On Perturbation bounds for HR decomposition
In 1, Bunser-Gerstner showed that almost every complex square matrix A cart be decomposed into a product of a so-called pseudo-Hermitian matrix H and an upper triangular matrix:A=HR and gave the necessary conditions for the existence and the uniqueness of HR decomposition. Let J = diag(±1 ) be a signature matrix. Let A be the set of those matrices A for which the leading principal minors of A* J A have the same signs as the corresponding minors of J, R. Bhatia in 2 showed that for any A∈A has a decomposition A= HR,where R is the upper triangular matrices whose diagonal entries are all positive, H satisfies H*JH = J.Let A and A = A + E have the decomposition A = HR and A=HR, R. Bhatia 2 obtained ‖(H)-H‖F≤√(2cond(H)‖R-1 ‖‖(A)-A‖F, (1)‖( R )-R‖F≤(2)cond ( R ) ‖H-1‖‖(A)-A‖F. (2)Where cond(X) =‖1X‖‖X-1‖,‖X‖is the spectral norm, ‖X‖F is the Frobenius norm. Adopting the technique used in 3,4, in this paper we first establish some new first-order bounds of the HR decomposition for A +tG = H(t)R(t), t≤ε if ε is small enough that the leading principal minors of (A + tG)T J(A + tG) have the same signs as the corresponding minors of J, and get H(O) = GR-1- H R(O)R-1, (3)R(0) = Jup(R-1 GT JH + Hr jGR-1)R. (4)In particular, when △A = εG, A +△A has the decomposition A +△A = (H + △H)(R + △R) with △H = εH(0) + O(ε2), △R =εk(0) + 0(ε2).Using the row scaling on the perturbation analysis for the sensitivity of R , the more tighter bound than(2) we obtain is‖R(O)‖F/‖R‖F≤KR(A) ‖G‖F/‖H‖F (5)where KR(A)inf D∈Dn KR(A,D)inf D∈Dn 1+ξ2 K2(D-1 R) ,D =diag(δ1,δ2,…,δn) ~ l) the set of all n×n real positive definitive diagonal matrices and δD - max δj/δi,δi> 0. And ‖H(0)‖F/‖H‖F ≤KH(A)‖G‖F/‖H‖F (6)is also more tighter than (1) for the sensitivity of H. Where K H(A) inf KH(A,D)inf 1+δD2 K2(HD-1) ‖R-1‖‖2‖G‖2.The condition estimates for the HR decomposition can also be derived by (5) and (6).
condition estimation HR decomposition perturbation analysis
Linzhang Lu Jiangzhou Lai
School of Mathematical Sciences, Xiamen University, Xiamen 361005, P.R.China
国际会议
The Third International Workshop on Applied Matriz Theory(第三届国际矩阵分析与应用会议)
杭州
英文
851-852
2009-07-09（万方平台首次上网日期，不代表论文的发表时间）