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 ${\displaystyle A}$ to be self-adjoint, but instead one needs to perform multiplications by the conjugate transpose A^*.

In the above formulation, the computed ${\displaystyle r_{k}\,}$ and ${\displaystyle r_{k}^{*}}$ satisfy

and thus are the respective residuals corresponding to ${\displaystyle x_{k}\,}$ and ${\displaystyle x_{k}^{*}}$, as approximate solutions to the systems

${\displaystyle x^{*}}$ is the adjoint, and ${\displaystyle {\overline {\alpha }}}$ 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 ${\displaystyle j<k}$ using the related projection

with