Generating a Sequence of Numbers

Problem

You want to iterate over a (possibly infinite) sequence of numbers the way you can iterate over an array or a range.

Solution

Write a generator function that yields each number in the sequence.

	def fibonacci(limit = nil)
	 seed1 = 0
	 seed2 = 1
	 while not limit or seed2 <= limit
	 yield seed2
	 seed1, seed2 = seed2, seed1 + seed2
	 end
	end

	fibonacci(3) { |x| puts x }
	# 1
	# 1
	# 2
	# 3

	fibonacci(1) { |x| puts x }
	# 1
	# 1

	fibonacci { |x| break if x > 20; puts x }
	# 1
	# 1
	# 2
	# 3
	# 5
	# 8
	# 13

Discussion

A generator for a sequence of numbers works just like one that iterates over an array or other data structure. The main difference is that iterations over a data structure usually have a natural stopping point, whereas most common number sequences are infinite.

One strategy is to implement a method called each that yields the entire sequence. This works especially well if the sequence is finite. If not, it's the responsibility of the code block that consumes the sequence to stop the iteration with the break keyword.

Range#each is an example of an iterator over a finite sequence, while Prime#each enumerates the infinite set of prime numbers. Range#each is implemented in C, but here's a (much slower) pure Ruby implementation for study. This code uses self.begin and self.end to call Range#begin and Range#end, because begin and end are reserved words in Ruby.

	class Range
	 def each_slow
	 x = self.begin
	 while x <= self.end
	 yield x
	 x = x.succ
	 end
	 end
	end

	(1..3).each_slow {|x| puts x} 
	# 1
	# 2
	# 3

The other kind of sequence generator iterates over a finite portion of an infinite sequence. These are methods like Fixnum#upto and Fixnum#step: they take a start and/ or an end point as input, and generate a finite sequence within those boundaries.

	class Fixnum
	 def double_upto(stop)
	 x = self
	 until x > stop
	 yield x
	 x = x * 2
	 end
	 end
	end
	10.double_upto(50) { |x| puts x }
	# 10
	# 20
	# 40

Most sequences move monotonically up or down, but it doesn't have to be that way:

	def oscillator
	 x = 1
	 while true
	 yield x
	 x *= -2
	 end
	end
	oscillator { |x| puts x; break if x.abs > 50; }
	# 1
	# -2
	# 4
	# -8
	# 16
	# -32
	# 64

Though integer sequences are the most common, any type of number can be used in a sequence. For instance, Float#step works just like Integer#step:

	1.5.step(2.0, 0.25) { |x| puts x }
	# => 1.5
	# => 1.75
	# => 2.0

Float objects don't have the resolution to represent every real number. Very small differences between numbers are lost. This means that some Float sequences you might think would go on forever will eventually end:

	def zeno(start, stop)
	 distance = stop - start
	 travelled = start
	 while travelled < stop and distance > 0
	 yield travelled
	 distance = distance / 2.0
	 travelled += distance
	 end 
	end

	steps = 0 
	zeno(0, 1) { steps += 1 }
	steps # => 54