随机三角矩阵条件数的数值实验及其分析

Experiments and analysis for the condition numbers of random triangular matrices

D .Viswanath和L .N .Trefethen在“Conditionnumbersofrandomtriangularmatrices”中证明了n阶随机三角矩阵的条件数kn,当n→∞时 ,nkn以概率 1收敛于 2 ;对于随机n阶单位三角矩阵的条件数kn,当n→∞时 ,nkn以概率 1收敛于 1.30 5 6834 10 ... 然而 ,通过数值实验得出的结果却似乎不相符 .实验结果还表明了随机三角矩阵的条件数趋于稳定 ,表明高斯消去法是几乎稳定的 ,理论证明尚得进一步探讨 .

D.Viswanath and L.N.Trefethen discussed the condition numbers of random triangular matrices in 1998.Two points were showed: the 2-norm condition numbers k n of random triangular matrices satisfy nk n->2 almost surely as n->∞,and the 2-norm condition numbers k n of random unit triangular matrices satisfy nk n->1.305 683 410... almost surely as n->∞.In this paper we point out that the conclusion seem not in according with the numerical experiments . The numerical experiments also indicate that the condition numbers of random matrices tend to be stable, and Gaussian elimination is almost stable. The strictly proof will be further researched.