Research On The Condition Numbers For Scaled Total Least Squares Problem

As an important branch of numerical algebra,the least squares problem is widely used in physics,signal processing,system theory,statistics and medicine.In the process of exploring the least squares problem,many scholars put forward different least squares models for different practical applications,such as ordinary least squares,data least squares,total least squares,scaled total least squares,etc.,where the scaled total least squares plays an important role since it unifies the ordinary least squares,data least squares and total least squares.Based on the scaled total least squares problem,the mixed least squares-scaled total least squares problem and matrix scaled total least squares problem with multiple right hands are proposed.Moreover,the condition numbers of these two kinds of least squares problems are studied.The main contents include the following three parts:In the first part,as a new model,mixed least squares-scaled total least squares problem is proposed,which can unify mixed least squares-total least squares and scaled total least squares.Firstly,the explicit expression of mixed least squares-scaled total least squares solution is given under certain conditions.Secondly,the normwise,componentwise,mixed condition numbers and their corresponding upper bounds for mixed least squares-scaled total least squares problem which reduced to the published results for mixed least squares-total least squares problem and scaled total least squares problem are studied.Finally,numerical examples show that the upper bound of the condition number is indeed a good estimate of the corresponding exact value.In the second part,the normwise condition number of linear function for mixed least squares-scaled total least squares problem is mainly considered.Firstly,the normwise condition number of Fréchet differentiability is given.Secondly,in order to overcome the problem encountered in the calculation process,a more effective calculation formula is given.As special cases,the normwise condition numbers of linear functions of mixed least squares-total least squares solution and scaled total least squares solution are given accordingly.Finally,numerical examples show that there is a certian difference in condition number with different linear operators.In the third part,the matrix scaled total least squares problem with single right hand is extended to the case with multiple right hands.Firstly,under the Golub-Van Loan condition,an explicit expression for the solution of matrix scaled total least squares with multiple right hands is given.Secondly,based on Kronecker product,the normwise,componentwise,mixed condition numbers of matrix scaled total least squares are given.For the convenience of estimation,the upper bounds of these condition numbers are also given.In addition,all of these results can be reduced to the total least squares problem with multiple right hands.Finally,numerical examples show the effectiveness of these upper bounds.