Class: Interval::Simple

Inherits:
Interval show all
Includes:
Enumerable
Defined in:
lib/interval.rb

Overview

Implements simple intervals. A simple interval is an interval with a single component, i.e., a connected set of real numbers (extended with plus and minus infinity.)

Constant Summary

Constants inherited from Interval

E, NewtonOptions, PI

Instance Attribute Summary collapse

Instance Method Summary collapse

Methods inherited from Interval

#&, #*, #+, #-, #/, [], #coerce, #empty?, eval, #hull, #inspect, #newton, #op, sum, #to_interval, #to_s, union, #|

Constructor Details

#initialize(a, b = a) ⇒ Simple

Constructs a simple interval, with given lower and upper bounds.

Interval::Simple.new(1)   # => Interval[1]
Interval::Simple.new(1,2) # => Interval[1,2]

Raises:

  • (ArgumentError) 


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# File 'lib/interval.rb', line 476

def initialize (a, b = a)
  raise ArgumentError, "Extrema must be numeric: #{[a,b].uniq.inspect}" unless
    Numeric === a && Numeric === b
  if (a.respond_to?(:nan?) && a.nan? ) || (b.respond_to?(:nan?) && b.nan?)
    @inf, @sup = -Infinity, Infinity
  else
    @inf, @sup = a, b
  end
  freeze
end

Instance Attribute Details

#inf ⇒ Object (readonly)

Returns the value of attribute inf.



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# File 'lib/interval.rb', line 455

def inf
  @inf
end

#sup ⇒ Object (readonly)

Returns the value of attribute sup.



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# File 'lib/interval.rb', line 456

def sup
  @sup
end

Instance Method Details

#**(n) ⇒ Object

Implements Interval’s power operator.



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# File 'lib/interval.rb', line 601

def ** (n)  # :nodoc:
  if n < 0
    (self ** -n).inverse
  elsif inf > 0
    self.class.new(
      FPU::down { FPU.power(inf, n) },
      FPU::up   { FPU.power(sup, n) })
  elsif sup < 0
    if (-1) ** n > 0
      (-self)**n
    else
      - (-self)**n
    end
  elsif (-1) ** n > 0
    self.class.new(0, FPU::up {[FPU.power(inf, n), FPU.power(sup, n)]}.max )
  else
    self.class.new(
      FPU::down { FPU.power(inf, n) },
      FPU::up   { FPU.power(sup, n) })
  end
end

#-@ ⇒ Object

Unitary minus.



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# File 'lib/interval.rb', line 555

def -@  # :nodoc:
  self.class.new(-sup,-inf)
end

#==(other) ⇒ Object

Two simple intervals are equal if they have same inf and sup.



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# File 'lib/interval.rb', line 498

def == (other)
  self.class == other.class && inf == other.inf && sup == other.sup
end

#add(other) ⇒ Object

Used to implement Interval’s plus operator (+).



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# File 'lib/interval.rb', line 548

def add (other)
  self.class.new(
    FPU.down {inf + other.inf},
    FPU.up {sup + other.sup})
end

#atan ⇒ Object

Implements Interval#atan



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# File 'lib/interval.rb', line 652

def atan  #:nodoc:
  self.class.new(FPU.atan_down(inf), FPU.atan_up(sup))
end

#components ⇒ Object

Return an array with one element, itself.



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# File 'lib/interval.rb', line 467

def components
  [self]
end

#construction ⇒ Object

Overrides Interval#construction



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# File 'lib/interval.rb', line 488

def construction # :nodoc:
  extrema.uniq
end

#cos ⇒ Object

Implements Interval#cos



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# File 'lib/interval.rb', line 684

def cos  #:nodoc:
  return Interval[-1,1] if inf.abs == Infinity || sup.abs == Infinity
  p = inf.divmod(Math::PI).first.to_i
  self.class.new(
    if include?((2*((p-1).div(2))+3) * Math::PI)
      -1
    else
      [FPU.cos_down(inf), FPU.cos_down(sup)].min
    end,
    if include?((2*(p.div(2))+2) * Math::PI)
      +1
    else
      [FPU.cos_up(inf), FPU.cos_up(sup)].max
    end)
end

#cosh ⇒ Object

Implements Interval#cosh



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# File 'lib/interval.rb', line 673

def cosh  #:nodoc:
  if inf >=0
    self.class.new(FPU.cosh_down(inf), FPU.cosh_up(sup))
  elsif sup <= 0
    self.class.new(FPU.cosh_down(sup), FPU.cosh_up(inf))
  else
    self.class.new(0,FPU.cosh_up([-inf,sup].max))
  end
end

#degenerate? ⇒ Boolean

Implements Interval#degenerate?

Returns:

  • (Boolean) 


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# File 'lib/interval.rb', line 642

def degenerate?  # :nodoc:
  inf == sup
end

#dsub(other) ⇒ Object

Dependent subtraction, i.e., the inverse of the addition.

Interval[3,4].dsub Interval[1,2]
                         # => Interval[2]
Interval[1,2] + Interval[2]
                         # => Interval[3,4]



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# File 'lib/interval.rb', line 565

def dsub(other)
  self.class.new(
    FPU.down{inf - other.inf},
    FPU.up  {sup - other.sup})
end

#each {|_self| ... } ⇒ Object

Implements Enumerable#each

Yields:

  • (_self)

Yield Parameters:



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# File 'lib/interval.rb', line 461

def each
  yield(self)
  self
end

#exp ⇒ Object

Implements Interval#exp



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# File 'lib/interval.rb', line 647

def exp  #:nodoc:
  self.class.new(FPU.exp_down(inf), FPU.exp_up(sup))
end

#extrema ⇒ Object

Returns the array with inf and sup.

Interval[1,2].extrema    # => [1, 2]



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# File 'lib/interval.rb', line 511

def extrema
  [inf,sup]
end

#include?(x) ⇒ Boolean

Implements Interval#include?.

Returns:

  • (Boolean) 


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# File 'lib/interval.rb', line 503

def include?(x)  # :nodoc:
  inf <= x  && x <= sup
end

#intersect(other) ⇒ Object

Implements Interval’s & operator.



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# File 'lib/interval.rb', line 624

def intersect (other)
  self.class.new([inf,other.inf].max, [sup,other.sup].min)
end

#inverse ⇒ Object

Implements Interval#inverse.



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# File 'lib/interval.rb', line 588

def inverse  # :nodoc:
  if include?(0) then
    Interval[
      [-Infinity, FPU::up { inf == 0 ? -Infinity : inf**-1 }],
      [FPU::down { sup == 0 ? Infinity  : sup**-1 }, Infinity]]
  else
    self.class.new(
      FPU::down { sup ** -1 },
      FPU::up   { inf ** -1 })
  end
end

#log ⇒ Object

Implements Interval#log



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# File 'lib/interval.rb', line 662

def log  #:nodoc:
  if inf >= 0
    self.class.new(FPU.log_down(inf), FPU.log_up(sup))
  elsif sup >= 0
    self.class.new(-Infinity, FPU.log_up(sup))
  else
    Interval[]
  end
end

#midpoint ⇒ Object

The point in the middle of the interval.

Interval[1,3].midpoint   # => 2.0



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# File 'lib/interval.rb', line 528

def midpoint
  (inf + sup) * 2 **-1
end

#multiply(other) ⇒ Object

Used to implement Interval’s multiplicative operator (*).



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# File 'lib/interval.rb', line 572

def multiply (other)
  self.class.new(
    FPU.down {
      [inf * other.inf, inf * other.sup,
       sup * other.inf, sup * other.sup]}.min,
    FPU.up {
      [inf * other.inf, inf * other.sup,
       sup * other.inf, sup * other.sup]}.max)
  # min and max fail if any of the products is a NaN, which can only
  # be generated by 0 * Infinity. In this case, the result is the
  # whole R-star.
  rescue ArgumentError
    self.class.new(-Infinity, Infinity)
end

#number ⇒ Object

If the interval is degenerate, return its only element, else raise an exception.

Interval.eval{1/2.0}.number
                         # => 0.5
Interval.eval{1/3.0}.number
                         # fails



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# File 'lib/interval.rb', line 539

def number
  if inf == sup
    inf
  else
    raise Exception::Nondegenerate, self
  end
end

#sharp? ⇒ Boolean

Implements Interval#sharp?

Returns:

  • (Boolean) 


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# File 'lib/interval.rb', line 629

def sharp?  # :nodoc:
  w = width
  if w == 0
    true
  elsif w.kind_of?(Float) && (inf >= 0  || sup <= 0)
    s1, s2 = extrema.map{|x| x.abs}.sort
    s1 + s1.to_f.ulp == s2
  else
    false
  end
end

#simple? ⇒ Boolean

Implements Interval#simple?, returning true.

Returns:

  • (Boolean) 


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# File 'lib/interval.rb', line 493

def simple?  # :nodoc:
  true
end

#sin ⇒ Object

Implements Interval#sin



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# File 'lib/interval.rb', line 701

def sin  #:nodoc:
  return Interval[-1,1] if inf.abs == Infinity || sup.abs == Infinity
  q = inf.divmod(Math::PI/2).first.to_i
  p = (q-1).div(2)
  self.class.new(
    if include?((2*((p-1).div(2))+3.5) * Math::PI)
      -1
    else
      [FPU.sin_down(inf), FPU.sin_down(sup)].min
    end,
    if include?((2*(p.div(2))+2.5) * Math::PI)
      +1
    else
      [FPU.sin_up(inf), FPU.sin_up(sup)].max
    end)
end

#sinh ⇒ Object

Implements Interval#sinh



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# File 'lib/interval.rb', line 657

def sinh  #:nodoc:
  self.class.new(FPU.sinh_down(inf), FPU.sinh_up(sup))
end

#tan ⇒ Object

Implement Interval#tan



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# File 'lib/interval.rb', line 719

def tan  #:nodoc:
  q = inf.divmod(Math::PI/2).first.to_i
  p = (q-1).div(2)
  if include?(Math::PI * (p+2.5))
    self.class.new(-Infinity,Infinity)
  elsif include?(Math::PI * (p+1.5))
    Interval[[-Infinity,FPU.tan_up(sup)],[FPU.tan_down(inf),Infinity]]
  else
    self.class.new(FPU.tan_down(inf),FPU.tan_up(sup))
  end
end

#width ⇒ Object

The interval’s width is sup - inf.

Interval[2].width        # => 0
Interval[2,3].width      # => 1



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# File 'lib/interval.rb', line 520

def width
  sup - inf
end