wz factorization via abay-broyden-spedicato algorithms

classes of‎ ‎abaffy-broyden-spedicato (abs) methods have been introduced for‎ ‎solving linear systems of equations‎. ‎the algorithms are powerful methods for developing matrix‎ ‎factorizations and many fundamental numerical linear algebra processes‎. ‎here‎, ‎we show how to apply the abs algorithms to devise algorithms to compute the wz and zw‎ ‎factorizations of a nonsingular matrix as well as the $w^tw$ and‎ ‎$z^tz$‎ ‎factorizations of a symmetric positives definite matrix‎. ‎we also describe the qz and the‎ ‎ qw  factorizations‎, ‎with q orthogonal‎, ‎and show how to appropriate the parameters of the abs algorithms to compute these factorizations‎.