A Subspace Error Estimate for Linear Systems

This paper proposes a new method for estimating the error in the solution of linear systems. A condition number is defined for a linear function of the solution components. This definition of the condition number is quite versatile. It reduces to the component condition number proposed by Chandrasekaran and Ipsen [SIAM J. Matrix Anal. Appl., 16 (1995), pp. 93–112] and to Skeel’s definition of condition number [J. ACM, 26 (1979), pp. 494–526] in some special cases, and it can be used to estimate the error in a subspace. The estimate is based on the adjoint equation in combination with small sample statistical theory. It can be implemented simply and is inexpensive to compute. Numerical examples are presented which illustrate the power and effectiveness of this error estimate.