Backward error analysis of the Lanczos bidiagonalization with reorthogonalization
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The k-step Lanczos bidiagonalization reduces a matrix A∈ℝ^m× n into a bidiagonal form B_k∈ℝ^(k+1)× k while generates two orthonormal matrices U_k+1∈ℝ^m× (k+1) and V_k+1∈ℝ^n×(k+1). However, any practical implementation of the algorithm suffers from loss of orthogonality of U_k+1 and V_k+1 due to the presence of rounding errors, and several reorthogonalization strategies are proposed to maintain some level of orthogonality. In this paper, by writing various reorthogonalization strategies in a general form we make a backward error analysis of the Lanczos bidiagonalization with reorthogonalization (LBRO). Our results show that the computed B_k by the k-step LBRO of A with starting vector b is the exact one generated by the k-step Lanczos bidiagonalization of A+E with starting vector b+δ_b (denoted by LB(A+E,b+δ_b)), where the 2-norm of perturbation vector/matrix δ_b and E depend on the roundoff unit and orthogonality levels of U_k+1 and V_k+1. The results also show that the 2-norm of U_k+1-U̅_k+1 and V_k+1-V̅_k+1 are controlled by the orthogonality levels of U_k+1 and V_k+1, respectively, where U̅_k+1 and V̅_k+1 are the two orthonormal matrices generated by the k-step LB(A+E,b+δ_b) in exact arithmetic. Thus the k-step LBRO is mixed forward-backward stable as long as the orthogonality of U_k+1 and V_k+1 are good enough. We use this result to investigate the backward stability of LBRO based SVD computation algorithm and LSQR algorithm. Numerical experiments are made to confirm our results.
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