Biconjugate gradient stabilized method

In numerical linear algebra, the biconjugate gradient stabilized method, often abbreviated as BiCGSTAB, is an iterative method developed by H. A. van der Vorst for the numerical solution of nonsymmetric linear systems. It is a variant of the biconjugate gradient method (BiCG) and has faster and smoother convergence than the original BiCG as well as other variants such as the conjugate gradient squared method (CGS). It is a Krylov subspace method.
Algorithmic steps.
Unpreconditioned BiCGSTAB.
To solve a linear system , BiCGSTAB starts with an initial guess and proceeds as follows:
Preconditioned BiCGSTAB.
Preconditioners are usually used to accelerate convergence of iterative methods. To solve a linear system with a preconditioner , preconditioned BiCGSTAB starts with an initial guess and proceeds as follows:
This formulation is equivalent to applying unpreconditioned BiCGSTAB to the explicitly preconditioned system
with , and . In other words, both left- and right-preconditioning are possible with this formulation.
Derivation.
BiCG in polynomial form.
In BiCG, the search directions and and the residuals and are updated using the following recurrence relations:
The constants and are chosen to be
where so that the residuals and the search directions satisfy biorthogonality and biconjugacy, respectively, i.e., for ,
It is straightforward to show that
where and are th-degree polynomials in . These polynomials satisfy the following recurrence relations:
Derivation of BiCGSTAB from BiCG.
It is unnecessary to explicitly keep track of the residuals and search directions of BiCG. In other words, the BiCG iterations can be performed implicitly. In BiCGSTAB, one wishes to have recurrence relations for
where with suitable constants instead of in the hope that will enable faster and smoother convergence in than .
It follows from the recurrence relations for and and the definition of that
which entails the necessity of a recurrence relation for . This can also be derived from the BiCG relations:
Similarly to defining , BiCGSTAB defines
Written in vector form, the recurrence relations for and are
To derive a recurrence relation for , define
The recurrence relation for can then be written as
which corresponds to
Determination of BiCGSTAB constants.
Now it remains to determine the BiCG constants and and choose a suitable .
In BiCG, with
Since BiCGSTAB does not explicitly keep track of or , is not immediately computable from this formula. However, it can be related to the scalar
Due to biorthogonality, is orthogonal to where is any polynomial of degree in . Hence, only the highest-order terms of and matter in the dot products and . The leading coefficients of and are and , respectively. It follows that
and thus
A simple formula for can be similarly derived. In BiCG,
Similarly to the case above, only the highest-order terms of and matter in the dot products thanks to biorthogonality and biconjugacy. It happens that and have the same leading coefficient. Thus, they can be replaced simultaneously with in the formula, which leads to
Finally, BiCGSTAB selects to minimize in -norm as a function of . This is achieved when
giving the optimal value
Generalization.
BiCGSTAB can be viewed as a combination of BiCG and GMRES where each BiCG step is followed by a GMRES() (i.e., GMRES restarted at each step) step to repair the irregular convergence behavior of CGS, as an improvement of which BiCGSTAB was developed. However, due to the use of degree-one minimum residual polynomials, such repair may not be effective if the matrix has large complex eigenpairs. In such cases, BiCGSTAB is likely to stagnate, as confirmed by numerical experiments.
One may expect that higher-degree minimum residual polynomials may better handle this situation. This gives rise to algorithms including BiCGSTAB2 and the more general BiCGSTAB(). In BiCGSTAB(), a GMRES() step follows every BiCG steps. BiCGSTAB2 is equivalent to BiCGSTAB() with .