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Fixed the sentence fragment. The missing definition of value query comes from the value query model of fixed-point computation where the algorithm is provided with an oracle that can be queried. |
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For Lipschitz-continuous functions, the absolute criterion is stronger than the residual criterion: If <math>f</math> is Lipschitz-continuous with constant <math>L</math>, then <math>|x-x_0|\leq \delta</math> implies <math>|f(x)-f(x_0)|\leq L\cdot \delta</math>. Since <math>x_0</math> is a fixed-point of <math>f</math>, this implies <math>|f(x)-x_0|\leq L\cdot \delta</math>, so <math>|f(x)-x|\leq (1+L)\cdot \delta</math>. Therefore, a δ-absolute fixed-point is also an {{mvar|ε}}-residual fixed-point with <math>\varepsilon = (1+L)\cdot \delta</math>.
The most basic step of a fixed-point computation algorithm is a '''value query''': given any <math>x</math> in <math>E^d</math>, the
The function <math>f</math> is accessible via '''evaluation''' queries: for any <math>x</math>, the algorithm can evaluate <math>f(x)</math>. The run-time complexity of an algorithm is usually given by the number of required evaluations.
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