It's a Krylov subspace method very similar to the much more popular conjugate gradient method, with similar construction and convergence properties.
This method is used to solve linear equations of the form where A is an invertible and Hermitian matrix, and b is nonzero.
It involves more numerical operations and requires more storage.
The only difference between this and the conjugate gradient method is the calculation of
Note: the above algorithm can be transformed so to make only one symmetric matrix-vector multiplication in each iteration.
By making a few substitutions and variable changes, a preconditioned conjugate residual method may be derived in the same way as done for the conjugate gradient method: The preconditioner