Biconjugate gradient method

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.

Unpreconditioned version of the algorithm

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. ${\displaystyle {\hat {p}}_{k+1}\leftarrow {\hat {r}}_{k+1}+\beta _{k}\cdot {\hat {p}}_{k}\,}$

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

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

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

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 ${\displaystyle \mathrm {deg} \left(P_{j'}\right)+j<k}$, 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 ${\displaystyle i+\mathrm {deg} \left(P_{i'}\right)<k}$, then ${\displaystyle v_{i}^{*}P_{i'}\left(AM^{-1}\right)r_{k}=0}$.

See also

• Biconjugate gradient stabilized method
• Conjugate gradient method

References

• Fletcher, R. (1976). Watson, G. Alistair (ed.). "Conjugate gradient methods for indefinite systems". Numerical Analysis. Lecture Notes in Mathematics. Springer Berlin / Heidelberg. 506: 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.