# Matrix enlarging methods and their application

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## Abstract

This paper explores several methods for matrix enlarging, where an enlarged matrix*Ã* is constructed from a given matrix*A*. The methods explored include matrix primitization, stretching and node splitting. Graph interpretations of these methods are provided. Solving linear problems using enlarged matrices yields the answer to the original*Ax=b* problem.*Ã* can exhibit several desirable properties. For example,*Ã* can be constructed so that the valence of any row and/or column is smaller than some desired number (≥4). This is beneficial for algorithms that depend on the square of the number of entries of a row or column. Most particularly, matrix enlarging can results in a reduction of the fill-in in the*R* matrix which occurs during orthogonal factorization as a result of dense rows. Numerical experiments support these conjectures.

## AMS subject classification

65F20## Key words

Sparse matrices node splitting orthogonal factorization least squares QR factorization matrix stretching matrix primitization## References

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