  We introduce two iterative methods, GPBiLQ and GPQMR, for solving unsymmetric
partitioned linear systems. The basic mechanism underlying GPBiLQ and GPQMR is
a novel simultaneous tridiagonalization via biorthogonality that allows for
short-recurrence iterative schemes. Similar to the biconjugate gradient method,
it is possible to develop another method, GPBiCG, whose iterate (if it exists)
can be obtained inexpensively from the GPBiLQ iterate. Whereas the iterate of
GPBiCG may not exist, the iterates of GPBiLQ and GPQMR are always well defined
as long as the biorthogonal tridiagonal reduction process does not break down.
We discuss connections between the proposed methods and some existing methods,
and give numerical experiments to illustrate the performance of the proposed
methods.
