In mathematics, more specifically in numerical linear algebra, the biconjugate gradient method is an algorithm to solve systems of linear equations Unlike the conjugate gradient method, this algorithm does not require the matrix
to be self-adjoint, but instead one needs to perform multiplications by the conjugate transpose A*.
In the above formulation, the computed
satisfy and thus are the respective residuals corresponding to
, as approximate solutions to the systems
is the complex conjugate.
The biconjugate gradient method is numerically unstable[citation needed] (compare to the biconjugate gradient stabilized method), but very important from a theoretical point of view.
Define the iteration steps by where
using the related projection with These related projections may be iterated themselves as A relation to Quasi-Newton methods is given by
, where The new directions are then orthogonal to the residuals: which themselves satisfy where
The biconjugate gradient method now makes a special choice and uses the setting With this particular choice, explicit evaluations of
and A−1 are avoided, and the algorithm takes the form stated above.