How well do the Hermite-Pad e approximants reduce the Gibbs - - PowerPoint PPT Presentation

The Gibbs phenomenon Hermite-Pad´ e approach Particular sequences of HP approximants Numerical experiments

How well do the Hermite-Pad´ e approximants reduce the Gibbs phenomenon?

Ana C. Matos

joint work with B. Beckermann, F. Wielonsky Laboratoire Paul Painlev´ e Universit´ e de Lille I Valery Kaliaguine University of Nishninovgorod

Luminy, octobre 2009

Ana C. Matos HP approximants and the reduction of the Gibbs phenomenon

The Gibbs phenomenon Hermite-Pad´ e approach Particular sequences of HP approximants Numerical experiments

OUTLINE OF THE TALK

the problem: accelerating partial sums of Fourier series of functions with jumps (spectral methods in PDE) definition of the Gibbs phenomenon and some classical approaches new approach: use of Hermite-Pad´ e forms

definition of Hermite-Pad´ e approximants and motivation rate of convergence for a model problem comparison with Pad´ e approximants numerical experiments

Ana C. Matos HP approximants and the reduction of the Gibbs phenomenon

The Gibbs phenomenon Hermite-Pad´ e approach Particular sequences of HP approximants Numerical experiments Definition Some different approaches

The Gibbs phenomenon

Problem Given a small number of coefficients of a real–valued Fourier series construct point values of f (t) = Re(∞

j=0 cjeijt)

If we consider the partial sums Sn(f )(t) = Re(

n

• j=0

cjeijt) = a0 2 +

n

• j=1

[aj cos(jt) + bj sin(jt)], how good is the approximation?

Ana C. Matos HP approximants and the reduction of the Gibbs phenomenon

The Gibbs phenomenon Hermite-Pad´ e approach Particular sequences of HP approximants Numerical experiments Definition Some different approaches

Convergence results

f smooth and periodic: exponential accuracy max

−π≤x≤π |f (x) − fN(x)| ≤ e−αN,

α > 0 f ∈ Cm−1([−π, π]) periodic (ˆ fk = O(|k|−(m+1)), k → ±∞ max

−π≤x≤π |f (x) − fN(x)| = O(N−m), N → ∞

f discontinuous or non periodic: nonuniform convergence of the Fourier series |f (x0) − fN(x0)| ∼ O 1 N

• away from discontinuity

max

[−π,π] |f (x) − fN(x)| doesn’t tend to 0

⇒ GIBBS PHENOMENON: oscillations and bad convergence

Ana C. Matos HP approximants and the reduction of the Gibbs phenomenon

The Gibbs phenomenon Hermite-Pad´ e approach Particular sequences of HP approximants Numerical experiments Definition Some different approaches

Example:the saw-tooth function

La fonction 10 20 30 Partial Fourier sums –3 –2 –1 1 2 3 –3 –2 –1 1 2 3 t

How to overcome this phenomenon?

Ana C. Matos HP approximants and the reduction of the Gibbs phenomenon

The Gibbs phenomenon Hermite-Pad´ e approach Particular sequences of HP approximants Numerical experiments Definition Some different approaches

Some different approaches

1 Linear summation methods: Ces`

aro means, De la Vall´ ee-Poussin means;

2 Gottlieb approach: to obtain exponential accuracy in the

maximum norm in any interval of analiticity of a discontinuous piecewise analytic function - uses Gegenbauer polynomials;

3 Eckhoff approach: split a singular function into two parts, one

presenting some regularity and the other corresponding to the singularities, modellized by some prototype functions (more details later)

4 Fourier-Pad´

e approach

Ana C. Matos HP approximants and the reduction of the Gibbs phenomenon

The Gibbs phenomenon Hermite-Pad´ e approach Particular sequences of HP approximants Numerical experiments Definition Some different approaches

Definition of the Fourier-Pad´ e approximants

Consider the following procedure (C. Brezinski, P. Wynn): construct Sn(f )(t) + i Sn(f )(t) = Gn(f )(eit), with

• Sn(f )(t) = n

j=1[aj sin(jt) − bj cos(jt)],

Gn(f ) the nth Taylor sum of the (formal) series G(f )(z) =

∞

• j=0

cj(f )zj, c0(f ) = a0 2 , cj(f ) = aj − ibj.

compute Pad´ e approximants of this power series [n + k/k]f (z)

Ana C. Matos HP approximants and the reduction of the Gibbs phenomenon

The Gibbs phenomenon Hermite-Pad´ e approach Particular sequences of HP approximants Numerical experiments Definition Some different approaches

Definition of the Fourier-Pad´ e approximants

use the real part for approaching f (t)= Re (G(f )(eit)). ǫ(n)

2k (t) = p q(t) = Re

• [n + k/k]G(f )(eit)
• where p and q are trigonometric polynomials of degrees n + k

and k respectively. we showed that for f ∈ L2 and Q(z) = 0 (denominator polynomial) for |z| ≤ 1 then f − p

q is orthogonal to sin(jt),

cos(jt) for j = 0, 1, ...., m + k.⇔ non linear Fourier-Pad´ e approximants numerical examples for functions with jumps show very good acceleration properties and strong reduction of Gibbs

• scillations

Ana C. Matos HP approximants and the reduction of the Gibbs phenomenon

The Gibbs phenomenon Hermite-Pad´ e approach Particular sequences of HP approximants Numerical experiments Definition Some different approaches

Numerical example

|function - approximant Re([n+k|k](exp(i*t)))| |function - partial sum of order n+2*k| |function - Cesaro/Fejer mean of partial sum| |function - de la Vallee Poussin mean of partial sum| n=0,k=8 1e–16 1e–15 1e–14 1e–13 1e–12 1e–11 1e–10 1e–09 1e–08 1e–07 1e–06 1e–05 .1e–3 .1e–2 .1e–1 .1 1. .1e2 –3 –2 –1 1 2 3 t

s(t) = π + t for t ∈ (−π, 0], s(t) = −π + t for t ∈ (0, π], the 2π periodic saw tooth function, having one jump of absolute value 2π at t = 0 in [−π, π), approximants computed with 18 Fourier coefficients

Ana C. Matos HP approximants and the reduction of the Gibbs phenomenon

The Gibbs phenomenon Hermite-Pad´ e approach Particular sequences of HP approximants Numerical experiments Definition Some different approaches

Convergence results (B. Beckermann,AM, F.Wielonsky, 2008)

we consider a class of test functions G (α,β)(z) = 2F1

• α + 1, 1

α + β + 2

• z,
• ,

α, β > −1 some examples are f (t) = sign(cos(t)) f (t) = | sin(t/2)| ∈ C0 \ C1 f (t) = (1 − cos(s))s(t)

Ana C. Matos HP approximants and the reduction of the Gibbs phenomenon

The Gibbs phenomenon Hermite-Pad´ e approach Particular sequences of HP approximants Numerical experiments Definition Some different approaches

Convergence results

Convergence for columns max

t∈I

• f (t) − Re
• [n|k](eit)
• = O(n−2k)n→∞

for fixed k even after perturbation of f with Cm function, m sufficiently large. Convergence of ray sequences: for some γ ≥ 1 lim

k→∞,n=γk max t∈I

• f (t) − Re
• [n|k](eit)
• 1/k

< 1.

Ana C. Matos HP approximants and the reduction of the Gibbs phenomenon

The Gibbs phenomenon Hermite-Pad´ e approach Particular sequences of HP approximants Numerical experiments Definition Some different approaches

Why does it work so well?

s(t) = −2

∞

• j=1

sin(jt) j = Re

• G(s)(eit)
• ,

with G(s)(z) = 2i

∞

• j=1

zj j = −2i log(1 − z) = 2iz 1 dx 1 − xz . s1(t) = | sin( t

2)|,

G(s1)(z) = 2 π − z π 1 1 − x √x dx 1 − xz , Stieltjes functions gσ(z) = 1

dσ(x) 1−zx are very well approximated by

Pad´ e approximants in compact subsets of C \ [1, +∞), in particular all poles are on (1, +∞)

Ana C. Matos HP approximants and the reduction of the Gibbs phenomenon

The Gibbs phenomenon Hermite-Pad´ e approach Particular sequences of HP approximants Numerical experiments Definition of Hermite-Pad´ e approximation Model problems: Nikishin systems Results on potential theory Rate of convergence

Why Hermite-Pad´ e forms ?

Hypothesis: We know location of singularity but not amplitude of jumps! If f ∈ Cn1+1([−π, 0) ∪ (0, π]) periodic has left- and right-hand side derivatives of order 0, 1, ..., n1 at t = 0: ∃dj ∈ R : e(t) = f (t) −

n1

• j=0

dj sinj(t)s(t) ∈ Cn1+1([−π, π]) In terms of z = eit: f (t) = Re (F(z)),

n1

• j=0

dj sinj(t) = Re ( i 2p1(z)), s(t) = Re 2 i log(1 − z)

• Ana C. Matos

HP approximants and the reduction of the Gibbs phenomenon

The Gibbs phenomenon Hermite-Pad´ e approach Particular sequences of HP approximants Numerical experiments Definition of Hermite-Pad´ e approximation Model problems: Nikishin systems Results on potential theory Rate of convergence

Why Hermite-Pad´ e forms ?

thus reasonable approximation: f (t) ≈ Re

• −p0(z) − p1(z) log(1 − z)
• with

p0 ∈ Pn0, p1 ∈ Pn1, p0(z) + p1(z) log(1 − z) + F(z) = O(zn0+n1+2)z→0. particular case of Hermite-Pad´ e approximants

Ana C. Matos HP approximants and the reduction of the Gibbs phenomenon

The Gibbs phenomenon Hermite-Pad´ e approach Particular sequences of HP approximants Numerical experiments Definition of Hermite-Pad´ e approximation Model problems: Nikishin systems Results on potential theory Rate of convergence

Hermite-Pad´ e approximation

Definition Find pj ∈ Pnj for j = 0, 1, 2 such that p0(z) + p1(z)g1(z) + p2(z)g2(z) = O(zn0+n1+n2+2)z→0, The Hermite–Pad´ e approximant of g2(z) (or in short HP approximant) of order n = (n0, n1, n2) is defined as Π

n(z) = −p0(z) + p1(z)g1(z)

p2(z) .

Ana C. Matos HP approximants and the reduction of the Gibbs phenomenon

The Gibbs phenomenon Hermite-Pad´ e approach Particular sequences of HP approximants Numerical experiments Definition of Hermite-Pad´ e approximation Model problems: Nikishin systems Results on potential theory Rate of convergence

Hermite-Pad´ e approximation

In our case g1(z) = log(1 − z), g2(z) = F(z) and approach f (t) = Re (g2(eit)) by f (t) ≈ Re

• Πn0,n1,n2(eit)
• Approximant with built-in singularity

Ana C. Matos HP approximants and the reduction of the Gibbs phenomenon

The Gibbs phenomenon Hermite-Pad´ e approach Particular sequences of HP approximants Numerical experiments Definition of Hermite-Pad´ e approximation Model problems: Nikishin systems Results on potential theory Rate of convergence

Hermite-Pad´ e approximation

approach first proposed by Driscoll and Fornberg - singular Fourier-Pad´ e approximant - with very convincing numerical experiments we can obtain rates of convergence by studying convergence

• f HP-approximants

n2 = 0: the Eckhoff approach: subtracting the singular part n1 = −1: Fourier-Pad´ e approximants (find themselves singularities)

Ana C. Matos HP approximants and the reduction of the Gibbs phenomenon

The Gibbs phenomenon Hermite-Pad´ e approach Particular sequences of HP approximants Numerical experiments Definition of Hermite-Pad´ e approximation Model problems: Nikishin systems Results on potential theory Rate of convergence

Model problems (logarithmic singularity at z = 1)

g1(z) = log(1 − z) = z 1

dx 1−xz .

g2(z) = z 1

u(x) dx 1−xz ,

u(x) = d

c dτ(y) x−y , [c, d] ∩ [0, 1] = ∅.

Property (1, g1(1/z), g2(1/z)) form a Nikishin system. ⇒ polynomials and residuals involved in their Hermite–Pad´ e approximants satisfy orthogonality relations with respect to varying weights ⇒ their n-th root asymptotics can be given in terms of the solution of a vector equilibrium problem in potential theory.

Ana C. Matos HP approximants and the reduction of the Gibbs phenomenon

The Gibbs phenomenon Hermite-Pad´ e approach Particular sequences of HP approximants Numerical experiments Definition of Hermite-Pad´ e approximation Model problems: Nikishin systems Results on potential theory Rate of convergence

Outline

some results of potential theory rate of convergence of Hermite-Pad´ e approximants error estimates for particular cases

diagonal sequences row sequences linear HP-approximants comparison with Pad´ e approximants

numerical experiments

Ana C. Matos HP approximants and the reduction of the Gibbs phenomenon

The Gibbs phenomenon Hermite-Pad´ e approach Particular sequences of HP approximants Numerical experiments Definition of Hermite-Pad´ e approximation Model problems: Nikishin systems Results on potential theory Rate of convergence

Definitions of potential theory

Mρ(∆) = {µ measure : supp (µ) ⊂ ∆, µ(C) = ρ} logarithmic potential Uµ(x) :=

• log(

1 |x−y|)dµ(y).

Consider σ ∈ M1([α, β]) and wn ∈ C([α, β]) sequence of weight functions. The corresponding orthonormal polynomials with varying weights {pk,n} , k, n ≥ 0 satisfy j, k = 0, 1, ... :

• wn(x)pj,n(x)pk,n(x) dσ(x) = δj,k,

with zero counting measures χk(pk,n) = 1

k

• pk,n(ξ)=0 δξ.

Ana C. Matos HP approximants and the reduction of the Gibbs phenomenon

The Gibbs phenomenon Hermite-Pad´ e approach Particular sequences of HP approximants Numerical experiments Definition of Hermite-Pad´ e approximation Model problems: Nikishin systems Results on potential theory Rate of convergence

Weak asymptotics for OP with varying weight

Lemma (Stahl & Totik ’92) If w1/n

n

→ exp(−2Q) uniformly in [α, β] and σ ∈ Reg then we have the weak star convergence χn(pn,n) → σµ where σµ ∈ M1([α, β]) is the unique minimizer in M1([α, β]) of IQ(ν) =

• log(

1 |x − y|) dν(x)dν(y) + 2

• Q dν

Ana C. Matos HP approximants and the reduction of the Gibbs phenomenon

The Gibbs phenomenon Hermite-Pad´ e approach Particular sequences of HP approximants Numerical experiments Definition of Hermite-Pad´ e approximation Model problems: Nikishin systems Results on potential theory Rate of convergence

Weak asymptotics for OP with varying weight

The minimizer σµ is uniquely characterized by the equilibrium conditions: ∃W ∈ R Uσµ(x) + Q(x) ≥ W if x ∈ [α, β], = W if x ∈ supp (σµ). Q = 0, supp (σ) = [α, β]: σ ∈ Reg iff σµ = ω[α,β] equilibrium measure.

Ana C. Matos HP approximants and the reduction of the Gibbs phenomenon

The Gibbs phenomenon Hermite-Pad´ e approach Particular sequences of HP approximants Numerical experiments Definition of Hermite-Pad´ e approximation Model problems: Nikishin systems Results on potential theory Rate of convergence

we consider sequences of Hermite-Pad´ e approximants satisfying the total degree n = n0 + n1 + n2 → ∞; n0 ≥ n1 ≥ n2; ray sequences n0, n1, n2 such that n0 n → ρ0, n1 n → ρ1, n2 n → ρ2. Then the n-th root asymptotic behavior for the error function is given by

Ana C. Matos HP approximants and the reduction of the Gibbs phenomenon

The Gibbs phenomenon Hermite-Pad´ e approach Particular sequences of HP approximants Numerical experiments Definition of Hermite-Pad´ e approximation Model problems: Nikishin systems Results on potential theory Rate of convergence

Rate of convergence

Theorem Assume that [c, d] is a compact interval and that τ ∈ Reg. Then, the error function (g2 − Π

n)(1/z) satisfies, locally uniformly for

z ∈ C \ ([0, 1] ∪ [c, d]), limn→∞ 1

n log |(g2 − Π n)(1/z)| =

(ρ1 + ρ2)Uµ(z) + ρ2Uν(z) + (ρ0 − ρ2)Uδ0(z) − W − w, the probability measures µ and ν, and the constants W and w, solve the following vector equilibrium problem in potential theory

Ana C. Matos HP approximants and the reduction of the Gibbs phenomenon

The Gibbs phenomenon Hermite-Pad´ e approach Particular sequences of HP approximants Numerical experiments Definition of Hermite-Pad´ e approximation Model problems: Nikishin systems Results on potential theory Rate of convergence

Rate of convergence

Theorem (cont.) µ, ν are the unique measures in M1([0, 1]), and M1([c, d]), respectively, satisfying the equilibrium conditions 2(ρ1 + ρ2)Uµ(x) − ρ2Uν(x) + (ρ0 − ρ1)Uδ0(x) ≥ W x ∈ [0, 1], = W x ∈ supp (µ) −(ρ1 + ρ2)Uµ(x) + 2ρ2Uν(x) + (ρ1 − ρ2)Uδ0(x) ≥ w x ∈ [c, d], = w x ∈ supp (ν). where W , w ∈ R.

Ana C. Matos HP approximants and the reduction of the Gibbs phenomenon

The Gibbs phenomenon Hermite-Pad´ e approach Particular sequences of HP approximants Numerical experiments Definition of Hermite-Pad´ e approximation Model problems: Nikishin systems Results on potential theory Rate of convergence

idea of the proof

Let An(z) = zn1p1(1/z), Bn(z) = zn2p2(1/z) monic, Cn(z) = An(z) + zn1−n2Bn(z)u(z)

• rthogonality relations for Cn(x):

1 xn0−n1Cn(x)xkdx = 0, k = 0, ..., n1 + n2 denote Hn ∈ Pn1+n2+1 monic with roots the simple zeros of Cn; orthogonality of Hn with respect to the varying weights nn0−n1Cn(x)/Hn(x);

• rthogonality relations for Bn(x):

k = 0, 1, ..., n2 − 1 : d

c

yn1−n2yk Bn(y) Hn(y)dτ(y) = 0.

Ana C. Matos HP approximants and the reduction of the Gibbs phenomenon

The Gibbs phenomenon Hermite-Pad´ e approach Particular sequences of HP approximants Numerical experiments Definition of Hermite-Pad´ e approximation Model problems: Nikishin systems Results on potential theory Rate of convergence

applying the lemma we obtain existence and uniqueness of two measures µ1 and µ2 χn(Hn) → µ, χn(Bn) → ν. satisfying the previous system of equilibrium conditions integral representation of the error: Bn(z)zn0−n2(g2(1/z) − Πn(1/z)) = 1 Hn(z) 1 xn0−n1Hn(x)Cn(x) z − x dx.

Ana C. Matos HP approximants and the reduction of the Gibbs phenomenon

The Gibbs phenomenon Hermite-Pad´ e approach Particular sequences of HP approximants Numerical experiments Diagonal HP-approximants Linear HP-approximants Fixed denominator degree HP-approximants

Comparison with Pad´ e approximants

Aim: compare the rate achieved by HP-approximants of type (n0, n1, n2) as (n → ∞), n = n0 + n1 + n2, n0 ≥ n1 ≥ n2 with that of Pad´ e approximants of type (m0, −1, m2) as m → ∞, m = m0 + m2, m0 ≥ m2 constructed from the same number of Taylor coefficients of g2 ⇒ m = n + 1 based on the solution of linear systems of equal dimensions ⇒ m0 = n0, m2 = n1 + n2 + 1

Ana C. Matos HP approximants and the reduction of the Gibbs phenomenon

The Gibbs phenomenon Hermite-Pad´ e approach Particular sequences of HP approximants Numerical experiments Diagonal HP-approximants Linear HP-approximants Fixed denominator degree HP-approximants

Rate of convergence of Pad´ e approximants

Let Θ

m = −

P0/ P2 is the Pad´ e approximant of type (m0, −1, m2)

• f the function g2 at the origin. We consider a ray sequence

m0 m → σ0 > 0, m2 m → σ2 > 0,

Theorem for z ∈ C \ [0, 1], lim

m→∞

1 m log |(g2 − Θ

m)(1/z)| = 2σ2U µ(z) + (σ0 − σ2)Uδ0(z) −

W . where µ measure, supported on [0, 1], solution of an equilibrium problem in potential theory 2σ2U

µ(x) + (σ0 − σ2)Uδ0(x)

• ≥

W , x ∈ [0, 1], = W , x ∈ supp ( µ).

Ana C. Matos HP approximants and the reduction of the Gibbs phenomenon

The Gibbs phenomenon Hermite-Pad´ e approach Particular sequences of HP approximants Numerical experiments Diagonal HP-approximants Linear HP-approximants Fixed denominator degree HP-approximants

Diagonal HP-approximants (n1 = n2)

ray sequences for Π

n (HP):

n = (n0, n1, n1) as (n → ∞) , ρ0 ≥ ρ1 = ρ2 for Θ

m (Pad´

e): m = (n0, −1, 2n1 + 1) , ρ0 ≥ 2ρ1 lim

z→1 |z|=1,z=1 limn→∞ |(g2 − Π n)(z)|1/n< 1 ,

lim

z→1 |z|=1,z=1 limm→∞ |(g2 − Θ m)(z)|1/m= 1.

⇒ there exists a neighborhood of 1 in C \ (0, 1) in which the Hermite–Pad´ e approximants achieve a rate of convergence which is better than the rate of the Pad´ e approximants.

Ana C. Matos HP approximants and the reduction of the Gibbs phenomenon

The Gibbs phenomenon Hermite-Pad´ e approach Particular sequences of HP approximants Numerical experiments Diagonal HP-approximants Linear HP-approximants Fixed denominator degree HP-approximants

Linear HP-approximants (n2 = 0)

for Π

n (HP):

n = (n0, n1, 0) as (n → ∞) , ρ0 ≥ ρ1 > 0 HP approximants without denominator (Eckhoff approximants) for Θ

m (Pad´

e): m = (n0, −1, n1 + 1) lim

z→1 |z|=1,z=1 limn→∞ |(g2 − Π n)(z)|1/n< 1 ,

lim

z→1 |z|=1,z=1 limm→∞ |(g2 − Θ m)(z)|1/m= 1.

⇒ there exists a neighborhood of 1 in C \ (0, 1) in which the Hermite–Pad´ e approximants achieve a rate of convergence which is better than the rate of the Pad´ e approximants.

Ana C. Matos HP approximants and the reduction of the Gibbs phenomenon

The Gibbs phenomenon Hermite-Pad´ e approach Particular sequences of HP approximants Numerical experiments Diagonal HP-approximants Linear HP-approximants Fixed denominator degree HP-approximants

Fixed denominator degree HP-approximants

row sequences for Π

n (HP):

n = (n0, n1, n1) such that n0 → ∞ while n1 remains constant , ρ0 = 1, ρ1 = ρ2 = 0 for Θ

m (Pad´

e): m = (n0, −1, 2n1 + 1) Theorem Assume that the measure dτ(y) in the definition of the function u(x) is regular and that its support [c, d] ⊂ (−∞, 0). Let |z − 1| ≤ 1/2. Then, for n0 sufficiently large so that C ≤ (n0 − 2n1 − 2)(1 − Re (z)), we have |(g2 − Π

n)(1/z)|

|(g2 − Θ

m)(1/z)| ≤

C|z − 1|2n1−1, where C and C are some constants that depend only on n1. ⇒ for z sufficiently close to 1, the Hermite–Pad´ e approximants do

Ana C. Matos HP approximants and the reduction of the Gibbs phenomenon

The Gibbs phenomenon Hermite-Pad´ e approach Particular sequences of HP approximants Numerical experiments

g2(z) = z 1

u(x) 1−xzdx,

u(x) = log x−c

d−x

• =

d

c dy x−y ,

g1(z) = i log(1 − z)

[c, d] = [−2, −0.3] 16 coefficients n1 + n2 + 2 = 6 unknowns partial sum n = 16 Pad´ e approximant n = (10, −1, 5) linear HP n = (10, 4, 0)

• n = (10, 3, 1)

“diagonal” approximant

• n = (10, 2, 2)
• Re(g2(eit) − Π

n(eit))

• close to singularity

from Pad´ e to linear HP: we gain 4 digits from linear HP to “diagonal” approximants: we gain 3 or 4 digits

Ana C. Matos HP approximants and the reduction of the Gibbs phenomenon

The Gibbs phenomenon Hermite-Pad´ e approach Particular sequences of HP approximants Numerical experiments

g2(z) = z 1

u(x) 1−xzdx,

u(x) = log x−c

d−x

• =

d

c dy x−y ,

g1(z) = i log(1 − z)

16 coefficients computed error curves versus theoretical error estimates

Ana C. Matos HP approximants and the reduction of the Gibbs phenomenon

The Gibbs phenomenon Hermite-Pad´ e approach Particular sequences of HP approximants Numerical experiments

g2(z) = z 1

u(x) 1−xzdx,

u(x) = log x−c

d−x

• =

d

c dy x−y ,

g1(z) = i log(1 − z)

increasing n0 ( n1, n2 fixed)

14 coefficients: n0 = 14 22 coefficients: n0 = 22

close to singularity from top to bottom:

partial sum n0 + 7, Pad´ e app n = (n0, −1, 7), linear HP n = (n0, 6, 0) and diagonal HP n = (n0, 3, 3)

Ana C. Matos HP approximants and the reduction of the Gibbs phenomenon

The Gibbs phenomenon Hermite-Pad´ e approach Particular sequences of HP approximants Numerical experiments

Bibliography

• B. Beckermann, A. Matos, F. Wielonsky, Reduction of the Gibbs phenomenon

for smooth functions with jumps by the ǫ-algorithm,JCAM 219 (2008), 329-349.

• C. Brezinski, Extrapolation algorithms for filtering series of functions, and

treating the Gibbs phenomenon, Numer. Algorithms 36(2004) 309-329.

• T. Driscoll, B. Fornberg,A Pad´

e-based algorithm for overcoming the Gibbs phenomenon, Numer. Algo. 26 (2001) 77-92.

• K. Eckhoff, Accurate and effcient reconstruction of discontinuous functions from

truncated series expansions, Maths. Comp. 204 (1993), 745-763.

• U. Fidalgo and G. Lopez, Rate of convergence of Generalized Hermite-Pad´

e Apporximants of Nikishin systems,Const. Approx. 23 (2006), 165-196

• E. Nikishin, V. Sorokin, Rational Approximations and Orthogonality, American

Mathematical Society.

Ana C. Matos HP approximants and the reduction of the Gibbs phenomenon