Research On The Total Least Squares Problem And Related Theories

This paper presents the convergence analysis,where the new theoretical error bounds on the convergence of the Lanczos methods are established based on results given by Saad,of the Lanczos bidiagonalization process to solve the total least squares(TLS)problem with the Gene-Golub condition.By considering the error between the approximate system generated by this process and the TLS system,we analyze how to effectively give the stopping criterion of this algorithm.In view of the minimality property of the core problem,we use the Lanczos bidiagonaliztion algorithm to solve the ill-conditioned TLS problem and estimate the error between the truncated total least squares(TTLS)solution and the projected TLS solution.In this paper,we find that the extended minimal backward errors and then the true minimal backward errors of the scaled total least squares(STLS)problem are equaled with those of its core problem.The same is true for the asymptotic estimate for the extended minimal backward errors.Thus it can effectively reduce the amount of computation of the backward errors due to the smaller size of the core problem.We also give practical and cheaply computable estimates of the backward errors for the STLS problem by Lanczos bidiagonalization process to it,and show how to use our results to easily obtain the corresponding results for the least squares(LS)and the data least squares(DLS)problems.We propose practical stopping criteria for the iterative solution of the STLS problems.In many linear parameter estimation problems,one can use the mixed least squares-total least squares(MTLS)approach to solve them.This paper is devoted to the perturbation analysis of MTLS problem which has not been considered before.Firstly,we present the normwise,mixed and componentwise condition numbers of the MTLS problem,and find that the normwise,mixed and componentwise condition numbers of the TLS problem and the LS problem are unified in the ones of the MTLS problem.In the analysis of the first order perturbation,we first provide an upper bound based on the normwise condition number.In order to overcome the problems encountered in calculating the normwise condition number,we give an upper bound for computing more effectively for the MTLS problem.As two estimation techniques for solving the linear parameter estimation problems,interesting connections between their solutions,their residuals for the MTLS problem and the LS problem are compared.In many applications in scientific computing and engineering one has to solve huge sparse linear systems of equations with several right-hand sides.Customarily,one uses the residual error as a stopping condition,but small residue does not imply accurate approximate solutions.Therefore,minimizing quantities other than the residual norm may be more adequate in this context.Based on the above considerations and the idea of solving minimal perturbations and corresponding solutions for TLS problems,we propose a block minimal joint backward perturbation algorithm(BMinPet).Combining the normwise backward error meets some optimality condition with minimizing the block joint backward perturbation norm of the matrix(A,B),a stopping criterion is given for consistent system with multiple right-hand sides.This process is a generalization of minimum perturbation algorithm(MinPert)by Kasenally&Simoncini to multiple right-hand sides.For the benefit of the amount of calculation,we give lower and upper bounds about block joint backward perturbation norm.As a byproduct,we come up with a conception of Function?_?(F,G)which is a generalization of Function?_?that is a symmetric gauge(SG)function.The relationships between BMinPet method and relative methods are investigated.Numerical examples show us that BMinPert shows its superiority compared with BFGMRES-S(m,p_f),GsGMRES,Bl-BiCG-rQ,BGMRES and BArnoldi in solving the large sparse ill-conditioned problems.Finally,Some numerical experiments are performed to illustrate our results.