xal.tools.math.Interval

All Implemented Interfaces:: Serializable

Direct Known Subclasses:: ClosedInterval, OpenInterval

public class Interval extends Object implements Serializable

Represents an interval on the real line. Intervals are identified by their end points, which may or may not be included depending upon whether the interval is open or closed. Nonetheless, we consider the left end the minimum value and the right end point the maximum value and refer to them as such.

Issues concerning boundary points are delegated to the derived classes.

Author:

Christopher K. Allen

See Also:

Serialized Form

Field Summary

Fields

Modifier and Type

Field

Description

static final Interval

REAL_LINE

The entire real line
Constructor Summary

Constructors

Constructor

Description

Interval()

Default constructor - creates a new instance of Interval with one point, the origin.

Interval(double min, double max)

Initializing constructor - create an interval with specified endpoints

Interval(Interval interval)

Copy constructor - create a new open interval initialized to the argument.
Method Summary

Modifier and Type

Method

Description

boolean

containsAE(Interval interval)

Contains Almost Everywhere - ⊂ a.e.

Interval

convexHull(Interval interval)

Compute and return the smallest interval containing both this interval and the argument interval (i.e., the union of they are intersected).

static Interval

createFromEndpoints(double dblMin, double dblMax)

Creates a new interval object according to the given endpoints.

static Interval

createFromMidpoint(double dblMidpt, double dblLng)

Creates an interval object given the midpoint (centroid) of the interval and the length of the interval (which is the Lebesgue measure).

boolean

equals(Object obj)

Are intervals equal

double

getMax()

Get maximum value of interval.

double

getMin()

Get minimum value of interval.

int

hashCode()

Interval

intersection(Interval interval)

Compute and return the largest interval contained in this interval and the argument interval (i.e., the intersection).

boolean

intersects(Interval interval)

Inspects for non-empty intersection with the given interval.

boolean

isBoundary(double x)

Is point a boundary element?

boolean

isRealLine()

Test if this interval represents the entire real line.

double

measure()

Compute the interval length.

boolean

membership(double x)

Is point a member of this interval I ⊂ R.

double

midpoint()

Compute the interval midpoint.

void

print(PrintWriter os)

Print out contents on an output stream

void

println(PrintWriter os)

Print out contents on an output stream, terminate in newline character

void

setMax(double max)

Set the right end point

void

setMin(double min)

Set the left end point

String

toString()

Return the contents of the interval as a String.

double[]

vertexCoordinates(double x)

Compute the local "vertex coordinates" of the argument x with respect to this interval I ⊂ R.

Methods inherited from class java.lang.Object
clone, finalize, getClass, notify, notifyAll, wait, wait, wait

Field Details
- REAL_LINE
  
  public static final Interval REAL_LINE
  
  The entire real line
Constructor Details
- Interval
  
  public Interval()
  
  Default constructor - creates a new instance of Interval with one point, the origin.
- Interval
  
  public Interval(double min, double max) throws IllegalArgumentException
  
  Initializing constructor - create an interval with specified endpoints
  
  Parameters:
  
  min - left endpoint
  
  max - right endpoint
  
  Throws:
  
  IllegalArgumentException - max is smaller than min
- Interval
  
  public Interval(Interval interval) throws MathException
  
  Copy constructor - create a new open interval initialized to the argument.
  
  Parameters:
  
  interval - interval to copy
  
  Throws:
  
  MathException - malformed interval to copy
Method Details
- createFromMidpoint
  
  public static Interval createFromMidpoint(double dblMidpt, double dblLng)
  
  Creates an interval object given the midpoint (centroid) of the interval and the length of the interval (which is the Lebesgue measure). We do not throw an exception assuming that both arguments are positive.
  
  Parameters:
  
  dblMidpt - the center location of the interval
  
  dblLng - the length of the interval
  
  Returns:
  
  new interval object with endpoints producing the given center and length
  
  Since:
  
  Apr 28, 2011
- createFromEndpoints
  
  public static Interval createFromEndpoints(double dblMin, double dblMax)
  
  Creates a new interval object according to the given endpoints. We do not throw an exception here, which would normally happen if the left-hand endpoint was larger than the right-hand endpoint. Instead we return a null value in this case.
  
  Parameters:
  
  dblMin - the left-hand endpoint
  
  dblMax - the right-hand endpoint
  
  Returns:
  
  the interval [dblMin,dblMax]
  
  Since:
  
  Apr 28, 2011
- setMin
  
  public void setMin(double min)
  
  Set the left end point
  
  Parameters:
  
  min - interval left-hand side
- setMax
  
  public void setMax(double max)
  
  Set the right end point
  
  Parameters:
  
  max - interval right-hand side
- getMin
  
  public double getMin()
  
  Get minimum value of interval.
  
  Returns:
  
  the left end point
- getMax
  
  public double getMax()
  
  Get maximum value of interval.
  
  Returns:
  
  the right end point
- measure
  
  public double measure()
  
  Compute the interval length. This is the difference in endpoints.
  
  Returns:
  
  length of the interval
- midpoint
  
  public double midpoint()
  
  Compute the interval midpoint. This is the average of the endpoints.
  
  Returns:
  
  interval center of mass
- membership
  
  public boolean membership(double x)
  
  Is point a member of this interval I ⊂ R. The test is done assuming I is closed, so the endpoints are included.
  
  Parameters:
  
  x - point to be tested as a member of this interval
  
  Returns:
  
  true if x ∈ I, false otherwise
- equals
  
  public boolean equals(Object obj)
  
  Are intervals equal
  
  Overrides:
  
  equals in class Object
  
  Parameters:
  
  I - interval object to be checked for equality
  
  Returns:
  
  true if both objects are equal as intervals
- hashCode
  
  public int hashCode()
  
  Overrides:
  
  hashCode in class Object
- isRealLine
  
  public boolean isRealLine()
  
  Test if this interval represents the entire real line. Such an interval has the structure [Double.NEGATIVE_INFINITY,Double.POSITIVE_INFINITY]].
  
  Returns:
  
  true if this interval represents the entire real line, false otherwise
  
  Since:
  
  Sep 25, 2015 by Christopher K. Allen
- intersects
  
  public boolean intersects(Interval interval)
  
  Inspects for non-empty intersection with the given interval. The interval is assumed closed.
  
  Parameters:
  
  interval - interval to inspect.
  
  Returns:
  
  true if this ∩ I ≠ {}, false otherwise
  
  Since:
  
  Jun 2, 2009
- containsAE
  
  public boolean containsAE(Interval interval)
  
  Contains Almost Everywhere - ⊂ a.e.
  
  Checks whether or not the given interval is a proper subset of this interval, modulo the endpoints. Specifically, I must be contained within this interval modulo the endpoints. We ignore the endpoints and, thus, the condition of whether or not either interval is closed or open.
  
  Parameters:
  
  interval - interval to be tested
  
  Returns:
  
  true if I ⊂ this, false otherwise
  
  Since:
  
  Apr 28, 2011
- isBoundary
  
  public boolean isBoundary(double x)
  
  Is point a boundary element?
  
  Parameters:
  
  x - point to be tested as boundary element
  
  Returns:
  
  true if x is a boundary element
- vertexCoordinates
  
  public double[] vertexCoordinates(double x) throws MathException
  
  Compute the local "vertex coordinates" of the argument x with respect to this interval I ⊂ R. These "local coordinates" {λ₁,λ₂} of a point x ∈ I can computed by solving the following linear equation for the {λ_i} :
  
  ⌈ x₁ x₂ ⌉
  
  ⌊ 1 1 ⌋
  
  ⌈ λ₁ ⌉
  
  | λ₂ |
  
  =
  
  ⌈ x ⌉
  
  ⌊ 1 ⌋
  
  where p = x ∈ R.
  
  NOTES:
  · After solving the above equation for the {λ₁,λ₂} if there is a λ_i such that λ_i ∉ [0,1], then x is not in T; that is, p∉ T.
  
  Parameters:
  
  x - point in I
  
  Returns:
  
  order 2-array vertex coordinates {λ₁,λ₂}
  
  Throws:
  
  MathException - argument is not a member of this set
- convexHull
  
  public Interval convexHull(Interval interval) throws MathException
  
  Compute and return the smallest interval containing both this interval and the argument interval (i.e., the union of they are intersected).
  
  Parameters:
  
  interval - right-hand-side argument
  
  Returns:
  
  union of this and I
  
  Throws:
  
  MathException - empty intersection
- intersection
  
  public Interval intersection(Interval interval)
  
  Compute and return the largest interval contained in this interval and the argument interval (i.e., the intersection). If the intersection of the two intervals contains no points (i.e., it is the empty set), then a null value is returned. Since the empty is a valid result of intersection, this value should be considered in any robust implementation.
  
  Parameters:
  
  interval - interval to be intersected with this interval
  
  Returns:
  
  intersection of this and I, or null if the empty set {}
- toString
  
  public String toString()
  
  Return the contents of the interval as a String. return string representation of interval
  Overrides:
  
  toString in class Object
  
  See Also:
  
  Object.toString()
- print
  
  public void print(PrintWriter os)
  
  Print out contents on an output stream
  
  Parameters:
  
  os - output stream receiving content dump
- println
  
  public void println(PrintWriter os)
  
  Print out contents on an output stream, terminate in newline character
  
  Parameters:
  
  os - output stream receiving content dump

Class Interval

Field Summary

Constructor Summary

Method Summary

Methods inherited from class java.lang.Object

Field Details

REAL_LINE

Constructor Details

Interval

Interval

Interval

Method Details

createFromMidpoint

createFromEndpoints

setMin

setMax

getMin

getMax

measure

midpoint

membership

equals

hashCode

isRealLine

intersects

containsAE

Contains Almost Everywhere - ⊂ a.e.

isBoundary

vertexCoordinates

NOTES:

convexHull

intersection

toString

print

println