A Partial Condition Number for Linear Least Squares Problems

Abstract. We consider here the linear least squares problem miny∈Rn ‖Ay− b‖2, where b ∈ Rm and A ∈ Rm×n is a matrix of full column rank n, and we denote x its solution. We assume that both A and b can be perturbed and that these perturbations are measured using the Frobenius or the spectral norm for A and the Euclidean norm for b. In this paper, we are concerned with the condition number of a linear function of x (LT x, where L ∈ Rn×k) for which we provide a sharp estimate that lies within a factor √ 3 of the true condition number. Provided the triangular R factor of A from ATA = RTR is available, this estimate can be computed in 2kn2 flops. We also propose a statistical method that estimates the partial condition number by using the exact condition numbers in random orthogonal directions. If R is available, this statistical approach enables us to obtain a condition estimate at a lower computational cost. In the case of the Frobenius norm, we derive a closed formula for the partial condition number that is based on the singular values and the right singular vectors of the matrix A.