# SPAI (SParse Approximate Inverse)

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## Definition

For a given sparse matrix A a sparse matrix $$M \approx {A}^{-1}$$ is computed by minimizing $$\Vert AM - {I\Vert }_{F}$$ in the Frobenius norm over all matrices with a certain sparsity pattern. In the SPAI algorithm the pattern of M is updated dynamically to improve the approximation until a certain stopping criterion is reached.

## Discussion

### Introduction

For applying an iterative solution method like the conjugate gradient method (CG), GMRES, BiCGStab, QMR, or similar algorithms, to a system of linear equations Ax = b with sparse matrix A, it is often crucial to include an efficient preconditioner. Here, the original problem Ax = b is replaced by the preconditioned system MAx = Mb or Ax = A(My) = b. In a parallel environment a preconditioner should satisfy the following conditions:

• M can be computed efficiently in parallel.

• Mc can be computed efficiently in parallel for any given vector c.

• The iterative solver applied on AMx = b or MAx...

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## Bibliography

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### Cite this entry

Huckle, T., Sedlacek, M. (2011). SPAI (SParse Approximate Inverse). In: Padua, D. (eds) Encyclopedia of Parallel Computing. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-09766-4_144