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MichaelBurr (talk | contribs) →Interval Newton: Editing interval Newton section |
MichaelBurr (talk | contribs) →Interval Newton: Working on interval Newton. Making various statements more precise. |
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# If <math>G</math> is [[contractive]] in a region containing <math>I</math>, then there is at most one root in <math>I</math>.
For instance, we observe that if <math>I</math> is a compact and convex region and <math>y\in I</math>, then, for any <math>x\in I</math>, there exist <math>c_1,\dots,c_n</math> such that
These properties can often be established using interval arithmetic. For example, the [[Interval_arithmetic#Interval_Newton_method|interval Newton operator]] replaces <math>x</math> with the region <math>I</math> containing <math>x</math>. For example, the interval Newton operator is▼
<math display="block">F(y)-F(x)=\begin{bmatrix}\nabla f_1(c_1)^T\\\vdots\\\nabla f_n(c_n)^T\end{bmatrix}(y-x).</math>
▲These properties can often be established using interval arithmetic. For example, the [[Interval_arithmetic#Interval_Newton_method|interval Newton operator]] replaces
<math>x</math> with the region <math>I</math> containing <math>x</math>. For example, the interval Newton operator is
<math display="block">N(I)=m(I)+D(I)^{-1}\cdot(I-m(I)),</math>
where <math>m(I)</math> is a point (often the [[center of mass]]) of <math>I</math> and <math>D(I)^{-1}</math> is an interval over-approximation to the set inverses of derivatives for <math>x\in I</math>. Then, one can show that if <math>x</math> is a fixed point of
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