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. Unlike the original BiCG method, it doesn't require multiplication by the transpose of the system matrix.
In the following sections, (x,y) = xT y denotes the dot product of vectors. To solve a linear system Ax = b, BiCGSTAB starts with an initial guess x0 and proceeds as follows:
In some cases, choosing the vector r̂0 randomly improves numerical stability.[1]
Preconditioners are usually used to accelerate convergence of iterative methods. To solve a linear system Ax = b with a preconditioner K = K1K2 ≈ A, preconditioned BiCGSTAB starts with an initial guess x0 and proceeds as follows:
This formulation is equivalent to applying unpreconditioned BiCGSTAB to the explicitly preconditioned system
with à = K −1
1 AK −1
2 , x̃ = K2x and b̃ = K −1
1 b. In other words, both left- and right-preconditioning are possible with this formulation.
In BiCG, the search directions pi and p̂i and the residuals ri and r̂i are updated using the following recurrence relations:
The constants αi and βi are chosen to be
where ρi = (r̂i−1, ri−1) so that the residuals and the search directions satisfy biorthogonality and biconjugacy, respectively, i.e., for i ≠ j,
It is straightforward to show that
where Pi(A) and Ti(A) are ith-degree polynomials in A. These polynomials satisfy the following recurrence relations:
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 Qi(A) = (I − ω1A)(I − ω2A)⋯(I − ωiA) with suitable constants ωj instead of ri = Pi(A)r0 in the hope that Qi(A) will enable faster and smoother convergence in r̃i than ri.
It follows from the recurrence relations for Pi(A) and Ti(A) and the definition of Qi(A) that
which entails the necessity of a recurrence relation for Qi(A)Ti(A)r0. This can also be derived from the BiCG relations:
Similarly to defining r̃i, BiCGSTAB defines
Written in vector form, the recurrence relations for p̃i and r̃i are
To derive a recurrence relation for xi, define
The recurrence relation for r̃i can then be written as
which corresponds to
Now it remains to determine the BiCG constants αi and βi and choose a suitable ωi.
In BiCG, βi = ρi/ρi−1 with
Since BiCGSTAB does not explicitly keep track of r̂i or ri, ρi is not immediately computable from this formula. However, it can be related to the scalar
Due to biorthogonality, ri−1 = Pi−1(A)r0 is orthogonal to Ui−2(AT)r̂0 where Ui−2(AT) is any polynomial of degree i − 2 in AT. Hence, only the highest-order terms of Pi−1(AT) and Qi−1(AT) matter in the dot products (Pi−1(AT)r̂0, Pi−1(A)r0) and (Qi−1(AT)r̂0, Pi−1(A)r0). The leading coefficients of Pi−1(AT) and Qi−1(AT) are (−1)i−1α1α2⋯αi−1 and (−1)i−1ω1ω2⋯ωi−1, respectively. It follows that
and thus
A simple formula for αi can be similarly derived. In BiCG,
Similarly to the case above, only the highest-order terms of Pi−1(AT) and Ti−1(AT) matter in the dot products thanks to biorthogonality and biconjugacy. It happens that Pi−1(AT) and Ti−1(AT) have the same leading coefficient. Thus, they can be replaced simultaneously with Qi−1(AT) in the formula, which leads to
Finally, BiCGSTAB selects ωi to minimize r̃i = (I − ωiA)si in 2-norm as a function of ωi. This is achieved when
giving the optimal value
BiCGSTAB can be viewed as a combination of BiCG and GMRES where each BiCG step is followed by a GMRES(1) (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 A 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[1] and the more general BiCGSTAB(l)[2]. In BiCGSTAB(l), a GMRES(l) step follows every l BiCG steps. BiCGSTAB2 is equivalent to BiCGSTAB(l) with l = 2.