This article includes a list of references, related reading, or external links, but its sources remain unclear because it lacks inline citations. Please help improve this article by introducing more precise citations. (September 2013) (Learn how and when to remove this message)

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

${\displaystyle Ax=b.\,}$

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^*.

The Algorithm

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.

Discussion

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.

Properties

• 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 ${\displaystyle v_{i}^{*}P_{i'}\left(AM^{-1}\right)r_{k}=0}$.

See also

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

References

• Fletcher, R. (1976). "Conjugate gradient methods for indefinite systems". In Watson, G. Alistair (ed.). Numerical analysis : proceedings of the Dundee Conference on Numerical Analysis. Lecture Notes in Mathematics. Vol. 506. Springer. pp. 73–89. doi:10.1007/BFb0080116. ISBN 978-3-540-07610-0.
• 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.