On theoretical and numerical aspects of symplectic Gram?Schmidt-like algorithms

Abstract

Gram-Schmidt-like orthogonalization process with respect to a given skew-symmetric scalar product is a key step in model reduction methods, structure-preserving, for large sparse Hamiltonian eigenvalue problem. Theoretical as well as numerical aspects of this step do not benefit of enough attention, compared to the one allowed to the classical Gram-Schmidt algorithm and its modified version. The aim of this paper is to revisit the symplectic GramSchmidt algorithms, to built some modified versions and to deal with their theoretical and numerical features.

Preserving geometric properties of the exponential matrix by block Krylov subspace methods

“…(3.4)), with the matrix D m having a convenient structure. This constraint conforms with the standard requirements used for building single vector Lanczos-type recurrences that preserve the symplectic structure of the problem [7,8,46,44].…”

“…Further developments include the derivation of a Hamiltonian matrix H m to effectively approximate eigenvalues of a Hamiltonian matrix A; see [7,8,46]. A more recent implementation of the Gram-Schmidt algorithm for the construction of a symplectic basis for p = 1 is given in [44], and expanded in [45] to devise a symplectic 'block-style' Lanczos method for the eigenvalue problem again for p = 1.…”

Block symplectic Gram–Schmidt method

“…In this section the bsgs with Algorithm 2 and Algorithm 3, csgs and msgs are tested [9]. The numerical environment is…”

“…Salam [10] proved that sr is equivalent to the modified symplectic Gram-Schmidt (msgs) method. Another method is sr factorizaton by the classical symplectic GramSchmidt (csgs) method [9]. However, there are fewer numerical experiments documented on the sgs compared to the gs decomposition for the qr, Arnoldi and Lanczos methods.…”

Symplectic Entropy as a Novel Measure for Complex Systems

“…The symplectic similarities in the symplectic space can preserve the structures of the Hamiltonian matrices [13][14][15]. Some numerical algorithms based on symplectic geometry-such as symplectic Householder transformations, symplectic QR-like decomposition [16], or the symplectic Gram-Schmidt algorithm [17]-are proposed and modified to solve the eigenvalues of the Hamiltonian matrices, particularly for sparse and large structured matrices [18]. Symplectic eigensolutions are proposed to perform the energy band analysis for a periodical waveguide by introducing symplectic mathematics into the electro-magnetic waveguide theory [19].…”