.. Even

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Even , Odd

u .. v u .: v

u .. v « (u + u&v) % 2:
u .: v « (u - u&v) % 2:

In the most commonly used case, v is arithmetic negation, and f=: u .. v is therefore f=: (u + u&-) % 2: ; that is, one-half the sum of u y and u -y . The resulting function is therefore even in the sense that f y « f -y for any y; its graph is reflected in the vertical axis. Similarly, u .: - is odd (f y « -f-y), and its graph is reflected in the origin. Less commonly, v is matrix transpose (|:), and may be any monadic function.

   y=: _2 _1 0 1 2
   1 2 3 4 5 & p. y            Polynomial with odd and even terms
57 3 1 15 129
 
   1 2 3 4 5 & p. .. - y       Even part of polynomial
93 9 1 9 93
   
   1 0 3 0 5 & p. y            Polynomial with even terms only
93 9 1 9 93

   E=: .. -                    Even adverb
   O=: .: -                    Odd adverb
   d=: 5j2&":@,.&.>            Display as columns with two digits

   d (5&o. ; ^O ; 6&o. ; ^E ; ^ ; (^E + ^O) ; 2&o. ; ^@j.E) y
+-----+-----+-----+-----+-----+-----+-----+-----+
|_3.63|_3.63| 3.76| 3.76| 0.14| 0.14|_0.42|_0.42|
|_1.18|_1.18| 1.54| 1.54| 0.37| 0.37| 0.54| 0.54|
| 0.00| 0.00| 1.00| 1.00| 1.00| 1.00| 1.00| 1.00|
| 1.18| 1.18| 1.54| 1.54| 2.72| 2.72| 0.54| 0.54|
| 3.63| 3.63| 3.76| 3.76| 7.39| 7.39|_0.42|_0.42|
+-----+-----+-----+-----+-----+-----+-----+-----+

   m=: ?. 4 4 $ 9
   (] ; (] .. |:) ; (] .: |:)) m
+-------+---------------+------------------+
|1 6 4 4|  1 3.5   6   2|   0  2.5   _2   2|
|1 0 6 6|3.5   0 4.5   3|_2.5    0  1.5   3|
|8 3 4 7|  6 4.5   4 5.5|   2 _1.5    0 1.5|
|0 0 4 6|  2   3 5.5   6|  _2   _3 _1.5   0|
+-------+---------------+------------------+

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