Consider symmetric saddle-point problems of the form
We show that many popular conjugate gradient-based methods for solving
() can be reformulated as applying the (preconditioned)
conjugate gradient method to (
) for some
and
We also provide conditions for guaranteeing that (
) is
positive definite. Using these conditions we propose new conjugate
gradient-based methods for solving (
) and give numerical
results for problems from optimization and fluid dynamics.