Block symplectic Gram–Schmidt method

Abstract

For large scale linear problems, it is common to use the symplectic Lanczos method which uses the symplectic Gram-Schmidt method to compute symplectic vectors. However, previous studies showed that the selection process of the parameter in the symplectic Gram-Schmidt method is flawed, as it results in a partially destroyed J-orthogonality of the J-orthogonal matrix. We explore a block type symplectic GramSchmidt method and a new condition for the reorthogonalization to maintain J-orthogonality and to more accurately compute symplectic factorization. Applying the block size scheme to this method, we develop a new procedure for computing symplectic vectors.