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, more specifically in numerical linear algebra, the biconjugate gradient method is an algorithm to solve systems of linear equations :

A x= b.\,

Unlike the conjugate gradient method, this algorithm does not require the

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A

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

The algorithm

# Choose initial guess

x_0\,

, two other vectors

x_0^*

and

b^*\,

and a preconditioner

M\,

r_0 \leftarrow b-A\, x_0\,

r_0^* \leftarrow b^*-x_0^*\, A

p_0 \leftarrow M^ r_0\,

p_0^* \leftarrow r_0^*M^\,

# for

k=0, 1, \ldots

do ##

\alpha_k \leftarrow \,

x_ \leftarrow x_k + \alpha_k \cdot p_k\,

x_^* \leftarrow x_k^* + \overline\cdot p_k^*\,

r_ \leftarrow r_k - \alpha_k \cdot A p_k\,

r_^* \leftarrow r_k^*- \overline \cdot p_k^*\, A

\beta_k \leftarrow \,

p_ \leftarrow M^ r_ + \beta_k \cdot p_k\,

p_^* \leftarrow r_^*M^  + \overline\cdot p_k^*\,

In the above formulation, the computed

r_k\,

and

r_k^*

satisfy :

r_k = b - A x_k,\,

r_k^* = b^* - x_k^*\, A

and thus are the respective residuals corresponding to

x_k\,

and

x_k^*

, as approximate solutions to the systems :

A x = b,\,

x^*\, A = b^*\,;

x^*

is the adjoint, and

\overline

is the

complex conjugate In mathematics, the complex conjugate of