The Rook's pivoting strategy

Abstract

Based on the geometric analysis of Gaussian elimination (GE) found in Neal and Poole (Linear Algebra Appl. 173 (1992) 239–264) and Poole and Neal (Linear Algebra Appl. 149 (1991) 249–272; 162–164 (1992) 309–324), a new pivoting strategy, Rook’ s pivoting (RP), was introduced in Neal and Poole (Linear Algebra Appl. 173 (1992) 239–264) which encourages stability in the back-substitution phase of GE while controlling the growth of round-off error during the sweep-out. In fact, Foster (J. Comput. Appl. Math. 86 (1997) 177–194) has previously shown that RP, as with complete pivoting, cannot have exponential growth error. Empirical evidence presented in Neal and Poole (Linear Algebra Appl. 173 (1992) 239–264) showed that RP produces computed solutions with consistently greater accuracy than partial pivoting. That is, Rook's pivoting is, on average, more accurate than partial pivoting, with comparable costs. Moreover, the overhead to implement Rook's pivoting in a scalar or serial environment is only about three times the overhead to implement partial pivoting. The theoretical proof establishing this fact is presented here, and is empirically confirmed in this paper and supported in Foster (J. Comput. Appl. Math. 86 (1997) 177–194).