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<p>So as to more efficiently represent this commonly occurring array, a new datatype — '''Arithmetic Progression Arrays''' ('''APAs''') — are created by the [[Index Generator]] function and may be reshaped arbitrarily (as long as no replication occurs) and still retain the datatype.</p> <p>For example, <apll>2 3 4β΄β³24</apll> has a very compact storage consisting of the array shape (<apll>2 3 4</apll>), the starting offset (<apll>βIO</apll>) and multiplier (<apll>1</apll>), plus the normal array overhead (which includes the number of elements (<apll>24</apll>)), for a total of <apll>96</apll> bytes. This datatype is also used to store reshaped Boolean or integer scalars, as in <apll>1e6β΄12</apll> or <apll>1e6/12</apll> both of which use <apll>80</apll> bytes (<apll>16</apll> bytes less than the preceding example because it has two fewer shape elements at eight bytes per).</p> <p>Within the implementation, this datatype is used to avoid treating elided coordinates for indexing (as in <apll>M[;1]</apll>) as a special case by substituting an APA for the elided coordinate (as in <apll>M[β³(β΄M)[1];1]</apll>).</p>
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