text: of this kind of “approximate interpolation” is controlled by the ε-truncation of the QTT rank parameters. In the present paper, we apply this approximation technique to long N-vectors representing the DOS sampled over the ﬁne representation grid Ω . (cid:48)h (cid:48) ∈ N The QTT-type approximation of an N-vector with N = qd , d , q = 2, 3, ..., is deﬁned as the tensor decomposition (approximation) in the TT or canonical format applied (cid:78) (cid:48) to a tensor obtained by the folding (reshaping) of the initial vector to a d -dimensional × · · · × q q data array. The latter is thought of as an element of the multi-dimensional Q (cid:48) K K ∈ {R C} (cid:48) quantized tensor space (cid:48) = d q, , , and d is the auxiliary dimension q,d j=1 (virtual, in contrary to the real space dimension d) parameter that measures the depth of ∈ R the quantization transform. A vector x = [x ] ∈ N, is reshaped to its multi-dimensional i i I Q quantized image in (cid:48) by q-adic folding, q,d F → ∈ Q { } (cid:48) : x X = [x(j)] (cid:48), j = j , . . . , j (cid:48) , q,d q,d 1 d ∈ { } (cid:48) with j 1, . . . , q for ν = 1, . . . , d . Here, for ﬁxed i, we have x(j) := x , and j = j (i) ν − i ν ν is deﬁned via q-coding, j 1 = C− , such that the coeﬃcients C− are found from the ν− 1+ν 1+ν q-adic representation of i 1 (binary coding for q = 2), (cid:88) (cid:48) − · · · (cid:48)− ≡ d − − i 1 = C + C q1 + + C (cid:48)− qd 1 (j 1)qν 1. 0 1 d 1 ν ν=1 Assuming that for the rank-r TT approximation of the quantized image X there holds ≤ (cid:48) r r, k = 1, . . . , d , the complexity of this representation for the tensor X reduces to the k logarithmic scale (cid:28) qr2 log N N. q The computational gain of the QTT approximation is justiﬁed by the perfect rank decom- position proven in [?] for a wide class of function-related tensors obtained by sampling the corresponding functions over a uniform or properly reﬁned grid. This class of functions includes complex exponentials, trigonometric functions, polynomials and Chebyshev poly- nomials, as well as wavelet basis functions. We refer to [?, ?, ?, ?] for further results on QTT approximation of functional vectors and various applications. In estimating the numerical complexity we use the average QTT rank further denoted (cid:118) (cid:117) by rqtt calculated as follows, (cid:117) (cid:88) (cid:116) − d 1 1 r = − r2, (4.1) qtt d 1 k k=1 where the QTT ranks r are the TT ranks of the quantized image X of a vector. k As an example we present the basic results on the rank-1 (resp. rank-2) QTT represen- (cid:48) tation (with q = 2) of the exponential (resp. trigonometric) vectors [?]. For given N = 2d , ∈ C { − } and z , the exponential N-vector, z := z = zn 1 N , can be reshaped by the dyadic ⊗ (cid:48) n n=1 folding to the rank-1, 2 d -tensor, F (cid:55)→ ⊗ (cid:48) − ∈ Q (cid:48) : z Z = d [1 z2p 1]T (cid:48). (4.2) 2,d p=1 2,d 16 ##LTLine## ##LTLine##

text: of

text: this

text: kind

text: of

text: “approximate

text: interpolation”

text: is

text: controlled

text: by

text: the

text: ε-truncation

text: of

text: the

text: QTT

text: rank

text: parameters.

text: In

text: the

text: present

text: paper,

text: we

text: apply

text: this

text: approximation

text: technique

text: to

text: long

text: N-vectors

text: representing

text: the

text: DOS

text: sampled

text: over

text: the

text: ﬁne

text: representation

text: grid

text: Ω

text: N

text: The

text: QTT-type

text: approximation

text: of

text: an

text: N-vector

text: with

text: N

text: qd

text: d

text: q

text: is

text: deﬁned

text: as

text: the

text: tensor

text: decomposition

text: (approximation)

text: in

text: the

text: TT

text: or

text: canonical

text: format

text: applied

text: to

text: a

text: tensor

text: obtained

text: by

text: the

text: folding

text: (reshaping)

text: of

text: the

text: initial

text: vector

text: to

text: a

text: d

text: -dimensional

text: q

text: q

text: data

text: array.

text: The

text: latter

text: is

text: thought

text: of

text: as

text: an

text: element

text: of

text: the

text: multi-dimensional

text: Q

text: K

text: K

text: {R

text: C}

text: quantized

text: tensor

text: space

text: d

text: q,

text: and

text: d

text: is

text: the

text: auxiliary

text: dimension

text: q,d

text: (virtual,

text: in

text: contrary

text: to

text: the

text: real

text: space

text: dimension

text: d)

text: parameter

text: that

text: measures

text: the

text: depth

text: of

text: R

text: the

text: quantization

text: transform.

text: A

text: vector

text: x

text: [x

text: N,

text: is

text: reshaped

text: to

text: its

text: multi-dimensional

text: i

text: i

text: I

text: Q

text: quantized

text: image

text: in

text: by

text: q-adic

text: folding,

text: q,d

text: F

text: Q

text: x

text: X

text: [x(j)]

text: j

text: j

text: j

text: q,d

text: q,d

text: d

text: with

text: j

text: q

text: for

text: ν

text: d

text: Here,

text: for

text: ﬁxed

text: i,

text: we

text: have

text: x(j)

text: x

text: and

text: j

text: j

text: (i)

text: ν

text: i

text: ν

text: ν

text: is

text: deﬁned

text: via

text: q-coding,

text: j

text: C−

text: such

text: that

text: the

text: coeﬃcients

text: C−

text: are

text: found

text: from

text: the

text: ν−

text: q-adic

text: representation

text: of

text: i

text: (binary

text: coding

text: for

text: q

text: d

text: i

text: C

text: C

text: C

text: qd

text: (j

text: d

text: ν

text: Assuming

text: that

text: for

text: the

text: rank-r

text: TT

text: approximation

text: of

text: the

text: quantized

text: image

text: X

text: there

text: holds

text: r

text: r,

text: k

text: d

text: the

text: complexity

text: of

text: this

text: representation

text: for

text: the

text: tensor

text: X

text: reduces

text: to

text: the

text: k

text: logarithmic

text: scale

text: log

text: N

text: N.

text: q

text: The

text: computational

text: gain

text: of

text: the

text: QTT

text: approximation

text: is

text: justiﬁed

text: by

text: the

text: perfect

text: rank

text: decom-

text: position

text: proven

text: in

text: for

text: a

text: wide

text: class

text: of

text: function-related

text: tensors

text: obtained

text: by

text: sampling

text: the

text: corresponding

text: functions

text: over

text: a

text: uniform

text: or

text: properly

text: reﬁned

text: grid.

text: This

text: class

text: of

text: functions

text: includes

text: complex

text: exponentials,

text: trigonometric

text: functions,

text: polynomials

text: and

text: Chebyshev

text: poly-

text: nomials,

text: as

text: well

text: as

text: wavelet

text: basis

text: functions.

text: We

text: refer

text: to

text: for

text: further

text: results

text: on

text: QTT

text: approximation

text: of

text: functional

text: vectors

text: and

text: various

text: applications.

text: In

text: estimating

text: the

text: numerical

text: complexity

text: we

text: use

text: the

text: average

text: QTT

text: rank

text: further

text: denoted

text: by

text: rqtt

text: calculated

text: as

text: follows,

text: d

text: r

text: qtt

text: d

text: k

text: where

text: the

text: QTT

text: ranks

text: r

text: are

text: the

text: TT

text: ranks

text: of

text: the

text: quantized

text: image

text: X

text: of

text: a

text: vector.

text: k

text: As

text: an

text: example

text: we

text: present

text: the

text: basic

text: results

text: on

text: the

text: (resp.

text: QTT

text: represen-

text: tation

text: (with

text: q

text: of

text: the

text: exponential

text: (resp.

text: trigonometric)

text: vectors

text: For

text: given

text: N

text: C

text: and

text: z

text: the

text: exponential

text: N-vector,

text: z

text: z

text: zn

text: N

text: can

text: be

text: reshaped

text: by

text: the

text: dyadic

text: n

text: folding

text: to

text: the

text: d

text: -tensor,

text: F

text: Q

text: z

text: Z

text: d

text: ##LTLine##

text: ##LTLine##


text: DOS via Lorentzians DOS via Lorentzians 2 5 exact DoS TDA exact DoS TDA QTT approx. QTT approx. 4 1.5 3 1 2 0.5 1 0 0 200 400 600 800 1000 5 10 15 20 25 30 eV eV Figure 4.3: DOS for H O molecule via Lorentzians (blue) and its QTT approximation (red) (left). 2 Zoom on the left most part of the spectrum (right). ε=0.04, r = 10.5. QTT DOS via Lorentzians DOS via Lorentzians 20 exact DoS TDA exact DoS TDA 15 QTT approx. QTT approx. 15 10 10 5 5 0 0 -5 0.5 1 1.5 2 200 400 600 800 eV 104 eV Figure 4.4: DOS for Glycine amino acid via Lorentzians (blue) and its QTT approximation (red), left; (left). Right: zoom of the ﬁrst part of the spectrum. ε=0.04, r = 16. QTT A comparison of Figures ?? and ?? indicates that the Lorentzian based DOS blurring is slightly smoother than Gaussian blurring. The moderate size of the QTT ranks in Figures ?? and ?? clearly shows the potentials of the QTT ε-interpolation for modeling the DOS of large lattice type clusters. We observe several gaps in the spectral densities, see Figure ??, ?? and ?? indicating that polynomial, rational or trigonometric interpolation can be applied only to some energy sub- intervals, but not in the whole interval [a, b]. Remarkably, the QTT approximant resolves well the DOS function in the whole energy interval including nearly zero values within the spectral gaps (hardly possible for polynomial/rational based interpolation). 4.3 Numerics for the QTT interpolation to the DOS function In the previous section we demonstrated that the QTT tensor approximation provides good resolution for the DOS function calculated for a number of molecules. In what follows, 19 ##LTFigure## ##LTFigure## ##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## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: DOS

label: paragraph

font: default

text: via

text: Lorentzians

text: DOS

text: via

text: Lorentzians

text: exact

text: DoS

text: TDA

text: exact

text: DoS

text: TDA

text: QTT

text: approx.

text: QTT

text: approx.

text: eV

text: eV

text: Figure

text: DOS

label: caption

text: for

text: H

text: O

text: molecule

text: via

text: Lorentzians

text: (blue)

text: and

text: its

text: QTT

text: approximation

text: (red)

text: (left).

text: Zoom

text: on

text: the

text: left

text: most

text: part

text: of

text: the

text: spectrum

text: (right).

text: r

text: QTT

text: DOS

text: via

text: Lorentzians

text: DOS

text: via

text: Lorentzians

text: exact

text: DoS

text: TDA

text: exact

text: DoS

text: TDA

text: QTT

text: approx.

text: QTT

text: approx.

text: eV

text: eV

text: Figure

text: DOS

text: for

text: Glycine

text: amino

text: acid

text: via

text: Lorentzians

text: (blue)

text: and

text: its

text: QTT

text: approximation

text: (red),

text: left;

text: (left).

text: Right:

text: zoom

text: of

text: the

text: ﬁrst

text: part

text: of

text: the

text: spectrum.

text: r

text: QTT

text: A

text: comparison

text: of

text: Figures

text: and

text: indicates

text: that

text: the

text: Lorentzian

text: based

text: DOS

text: blurring

text: is

text: slightly

text: smoother

text: than

text: Gaussian

text: blurring.

text: The

text: moderate

text: size

text: of

text: the

text: QTT

text: ranks

text: in

text: Figures

text: and

text: clearly

text: shows

text: the

text: potentials

text: of

text: the

text: QTT

text: ε-interpolation

text: for

text: modeling

text: the

text: DOS

text: of

text: large

text: lattice

text: type

text: clusters.

text: We

text: observe

text: several

text: gaps

text: in

text: the

text: spectral

text: densities,

text: see

text: Figure

text: and

text: indicating

text: that

text: polynomial,

text: rational

text: or

text: trigonometric

text: interpolation

text: can

text: be

text: applied

text: only

text: to

text: some

text: energy

text: sub-

text: intervals,

text: but

text: not

text: in

text: the

text: whole

text: interval

text: [a,

text: b].

text: Remarkably,

text: the

text: QTT

text: approximant

text: resolves

text: well

text: the

text: DOS

text: function

text: in

text: the

text: whole

text: energy

text: interval

text: including

text: nearly

text: zero

text: values

text: within

text: the

text: spectral

text: gaps

text: (hardly

text: possible

text: for

text: polynomial/rational

text: based

text: interpolation).

text: Numerics

text: for

text: the

text: QTT

text: interpolation

text: to

text: the

text: DOS

text: function

text: In

text: the

text: previous

text: section

text: we

text: demonstrated

text: that

text: the

text: QTT

text: tensor

text: approximation

text: provides

text: good

text: resolution

text: for

text: the

text: DOS

text: function

text: calculated

text: for

text: a

text: number

text: of

text: molecules.

text: In

text: what

text: follows,

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text: sites for the description of our method including the rank-structured approximation to the BSE/TDA matrix, basic notions of the regularization of DOS by Lorentzians and a short summary on the existing methods for matrices of general structure. Section discusses the main techniques of the presented method and the corresponding analysis in Theorems 3.1 and 3.2. The numerical tests conﬁrm the linear scaling of our algorithm in the size of the grid on which the DOS is evaluated. Section ?? presents a summary of the QTT tensor approxi- mation of function related vectors and the analysis of the QTT tensor ranks of the DOS, see Theorem ??. In Section ?? the ACA based QTT interpolation is applied to the discretized DOS, where the quality of the interpolation is illustrated numerically. The beneﬁcial features of the new computational schemes are veriﬁed by extensive numerical experiments on the examples of various molecular systems. Section ?? outlines the extension of the approach to the case of full BSE system. Conclusions summarize the main results and address the application perspectives. 2 Main prerequisites and outline of initial applications 2.1 Rank-structured approximation to BSE matrix In this paper we describe a method for eﬃcient and accurate approximation of the DOS for large rank-structured symmetric matrices. Our basic application is concerned with es- timating the DOS and the absorption spectrum for the Bethe-Salpeter problem describing the excitation energies of molecules. × The 2 2-block matrix(cid:18)rep(cid:19)resen(cid:18)tation of the(cid:19)B(cid:18)ethe(cid:19)-Salpete(cid:18)r Ha(cid:19)miltonian (BSH) leads to the following eigenvalue problem. x ≡ A B x x H k − ∗ − ∗ k = ω k , (2.1) y B A y k y k k k × − where the matrix blocks of size n n, wi(cid:99)th n = N = N(cid:102)(N N ), are deﬁned by ov o b o − − A = ∆ε + V W , B = V W , (2.2) and eigenvalues ω correspond to the excitation energies. Here ∆ε is a diagonal matrix k and ∈ I { } ∈ I { } V = [v ] a, b := N + 1, . . . , N , i, j := 1, . . . , N , ia,jb v o b o o is the rank-R two-electron integrals (TEI) matrix projected onto the Hartree-Fock molec- B ular orbital basis, where N is the number of Gaussian type orbital (GTO) basis functions b and N denotes the number of occupied orbitals [?]. o The method for solving the Bethe-Salpeter equation (BSE) using low-rank factorizations of the generating matrices has been introduced in [?]. It is based on a tensor-structured grid-based Hartree-Fock (HF) solver which provides not only the full set of eigenvalues and HF orbitals, but also the two-electron integrals tensor in the form of a low-rank (cid:102)Cholesky factorization, see [?] and references therein. The matrix V inherits its low rank from the two-electron integrals tensor, and W is also proven to have a small (cid:15)-rank (see [?]). In particular, there holds 4

text: sites

text: for

text: the

text: description

text: of

text: our

text: method

text: including

text: the

text: rank-structured

text: approximation

text: to

text: the

text: BSE/TDA

text: matrix,

text: basic

text: notions

text: of

text: the

text: regularization

text: of

text: DOS

text: by

text: Lorentzians

text: and

text: a

text: short

text: summary

text: on

text: the

text: existing

text: methods

text: for

text: matrices

text: of

text: general

text: structure.

text: Section

text: discusses

text: the

text: main

text: techniques

text: of

text: the

text: presented

text: method

text: and

text: the

text: corresponding

text: analysis

text: in

text: Theorems

text: and

text: The

text: numerical

text: tests

text: conﬁrm

text: the

text: linear

text: scaling

text: of

text: our

text: algorithm

text: in

text: the

text: size

text: of

text: the

text: grid

text: on

text: which

text: the

text: DOS

text: is

text: evaluated.

text: Section

text: presents

text: a

text: summary

text: of

text: the

text: QTT

text: tensor

text: approxi-

text: mation

text: of

text: function

text: related

text: vectors

text: and

text: the

text: analysis

text: of

text: the

text: QTT

text: tensor

text: ranks

text: of

text: the

text: DOS,

text: see

text: Theorem

text: In

text: Section

text: the

text: ACA

text: based

text: QTT

text: interpolation

text: is

text: applied

text: to

text: the

text: discretized

text: DOS,

text: where

text: the

text: quality

text: of

text: the

text: interpolation

text: is

text: illustrated

text: numerically.

text: The

text: beneﬁcial

text: features

text: of

text: the

text: new

text: computational

text: schemes

text: are

text: veriﬁed

text: by

text: extensive

text: numerical

text: experiments

text: on

text: the

text: examples

text: of

text: various

text: molecular

text: systems.

text: Section

text: outlines

text: the

text: extension

text: of

text: the

text: approach

text: to

text: the

text: case

text: of

text: full

text: BSE

text: system.

text: Conclusions

text: summarize

text: the

text: main

text: results

text: and

text: address

text: the

text: application

text: perspectives.

text: Main

text: prerequisites

text: and

text: outline

text: of

text: initial

text: applications

text: Rank-structured

text: approximation

text: to

text: BSE

text: matrix

text: In

text: this

text: paper

text: we

text: describe

text: a

text: method

text: for

text: eﬃcient

text: and

text: accurate

text: approximation

text: of

text: the

text: DOS

text: for

text: large

text: rank-structured

text: symmetric

text: matrices.

text: Our

text: basic

text: application

text: is

text: concerned

text: with

text: es-

text: timating

text: the

text: DOS

text: and

text: the

text: absorption

text: spectrum

text: for

text: the

text: Bethe-Salpeter

text: problem

text: describing

text: the

text: excitation

text: energies

text: of

text: molecules.

text: The

text: of

text: (BSH)

text: leads

text: to

text: the

text: following

text: eigenvalue

text: problem.

text: x

text: A

text: B

text: x

text: x

text: H

text: k

text: k

text: ω

text: k

text: y

text: B

text: A

text: y

text: k

text: y

text: k

text: k

text: k

text: where

text: the

text: matrix

text: blocks

text: of

text: size

text: n

text: n,

text: n

text: N

text: N

text: are

text: deﬁned

text: by

text: ov

text: o

text: b

text: o

text: A

text: ∆ε

text: V

text: W

text: B

text: V

text: W

text: and

text: eigenvalues

text: ω

text: correspond

text: to

text: the

text: excitation

text: energies.

text: Here

text: ∆ε

text: is

text: a

text: diagonal

text: matrix

text: k

text: and

text: I

text: I

text: V

text: [v

text: a,

text: b

text: N

text: N

text: i,

text: j

text: N

text: ia,jb

text: v

text: o

text: b

text: o

text: o

text: is

text: the

text: rank-R

text: two-electron

text: integrals

text: (TEI)

text: matrix

text: projected

text: onto

text: the

text: Hartree-Fock

text: molec-

text: B

text: ular

text: orbital

text: basis,

text: where

text: N

text: is

text: the

text: number

text: of

text: Gaussian

text: type

text: orbital

text: (GTO)

text: basis

text: functions

text: b

text: and

text: N

text: denotes

text: the

text: number

text: of

text: occupied

text: orbitals

text: o

text: The

text: method

text: for

text: solving

text: the

text: Bethe-Salpeter

text: equation

text: (BSE)

text: using

text: low-rank

text: factorizations

text: of

text: the

text: generating

text: matrices

text: has

text: been

text: introduced

text: in

text: It

text: is

text: based

text: on

text: a

text: tensor-structured

text: grid-based

text: Hartree-Fock

text: (HF)

text: solver

text: which

text: provides

text: not

text: only

text: the

text: full

text: set

text: of

text: eigenvalues

text: and

text: HF

text: orbitals,

text: but

text: also

text: the

text: two-electron

text: integrals

text: tensor

text: in

text: the

text: form

text: of

text: a

text: low-rank

text: factorization,

text: see

text: and

text: references

text: therein.

text: The

text: matrix

text: V

text: inherits

text: its

text: low

text: rank

text: from

text: the

text: two-electron

text: integrals

text: tensor,

text: and

text: W

text: is

text: also

text: proven

text: to

text: have

text: a

text: small

text: (see

text: In

text: particular,

text: there

text: holds


text: ≈ ∈ R × ≤ V L LT , L n RV , R R , (2.3) V V V V B (cid:102) (cid:99) O | | ≤ with the rank estimates R = R (ε) = (N log ε ), and rank(W ) rank(V ). V V b In [?], it was shown that the matrix W , which does not exhibit an accurate low rank (cid:99) (cid:98) representation, can be well approximated by a block diagonal matrix ≈ (cid:98) W blockdiag[B, D], × where B is a N N dense block with N = O(nα), α < 1. The size of N is nearly the W W W W same as the rank parameter of L . As a result, the TDA matrix A can be approximated by V (cid:98) (cid:98) a sum of a block-diagonal matrix and a low rank matrix shown in Figure 2.1, i.e., ≈ − ≡ A A = ∆ε + QQT blockdiag[B, D] blockdiag[B , D ] + QQT . 0 0 Rv Nov Nw (cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1) (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) A = + (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) − (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) (cid:98) N ov Figure 2.1: Diagonal plus low-rank plus reduced-block structure of the matrix A. An eﬃcient structured solver designed to calculate a number of minimal eigenvalues of the block-diagonal plus low-rank representation of the BSE/TDA matrices is described in [?]. It is based on an eﬃcient subspace iteration of the matrix inverse, which for rank-structured matrix formats can be evaluated eﬃciently by using the Sherman-Morrison-Woodbury for- O mula, thus reducing the numerical expense of the direct diagonalization down to (N2) b in the size of the atomic orbitals basis set, N . Furthermore, this solver also includes a b QTT-based compression scheme, where both eigenvectors and the rank-structured BSE ma- trix blocks are represented by block-QTT tensors. The block-QTT representation of the eigenvector is determined by an alternating least squares (ALS) iterative algorithm. The overall asymptotic complexity for computing several smallest in modulo eigenvalues in the O BSE spectral problem by using the QTT approximation is estimated to be (log(N )N2), o o where N is the number of occupied orbitals. o Matrices in the form (2.1) are called J-symmetric or Hamiltonian, see [?] for implications on the algebraic properties of the BSE matrix. In particular, solutions of equation (2.1) come − in pairs: excitation energies ω with eigenvectors (x , y ), and de-excitation energies ω ∗ ∗ k k k k with eigenvectors (y , x ). k k × The simpliﬁcation in the BSH, H, deﬁned by the n n symmetric diagonal block A is called the Tamm-Dancoﬀ (TDA) approximation. In what follows, we are interested in the TDA spectral problem, Au = λ u , k = 1, . . . , n, k k k providing good approximations to ω , x . k k 5 ##LTFigure##

label: equation

text: R

text: V

text: L

text: LT

text: L

text: n

text: RV

text: R

text: R

text: V

text: V

text: V

text: V

text: B

text: O

text: with

text: the

text: rank

text: estimates

text: R

text: R

text: (ε)

text: (N

text: log

text: ε

text: and

text: rank(W

text: rank(V

text: V

text: V

text: b

text: In

text: it

text: was

text: shown

text: that

text: the

text: matrix

text: W

text: which

text: does

text: not

text: exhibit

text: an

text: accurate

text: low

text: rank

text: representation,

text: can

text: be

text: well

text: approximated

text: by

text: a

text: block

text: diagonal

text: matrix

text: W

text: blockdiag[B,

text: D],

text: where

text: B

text: is

text: a

text: N

text: N

text: dense

text: block

text: with

text: N

text: O(nα),

text: α

text: The

text: size

text: of

text: N

text: is

text: nearly

text: the

text: W

text: W

text: W

text: W

text: same

text: as

text: the

text: rank

text: parameter

text: of

text: L

text: As

text: a

text: result,

text: the

text: TDA

text: matrix

text: A

text: can

text: be

text: approximated

text: by

text: V

text: a

text: sum

text: of

text: a

text: block-diagonal

text: matrix

text: and

text: a

text: low

text: rank

text: matrix

text: shown

text: in

text: Figure

text: i.e.,

text: A

text: A

text: ∆ε

text: QQT

text: blockdiag[B,

text: D]

text: blockdiag[B

text: D

text: QQT

text: Rv

text: Nov

text: Nw

text: A

text: N

text: ov

text: Figure

text: Diagonal

text: plus

text: low-rank

text: plus

text: reduced-block

text: structure

text: of

text: the

text: matrix

text: A.

text: An

text: eﬃcient

text: structured

text: solver

text: designed

text: to

text: calculate

text: a

text: number

text: of

text: minimal

text: eigenvalues

text: of

text: the

text: block-diagonal

text: plus

text: low-rank

text: representation

text: of

text: the

text: BSE/TDA

text: matrices

text: is

text: described

text: in

text: It

text: is

text: based

text: on

text: an

text: eﬃcient

text: subspace

text: iteration

text: of

text: the

text: matrix

text: inverse,

text: which

text: for

text: rank-structured

text: matrix

text: formats

text: can

text: be

text: evaluated

text: eﬃciently

text: by

text: using

text: the

text: Sherman-Morrison-Woodbury

text: for-

text: O

text: mula,

text: thus

text: reducing

text: the

text: numerical

text: expense

text: of

text: the

text: direct

text: diagonalization

text: down

text: to

text: b

text: in

text: the

text: size

text: of

text: the

text: atomic

text: orbitals

text: basis

text: set,

text: N

text: Furthermore,

text: this

text: solver

text: also

text: includes

text: a

text: b

text: QTT-based

text: compression

text: scheme,

text: where

text: both

text: eigenvectors

text: and

text: the

text: rank-structured

text: BSE

text: ma-

text: trix

text: blocks

text: are

text: represented

text: by

text: block-QTT

text: tensors.

text: The

text: block-QTT

text: representation

text: of

text: the

text: eigenvector

text: is

text: determined

text: by

text: an

text: alternating

text: least

text: squares

text: (ALS)

text: iterative

text: algorithm.

text: The

text: overall

text: asymptotic

text: complexity

text: for

text: computing

text: several

text: smallest

text: in

text: modulo

text: eigenvalues

text: in

text: the

text: O

text: BSE

text: spectral

text: problem

text: by

text: using

text: the

text: QTT

text: approximation

text: is

text: estimated

text: to

text: be

text: (log(N

text: o

text: o

text: where

text: N

text: is

text: the

text: number

text: of

text: occupied

text: orbitals.

text: o

text: Matrices

text: in

text: the

text: form

text: are

text: called

text: J-symmetric

text: or

text: Hamiltonian,

text: see

text: for

text: implications

text: on

text: the

text: algebraic

text: properties

text: of

text: the

text: BSE

text: matrix.

text: In

text: particular,

text: solutions

text: of

text: equation

text: come

text: in

text: pairs:

text: excitation

text: energies

text: ω

text: with

text: eigenvectors

text: (x

text: y

text: and

text: de-excitation

text: energies

text: ω

text: k

text: k

text: k

text: k

text: with

text: eigenvectors

text: (y

text: x

text: k

text: k

text: The

text: simpliﬁcation

text: in

text: the

text: BSH,

text: H,

text: deﬁned

text: by

text: the

text: n

text: n

text: symmetric

text: diagonal

text: block

text: A

text: is

text: called

text: the

text: Tamm-Dancoﬀ

text: (TDA)

text: approximation.

text: In

text: what

text: follows,

text: we

text: are

text: interested

text: in

text: the

text: TDA

text: spectral

text: problem,

text: Au

text: λ

text: u

text: k

text: n,

text: k

text: k

text: k

text: providing

text: good

text: approximations

text: to

text: ω

text: x

text: k

text: k

text: ##LTFigure##


text: In general, methods for solving partial eigenvalue problems for matrices with a special structure as in the BSE setting are conceptually related to the approaches for Hamiltonian matrices [?, ?, ?, ?], particularly to those based on minimization principles [?, ?]. A struc- tured Lanczos algorithm for estimation of the optical absorption spectrum was described in [?]. Various structured eigensolvers tailored for electronic structure calculations are discussed in [?, ?, ?, ?, ?, ?]. 2.2 Density of states for symmetric matrices To ﬁx the idea, we ﬁrst consider the case of symmetric matrices. Following [?], we use (cid:88) the simple deﬁnition of the DOS for symmetric matrices 1 n − ∈ φ(t) = δ(t λ ), t, λ [0, a], (2.4) j j n j=1 where δ is the Dirac distribution and the λ ’s are the eigenvalues of A = AT ordered as ≤ ≤ · · · ≤ j λ λ λ . 1 2 n Several classes of blurring approximations to φ(t)(cid:18)are us(cid:19)ed in the literature. One can replace each Dirac-δ by a Gaussian function with width η > 0, i.e., (cid:32) √ 1 − t2 δ(t) g (t) = exp , η 2πη 2η2 where the choice of the regularization parameter η depends on the particular problem setting. (cid:88) As a result, (2.4) can be approximated by ≈ 1 n − φ(t) φ (t) := g (t λ ), (2.5) η η j n j=1 on the whole energy interval [0, a]. (cid:18) (cid:19) We may also replace each Dirac-δ by a Lorentzian, i.e., (cid:32) 1 η 1 1 δ(t) L (t) := = Im − , (2.6) η π t2 + η2 π t iη (cid:88) so that an approximate DOS can be written as ≈ 1 n − φ(t) φ (t) := L (t λ ). (2.7) η η j n j=1 → When η 0 , both Gaussians and Lorentzians converge to the Dirac distribution, i.e., + lim g (t) = lim L (t) = δ(t). → η → η η 0+ η 0+ However, they exhibit diﬀerent features of the approximant for small η > 0. In the case of Gaussians, one expects a sharp resolution of the spectral peaks, while the Lorentzian based representation aims to resolve better the global landscape of φ(t). 6 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: In

text: general,

text: methods

text: for

text: solving

text: partial

text: eigenvalue

text: problems

text: for

text: matrices

text: with

text: a

text: special

text: structure

text: as

text: in

text: the

text: BSE

text: setting

text: are

text: conceptually

text: related

text: to

text: the

text: approaches

text: for

text: Hamiltonian

text: matrices

text: particularly

text: to

text: those

text: based

text: on

text: minimization

text: principles

text: A

text: struc-

text: tured

text: Lanczos

text: algorithm

text: for

text: estimation

text: of

text: the

text: optical

text: absorption

text: spectrum

text: was

text: described

text: in

text: Various

text: structured

text: eigensolvers

text: tailored

text: for

text: electronic

text: structure

text: calculations

text: are

text: discussed

text: in

label: section

text: Density

text: of

text: states

text: for

text: symmetric

text: matrices

text: To

text: ﬁx

text: the

text: idea,

text: we

text: ﬁrst

text: consider

text: the

text: case

text: of

text: symmetric

text: matrices.

text: Following

text: we

text: use

text: the

text: simple

text: deﬁnition

text: of

text: the

text: DOS

text: for

text: symmetric

text: matrices

text: n

text: φ(t)

text: δ(t

text: λ

text: t,

text: λ

text: a],

text: j

text: j

text: n

text: where

text: δ

text: is

text: the

text: Dirac

text: distribution

text: and

text: the

text: λ

text: ’s

text: are

text: the

text: eigenvalues

text: of

text: A

text: AT

text: ordered

text: as

text: j

text: λ

text: λ

text: λ

text: n

text: Several

text: classes

text: of

text: blurring

text: approximations

text: to

text: in

text: the

text: literature.

text: One

text: can

text: replace

text: each

text: Dirac-δ

text: by

text: a

text: Gaussian

text: function

text: with

text: width

text: η

text: i.e.,

text: δ(t)

text: g

text: (t)

text: exp

text: η

text: 2πη

text: where

text: the

text: choice

text: of

text: the

text: regularization

text: parameter

text: η

text: depends

text: on

text: the

text: particular

text: problem

text: setting.

text: As

text: a

text: result,

text: can

text: be

text: approximated

text: by

text: n

text: φ(t)

text: φ

text: (t)

text: g

text: (t

text: λ

text: η

text: η

text: j

text: n

text: on

text: the

text: whole

text: energy

text: interval

text: a].

text: We

text: may

text: also

text: replace

text: each

text: Dirac-δ

text: by

text: a

text: Lorentzian,

text: i.e.,

text: η

text: δ(t)

text: L

text: (t)

text: Im

text: η

text: π

text: η2

text: π

text: t

text: iη

text: so

text: that

text: an

text: approximate

text: DOS

text: can

text: be

text: written

text: as

text: n

text: φ(t)

text: φ

text: (t)

text: L

text: (t

text: λ

text: η

text: η

text: j

text: n

text: When

text: η

text: both

text: Gaussians

text: and

text: Lorentzians

text: converge

text: to

text: the

text: Dirac

text: distribution,

text: i.e.,

text: lim

text: g

text: (t)

text: lim

text: L

text: (t)

text: δ(t).

text: η

text: η

text: η

text: η

text: However,

text: they

text: exhibit

text: diﬀerent

text: features

text: of

text: the

text: approximant

text: for

text: small

text: η

text: In

text: the

text: case

text: of

text: Gaussians,

text: one

text: expects

text: a

text: sharp

text: resolution

text: of

text: the

text: spectral

text: peaks,

text: while

text: the

text: Lorentzian

text: based

text: representation

text: aims

text: to

text: resolve

text: better

text: the

text: global

text: landscape

text: of

text: φ(t).

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: a relatively large number m of stochastic realizations to obtain satisfactory m√ean value r approximation. The convergence rate is expected to be on the order of O(1/ m ). On r the other hand, with the limited number of interpolation points, the polynomial type of interpolation schemes applied to highly non-regular shapes as shown, say, in Figure 2.4 (left), can only provide limited resolution and is unlikely to reveal spectral gaps and many local peaks of interest. 3 Fast evaluation of DOS for rank-structured matrices 3.1 DOS by the trace of rank-structured matrix inverse In what follows, we propose an approach that is based on evaluating the trace term in (2.8) directly (without stochastic sampling). This approach relies on the following two techniques: ∈ (A) using the low-rank BSE matrix structure as in [?], which allows for each ﬁxed t [0, a] the direct matrix inversion and computation of the respective traces, (B) the low-rank QTT tensor interpolation of the function L (t) sampled on a ﬁne uniform { } η grid t , . . . , t in the whole spectral interval [0, a] or on some subinterval of [0, a]. 1 M For the class of block-diagonal plus low-rank matrices arising in the reduced model ap- § proach for BSE problem [?, ?], we have (see for more details) ∈ R × { } A = E + P QT , with P, Q n R, E = blockdiag B , D , (3.1) 0 0 × where the rank parameter R is small compared to n, the full n n matrix block B is B −B 0 of size n = O(nα), 0 < α < 1, and D is a diagonal matrix of size n n . B 0 B Notice that even in the case of structured matrices in (3.1) the traditional approach by (2.10) leads to a sequence of linear systems (2.12) to be solved many times in the course of ∈ stochastic sampling, for each of many interpolation points t [0, a]. In our approach, for the class of rank-structured matrices (3.1), we propose to avoid stochastic sampling in (2.10) by introducing a direct scheme that allows us to evaluate the trace of matrices f (A) or f (A) deﬁned in (2.11), corresponding to the matrix resolvent in 1 2 (2.8) and (2.9), respectively, by one-step straightforward matrix calculation. To that end, let us ﬁrst construct the reduced-model approximation to the matrix inverse − { } A 1 for the matrix in (3.1), where the block-diagonal part E(t) = blockdiag B(t), D(t) corresponds to the case of (2.8), i.e., − − B(t) = tI iηI + B , D(t) = tI iηI + D . (3.2) B B 0 D D 0 Here B and D denote the corresponding matrix blocks in the representation of the 0 0 diagonal block A in the initial BSE matrix, see (3.1), and I , I denote the identity matrices B D corresponding to the respective index subsets. For the ease of exposition, we further assume ≤ that the matrix size of the block B in (3.2) is bounded by n = O(nα) with α 1/3. This B assumption on the block size ensures the linear complexity scaling of our algorithm in the matrix size n. 10 ##LTLine##

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text: 7 netotransport. isothermal magnetoresistance (MR) of (001) strained pc D. Zero ﬁeld transport properties SCIO thin ﬁlms. As discussed previously in section C, the magnetic easy axis in this series changes from the ”in-plane” We now turn to the electrical resistivity of SCIO thin to the ”out-of-plane” as the strain changes from tensile to ﬁlms. The temperature dependent ρ(T) measurements for compressive. This change of the magnetic anisotropy has a (111) oriented d(SCIO) = 30 nm thick SCIO ﬁlm (with a direct inﬂuence on the magnetoresistance. As shown in pc and without STO protective top layer) are shown in Figure Figure 9, MR below 6 K is positive for thin ﬁlms grown un- 8. Similar to the bulk SCIO (Ref.22), thin ﬁlms also show der compressive strain on LSAT substrate and negative for an insulating behavior. The values of room temperature those under tensile strain on GSO substrate. This indicates resistivity for the air protected and unprotected ﬁlms were a direct relation between MR sign and magnetic anisotropy. 3.08(1)mΩcm and 6.66(1)mΩcm, respectively. The re- This is further corroborated by the comparison of the MR at ported bulk value (Ref.22) was about 3 and 1.5 times larger, 2 K of all strained SCIO thin ﬁlms, which, as shown in Figure respectively. Upon cooling to 4 K, the electrical resistance 10, follow this trend. This observation suggests a strong in- of the unprotected ﬁlm strongly increases, becoming sev- ﬂuence of the anisotropic magnetoresistance (AMR) effect eral orders of magnitude larger than the resistance of the in combination with the reorientation of the AFM ordered protectedﬁlm; the ρ(T) of the bulk SCIO lies in between. Co sublattice in SCIO in an external transverse magnetic This indicates the importance of the STO capping layer ﬁeld. In addition, structural disorder in combination with to protect the (surface) properties of SCIO. As shown in large spin orbit coupling will generate a quantum correc- the inset of Figure 8 the room temperature resist−ivity is tion to MR due to weak anti-localization (WAL). Based on almost independent on the ﬁlm thickness for d > 5 8nm. the Dresselhaus effekt41 we used for the description of the Unfortunately, it was impossible to probe the thickness de- WAL term of the MR a Dresselhaus like contribution with pendence at low temperatures. Fitting the ρ(T) data of the isotropic spin orbit scattering, see supplemental material27. capped SCIO thin ﬁlm between 250 and 300 K results in For the AMR contribution the orientation of the moments a charge gap of 32(5) meV27. Below 70 K, ρ(T) indicates of Co sublattices in their AF ground state with respect to variable range hopping behavior27. the ﬁeld direction is important. For tensile strain at zero The electrical resistivity of the various ﬁlms, strained ﬁeld the moments are oriented in the ﬁlm plane and appli- along the (001) direction, is discussed in supplemen- cation of a transverse ﬁeld leads to a continuous rotation pc tal material27. The overall insulating behavior does not of both sublattice moments out of the plane. This leads to a negative MR. For compressive strain, the moments of change, though a systematic enhancement of ρ(300 K) with strain is observed27. the two sublattices initially point parallel and antiparallel to the applied ﬁeld. If there would be very weak coupling E. Magnetotransport within the (001) strain series pc between the two sublattices, only the moments of the an- Considering the information on the magnetic anisotropy, 1 0 obtained in the ”Spectroscopy” section, we can now address S C I O / L S A T the magnetotransport properties of the SCIO thin ﬁlms un- 2 K der different strains. We focus on the low-temperature 5 1 0 8 S C I O o n ( 1 1 1 ) S T O 1 0 7 S C I O o n ( 1 1 1 ) S T O 1 0 0 w i t h S T O c a p p i n g 1 0 6 0 6 K 1 0 1 0 5 - 5 1 0 4 1 1 1 0 1 0 0 - 1 0 1 0 3 d ( S C I O ) ( n m ) - 1 5 6 K 1 0 2 - 2 0 1 0 1 Narayanan et al.22 - 2 5 S C I O / G d S c O 2 K 1 0 0 - 3 0 3 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 0 5 1 0 1 5 T ( K ) µ H ( T ) 0 Figure 8. (Color online) Temperature dependence of the electrical Figure−9. Isothermal transverse magnetoresistance MR = resistivity of SCIO for a sample with STO protection (blue) and (ρ(H) ρ(0))/ρ(0) of (001) strained SCIO thin ﬁlms grown on pc without protection (red) in comparision to bulk data (black line) LSAT (compressive strain) and GdScO (tensile strain) at various 3 obtained by Narayanan et al.22. Inset: Thickness dependence of temperatures between 2 and 6 K (ﬁelds always applied transverse speciﬁc resistance at 300K (dashed line as guide for eyes). to the ﬁlm planes). ) K 0 0 3 ( / ) T ( (cid:1) (cid:1) ) m c m ( ) K 0 0 3 ( (cid:1) (cid:1) ) % ( R M ##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## 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text: D.

text: Zero

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text: this

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text: We

text: now

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text: the

text: ”out-of-plane”

text: as

text: the

text: strain

text: changes

text: from

text: tensile

text: to

text: ﬁlms.

text: The

text: temperature

text: dependent

text: ρ(T)

text: measurements

text: for

text: compressive.

text: This

text: change

text: of

text: the

text: magnetic

text: anisotropy

text: has

text: a

text: oriented

text: d(SCIO)

text: nm

text: thick

text: SCIO

text: ﬁlm

text: (with

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text: on

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text: shown

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text: an

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text: The

text: values

text: of

text: room

text: temperature

text: those

text: under

text: tensile

text: strain

text: on

text: GSO

text: substrate.

text: This

text: indicates

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text: the

text: air

text: protected

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text: unprotected

text: ﬁlms

text: were

text: a

text: direct

text: relation

text: between

text: MR

text: sign

text: and

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text: and

text: respectively.

text: The

text: re-

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text: is

text: further

text: corroborated

text: by

text: the

text: comparison

text: of

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