Bi-Conjugate Gradient Stabilized

The multi-shifted bi-conjugate gradient stabilized method (BICGStab), computes the solution to the problem \( M\textbf{x} = \textbf{b}\) and, with some additional linear algebra, computes also the solution to a set of shifted linear systems of the form

\((M+\sigma_i) \textbf{x} =\textbf{b} \quad i=1,2,... ,N \)

where \( \sigma_i\) are the shift values. The main advantage of this method is that the shifted solutions do not require any additional matrix-vector multiplications. If the matrix-vector multiplication cost is much larger than the linear algebra cost, this routine can speed up the calculation dramatically for scenarios where the shifted solutions are required.

Definitions:
\( \text{residue} = || ( M+\sigma_i) - \textbf{b} || \)

\( \text{error} = || ( M+\sigma_i) - \textbf{b} || / ||\textbf{b}|| \)

Usage

int qcd::bicgstabm_device(device_wilson_field &src, int noshifts, double shifts[],device_wilson_field* sol[], double dError, int maxiters, matmult<device_wilson_field>& pfncMatMult, device_random_field& rnd);

1. src = b.
2. noShifts = N the number of shifts.
3. shifts[] = a pointer to the array of shift values.
4. sol = a pointer to an array of (N+1) pointers pointing to the solutions (the first one is the no-shift solution and the rest are the shifted solutions).
5. dError = the maximum error in the solution (see definition).
6. maxiters = the maximum number of iterations allowed.
7. pfncMatMult = pointer to the matrix multiplication.
8. rnd = a field of random vector generators -- this is used to initialize the \( w_0 \) vector in the BiCGstab routine.
9. the function returns the number of iterations required for convergence.

NOTES:

1. The BICGStabM only guarantees the convergence of the linear system associated with the pfncMatMult you give it. You need to check this in your code. In fact, you should have a segment of your code check for this. If one or more shifts fails, for example, you could restart them using a single shift inverter. They should converge quickly since they probably already have almost converged to the desired accuracy.
2. The example explains in more detail how to correctly use BICGStabM through an example.

You can find the example here
The code implementing the example is "control_test_bicgstabm.cu". It is located in in /gwu_qcd/inverters/ directory. You can find a .pdf of it here