A variable metric method for approximating generalized inverses of matrices (Q2747483)

Two new methods of the variable metric update family are analysed for the solution of a linear system \(Ax=b\). They generalize the Dennis-Fletcher-Powell and the Broyden-Fletcher-Goldfarb-Shannon methods to nonsymmetric matrices \(A\) of size \(m\times n\). Matrices \(H_k\) are updated that approximate a right inverse for \(A\) if \(m\leq n\) or a Moore-Penrose inverse if \(m\geq n\). Moreover \(AH_k\) is symmetric positive semidefinite. The residuals are minimal and belong to the Krylov space \({\mathcal K}(AH_0,r_0)\) generated by the inital \(AH_0\) and the initial residual \(r_0\). The correction vectors for the residuals \(z_k=r_{k+1}-r_k\) form an orthogonal basis for \(AH_0{\mathcal K}(AH_0,r_0)\). NEWLINENEWLINENEWLINEThese and other relations with more general conjugate gradient methods are proved. The update of \(H_k\) is of rank 2, while several other methods in the variable metric family are rank 1 updates. A memory efficient implementation for sparse matrices and numerical comparison with several variable metric methods are included.