An Introduction to Interval Methods

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An Introduction to Interval Methods

• Zach Voller

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Basic Notation

• An interval is defined to be
• where is the infimum of and
  is the supremum of
• An interval is thin if
• An interval is thick if

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Order Relations for Intervals

• Let then
• Thus

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Examples

• Some clear relations

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Elementary Operations

• Let then
• Notice that this restricts our definition of
  t o be
• defined on intervals such that
• Most elementary operations can be defined just by
  using the endpoints of the intervals

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Examples of elementary Operations

• Note the following examples

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Elementary Functions

• Elementary Functions are members of a predefined
  set of real function that are continuous on every
  closed interval that they are defined on.
• Some examples of elementary functions are
• (this is magnitude, not absolute
  value)

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Rounding Error and Interval Methods

• The way a function is defined plays a critical
  role in determining if the calculated interval is
  an accurate estimation.
• This is due to the memoryless property of
  interval methods

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Example

• Let
• Then,
• but,
• where is the outward rounded value of

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Rounding Error (ctd.)

• This error is due to the fact that the variable
• occurs more than once in the function
• Theorem
• Let be an arithmetical expression in
• if each variable occurs only once then,
• for all
  intervals in the domain of where
  is the tightest interval containing
  .

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Rounding Error (ctd.)

• Thus the challenge is to change the expression to
  an equivalent form with only one copy of each
  variable.
• This can be done by methods like completing the
  square or other transformations.
• These errors leads to an overestimation of the
  range of the function values may become extremely
  large over the set of intervals

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Error (ctd.)

• While the overestimation may blow up over a set
  of intervals, it will remain small for
  sufficiently narrow intervals.
• We can bound the range by considering the
  deviation of from a fixed center (often
  the midpoint of the interval).

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Computing a bound for the overestimation of the
range
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Mean Value Form

• Let be the midpoint of the
  interval .
• then by mean value theorem we get that
• () for
  some , thus
• and we get that ()
• Thus the Mean Value Form
• is obtained.

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Example

• Let then we get the
  following values

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Example (ctd.)

• Thus we see that the Mean Value Form is a more
  accurate approximation for narrower intervals.
• Note In the previous example I did not take into
  account the rounding error for . Since
  computers can only calculate a finite number of
  digits, the approximation will be slightly worse.

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Mean Value Form (ctd.)

• It can be shown that for narrow intervals the
  Mean Value Form gives a better enclosure for the
  range of the function than simple interval
  evaluation.
• There are other very similar methods that all
  involve a fixed center value in the interval. Any
  of these methods satisfy an overestimation bound
  that can be easily checked for acuracy.

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The Error Bound

• Let be a real function,
  an interval in and let . Suppose
  that is an n-dimensional vector such that
• 
  for some
• then, the interval encloses
  the range of and the error bound is
• I need to work on this to further understand the
  proof and how to efficiently calculate this!!!!!!

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Interval Matrices

• Let
• be a (m x n) interval matrix where each is
  an interval. We now define

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Interval Matrices (ctd.)

• Now note that we can redefine the matrix interval
  as
• Interval Matrix Addition and Subtraction are
  defined as before where the entries of the
  interval matrix are added or subtracted component
  wise.
• All the normal properties of matrix addition
  hold. (i.e associative, commutative)

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Interval Matrices (ctd.)

• We need to be careful with scalar multiplication
  since the scalars can be either a constant or an
  interval. Scalar distribution does not hold when
  the scalar is an interval.
• Let be interval matrices, and
  an interval over the reals then,

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Example

• Let
• then,
• And for scalars

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Example (ctd)

• Note but is
  not in
• thus,

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Linear Interval Equations

• A linear interval equation with (m x n)
  coefficient matrix whose entries are
  intervals and right hand side an (m x 1)
  interval vector is the family of linear
  equations
• such that
• The enclosure of the solution set is denoted by

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Linear Interval Equations (ctd)

• Typically, the structure of the solution is
  complicated and is usually not an interval
  vector.
• We denote the hull of the solution set
• This is the interval that all solution are
  contained in.
• If is an invertible square matrix, then
• has a solution

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Solving a basic linear system

• Let be a (m x n) interval matrix over the
  reals, then

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Solving a basic linear system (ctd)

• Theorem
• Let be (m x n) interval matrix over the
  reals and a (m x 1) interval vector over the
  reals. Then

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Example

• Let
• then,

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Example (ctd)

• Now,

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Example (ctd)

• Thus,
• Plotting the solution we get

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Graph