  The conjugate gradient (CG) method is a classic Krylov subspace method for
solving symmetric positive definite linear systems. We introduce an analogous
semi-conjugate gradient (SCG) method for unsymmetric positive definite linear
systems. Unlike CG, SCG requires the solution of a lower triangular linear
system to produce each semi-conjugate direction. We prove that SCG is
theoretically equivalent to the full orthogonalization method (FOM), which is
based on the Arnoldi process and converges in a finite number of steps. Because
SCG's triangular system increases in size each iteration, we study a sliding
window implementation (SWI) to improve efficiency, and show that the directions
produced are still locally semi-conjugate. A counterexample illustrates that
SWI is different from the direct incomplete orthogonalization method (DIOM),
which is FOM with a sliding window. Numerical experiments from the
convection-diffusion equation and other applications show that SCG is robust
and that the sliding window implementation SWI allows SCG to solve large
systems efficiently.
