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\title{The Improved Biconjugate Gradient Method on Massively Parallel
     Distributed Memory Computers}

\author{\em Tianruo Yang $\dagger$, Hai-Xiang Lin$\ddagger$ \\ \em $\dagger$
  Department of Computer and Information Science \\ \em  Link\"oping
  University,  S-581 83, Link\"oping, Sweden \\
  \em $\ddagger$ Department of Technical Mathematics and Computer
  Science \\ \em TU Delft, P.O. Box 356, 2600 GA Delft, The Netherlands}

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\begin{abstract}
For the solutions of linear systems of equations with unsymmetric
coefficient sparse matrices, we propose an improved version of the
biconjugate gradient (IBiCG) method by using the Lanczos process as a
major component combining elements of numerical stability and parallel
algorithm design. For Lanczos process, stability is obtained by a coupled
two-term procedure that generates Lanczos vectors scaled to unit
length. The algorithm, which is shown to be mathematically equivalent
to the original biconjugate gradient method, is derived such that all
inner products and matrix-vector multiplications of a single iteration step are
independent and communication time required for inner product can
be overlapped efficiently with computation time. Therefore, the cost of
global communication on parallel distributed memory computers can be
significantly reduced. The resulting IBiCG algorithm maintains the
favorable properties of the Lanczos process while not increasing
computational costs. Additionally, the data distribution and the
communication scheme for the sparse operations are designed particularly
based on the analysis of the non-zero elements which are crucial for
efficient execution. The efficiency of this method is demonstrated by
numerical experimental results with regards to both scaling behavior
and execution time from real applications
carried out on a massively parallel distributed memory computer, the
Parsytec GC/PowerPlus.
\end{abstract}

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