The SPIKE algorithm deals with a linear system, where is a banded matrix of bandwidthmuch less than, and is an matrix containing right-hand sides. It is divided into a preprocessing stage and a postprocessing stage.
Preprocessing stage
In the preprocessing stage, the linear system is partitioned into a block tridiagonal form Assume, for the time being, that the diagonal blocks are nonsingular. Define a block diagonal matrix then is also nonsingular. Left-multiplying to both sides of the system gives which is to be solved in the postprocessing stage. Left-multiplication by is equivalent to solving systems of the form , which can be carried out in parallel. Due to the banded nature of, only a few leftmost columns of each and a few rightmost columns of each can be nonzero. These columns are called the spikes.
Postprocessing stage
, assume that each spike contains exactly columns . Partition the spikes in all and into where,, and are of dimensions. Partition similarly all and into Notice that the system produced by the preprocessing stage can be reduced to a block pentadiagonal system of much smaller size which we call the reduced system and denote by. Once all and are found, all can be recovered with perfect parallelism via
SPIKE as a polyalgorithmic banded linear system solver
Despite being logically divided into two stages, computationally, the SPIKE algorithm comprises three stages:
Each of these stages can be accomplished in several ways, allowing a multitude of variants. Two notable variants are the recursive SPIKE algorithm for non-diagonally-dominant cases and the truncated SPIKE algorithm for diagonally-dominant cases. Depending on the variant, a system can be solved either exactly or approximately. In the latter case, SPIKE is used as a preconditioner for iterative schemes like Krylov subspace methods and iterative refinement.
Recursive SPIKE
Preprocessing stage
The first step of the preprocessing stage is to factorize the diagonal blocks. For numerical stability, one can use LAPACK's XGBTRF routines to LU factorize them with partial pivoting. Alternatively, one can also factorize them without partial pivoting but with a "diagonal boosting" strategy. The latter method tackles the issue of singular diagonal blocks. In concrete terms, the diagonal boosting strategy is as follows. Let denote a configurable "machine zero". In each step of LU factorization, we require that the pivot satisfy the condition If the pivot does not satisfy the condition, it is then boosted by where is a positive parameter depending on the machine's unit roundoff, and the factorization continues with the boosted pivot. This can be achieved by modified versions of ScaLAPACK's XDBTRF routines. After the diagonal blocks are factorized, the spikes are computed and passed on to the postprocessing stage.
Postprocessing stage
The two-partition case
In the two-partition case, i.e., when, the reduced system has the form An even smaller system can be extracted from the center: which can be solved using the block LU factorization Once and are found, and can be computed via
The multiple-partition case
Assume that is a power of two, i.e.,. Consider a block diagonal matrix where for. Notice that essentially consists of diagonal blocks of order extracted from. Now we factorize as The new matrix has the form Its structure is very similar to that of, only differing in the number of spikes and their height. Thus, a similar factorization step can be performed on to produce and Such factorization steps can be performed recursively. After steps, we obtain the factorization where has only two spikes. The reduced system will then be solved via The block LU factorization technique in the two-partition case can be used to handle the solving steps involving,..., and for they essentially solve multiple independent systems of generalized two-partition forms. Generalization to cases where is not a power of two is almost trivial.
Truncated SPIKE
When is diagonally-dominant, in the reduced system the blocks and are often negligible. With them omitted, the reduced system becomes block diagonal and can be easily solved in parallel. The truncated SPIKE algorithm can be wrapped inside some outer iterative scheme to improve the accuracy of the solution.
SPIKE for tridiagonal systems
The first SPIKE partitioning and algorithm was presented in and was designed as the means to improve the stability properties of a parallel Givens rotations-based solver for tridiagonal systems. A version of the algorithm, termed g-Spike, that is based on serial Givens rotations applied independently on each block was designed for the NVIDIA GPU. A SPIKE-based algorithm for the GPU that is based on a special block diagonal pivoting strategy is described in.
SPIKE as a preconditioner
The SPIKE algorithm can also function as a preconditioner for iterative methods for solving linear systems. To solve a linear system using a SPIKE-preconditioned iterative solver, one extracts center bands from to form a banded preconditioner and solves linear systems involving in each iteration with the SPIKE algorithm. In order for the preconditioner to be effective, row and/or column permutation is usually necessary to move "heavy" elements of close to the diagonal so that they are covered by the preconditioner. This can be accomplished by computing the weighted spectral reordering of. The SPIKE algorithm can be generalized by not restricting the preconditioner to be strictly banded. In particular, the diagonal block in each partition can be a general matrix and thus handled by a direct general linear system solver rather than a banded solver. This enhances the preconditioner, and hence allows better chance of convergence and reduces the number of iterations.
Implementations
offers an implementation of the SPIKE algorithm under the name Intel Adaptive Spike-Based Solver. Tridiagonal solvers have also been developed for the NVIDIA GPU and the Xeon Phi co-processors. The method in is the basis for a tridiagonal solver in the cuSPARSE library. The Givens rotations based solver was also implemented for the GPU and the Intel Xeon Phi.
