On functional iteration and the calculation of roots

This paper has the dual objectives of (1) setting theoretical limits to the rates of convergence of iteration processes towards the zeros of a function when the values of the function, or the values of the function and its derivatives, are available and (2) suggesting new families of computationally effective iteration formulas. The proofs of the theorems stated, numerical verification of theoretical error estimates, various specific applications, and results concerning work on variations of the themes reported here, will appear later.

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