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 A to be self-adjoint, but instead one needs to perform multiplications by the conjugate transpose A * .

The Algorithm

In the above formulation, the computed r_k, and r_k^* satisfy and thus are the respective residuals corresponding to x_k, and x_k^, as approximate solutions to the systems x^ is the adjoint, and is the complex conjugate.

Unpreconditioned version of the algorithm

Discussion

The biconjugate gradient method is numerically unstable (compare to the biconjugate gradient stabilized method), but very important from a theoretical point of view. Define the iteration steps by where j<k using the related projection with These related projections may be iterated themselves as A relation to Quasi-Newton methods is given by and, where The new directions are then orthogonal to the residuals: which themselves satisfy where i,j<k. The biconjugate gradient method now makes a special choice and uses the setting With this particular choice, explicit evaluations of P_k and A −1 are avoided, and the algorithm takes the form stated above.

Properties