### Quadrant Interlocking Factorization Algorithm of Hourglass Matrix from Nonsingular Matrix

#### Abstract

This paper presents the quadrant interlocking factorization ($QIF$) of nonsingular matrix, alternatively called $WH$ factorization, to yield hourglass matrix ($H$-matrix). The $WH$ factorization algorithm of $H$-matrix is synonymous to $WZ$ factorization algorithm of $Z$-matrix, unlike $LU$ factorization. We examine the conditions to generate the zero and nonzero entries of $H$-matrix from the factorization algorithm, and compare the $H$-matrix and $Z$-matrix. Then we conclude that the existence of $WH$ factorization implies $WZ$ factorization.

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