A Gaussian Method for the Square Root of Accretive Operators

Eleonora Denich; Paolo Novati

doi:10.1515/cmam-2022-0033

Article

A Gaussian Method for the Square Root of Accretive Operators

Eleonora Denich and Paolo Novati

Published/Copyright: August 30, 2022

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From the journal Computational Methods in Applied Mathematics Volume 23 Issue 1

Abstract

We consider the approximation of the inverse square root of regularly accretive operators in Hilbert spaces. The approximation is of rational type and comes from the use of the Gauss–Legendre rule applied to a special integral formulation of the fractional power. We derive sharp error estimates, based on the use of the numerical range, and provide some numerical experiments. For practical purposes, the finite-dimensional case is also considered. In this setting, the convergence is shown to be of exponential type. The method is also tested for the computation of a generic fractional power.

Keywords: Fractional Powers; Regularly Accretive Operators; Gaussian Quadrature Rule

MSC 2010: 47A58; 65F60; 65D32

Funding source: Gruppo Nazionale per il Calcolo Scientifico

Award Identifier / Grant number: 2019-04

Funding statement: This work was partially supported by GNCS-INdAM, FRA-University of Trieste and CINECA under HPC-TRES program award number 2019-04. The authors are members of the INdAM research group GNCS.

A Approximation of β ⋆

Let Ψ s 0 ∈ E be the ellipse passing through the point 2 τ ⁢ i - 1 (cf. (4.11)). The value β ⋆ , such that the function Φ ( 2 ) ⁢ ( 1 + ρ ⁢ e - i ⁢ β ⁢ π ) possesses a local maximum for β > β ⋆ , is the one for which Ψ s 0 is also tangent to the curve χ ( 2 ) ⁢ ( Γ β ⋆ - ) at 2 τ ⁢ i - 1 (Figure 3). In order to compute β ⋆ , we consider the tangents at 2 τ ⁢ i - 1 to the ellipse and to the curve, and impose them to have the same slope. Before starting, we need to derive the semi-width 𝛾 and the semi-height 𝛿 of the ellipse. By geometrical evidence, we have that

(A.1) δ = ℑ ⁡ { 1 2 ⁢ ( s 0 ⁢ e i ⁢ π 2 + 1 s 0 ⁢ e - i ⁢ π 2 ) } = 1 2 ⁢ ( s 0 - 1 s 0 )

and γ 2 = δ 2 + 1 . At this point, we remind that the slope of the tangent at 2 τ ⁢ i - 1 to the ellipse is

(A.2) m = - δ 2 γ 2 ⁢ ℜ ⁡ ( 2 τ ⁢ i - 1 ) ℑ ⁡ ( 2 τ ⁢ i - 1 ) = δ 2 δ 2 + 1 ⁢ ( τ 2 ) ∼ τ 2 + τ ∼ 1 , τ → + ∞ ,

where we have used (A.1) and

s 0 - 1 s 0 ∼ 4 2 ⁢ τ - 1 4 ,

which comes from (4.11). Now, it is not difficult to show that the angle between the tangent to the curve χ ( 2 ) ⁢ ( Γ β - ) at 2 τ ⁢ i - 1 and the line ℑ ⁡ ( z ) = 2 τ is given by 1 2 ⁢ ( π - 2 ⁢ β ⁢ π ) . Hence, in order to find an approximation of β ⋆ , by (A.2), we impose the condition

tan ⁡ [ π 2 ⁢ ( 1 - 2 ⁢ β ) ] = 1

that leads to β ⋆ ∼ 1 4 as τ → + ∞ .

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Received: 2022-02-03

Revised: 2022-05-09

Accepted: 2022-08-17

Published Online: 2022-08-30

Published in Print: 2023-01-01

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Articles in the same Issue

https://doi.org/10.1515/cmam-2022-0033

Keywords for this article

Fractional Powers; Regularly Accretive Operators; Gaussian Quadrature Rule