14. Partitions
The function sum=: +/ applies to an entire list argument;
to compute partial sums or subtotals,
it is necessary to apply it to each prefix of the argument. For example:
sum=: +/ a=: 1 2 3 4 5 6 (sum a) ; (sum a) +--+--------------+ |21|1 3 6 10 15 21| +--+--------------+
The symbol denotes the prefix adverb,
which applies its argument (in this case sum)
to each prefix of the eventual argument. The adverb . applies
similarly to suffixes:
sum. a 21 20 18 15 11 6
The monad < simply boxes its arguments,
and the verbs < and <. therefore show
the effects of partitions with great clarity. For example:
<1 2 3 +-----+ |1 2 3| +-----+ (<1),(<1 2),(<1 2 3) +-+---+-----+ |1|1 2|1 2 3| +-+---+-----+ < a +-+---+-----+-------+---------+-----------+ |1|1 2|1 2 3|1 2 3 4|1 2 3 4 5|1 2 3 4 5 6| +-+---+-----+-------+---------+-----------+ <. a +-----------+---------+-------+-----+---+-+ |1 2 3 4 5 6|2 3 4 5 6|3 4 5 6|4 5 6|5 6|6| +-----------+---------+-------+-----+---+-+
The oblique adverb /. partitions a table
along diagonal lines. Thus:
</. t=: 1 2 1 */ 1 3 3 1 +-+---+-----+-----+---+-+ |1|3 2|3 6 1|1 6 3|2 3|1| +-+---+-----+-----+---+-+ t ; (sum/. t) ; (10 #. sum/. t) ; (121*1331) +-------+-------------+------+------+ |1 3 3 1|1 5 10 10 5 1|161051|161051| |2 6 6 2| | | | |1 3 3 1| | | | +-------+-------------+------+------+
Exercises
| 14.1 | Define programs analogous to sum=:+/ for progressive products, progressive maxima, and progressive minima. |
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| 14.1 | Treat the following programs and comments like those of Section 12, that is, as exercises in reading and writing. Experiment with expressions such as c pol x and c pp d and (c pp d) pol x with c=:1 3 3 1 and d=:1 2 1 and x=:i.5 . See the dictionary or Section 20 for the use of rank:
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