Accelerating with rank-one updates
One considers the iteration \(x_{k+1}=x_ k+H_ k(b-Ax_ k)\) for solving systems of linear algebraic equations \(Ax=b\) with a square non- singular matrix A. Special emphasis is put on the choice of the matrix \(H_ k\). One improves \(H_ k\) during each step in such a way that with increasing number of iterations the sequence of matrices \(H_ k\) tends to \(A^{-1}\). This is obtained by using the rank-one updates \(H_ k\) on each step of modification. Sufficient conditions are derived under which the matrices \(H_ k\) are not singular for any \(x_ 0\) and \(H_ 0\). Necessary and sufficient conditions are obtained under which the matrix \(H_ k+I\) becomes singular under the assumption that the matrix \(H_ k\) is not singular. One presents an algorithm which realizes the iterative process.
scientific article; zbMATH DE number 4119486