1. Choose initial guess ${\displaystyle x_{0}\,}$ , two other vectors ${\displaystyle x_{0}^{*}}$  and ${\displaystyle b^{*}\,}$  and a preconditioner ${\displaystyle M\,}$ 
2. ${\displaystyle r_{0}\leftarrow b-A\,x_{0}\,}$ 
3. ${\displaystyle r_{0}^{*}\leftarrow b^{*}-x_{0}^{*}\,A^{*}}$ 
4. ${\displaystyle p_{0}\leftarrow M^{-1}r_{0}\,}$ 
5. ${\displaystyle p_{0}^{*}\leftarrow r_{0}^{*}M^{-1}\,}$ 
6. for ${\displaystyle k=0,1,\ldots }$  do
   1. ${\displaystyle \alpha _{k}\leftarrow {r_{k}^{*}M^{-1}r_{k} \over p_{k}^{*}Ap_{k}}\,}$ 
   2. ${\displaystyle x_{k+1}\leftarrow x_{k}+\alpha _{k}\cdot p_{k}\,}$ 
   3. ${\displaystyle x_{k+1}^{*}\leftarrow x_{k}^{*}+{\overline {\alpha _{k}}}\cdot p_{k}^{*}\,}$ 
   4. ${\displaystyle r_{k+1}\leftarrow r_{k}-\alpha _{k}\cdot Ap_{k}\,}$ 
   5. ${\displaystyle r_{k+1}^{*}\leftarrow r_{k}^{*}-{\overline {\alpha _{k}}}\cdot p_{k}^{*}\,A^{*}}$ 
   6. ${\displaystyle \beta _{k}\leftarrow {r_{k+1}^{*}M^{-1}r_{k+1} \over r_{k}^{*}M^{-1}r_{k}}\,}$ 
   7. ${\displaystyle p_{k+1}\leftarrow M^{-1}r_{k+1}+\beta _{k}\cdot p_{k}\,}$ 
   8. ${\displaystyle p_{k+1}^{*}\leftarrow r_{k+1}^{*}M^{-1}+{\overline {\beta _{k}}}\cdot p_{k}^{*}\,}$ 

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

${\displaystyle r_{k}=b-Ax_{k},\,}$ 

${\displaystyle r_{k}^{*}=b^{*}-x_{k}^{*}\,A^{*}}$ 

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

${\displaystyle Ax=b,\,}$ 

${\displaystyle x^{*}\,A^{*}=b^{*}\,;}$ 

${\displaystyle x^{*}}$  is the adjoint, and ${\displaystyle {\overline {\alpha }}}$  is the complex conjugate.

Unpreconditioned version of the algorithm edit

1. Choose initial guess ${\displaystyle x_{0}\,}$ ,
2. ${\displaystyle r_{0}\leftarrow b-A\,x_{0}\,}$ 
3. ${\displaystyle {\hat {r}}_{0}\leftarrow {\hat {b}}-{\hat {x}}_{0}A}$ 
4. ${\displaystyle p_{0}\leftarrow r_{0}\,}$ 
5. ${\displaystyle {\hat {p}}_{0}\leftarrow {\hat {r}}_{0}\,}$ 
6. for ${\displaystyle k=0,1,\ldots }$  do
   1. ${\displaystyle \alpha _{k}\leftarrow {{\hat {r}}_{k}r_{k} \over {\hat {p}}_{k}Ap_{k}}\,}$ 
   2. ${\displaystyle x_{k+1}\leftarrow x_{k}+\alpha _{k}\cdot p_{k}\,}$ 
   3. ${\displaystyle {\hat {x}}_{k+1}\leftarrow {\hat {x}}_{k}+\alpha _{k}\cdot {\hat {p}}_{k}\,}$ 
   4. ${\displaystyle r_{k+1}\leftarrow r_{k}-\alpha _{k}\cdot Ap_{k}\,}$ 
   5. ${\displaystyle {\hat {r}}_{k+1}\leftarrow {\hat {r}}_{k}-\alpha _{k}\cdot {\hat {p}}_{k}A}$ 
   6. ${\displaystyle \beta _{k}\leftarrow {{\hat {r}}_{k+1}r_{k+1} \over {\hat {r}}_{k}r_{k}}\,}$ 
   7. ${\displaystyle p_{k+1}\leftarrow r_{k+1}+\beta _{k}\cdot p_{k}\,}$ 
   8. ${\displaystyle {\hat {p}}_{k+1}\leftarrow {\hat {r}}_{k+1}+\beta _{k}\cdot {\hat {p}}_{k}\,}$ 

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

${\displaystyle x_{k}:=x_{j}+P_{k}A^{-1}\left(b-Ax_{j}\right),}$ 

${\displaystyle x_{k}^{*}:=x_{j}^{*}+\left(b^{*}-x_{j}^{*}A\right)P_{k}A^{-1},}$ 

where ${\displaystyle j<k}$  using the related projection

${\displaystyle P_{k}:=\mathbf {u} _{k}\left(\mathbf {v} _{k}^{*}A\mathbf {u} _{k}\right)^{-1}\mathbf {v} _{k}^{*}A,}$ 

with

${\displaystyle \mathbf {u} _{k}=\left[u_{0},u_{1},\dots ,u_{k-1}\right],}$ 

${\displaystyle \mathbf {v} _{k}=\left[v_{0},v_{1},\dots ,v_{k-1}\right].}$ 

These related projections may be iterated themselves as

${\displaystyle P_{k+1}=P_{k}+\left(1-P_{k}\right)u_{k}\otimes {v_{k}^{*}A\left(1-P_{k}\right) \over v_{k}^{*}A\left(1-P_{k}\right)u_{k}}.}$ 

A relation to Quasi-Newton methods is given by ${\displaystyle P_{k}=A_{k}^{-1}A}$  and ${\displaystyle x_{k+1}=x_{k}-A_{k+1}^{-1}\left(Ax_{k}-b\right)}$ , where

${\displaystyle A_{k+1}^{-1}=A_{k}^{-1}+\left(1-A_{k}^{-1}A\right)u_{k}\otimes {v_{k}^{*}\left(1-AA_{k}^{-1}\right) \over v_{k}^{*}A\left(1-A_{k}^{-1}A\right)u_{k}}.}$ 

The new directions

${\displaystyle p_{k}=\left(1-P_{k}\right)u_{k},}$ 

${\displaystyle p_{k}^{*}=v_{k}^{*}A\left(1-P_{k}\right)A^{-1}}$ 

are then orthogonal to the residuals:

${\displaystyle v_{i}^{*}r_{k}=p_{i}^{*}r_{k}=0,}$ 

${\displaystyle r_{k}^{*}u_{j}=r_{k}^{*}p_{j}=0,}$ 

which themselves satisfy

${\displaystyle r_{k}=A\left(1-P_{k}\right)A^{-1}r_{j},}$ 

${\displaystyle r_{k}^{*}=r_{j}^{*}\left(1-P_{k}\right)}$ 

where ${\displaystyle i,j<k}$ .

The biconjugate gradient method now makes a special choice and uses the setting

${\displaystyle u_{k}=M^{-1}r_{k},\,}$ 

${\displaystyle v_{k}^{*}=r_{k}^{*}\,M^{-1}.\,}$ 

With this particular choice, explicit evaluations of ${\displaystyle P_{k}}$  and A⁻¹ are avoided, and the algorithm takes the form stated above.

• If ${\displaystyle A=A^{*}\,}$  is self-adjoint, ${\displaystyle x_{0}^{*}=x_{0}}$  and ${\displaystyle b^{*}=b}$ , then ${\displaystyle r_{k}=r_{k}^{*}}$ , ${\displaystyle p_{k}=p_{k}^{*}}$ , and the conjugate gradient method produces the same sequence ${\displaystyle x_{k}=x_{k}^{*}}$  at half the computational cost.
• The sequences produced by the algorithm are biorthogonal, i.e., ${\displaystyle p_{i}^{*}Ap_{j}=r_{i}^{*}M^{-1}r_{j}=0}$  for ${\displaystyle i\neq j}$ .
• if ${\displaystyle P_{j'}\,}$  is a polynomial with ${\displaystyle \deg \left(P_{j'}\right)+j<k}$ , then ${\displaystyle r_{k}^{*}P_{j'}\left(M^{-1}A\right)u_{j}=0}$ . The algorithm thus produces projections onto the Krylov subspace.
• if ${\displaystyle P_{i'}\,}$  is a polynomial with ${\displaystyle i+\deg \left(P_{i'}\right)<k}$ , then ${\displaystyle v_{i}^{*}P_{i'}\left(AM^{-1}\right)r_{k}=0}$ .

• Biconjugate gradient stabilized method (BiCG-Stab)
• Conjugate gradient method (CG)
• Conjugate gradient squared method (CGS)

• Fletcher, R. (1976). Watson, G. Alistair (ed.). "Conjugate gradient methods for indefinite systems". Numerical Analysis. Lecture Notes in Mathematics. 506. Springer Berlin / Heidelberg: 73–89. doi:10.1007/BFb0080109. ISBN 978-3-540-07610-0. ISSN 1617-9692.
• Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007). "Section 2.7.6". Numerical Recipes: The Art of Scientific Computing (3rd ed.). New York: Cambridge University Press. ISBN 978-0-521-88068-8.