Optimal orthogonalization processes  

作　　者：Marko Huhtanen Pauliina Uusitalo 

机构地区：[1]Department of Electrical and Information Engineering,University of Oulu,Oulu 90570,Finland

出　　处：《Science China Mathematics》2022年第1期203-220,共18页中国科学：数学（英文版）

基　　金：supported by the Academy of Finland(Grant No.288641)。

摘　　要：Two optimal orthogonalization processes are devised toorthogonalize,possibly approximately,the columns of a very large and possiblysparse matrix A∈C^(n×k).Algorithmically the aim is,at each step,to optimallydecrease nonorthogonality of all the columns of A.One process relies on using translated small rank corrections.Another is a polynomial orthogonalization process forperforming the Löwdin orthogonalization.The steps rely on using iterative methods combined,preferably,with preconditioning which can have a dramatic effect on how fast thenonorthogonality decreases.The speed of orthogonalization depends on howbunched the singular values of A are,modulo the number of steps taken.These methods put the steps of the Gram-Schmidt orthogonalizationprocess into perspective regardingtheir(lack of)optimality.The constructions are entirely operatortheoretic and can be extended to infinite dimensional Hilbert spaces.

关 键 词：optimal orthogonalization sparse matrix Gram-Schmidt orthogonalization Lowdin orthogonalization polynomial orthogonalization implicit orthogonalization PRECONDITIONING Gram matrix frame inequality 

分 类 号：O151.21[理学—数学]