page_title: Interval arithmetic
section_title: Interval Newton method
hierarchical_section_title: Interval arithmetic / Interval methods / Interval Newton method
context_page_description: Interval arithmetic is a mathematical technique used to put bounds on rounding errors and measurement errors in mathematical computation. Numerical methods using interval arithmetic can guarantee reliable, mathematically correct results. Instead of representing a value as a single number, interval arithmetic represents each value as a range of possibilities. For example, instead of estimating the height of someone as exactly 2.0 metres, using interval arithmetic one might be certain that that person is somewhere between 1.97 and 2.03 metres.
  Mathematically, instead of working with an uncertain real, one works with the ends of an interval that contains. In interval arithmetic, any variable lies in the closed interval between and. A function, when applied to, yields an uncertain result; produces an interval which includes all the possible values for for all.
  Interval arithmetic is suitable for a variety of purposes. The most common use is in software, to keep track of rounding errors in calculations and of uncertainties in the knowledge of the exact values of physical and technical parameters.
context_section_description: An interval variant of Newton's method for finding the zeros in an interval vector  can be derived from the average value extension. For an unknown vector  applied to , gives
  .
  For a zero , that is , and thus must satisfy
  .
  This is equivalent to
  .
  An outer estimate of  can be determined using linear methods.
  In each step of the interval Newton method, an approximate starting value  is replaced by  and so the result can be improved iteratively. In contrast to traditional methods, the interval method approaches the result by containing the zeros. This guarantees that the result produces all zeros in the initial range. Conversely, it proves that no zeros of  were in the initial range  if a Newton step produces the empty set.
  The method converges on all zeros in the starting region. Division by zero can lead to separation of distinct zeros, though the separation may not be complete; it can be complemented by the bisection method.
  As an example, consider the function , the starting range , and the point . We then have  and the first Newton step gives
  .
  More Newton steps are used separately on  and . These converge to arbitrarily small intervals around  and .
  The Interval Newton method can also be used with thick functions such as  , which would in any case have interval results. The result then produces intervals containing .
caption: Reduction of the search area in the interval Newton step in "thick" functions
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