Subsections:
The conjugate gradient algorithm can be interpreted as the following optimization problem: minimize 
defined by
![\[ \phi (x) = 1/2 x^ T Ax - x^ T b \]](images/imlug_sma0008.png){kind=small}
where 
and 
are symmetric and positive definite.
At each iteration is minimized along an 
-conjugate direction, constructing orthogonal residuals:
![\[ r_ i \; \bot \; {\mathcal K}_ i(A;\, r_0),\quad r_ i = A x_ i - b \]](images/imlug_sma0011.png){kind=small}
where 
is a Krylov subspace:
![\[ {\mathcal K}_ i\, (A; r) = \mbox{\textrm{span}} \{ r,\, Ar,\, A^2r,\, \ldots ,\, A^{i - 1}r\} \]](images/imlug_sma0013.png){kind=small}
Minimum residual algorithms work by minimizing the Euclidean norm 
over . At each iteration, 
is the vector in that gives the smallest residual.
The biconjugate gradient algorithm belongs to a more general class of Petrov-Galerkin methods, where orthogonality is enforced
in a different 
-dimensional subspace ( remains in ):
![\[ r_ i \; \bot \; \{ w,\, A^ T w,\, (A^ T)^{2} w,\, \ldots ,\, (A^ T)^{\, i - 1} w\} \]](images/imlug_sma0017.png){kind=small}