values. In Table 1 we listed selected (rounded) values of θ for various m and certain m m tolerances. We next describe the choice of the parameters used in our implementation. For each m the optimal value of the integer s is given by s = (cid:100)(cid:107)A(cid:107)/θ (cid:101). (16) m We recall that we have chosen m = 100. The cost of the interpolation is domi- max nated by the number of matrix-vector products computed during Newton interpolation. Therefore, the cost is at most C (A) := sm = m(cid:100)(cid:107)A(cid:107)/θ (cid:101), (17) m m 8

text: We

text: choose

text: θ

text: according

text: to

text: the

text: third

text: option,

text: which

text: favors

text: normal

text: matrices

text: where

text: all

text: m,c

text: eigenvalues

text: lie

text: in

text: an

text: interval.

text: More

text: precisely,

text: we

text: select

text: θ

text: min{c:

text: θ

text: c}.

text: m

text: m,c

text: This

text: means

text: that

text: θ

text: is

text: the

text: ﬁrst

text: intersection

text: point

text: of

text: the

text: graph

text: (c,

text: θ

text: with

text: the

text: m

text: m,c

text: diagonal,

text: cf.

text: Figure

text: The

text: behavior

text: of

text: the

text: curves

text: c

text: (c,

text: θ

text: is

text: not

text: unexpected.

text: Let

text: us

text: consider

text: the

text: m,c

text: approximation

text: of

text: eθ

text: by

text: L

text: (θ)

text: for

text: θ

text: and

text: s

text: Then

text: h˜

text: (θ)/θ

text: is

text: an

text: overestimate

text: m,c

text: of

text: the

text: relative

text: backward

text: error

text: h

text: (θ)/θ.

text: The

text: value

text: θ

text: represents

text: the

text: maximum

text: m,c

text: value

text: for

text: which

text: Lm,c(θm,c)

text: is

text: an

text: acceptable

text: approximation

text: of

text: eθm,c.

text: The

text: value

text: corresponds

text: to

text: interpolation

text: at

text: a

text: set

text: of

text: conﬂuent

text: points

text: at

text: c

text: i.e.

text: the

text: truncated

text: Taylor

text: series

text: approximation.

text: If

text: we

text: slightly

text: increase

text: the

text: interpolation

text: interval

text: [−c,

text: c],

text: about

text: half

text: of

text: the

text: interpolation

text: points

text: lie

text: in

text: c].

text: Therefore,

text: it

text: is

text: possible

text: to

text: have

text: an

text: acceptable

text: interpolation

text: up

text: to

text: θ

text: θ

text: If

text: we

text: continue

text: to

text: increase

text: c,

text: the

text: mutual

text: m,c

text: distance

text: between

text: interpolation

text: points

text: increases

text: as

text: well.

text: When

text: the

text: interval

text: gets

text: too

text: large,

text: the

text: number

text: of

text: interpolation

text: points

text: is

text: too

text: small

text: to

text: achieve

text: the

text: desired

text: accuracy

text: and

text: the

text: value

text: θ

text: starts

text: to

text: decrease.

text: m,c

text: Figure

text: shows

text: that

text: θ

text: θ

text: for

text: all

text: c

text: θ

text: Therefore,

text: we

text: can

text: safely

text: interpolate

text: m,c

text: m

text: m

text: with

text: degree

text: m

text: for

text: all

text: intervals

text: [−c,

text: c]

text: with

text: c

text: θ

text: We

text: will

text: use

text: this

text: fact

text: in

text: Section

text: m

text: We

text: compute

text: θ

text: by

text: a

text: combination

text: of

text: two

text: root

text: ﬁnding

text: algorithms

text: based

text: on

text: Newton’s

text: m

text: method.

text: The

text: inner

text: equation

text: for

text: computing

text: θ

text: is

text: solved

text: by

text: an

text: exact

text: Newton

text: m,c

text: iteration

text: with

text: an

text: accuracy

text: of

text: The

text: outer

text: equation

text: θ

text: c

text: is

text: also

text: solved

text: by

text: m,c

text: Newton’s

text: method.

text: This

text: time,

text: however,

text: the

text: necessary

text: derivative

text: is

text: approximated

text: by

text: numerical

text: diﬀerentiation.

text: We

text: compute

text: the

text: result

text: up

text: to

text: an

text: accuracy

text: of

text: The

text: resulting

text: values

text: are

text: truncated

text: to

text: digits

text: (double

text: precision)

text: and

text: used

text: henceforth

text: as

text: the

text: θ

text: values.

text: In

text: Table

text: we

text: listed

text: selected

text: (rounded)

text: values

text: of

text: θ

text: for

text: various

text: m

text: and

text: certain

text: m

text: m

text: tolerances.

text: We

text: next

text: describe

text: the

text: choice

text: of

text: the

text: parameters

text: used

text: in

text: our

text: implementation.

text: For

text: each

text: m

text: the

text: optimal

text: value

text: of

text: the

text: integer

text: s

text: is

text: given

text: by

text: s

text: m

text: We

text: recall

text: that

text: we

text: have

text: chosen

text: m

text: The

text: cost

text: of

text: the

text: interpolation

text: is

text: domi-

text: max

text: nated

text: by

text: the

text: number

text: of

text: matrix-vector

text: products

text: computed

text: during

text: Newton

text: interpolation.

text: Therefore,

text: the

text: cost

text: is

text: at

text: most

text: C

text: (A)

text: sm

text: m

text: m


text: ﬂexibility as an ellipse can vary its shape from an interval to a circle. By this we can better capture the distribution of the eigenvalues of a matrix A than by simply taking the number (cid:107)A(cid:107). Again our aim is to ﬁnd a bound for h . Due to the fact that zero is among the m+1,c interpolation points for m ≥ 2 it is convenient to write h as m+1,c h (X) = Xg (X). (19) m+1,c m+1,c For ﬁxed ε > 0 we can rewrite (10) in the Euclidean norm as (cid:107)∆A(cid:107) (cid:107)h (s−1A)(cid:107) 2 m+1,c 2 = (cid:107)A(cid:107) (cid:13) (cid:107)(cid:90)s−1A(cid:107) (cid:13) 2 (cid:13) 2 (cid:13) ≤ (cid:13)(cid:13)(cid:107)gm+1,c(s−1A)(cid:107)2 (cid:13)(cid:13) (20) 1 = g (z)(zI − s−1A)−1 dz m+1,c 2πi Γ 2 L(Γ) ≤ (cid:107)g (cid:107) . m+1,c Γ 2πε Here Γ = ∂K denotes the boun(cid:8)dary(cid:13) of the dom(cid:13)ain K th(cid:9)at contains Λε(s−1A), the ε-pseudospectrum of s−1A. The ε-pseu(cid:13)dospectrum (cid:13)of a matrix X is given by Λ (X) = z : (zI − X)−1 ≥ ε−1 . ε 2 The length of Γ is denoted by L(Γ) and (cid:107) · (cid:107) is the maximum norm on Γ. For given Γ m, c, and K the last term in (20) can be computed in high precision. We use 300 digits and sample the contour in any performed computation. For the time being, let us ﬁx m. Furthermore, we assume that Γ is an ellipse, with focal interval equal to the interpolation interval [−c, c], and its convex hull K encloses Λ (s−1A). As a result of these assumptions, there is not only one ellipse but rather a ε two parameter family of ellipses Γ . The parameters are the right endpoint c of the γ,c interpolation (focal) interval and the capacity γ of the ellipse, that is the half sum of the semi-axes. In the following we describe how to extract a discrete set of ellipses from the two parameter family of ellipses. This discrete set can then be stored and used in the algorithm. We start by reducing the two parameter family of ellipses Γ to a one parameter γ,c family, only depending on the focal interval [−c, c]. For ﬁxed c we have a family of confocal ellipses that are desc(cid:26)ribed by (cid:27) C c2 Γ = z ∈ : z = γw + , |w| = 1 . (21) γ,c 4γw 10 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ﬂexibility

text: as

text: an

text: ellipse

text: can

text: vary

text: its

text: shape

text: from

text: an

text: interval

text: to

text: a

text: circle.

text: By

text: this

text: we

text: can

text: better

text: capture

text: the

text: distribution

text: of

text: the

text: eigenvalues

text: of

text: a

text: matrix

text: A

text: than

text: by

text: simply

text: taking

text: the

text: number

text: Again

text: our

text: aim

text: is

text: to

text: ﬁnd

text: a

text: bound

text: for

text: h

text: Due

text: to

text: the

text: fact

text: that

text: zero

text: is

text: among

text: the

text: interpolation

text: points

text: for

text: m

text: it

text: is

text: convenient

text: to

text: write

text: h

text: as

text: h

text: (X)

text: Xg

text: (X).

text: For

text: ﬁxed

text: ε

text: we

text: can

text: rewrite

text: in

text: the

text: Euclidean

text: norm

text: as

text: g

text: (z)(zI

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