Is there a specific date when Laguerre published his root finding method? I found his 1880 note Résolution des équations numériques, but I am not sure if this is the source because I can not read French.


The scanned pages might be more readily available here.

Indeed the method is presented as a novel; it is only compared to Newton's method. It is first introduced as a method for finding root enclosures and approximations of the largest and smallest roots of polynomials

under the condition that all roots are real.

Under this condition the usual "trick" in the derivation also makes sense, as when $\alpha$ is the largest root, and $x$ is a close approximation, then $\frac1{x-\alpha}$ is large in $$ G=\frac{f'(x)}{f(x)}=\frac1{x-\alpha}+\frac1{x-\beta}+... $$ and $$ H=\frac{f'(x)^2-f(x)f''(x)}{f(x)^2}=\frac1{(x-\alpha)^2}+\frac1{(x-\beta)^2}+... $$ and the remaining terms are small and of the same sign and can thus be considered as approximately equal. Setting them equal to zero, as Laguerre discusses first, gives in the first equation Newton's method and in the second $$ 0<\frac1{(x-\alpha)^2}\le H \\ \implies \alpha\le x-\frac{|f(x)|}{\sqrt{f'(x)^2-f(x)f''(x)}} ~\text{ or }~ x+\frac{|f(x)|}{\sqrt{f'(x)^2-f(x)f''(x)}}\le \alpha $$ giving a root exclusion region around $x$.


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