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For the [[Function of a real variable|basic real number function]] <math>\ f\ ,</math> given in the first section, the function simply takes in and puts out real numbers. There, the function <math>\ g\ </math> is a ''[[divided difference]]''. In the generalized form here, the operator <math>\ G\ </math> is the analogue of a divided difference for use in the [[Banach space]]. The operator <math>G</math> is roughly equivalent to a [[Matrix (mathematics)|matrix]] whose entries are all functions of [[vector (mathematics)|vector]] [[Argument of a function|arguments]] <math>\ u\ </math> and <math>\ v ~.</math>
If division is possible in the [[Banach space]], then the linear operator <math>\ G\ </math> can be obtained from
:<math> G\left( u, v \right) = \bigl[\ F\left( u \right)- F\left( v \right)\ \bigr]\ \bigl(\ u - v\ \bigr)^{-1}\ ,</math>
which may provide some insight: Expressed in this way, the linear operator <math>\ G\ </math> can be more easily seen to be an elaborate version of the [[divided difference]] <math>\ g\ </math> discussed in the first section, above. The quotient form is shown here for orientation only; it is ''not'' required ''per se''. Note also that division within the Banach space is not necessary for the elaborated Steffensen's method to be viable; the only requirement is that the operator <math>\ G\ </math> satisfy ({{EquationNote|1}}).
Steffensen's method is then very similar to the Newton's method, except that it uses the divided difference <math>\ G \bigl( F\left( x \right), x \bigr)\ </math> instead of the derivative <math>\ F'(x) ~.</math> Note that for arguments <math>\ x\ </math> close to some fixed point <math>\ x_\star\ ,</math>
In the case that division is possible in the Banach space, the generalized iteration formula is given by
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