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{{NumBlk|:|<math>F\left( u \right) - F\left( v \right) = G\left( u, v \right) \bigl[u - v\bigr] </math>|{{EquationRef|1}}}}
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}}).
▲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>.
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>, fixed point functions <math>F</math> and their linear operators <math>G</math> meeting condition ({{EquationNote|1}}), <math>F'(x) \approx G \bigl( F\left( x \right), x \bigr) \approx I</math>, where <math>I</math> is the identity operator.
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