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u/. y Oblique Adverb
Rank Infinity -- operates on x and y as a whole, by items of y -- WHY IS THIS IMPORTANT?
(u/.y) applies verb u to the oblique diagonals of table y
] y=: 4 4 $ i.3 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 </. y +-+---+-----+-------+-----+---+-+ |0|1 1|2 2 2|0 0 0 0|1 1 1|2 2|0| +-+---+-----+-------+-----+---+-+
Common Uses
1. As part of the idiom for polynomial multiplication
(1 2) */ (3 4 2) NB. consider this product 3 4 2 6 8 4 +/ /. (1 2) */ (3 4 2) NB. sum up the oblique diagonals 3 10 10 4 pmul =: +//.@(*/) NB. define a dyadic verb to calculate the coefficients 1 2 pmul 3 4 2 NB. of e.g. (2x+1)(2x^2+4x+3)=4x^3+10x^2+10x+3 3 10 10 4
More Information
1. u can be a cyclic gerund.
Details
1. The arguments to the invocations of u are lists of (_2)-cells of y
2. If y has no items, the result of u/. y will also have no items.
However, u will be executed once on a cell of 0 _2-cells to establish the shape of a result of u, and then the result of u/. y will be a list of 0 of that shape
$ ]/. i. 0 3 NB. u is executed on a cell of shape 0 0 0 $ ]/. i. 0 3 4 NB. u is executed on a cell of shape 0 4 0 0 4
Use These Combinations
Combinations using u/. y that have exceptionally good performance are shown in Diagonals and Polynomials.
x u/. y Key Adverb x u/.. y Key dyad Adverb
Rank Infinity -- operates on x and y as a whole, by items -- WHY IS THIS IMPORTANT?
Classifies items of x into partitions of identical items. Then y is partitioned using the partitioning that was calculated for x, and u is applied on each partition of y.
Note: x and y must have the same number of items.
Each item of x is the key of the corresponding item of y, and u is applied to each set of items of y with equal keys.
For /., noun x is used only for the partitioning and is not an argument to operand u. For /.., the key value from x becomes the x argument to u.
x is classified by tolerant comparison. The value used for the x argument to u is the first one in the partition.
y=: 'AbcDeFGhijk' ] x=: y e. 'ABCDEFGHIJKLMNOPQRSTUVWXYZ' NB. 1=uppercase, 0=lowercase 1 0 0 1 0 1 1 0 0 0 0 x ]/. y NB. partitions of y ADFG bcehijk x </. y +----+-------+ |ADFG|bcehijk| +----+-------+ x ;/.. y NB. key from x and partition of y +-+-------+ |1|ADFG | +-+-------+ |0|bcehijk| +-+-------+
Common Uses
1. Count the numbers of different items, using #/.
a =: 3 1 4 1 5 9 2 6 5 3 5 8 9 7 9 (~.a) ,: (a #/. a) NB. unique digits, then frequency of each 3 1 4 5 9 2 6 8 7 2 2 1 3 3 1 1 1 1
2. Collect information from records with identical keys
]data =: _2 ]\ 'Fred';90;'John';75;'Fred';95;'John';85;'Susan';100;'John';86 NB. A table
+-----+---+
|Fred |90 |
+-----+---+
|John |75 |
+-----+---+
|Fred |95 |
+-----+---+
|John |85 |
+-----+---+
|Susan|100|
+-----+---+
|John |86 |
+-----+---+
]s =: ({."1 data) <@;/. ({:"1 data) NB. Use first column as keys; collect second-column values
+-----+--------+---+
|90 95|75 85 86|100|
+-----+--------+---+
(~. {."1 data) ,. s NB. Reinstall the name
+-----+--------+
|Fred |90 95 |
+-----+--------+
|John |75 85 86|
+-----+--------+
|Susan|100 |
+-----+--------+
More Information
1. u can be a cyclic gerund
x tolower`toupper/. y adfg BCEHIJK
Details
1. x u/. y is a member of the i.-family.
2. The argument to each invocation of u is a list of items of y (this has the same rank as y unless y is an atom).
3. The partitions are processed in the same order as the items in ~. x .
4. The partitioning uses tolerant comparison. Use u/.!.0 for intolerant comparison.
5. If y has no items, the result of x u/. y will also have no items.
However, u will be executed once on a cell of 0 items to establish the shape of a result of u, and then the result of x u/. y will be a list of 0 of that shape
$ '' ]/. '' NB. u is executed on cell of shape 0
0 0
$ '' {./. '' NB. u produces an atomic result
0
$ '' {./. i. 0 3 NB. u is executed on cell of shape 0 3
0 3
---
Oddities
If a tolerance is specified using u/.!.f, that tolerance is used as the default tolerance during the execution of u. ---
Use These Combinations
Combinations using x u/. y that have exceptionally good performance are shown in Partitions and Diagonals and Polynomials.