Engee documentation

BiCGStab(l)

BiCGStab(l) solves the problem approximately for where is a general, linear operator and the right-hand side vector. The methods combines BiCG with GMRES iterations, resulting in a short-recurrence iteration. As a result the memory is fixed as well as the computational costs per iteration.

Usage

bicgstabl(A, b, l; kwargs...) -> x, [history]

Same as bicgstabl!, but allocates a solution vector x initialized with zeros.

bicgstabl!(x, A, b, l; kwargs...) -> x, [history]

Arguments

  • A: linear operator;

  • b: right hand side (vector);

  • l::Int = 2: Number of GMRES steps.

Keywords

  • max_mv_products::Int = size(A, 2): maximum number of matrix vector products.

For BiCGStab(l) this is a less dubious term than "number of iterations";

  • Pl = Identity(): left preconditioner of the method;

  • abstol::Real = zero(real(eltype(b))), reltol::Real = sqrt(eps(real(eltype(b)))): absolute and relative tolerance for the stopping condition |r_k| ≤ max(reltol * |r_0|, abstol), where r_k ≈ A * x_k - b is the approximate residual in the kth iteration;

    1. The true residual norm is never computed during the iterations, only an approximation; 2. If a left preconditioner is given, the stopping condition is based on the preconditioned residual.

Return values

if log is false

  • x: approximate solution.

if log is true

  • x: approximate solution;

  • history: convergence history.

Implementation details

The method is based on the original article [1], but does not implement later improvements. The normal equations arising from the GMRES steps are solved without orthogonalization. Hence the method should only be reliable for relatively small values of .

The r and u factors are pre-allocated as matrices of size , so that BLAS2 methods can be used. Also the random shadow residual is pre-allocated as a vector. Hence the storage costs are approximately vectors.

BiCGStabl(l) can be used as an iterator.
1. Sleijpen, Gerard LG, and Diederik R. Fokkema. "BiCGstab(l) for linear equations involving unsymmetric matrices with complex spectrum." Electronic Transactions on Numerical Analysis 1.11 (1993): 2000.