# A new interpretation of the WZ factorization using block scaled ABS algorithms

### Abstract

The WZ factorization suitable for parallel computing, was introduced by Evans. A block generalization of the ABS class of methods for the solution of linear system of equations is given and it is shown that it covers the whole class of conjugate direction methods dened by Stewart. The methods produce a factorization of the coecient matrix implicitly, generating well known matrix factorizations. Here, we show how to set the parameters of a block ABS algorithm to compute the WZ and ZW fac- torizations of a nonsingular matrix as well as the WTW and ZTZ factorizations of a symmetric positives denite matrix. We also show how to appropriate the pa- rameters to construct algorithms for computing the QZ and the QW factorizations, where QTQ is an X-matrix. We also provide a new interpretation of the necessary and sucient condition for the existence of the WZ and the ZW factorizations of a nonsingular matrix.### References

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