Weighted LSQR for solving Nystr¨om discretization of first kind Fredholm integral equations

A variant of LSQR method called weighted LSQR(WLSQR) method is given. The WLSQR method uses a new inner product instead of the Euclidean one. We apply the new algorithm to various discrete Fredholm integral equations with smooth kernels. In this case, the new inner product arises from numerical integration used to the discrete equation. On this basis, the proposed algorithm generates a sequence of vectors which are approximations of solution at quadrature points. Moreover, we give an example to illustrate the performance of the WLSQR compared with the LSQR method