Condition Numbers For Indefinite Least Squares Problem With Multiple Right-hand Sides

The indefinite least squares problem with multiple right-hand sides is a generalization of the indefinite least squares problem (?)(b-Ax)T J(b-Ax)on matrix equation AX = B.The problem was first proposed by Ou yangjun and Peng zhenyun and its definition is(?) where the tr is the trace,A ? Rm×n,B?Rm×s are m×n and m×s real matrices and J is the signature matrix,that is,it is a diagonal matrix which the diagonal elements are±1.The condition number is a basic concept in numerical algebra,which has some important applications in numerical algorithm.For example,we can estimate the forward error of a backward stable algorithm when combining the condition number of the problem and the backward error of the algorithm.So the condition number theory plays a very important role in error analysis.In this paper,we investigate the condition numbers for indefinite least squares problem with multiple right-hand sides.The normwise,mixed and componentwise condition numbers and the corresponding structured condition numbers are presented.The structured matrices under consideration include the linear structured matrices,such as the Toeplitz,Hankel,symmetric,and tridiagonal matrices,and the nonlinear structured matrices,such as the Vandermonde and Cauchy matrices.Numerical examples show that the structured condition numbers are tighter than the unstructured ones.Besides we exhibit explicit expressions for normwise,mixed and componentwise condition numbers for the residuals of indefinite least squares problem with multiple right-hand sides.Considering the connections between the indefinite least squares problem and total least squares problem,we derive the condition numbers for the residuals of total least squares problem with multiple right-hand sides.