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Rational Krylov for Stieltjes matrix functions: convergenceand pole selection
Leonardo Robol, UniPI (joint work with S. Massei, EPFL)13 Febbraio 2020, Montecatini, Convegno GNCS
0
Progetto GNCS Giovani ricercatori
• Progetto GNCS “Giovani ricercatori” 2018/2019.• Titolo: “Metodi di proiezione per equazioni di matrici e sistemi lineari con operatori
definiti tramite somme di prodotti di Kronecker, e soluzioni con struttura di rango.”• Supporto per la partecipazione alle conferenze ILAS2019 (Rio de Janeiro) e ICIAM2019
(Valencia).
1
Motivation: Fractional diffusion equations
We are concerned with the “fractional equivalent” of the 1D/2D Laplacian. That is, instead ofconsidering
∂2u∂x2 = f (x), ∆u = f (x , y)
we deal with∂αu∂yα = f (x), ∆αu = f (x , y)
for 1 < α < 2.
Fractional diffusion allow to model nonlocal behavior.
• Useful in describing anomalous diffusion phenomena (plasma physics, financial markets,. . . )
• Several (non-equivalent) definitions in the literature: Riemann-Liouville, Caputo,Grunwald-Letkinov, Matrix Transform Method
2
Motivation: Matrix Transform Method
Let A be the usual discretization of the Laplacian operator.
One way to define the discrete operator M representing the derivative of order α is to imposethat
M 2α = A ⇐⇒ M = Aα
2 (1 < α < 2)
• This follows by “composition” of derivatives• The problem is recast as computing x := f (A)v , with f (z) = z−α2 .
3
Evaluation of matrix functions
Let f (z) : Ω→ R be an analytic function on Ω ⊆ R+.
Problem 1. Given a symmetric positive definite matrix A ∈ Rn×n with eigenvalues in[λmin, λmax] ⊂ Ω and v ∈ Rn compute
x := f (A)v .
Applications: system of ODEs, exponential integrators, fractional diffusion problems, networkcommunicability measures, control theory, ... .
4
Subspace projection methods
Let U ⊂ Rn be a `-dimensional subspace (` n) with an orthogonal basis U = [u1| . . . |u`] andA` = U∗AU, v` = U∗v be the projections of A and v on U .
• Linear systemsx = A−1v ≈ x` := UA−1
` v`.
• Matrix functionsx = f (A)v ≈ x` := Uf (A`)v`.
If we use the Krylov subspace U = K`(A, v) := spanv ,Av , . . . ,A`−1v then,
‖x − x`‖2 ≤ C · minp(z)∈P`−1
maxz∈[λmin,λmax]
|p(z)− f (z)|,
where P` := poly of degree ≤ ` and C is independent on A and f .
5
Subspace projection methods
Let U ⊂ Rn be a `-dimensional subspace (` n) with an orthogonal basis U = [u1| . . . |u`] andA` = U∗AU, v` = U∗v be the projections of A and v on U .
• Linear systemsx = A−1v ≈ x` := UA−1
` v`.
• Matrix functionsx = f (A)v ≈ x` := Uf (A`)v`.
If we use the Krylov subspace U = K`(A, v) := spanv ,Av , . . . ,A`−1v then,
‖x − x`‖2 ≤ C · minp(z)∈P`−1
maxz∈[λmin,λmax]
|p(z)− f (z)|,
where P` := poly of degree ≤ ` and C is independent on A and f .
5
Polynomial approximation does not always work...
Sometimes the convergence of polynomial approximation struggles, e.g. x = A− 12 v ,
[λmin, λmax] ≈ [10−5, 4], n = 1000.
0 50 100 150 200
10−3
10−1
101
`
‖x − x`‖2
6
Rational Krylov method
Rational Krylov subspace. Given Σ` := σ1, . . . , σ` ⊂ C it is defined as1
RK`(A, v ,Σ`) : = q`(A)−1K`+1(A, v) =
p(A)q`(A) v : p(z) ∈ P`
= spanv , (σ1I − A)−1v , . . . , (σ`I − A)−1v
where q`(z) :=∏
j(z − σj)−1.
If U = RK`(A, v ,Σ`) we get a problem of rational approximation with fixed poles
‖x − x`‖2 ≤ C · minr(z)∈ P`q`(z)
maxz∈[λmin,λmax]
|r(z)− f (z)|.
1last equality is valid only for distinct poles
7
Rational approximation does not always work...
• Uniform approximations of highly oscillatory functions on the whole positive real line arenot possible for polynomials or rational functions
• In general a number of steps dependent on the norm of the operator is needed forconvergence2.
• E.g. x = cos(A)v , where ‖A‖2 ≈ 105, n = 1000
0 20 40 60 80 100
10−3
10−1
`
‖x−
x `‖ 2
Ext. KrylovRat. Krylov
2V. Grimm, M. Hochbruck. Rational approximation to trigonometric operators, BIT 2006.
8
Stieltjes functions
A favourable case is when f (z) : R+ → R if defined via a Stieltjes integral:
1. f (z) =∫ ∞
0
1z + t dµ(t) Cauchy-Stieltjes/Markov function
2. f (z) =∫ ∞
0e−ztdµ(t) Laplace-Stieltjes function
where dµ(t) is a non negative measure on R+.
Examples of functions in these classes are
1. z−α = sin(απ)π
∫ ∞0
t−αz + t dt α ∈ (0, 1), log(1 + z)
z ,e−t√
z − 1z .
2. e−z ,1− e−z
z , ϕj(z) :=∫ ∞
0e−tz [max1− t, 0]j−1
(j − 1)! dt.
9
Stieltjes functions
A favourable case is when f (z) : R+ → R if defined via a Stieltjes integral:
1. f (z) =∫ ∞
0
1z + t dµ(t) Cauchy-Stieltjes/Markov function
2. f (z) =∫ ∞
0e−ztdµ(t) Laplace-Stieltjes function
where dµ(t) is a non negative measure on R+.
Examples of functions in these classes are
1. z−α = sin(απ)π
∫ ∞0
t−αz + t dt α ∈ (0, 1), log(1 + z)
z ,e−t√
z − 1z .
2. e−z ,1− e−z
z , ϕj(z) :=∫ ∞
0e−tz [max1− t, 0]j−1
(j − 1)! dt.
9
Stieltjes functions
• Laplace Stieltjes functions on (0,∞) are also known as completely monotone functions,i.e. those such that (−1)j f (j)(z) > 0 ∀z > 0,
• By considering dµ(t) to be a finite sum of Dirac deltas we have that rational functions ofthe form
f (z) =h∑
j=1
αjz − βj
, αj > 0, βj < 0,
are Cauchy-Stieltjes.• It holds (Bernstein’s theorem3)
z−1 ∈ Cauchy − Stieltjes ⊂ Laplace − Stieltjes.
3S. Bernstein. Sur les fonctions absolument monotones, Acta Mathematica 1929
10
Evaluation of Stieltjes matrix functions
Let f (z) : R+ → R be an analytic function defined via a Stieltjes integral:
f (z) =∫ ∞
0g(t, z)dµ(t) g(t, z) ∈
1
z + t , e−zt.
Problem 1. Given a symmetric positive definite matrix A ∈ Rn×n with eigenvalues in[λmin, λmax] ⊂ Ω and v ∈ Rn compute
x := f (A)v .
Goal. Provide selection strategies for Σ` and estimates of the error ‖x − x`‖2.
11
Outline
• Problem 1:x = f (A)v
• Cauchy-Stieltjes case• Laplace-Stieltjes case
• Problem 2:x = f (I ⊗ A + B ⊗ I)vec(low-rank matrix)
• Laplace-Stieltjes case• Cauchy-Stieltjes case
12
Cauchy-Stieltjes functions: Sketching the idea
• By writing the integral formulation of f we get:
f (A)v =∫ ∞
0(A + tI)−1v dµ(t) ≈
∫ ∞0
U(A` + tI)−1v` dµ(t).
• So we need a space RK(A, v ,Σ`) that approximates simultaneously well (A + tI)−1v forany t > 0.
• Problem: Krylov subspaces are not shift invariant (apart from polynomial Krylov).
13
Cauchy-Stieltjes functions: Sketching the idea
• Approximating (A + tI)−1 ∀t ≥ 0, is linked to approximate 1λ+t in the strip
[λmin, λmax]× [0,∞].• We consider a Skeleton approximation of the form
1λ+ t ≈ fskel (λ, t) :=
[1
λ+ t1, . . . ,
1λ+ t`
]M−1
1
λ1+t...1
λ`+t
, Mij =(
1λi + tj
).
14
Cauchy-Stieltjes functions: Sketching the idea
fskel (λ, t) :=[
1λ+ t1
, . . . ,1
λ+ t`
]M−1
1
λ1+t...1
λ`+t
Skeleton approximations have 2 crucial properties:
1 Explicit expression of the residual error4:
1λ+ t − fskel (λ, t) = 1
λ+ t ·r(λ)
r(−t) , r(z) :=∏
j
z − λjz + tj
.
2 fskel (λ, t) is a rational function in λ with poles −t1, . . . ,−t`.If we set Σ` = −t1, . . . ,−t`, then
fskel (A, t)v ∈ RK`(A, v ,Σ`) ∀t ∈ [0,∞].
4Oseledets. Lower bounds for separable approximations of the Hilbert kernel, Sbornik 2007.
15
Cauchy-Stieltjes functions: Sketching the idea
fskel (λ, t) :=[
1λ+ t1
, . . . ,1
λ+ t`
]M−1
1
λ1+t...1
λ`+t
Skeleton approximations have 2 crucial properties:
1 Explicit expression of the residual error4:
1λ+ t − fskel (λ, t) = 1
λ+ t ·r(λ)
r(−t) , r(z) :=∏
j
z − λjz + tj
.
2 fskel (λ, t) is a rational function in λ with poles −t1, . . . ,−t`.If we set Σ` = −t1, . . . ,−t`, then
fskel (A, t)v ∈ RK`(A, v ,Σ`) ∀t ∈ [0,∞].4Oseledets. Lower bounds for separable approximations of the Hilbert kernel, Sbornik 2007.
15
Cauchy-Stieltjes functions: Sketching the idea
• We have the following point-wise estimate of the error 5
‖(tI + A)−1v − U(tI + A`)−1v`‖2 ≤2‖v‖2λmin + t min
r(z)∈P`Σ`
maxz∈[λmin,λmax] |r(z)|minz∈[−∞,0] |r(z)| .
• Minimizing the right hand side over Σ` means solving
minr(z)∈R`,`
maxz∈[λmin,λmax] |r(z)|minz∈[−∞,0] |r(z)| ,
where R`,` is the set of (`, `) rational functions.
5An estimate of the L2-norm in:Druskin, Knizhnerman, Zaslavsky. Solution of large scale evolutionary problems using rational Krylov subspaceswith optimized shifts, SISC 2009.
16
Cauchy-Stieltjes functions: Sketching the idea
• We have the following point-wise estimate of the error 5
‖(tI + A)−1v − U(tI + A`)−1v`‖2 ≤2‖v‖2λmin + t min
r(z)∈P`Σ`
maxz∈[λmin,λmax] |r(z)|minz∈[−∞,0] |r(z)| .
• Minimizing the right hand side over Σ` means solving
minr(z)∈R`,`
maxz∈[λmin,λmax] |r(z)|minz∈[−∞,0] |r(z)| ,
where R`,` is the set of (`, `) rational functions.
5An estimate of the L2-norm in:Druskin, Knizhnerman, Zaslavsky. Solution of large scale evolutionary problems using rational Krylov subspaceswith optimized shifts, SISC 2009.
16
Cauchy-Stieltjes functions: Sketching the idea
• This problem can be transformed via a Moebius map T (z) = αz+βγz+δ into
minr(z)∈R`,`
maxz∈[a,b] |r(z)|minz∈[−b,−a] |r(z)| . (1)
that Zolotarev solved ≈ 140 years ago.
• In particular, we know explicitly the optimal poles Σ` of the rational function that solves(1).So the optimal poles Σ∗` for our starting problem are given by
Σ∗` := T−1(Σ`).
17
Theoretical result
TheoremLet f (z) be a Cauchy-Stieltjes function, U be an orthogonal basis of RK`(A, v ,Σ∗` ) andx` = Uf (A`)v`. Then
‖f (A)v − x`‖2 ≤ 8f (λmin)‖v‖2ρ`,
where ρ := exp(− π2
log(
16λmaxλmin
)).
Similar results by Beckermann and Reichel6.
6Beckerman, Reichel. Error estimate and evaluation of matrix functions via the Faber transform, SINUM 2009
18
Numerical results
x = A− 12 v , A = trid(−1, 2,−1) ∈ R1000×1000
0 10 20 30 40 5010−11
10−6
10−1
104
Iterations (`)
‖x−
x `‖ 2
BoundExt. KrylovPol. KrylovRat. Krylov
markovfunmv7
7Guttel, Knizhnerman. A black-box rational Arnoldi variant for Cauchy-Stieltjes matrix functions, BIT 2013
19
Laplace-Stieltjes functions
f (λ) =∫ ∞
0e−λtdµ(t)
• The core idea is to exploit the relation:
e−λt = 12πi
∫iR
est
λ+ s ds
to link (parameter dependent) resolvents with exponentials.
• Then, using the approximation
e−λt ≈∫
iRfskel (λ, s)est ds
yields‖e−tAv − Ue−tA`v`‖2 ≤ 4γ` max
z∈[λmin,λmax]|r(z)|, r(z) :=
∏j
z − λjz + λj
with U orthogonal basis of RK`(A, v , −λ1, . . . ,−λ`).
20
Laplace-Stieltjes functions
f (λ) =∫ ∞
0e−λtdµ(t)
• The core idea is to exploit the relation:
e−λt = 12πi
∫iR
est
λ+ s ds
to link (parameter dependent) resolvents with exponentials.
• Then, using the approximation
e−λt ≈∫
iRfskel (λ, s)est ds
yields‖e−tAv − Ue−tA`v`‖2 ≤ 4γ` max
z∈[λmin,λmax]|r(z)|, r(z) :=
∏j
z − λjz + λj
with U orthogonal basis of RK`(A, v , −λ1, . . . ,−λ`).
20
Laplace-Stieltjes functions
TheoremLet f (z) be a Laplace-Stieltjes function, U be an orthogonal basis of RK`(A, v ,Σ`) andx` = Uf (A`)v`. Then there exists a choice of Σ` such that
‖f (A)v − x`‖2 ≤ 8γ`f (0)‖v‖2ρ`2 ρ := exp
− π2
log(
4λmaxλmin
) ,
where γ` := 2.23 + 2π log(4`λmax
λmin).
Conjecture: The result holds with γ` = 1.
21
2D problems with tensor structure
• When considering discretization (say, finite differences) of
∂2u∂x2 + ∂2u
∂y 2 = f (x , y)
on a rectangular domain one get a linear system of the form
(A⊗ I + I ⊗ A)︸ ︷︷ ︸M
x = v .
• Moreover, if f (x , y) is a regular function or has a small support (e.g. a point source) then
v ≈ vec(C)
where C is a low-rank matrix, i.e. C = WZ T for some tall and skinny matrices W ,Z .• Applying the matrix transform method to the fractional analogous of this problem requires
to compute f (M)vec(C).
22
Function evaluation and Kronecker structure
Problem 2. Given two symmetric positive definite matrices A,B ∈ Rn×n with eigenvalues in[λmin, λmax] ⊂ Ω and v = vec(C) = vec(low-rank matrix) ∈ Rn2 , compute
x := f (I ⊗ A + B ⊗ I)v , or equivalently X := vec−1(f (I ⊗ A + B ⊗ I)v).
• If f (z) = z−1, then this is equivalent to solve the Sylvester equation: we know thelow-rank structure is transferred from C to X .
• Does this hold in more general cases? Yes, it does.• Once again, we focus on the case where f (z) is a Stieltjes function:
f (z) =∫ ∞
0g(t, z)dµ(t), g(t, z) ∈
1
z + t , e−zt.
23
Function evaluation and Kronecker structure
Problem 2. Given two symmetric positive definite matrices A,B ∈ Rn×n with eigenvalues in[λmin, λmax] ⊂ Ω and v = vec(C) = vec(low-rank matrix) ∈ Rn2 , compute
x := f (I ⊗ A + B ⊗ I)v , or equivalently X := vec−1(f (I ⊗ A + B ⊗ I)v).
• If f (z) = z−1, then this is equivalent to solve the Sylvester equation: we know thelow-rank structure is transferred from C to X .
• Does this hold in more general cases? Yes, it does.• Once again, we focus on the case where f (z) is a Stieltjes function:
f (z) =∫ ∞
0g(t, z)dµ(t), g(t, z) ∈
1
z + t , e−zt.
23
Function evaluation and Kronecker structure
Problem 2. Given two symmetric positive definite matrices A,B ∈ Rn×n with eigenvalues in[λmin, λmax] ⊂ Ω and v = vec(C) = vec(low-rank matrix) ∈ Rn2 , compute
x := f (I ⊗ A + B ⊗ I)v , or equivalently X := vec−1(f (I ⊗ A + B ⊗ I)v).
• If f (z) = z−1, then this is equivalent to solve the Sylvester equation: we know thelow-rank structure is transferred from C to X .
• Does this hold in more general cases? Yes, it does.
• Once again, we focus on the case where f (z) is a Stieltjes function:
f (z) =∫ ∞
0g(t, z)dµ(t), g(t, z) ∈
1
z + t , e−zt.
23
Function evaluation and Kronecker structure
Problem 2. Given two symmetric positive definite matrices A,B ∈ Rn×n with eigenvalues in[λmin, λmax] ⊂ Ω and v = vec(C) = vec(low-rank matrix) ∈ Rn2 , compute
x := f (I ⊗ A + B ⊗ I)v , or equivalently X := vec−1(f (I ⊗ A + B ⊗ I)v).
• If f (z) = z−1, then this is equivalent to solve the Sylvester equation: we know thelow-rank structure is transferred from C to X .
• Does this hold in more general cases? Yes, it does.• Once again, we focus on the case where f (z) is a Stieltjes function:
f (z) =∫ ∞
0g(t, z)dµ(t), g(t, z) ∈
1
z + t , e−zt.
23
Galerkin projection for evaluating f (M)vec(C)
Inspired by the Galerkin projection for Sylvester equations, we may consider the followingalgorithm8:
• Choose subspaces of Rn spanned by orthogonal matrices U,V .• Evaluate the projected matrix function
vec(Y ) = f ((V ⊗ U)∗M(V ⊗ U))vec(U∗CV ) = f (M)vec(U∗CV ),
where M := V ∗BV ⊗ I + I ⊗ U∗AU.• Use UYV ∗ as approximation for X .
8Benzi, Simoncini. Approximation of functions of large matrices with Kronecker structure, NumerischeMathematik, 2017.
24
Galerkin projection for evaluating f (M)vec(C)
Inspired by the Galerkin projection for Sylvester equations, we may consider the followingalgorithm8:
• Choose subspaces of Rn spanned by orthogonal matrices U,V .
• Evaluate the projected matrix function
vec(Y ) = f ((V ⊗ U)∗M(V ⊗ U))vec(U∗CV ) = f (M)vec(U∗CV ),
where M := V ∗BV ⊗ I + I ⊗ U∗AU.• Use UYV ∗ as approximation for X .
8Benzi, Simoncini. Approximation of functions of large matrices with Kronecker structure, NumerischeMathematik, 2017.
24
Galerkin projection for evaluating f (M)vec(C)
Inspired by the Galerkin projection for Sylvester equations, we may consider the followingalgorithm8:
• Choose subspaces of Rn spanned by orthogonal matrices U,V .• Evaluate the projected matrix function
vec(Y ) = f ((V ⊗ U)∗M(V ⊗ U))vec(U∗CV ) = f (M)vec(U∗CV ),
where M := V ∗BV ⊗ I + I ⊗ U∗AU.
• Use UYV ∗ as approximation for X .
8Benzi, Simoncini. Approximation of functions of large matrices with Kronecker structure, NumerischeMathematik, 2017.
24
Galerkin projection for evaluating f (M)vec(C)
Inspired by the Galerkin projection for Sylvester equations, we may consider the followingalgorithm8:
• Choose subspaces of Rn spanned by orthogonal matrices U,V .• Evaluate the projected matrix function
vec(Y ) = f ((V ⊗ U)∗M(V ⊗ U))vec(U∗CV ) = f (M)vec(U∗CV ),
where M := V ∗BV ⊗ I + I ⊗ U∗AU.• Use UYV ∗ as approximation for X .
8Benzi, Simoncini. Approximation of functions of large matrices with Kronecker structure, NumerischeMathematik, 2017.
24
Evaluating the small scale matrix function
For f (z) = z−1, evaluating f (M)vec(C) can be done in O(n3) using Bartels-Stewart.
For a more general f (z) and M = I ⊗ A + B ⊗ I:
• The projected matrix retains the same structure.• M can be diagonalized in O(n3) by diagonalizing A and B.
If Q∗AAQA = DA, Q∗BBQB = DB , then
vec(X ) = f (M)vec(C) = (QA ⊗ QB)f (I ⊗ DA + DB ⊗ I)vec(Q∗ACQB)
which can be written as
X = QA(F (Q∗ACQB))Q∗B , Fij := f (λA,i + λB,j).
Again, total cost is about O(n3).
25
Galerkin projection for evaluating f (M)vec(C)
• Algorithm reduces to Galerkin for Sylvester eqs. if f (z) = z−1.• Proposed by Benzi & Simoncini for more general f (z) using polynomial Krylov subspaces.• Using rational Krylov subspaces yields improved convergence in ill-conditioned cases;
major difficulty: how to choose the poles?
26
Determining the poles: Laplace-Stieltjes case
Here the situation is straightforward because e−tM = eI⊗(−tA)e(−tB)⊗I , so that
vec(X ) = f (M)vec(C)
=∫ ∞
0e−tMvec(C)dµ(t)
= vec(∫ ∞
0e−tACe−tBdµ(t)
)= vec
(∫ ∞0
(e−tAW
) (e−tBZ
)T dµ(t)).
Hence we can choose the same set of poles used in 1D case, for both Krylov subspaces.
27
Convergence result: Laplace Stieltjes case
TheoremThere exists a choice of poles that, for any Laplace-Stieltjes function f applied toM = I ⊗ A + B ⊗ I, where A,B are symmetric positive definite with spectrum contained in[λmin, λmax], yields the convergence
‖X − X`‖2 ≤ 16γ`,κf (0) · ‖C‖2 · ρ`2 , ρ = exp
(π2
log(4λmaxλmin
)
).
28
Numerical results
0 10 20 30 40 5010−15
10−9
10−3
103
Iterations (`)
‖X−
X `‖ 2
Evaluation of vec(X ) = ϕ1(I ⊗ A + A⊗ I)vec(C)
BoundExt. Kryl.Pol. Kryl.Rat. Kryl.
29
Determining the poles: Cauchy Stieltjes case
If f (z) =∫∞
0 dµ(t)/(z + t) then, f (M) =∫∞
0 dµ(t)(tI +M)−1.
Since tI +M = A⊗ I + I ⊗ (B + tI) we have
vec(X ) = f (M)vec(C) ⇐⇒ X =∫ ∞
0Xtdµ(t), AXt + Xt(B + tI) = C .
• Can we design a projection space that is good for all values of t?• This can be related to the Zolotarev problem:
max z ∈ [λmin, λmax]|r(z)|minz∈(−∞,−λmin] |r(z)|
• Using a Mobius transform, this can be mapped (approximately) into a Zolotarev problemon [−2λmax,−λmin] ∪ [λmin, 2λmax].
• We can compute optimal poles there and go back.
30
Convergence result: Cauchy Stieltjes case
TheoremThere exists a choice of poles that, for any Cauchy-Stieltjes function f applied toM = I ⊗ A + B ⊗ A, where A,B are symmetric positive definite with spectrum contained in[λmin, λmax], yields the convergence
‖X − X`‖2 ≤ 4 · f (2λmin) ·(
1 + λmaxλmin
)· ‖C‖2 · ρ−`, ρ = exp
(π2
log(8λmaxλmin
)
).
Note: the exponent for f (z) = z−1 is exp(
π2
log(4λmaxλmin
)
).
31
Decaying singular values
The theory predicts the decay in the singular values as well.
0 10 20 30 40 5010−17
10−12
10−7
10−2
`
Evaluation of vec(X ) = (I ⊗ A + A⊗ I)−0.7vec(C)
Bound‖X` − X‖2σ`(X )
32
Conclusions and outlook
• Practically, nested sequences of poles with the same asymptotic behavior can be used.
Possible extensions:
• Similar result for more general spectra configuration (and normal matrices) — implicitlydepends on the Zolotarev rational approximation problem.
• For non-normal cases, one can resort to using the field-of-values instead of the spectrum.• Divide and conquer methods for right hand sides obtained as vectorizations of banded or
hierarchical matrices.• Higher dimensional Laplace-like operators.
Full story:
• Rational Krylov for Stieltjes matrix functions: convergence and pole selection. S. Massei.,L.R., arXiv, 2019.
33
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