⌽"trams"smartRearranging and transforming arrays
Reverse, rotate and transpose arrays; work with windows, selections and choices
Read Searching and indexing first.
Rotate / transpose / flip
⌽ (Circle Stile)
monadic ⌽ (Reverse)
Reverse puts items in the opposite order along the last axis. On a matrix, it reverses each row:
mat←[0 1 2 3⋄4 5 6 7⋄8 9 10 11]
mat0 1 2 3
4 5 6 7
8 9 10 11⌽mat 3 2 1 0
7 6 5 4
11 10 9 8Reverse undoes itself: its inverse is Reverse again.
⌽⁻¹"smart"tramsdyadic ⌽ (Rotate)
A positive count rotates left; a negative count rotates right. Items that leave one end return at the other.
1⌽"HatStand"atStandH3⌽"HatStand"StandHat¯2⌽1 2 3 4 5 65 6 1 2 3 4mat0 1 2 3
4 5 6 7
8 9 10 11¯1⌽mat 3 0 1 2
7 4 5 6
11 8 9 10A vector of counts can rotate each row by a different amount:
1 ¯1 2⌽mat 1 2 3 0
7 4 5 6
10 11 8 9The inverse of Rotate turns in the opposite direction, using the same count:
1⌽⁻¹"atStandH"HatStand⊖ (Circle Bar)
monadic ⊖ (Reverse First)
⊖ acts along the first axis. A vector has only one axis, so ⌽ and ⊖ do the same thing there. On a matrix, ⊖ reverses or rotates the rows.
⊖"trams"smart⊖mat8 9 10 11
4 5 6 7
0 1 2 3Reverse First also undoes itself:
⊖⁻¹⊖mat0 1 2 3
4 5 6 7
8 9 10 11dyadic ⊖ (Rotate First)
3⊖"HatStand"StandHat¯1⊖mat8 9 10 11
0 1 2 3
4 5 6 7Its dyadic inverse reverses the rotation, using the same count:
¯1⊖⁻¹¯1⊖mat0 1 2 3
4 5 6 7
8 9 10 11⍉ (Circle Backslash)
monadic ⍉ (Transpose)
Transpose reverses the order of the axes. A matrix’s rows become its columns:
⍉mat0 4 8
1 5 9
2 6 10
3 7 11A 2 3 4 array becomes a 4 3 2 array:
cube←2 3 4⍴⍳24
cube 0 1 2 3
4 5 6 7
8 9 10 11
12 13 14 15
16 17 18 19
20 21 22 23⍉cube 0 12
4 16
8 20
1 13
5 17
9 21
2 14
6 18
10 22
3 15
7 19
11 23⍴⍉cube[4 3 2]ₓTranspose undoes itself too:
⍉⁻¹⍉mat0 1 2 3
4 5 6 7
8 9 10 11The superscript ᵀ writes monadic Transpose after an array. It is for arrays, not functions. It binds before functions, as the digit superscripts do. After a space it applies to the whole run on its left:
matᵀ0 4 8
1 5 9
2 6 10
3 7 11Without a space, Index reads row 1 of the transpose. With a space before ᵀ, it first reads row 1, then transposes that vector:
1⌷matᵀ1 5 91⌷mat ᵀ4 5 6 7dyadic ⍉ (Transpose)
Each item on the left gives the destination of the corresponding input axis. 1 0 sends the rows to axis 1 and the columns to axis 0:
1 0⍉mat0 4 8
1 5 9
2 6 10
3 7 11A diagonal takes matching positions along two axes: row 0 with column 0, row 1 with column 1, and so on. Sending two axes to the same destination selects their diagonal. The shorter axis determines its length:
0 0⍉mat0 5 10With three axes, 1 0 2 swaps the first two. Repeated destinations again select diagonals:
⍴cube[2 3 4]ₓ⍴1 0 2⍉cube[3 2 4]ₓ0 0 0⍉cube0 171 0 0⍉cube 0 12
5 17
10 22When each axis has a different destination, the inverse undoes the routing. Repeated destinations discard items, so a diagonal has no standalone inverse:
(2 0 1⍉⁻¹2 0 1⍉cube)≡cube$tSplitting, naming and merging axes
A nested item splits one input axis into several axes. Here the last axis of length 8 becomes two axes of lengths 2 and 4, named "group" and "offset". Each pair gives a destination and length. ∞ computes the length needed to keep the number of items unchanged. The key on the group, "packed", names the axis when it is merged back:
packed←["batch":2 "row":3 "packed":8]⍴⍳48
split←["batch":0 "row":2 "packed":["group":1 2;"offset":3 ∞]]⍉
⍴split packed["batch":2 "group":2 "row":3 "offset":4]ₓThe inverse form, ⍉⁻¹, lists input axes in output order. A nested group merges its axes into one. This puts the group and offset axes together again:
merged←["batch" "row" "packed":["group" "offset"]]⍉⁻¹split packed
merged≡packed$tThe inverse of a split merges the factors back into their original axis. This bound function remembers the numeric routing:
(split⁻¹split packed)≡packed$tA named output pattern in ⍉⁻¹ does not record the previous axis order, so it has no standalone inverse.
Other operators
⌺ (Quad Diamond)
Dyadic operator ⌺ (Stencil)
A window is a section of an array around one position. Stencil applies its left operand to each window, with that position at the centre. Its right operand gives the window’s size. At the edges, missing items use fill: zero for these numeric arrays. Enclosing each window shows what the operand receives:
⊂⌺3[1 2 3 4](0 1 2⋄1 2 3⋄2 3 4⋄3 4 0)Replace enclosure with summation to add the neighbours:
+/⌺3[1 2 3 4]3 6 9 7Two sizes make a rectangular window. Here we sum each three-by-three window:
{+/,⍵}⌺[3 3] [1 2 3⋄4 5 6⋄7 8 9]12 21 16
27 45 33
24 39 28A two-row specification gives the sizes above the movements. This uses three-item windows centred two positions apart:
+/⌺[[3]⋄[2]]⍳81 6 12 18⊘ (Valences)
A function’s valence is the number of arguments it takes. Valences supplies separate functions for calls with one and two arguments. This function negates with one argument and adds with two:
f←-⊘+
f3¯310 f 313The inverse uses the inverse of whichever function that valence selects. Here it negates with one argument and subtracts the fixed left argument with two:
f⁻¹¯3310 f⁻¹ 133@ (Under)
f@g applies f to the selection that g makes, and writes it back. The other items stay unchanged. A list of positions selects along the first axis. A value in the function position replaces the selection:
0@[1 3] ⍳60 0 2 0 4 5[30 10]@[1 3] ⍳60 30 2 10 4 5Exact input keeps the reciprocals rational. Positions 1 and 3 hold 2 and 4, whose reciprocals are 1ᵣ2 and 1ᵣ4:
÷@[1 3] ↦6ₓ1ₓ 1ᵣ2 3ₓ 1ᵣ4 5ₓ 6ₓ10 ×@[1 3] ⍳50 10 2 30 4A function inside a one-item list selects where its result is true. Here 2| marks the odd numbers:
0@[2|] ⍳60 0 2 0 4 0⌽@[2|] ⍳60 5 2 3 4 1A selecting function can also supply the right operand. Drop skips the first item; Under changes only the rest:
-@ 1↓ 1 2 31 ¯2 ¯3With an invertible transformation, Under transforms, applies its left operand, then reverses the transformation. To round down to tenths, multiply by ten, floor, then divide by ten:
⌊@ 10× 1.25 2.781.2 2.7Dot selections after Same can supply Under’s selector without a dfn. Here the selected positions are multiplied by ten; a field selector changes just that field:
v←10 20 30 40
{⍵×10}@ ⊢.[[1 3]] v10 200 30 400order←["price":10 "qty":2]
≥@ ⊢.qty order["price":10 "qty":3]Under’s inverse keeps the selection and inverts its left function. This decrements the selected items:
(≥@[1 3])⁻¹10 20 30 4010 19 30 39⍚ (Agenda)
Agenda chooses a function from a list. Each function is a case. Its left operand supplies the case’s position, or a function that computes that position. A false Boolean chooses case 0, and a true Boolean chooses case 1:
abs←(0≤)⍚[- ⊢]
abs¯33abs33A vector of choices picks one case for each cell. Here negative numbers become zero, while nonnegative numbers take their square roots:
(0≤)⍚[0 √] ¯4 9 160 3 4Execute and Format
⍎ (Hydrant)
Monadic ⍎ (Execute expression)
Execute reads a string as code and returns its last result. The string can refer to names already defined here.
⍎"1+1"2v←1 2 3
⍎"v"1 2 3Intermediate statements still display their results, even when the last result is assigned. This prints 2 and stores 4 in answer:
answer←⍎"1+1 ⋄ 2+2"2answer4Execute’s inverse writes a value as BPL source text that Execute can read back. Exact numbers keep their marks:
source←⍎⁻¹[1 2 3]ₓ
[source]["[1 2 3]ₓ"]⍎source[1 2 3]ₓdyadic ⍎ (Execute in)
A keyed left argument supplies names for the expression:
["price":10.5 "qty":2]⍎"price×qty"21⍕ (Thorn)
Monadic ⍕ (Format)
Format turns values into text. The numeric vector and its character representation display alike:
4 5 64 5 6⍕4 5 64 5 6The vector has three numbers; the text has five characters, including its two spaces:
[≢4 5 6;≢⍕4 5 6][3 5]ₓA single number on the left sets the decimal places, the number of digits after the decimal point:
2⍕3.125 0.0023.13 0.00Dyadic ⍕ (Format By Specification)
A two-item vector gives the field width and decimal places. The width is the number of characters for each number, padded with spaces:
6 2⍕3.125 0.002 3.13 0.00Indexing covers more selections, and each entry in the glyph reference links to its full definitions.
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