关于李锐“步法”与方程的多个正根求法的讨论

On LI Rui's“Bufa”and the Methods of Solving Equation with the Several Positive Roots

中国古代在解高次方程时,只根据需要求出一个正根,没有统一的求解多个正根的方法。清代数学家李锐拓展了传统开方术中确定初商的“步法”使用,通过方程的系数间不同项之间的“步法”获得不同的正根,并确定所得正根在方程的所有正根中的大小关系。他设计了“代开法”,在求得一根之后,通过求解低一次幂的减根变换后的新方程来继续求得其他各正根。

Traditional Chinese mathematicians sought only one positive root for an equation,no matter how high the degree of the equation was.There was no effective method for finding all solutions of an equation with multiple positive roots before the time of Li Rui who was a famous mathematician in the early 19th century.Li Rui expanded the use of the bufa,a step for determining the first digit and its numerical place of one root in traditional extraction procedure.He applied the bufa to different coefficients of an equation to find different root and to determine its size order among all positive roots of the equation.And he also designed daikafa,a new procedure of extracting another root by solving a reduced low degree equation derived from the original equation after one root has been extracted.