[Bug-apl] APL2 reference sheet

[McGuinness, Brian]

This is a quick reference sheet on APL2
that I put together for my own use.
Please feel free to use it in your
documentation if you feel that it is
useful.

--- Brian

Title: APL2 Quick Reference Sheet

APL2 Quick Reference Sheet

Complex Conjugate	+X	+23 0 ¯31 1J1 ⇔ 23 0 ¯31 1J¯1		Add	A+B	75 3 46 + 5 ¯2 8 ⇔ 80 1 54
Negate	-X	-1 ¯17 44.8 ⇔ ¯1 17 ¯44.8		Subtract	A-B	75 3 46 - 5 ¯2 8 ⇔ 70 5 38
Signum	×X	×56.2 ¯1.4 0 0J¯3 ⇔ 1 ¯1 0 0J¯1		Multiply	A×B	75 3 46 × 5 ¯2 8 ⇔ 375 ¯6 368
Reciprocal	÷X	÷1 2 3 4 ⇔ 1 0.5 0.3333333333 0.25		Divide	A÷B	75 3 46 ÷ 5 ¯2 8 ⇔ 15 ¯1.5 5.75
Magnitude	|X	|23 0 ¯31 1J1 ⇔ 23 0 31 1.414213562		Residue (Remainder)	A|B	3|15.4 ¯21 ¯23 9 8 ⇔ 0.4 0 1 0 2
Exponential (eX)	*X	*1 2 ⇔ 2.718281828 7.389056099		Power	A*B	3 7 16 * 3 2 0.5 ⇔ 27 49 4
Natural Logarithm	⍟X	⍟2.718281828 10 ⇔ 1 2.302585093		Logarithm (Base A)	A⍟B	2⍟1023 ⇔ 9.99859043
Ceiling	⌈X	⌈ ¯2.8 ¯1.1 0 1.1 2.5 ⇔ ¯2 ¯1 0 2 3		Maximum	A⌈B	4 17 ¯2 ⌈ 3 64.8 1 ⇔ 4 64.8 1
Floor	⌊X	⌊ ¯2.8 ¯1.1 0 1.1 2.5 ⇔ ¯3 ¯2 0 1 2		Minimum	A⌊B	4 17 ¯2 ⌊ 3 64.8 1 ⇔ 3 17 ¯2
Factorial	!X	!14 ⇔ 87178291200		Binomial Coefficient	A!B	Number of combinations of B items taken A at a time
Pi Times	○X	○÷180 ⇔ 0.01745329252		Circular Functions	A○B	See table below
Roll	?X	Randomly choose an item from ⍳X		Deal	A?B	Randomly choose A items from ⍳B without replacement
				Base Value	A⊥B	2⊥0 1 0 0 0 1 ⇔ 17
				Representation	A⊤B	2 2 2 2 2 2 ⊤ 17 ⇔ 0 1 0 0 0 1
Grade Up	⍋X	List of indices that will sort X in ascending order		Grade Up	A⍋B	Alphabetic sort: A gives the collating sequence
Grade Down	⍒X	List of indices that will sort X in descending order		Grade Down	A⍒B	Alphabetic sort: A gives the collating sequence
Matrix Inverse	⌹X			Matrix Divide	A⌹B	Multiply A by the inverse of matrix B
Execute	⍎X	Evaluate APL2 _expression_ given as character vector X
Format	⍕X	⍕¯17.5 ⇔ '¯17.5'		Format	A⍕B	0 3 4 0 6 ¯2⍕ ¯12.14 30 17.1 ⇔ '¯12.140  30 1.7E1' '06/06/0006' ⍕ 7 20 1969 ⇔ '07/20/1969'
				Expand	A\B	1 0 0 1 1 0 1\3 1 4 2 ⇔ 3 0 0 1 4 0 2
				Expand	A⍀B	Expand along the first axis of B
				Replicate	A/B	2 0 ¯5 3 1/3 1 4 2 ⇔ 3 3 0 0 0 0 0 4 4 4 2
				Replicate	A⌿B	Replicate along the first axis of B

Boolean Functions

Less Than	A<B	1 2 3 < 3 2 1 ⇔ 1 0 0
Less Than or Equal To	A≤B	1 2 3 ≤ 3 2 1 ⇔ 1 1 0
Equal To	A=B	1 2 3 = 3 2 1 ⇔ 0 1 0
Not Equal To	A≠B	1 2 3 ≠ 3 2 1 ⇔ 1 0 1
Greater Than or Equal To	A≥B	1 2 3 ≥ 3 2 1 ⇔ 0 1 1
Greater Than	A>B	1 2 3 > 3 2 1 ⇔ 0 0 1
Not	~X	~0 1 ⇔ 1 0
And	A∧B	1 1 0 0 ∧ 1 0 1 0 ⇔ 1 0 0 0
Or	A∨B	1 1 0 0 ∨ 1 0 1 0 ⇔ 1 1 1 0
Nand	A⍲B	1 1 0 0 ⍲ 1 0 1 0 ⇔ 0 1 1 1
Nor	A⍱B	1 1 0 0 ⍱ 1 0 1 0 ⇔ 0 0 0 1
Match	A≡B	1 if A and B have the same dimensions, nesting levels, and contents, 0 otherwise
Membership	A∈B	'ace'∈'A Programming Language' ⇔ 1 0 1
Find	A⍷B	'ISSI'⍷'MISSISSIPPI' ⇔ 0 1 0 0 1 0 0 0 0 0 0

Circular Functions

0○X	(1-X*2)*0.5		0○X	(1-X*2)*0.5
1○X	Sine		¯1○X	Arcsine
2○X	Cosine		¯2○X	Arccosine
3○X	Tangent		¯3○X	Arctangent
4○X	(1+X*2)*0.5		¯4○X	(¯1+X*2)*0.5
5○X	Hyperbolic Sine		¯5○X	Inverse Hyperbolic Sine
6○X	Hyperbolic Cosine		¯6○X	Inverse Hyperbolic Cosine
7○X	Hyperbolic Tangent		¯7○X	Inverse Hyperbolic Tangent
8○X	(¯1-X*2)*0.5 for X<0 -(¯1-X*2)*0.5 for X≥0		¯8○X	-8○X
9○X	Real part of X		¯9○X	X
10○X	|X		¯10○X	+X (complex conjugate)
11○X	Imaginary part of X		¯11○X	0J1×X
12○X	Phase of X		¯12○X	*0J1×X

Note: one way to get the arctangent of Y÷X in the proper quadrant (-π to +π) is 12○X+11○Y

Structural Functions

				Index	A⌷B	Select items from B using indices A
Index Generator	⍳X	⍳8 ⇔ 1 2 3 4 5 6 7 8		Index Of	A⍳B	'HELLO'⍳'ELP' ⇔ 2 3 6
Not (see Boolean Functions)				Without	A~B	(⍳8)~1 4 5 ⇔ 2 3 6 7 8
Shape of	ρX	List of dimensions of X		Reshape	AρB	3 4ρ'ABCDEFGHIJKL' ⇔  ABCD EFGH IJKL
Reverse	⌽X	⌽⍳8 ⇔ 8 7 6 5 4 3 2 1		Rotate	A⌽B	3⌽⍳8 ⇔ 4 5 6 7 8 1 2 3
Reverse Vertically	⊖X	⊖3 4ρ'ABCDEFGHIJKL' ⇔  IJKL EFGH ABCD		Rotate Vertically	A⊖B	1⊖3 4ρ'ABCDEFGHIJKL' ⇔  EFGH IJKL ABCD
Transpose	⍉X	⍉3 4ρ'ABCDEFGHIJKL' ⇔  AEI BFJ CGK DHL		Transpose	A⍉B	The elements of A give the new order for the axes of B
First	↑X	The first element of X		Take	A↑B	¯3↑⍳8 ⇔ 6 7 8
				Drop	A↓B	3↓⍳8 ⇔ 4 5 6 7 8
Enclose	⊂X	Represent array X as a scalar		Partition	A⊂B	1 0 1 1 2 2 2 3 ⊂ 'ABCDEFGH' ⇔ ┌→─────────────────┐ │┌→┐ ┌→─┐ ┌→──┐ ┌→┐│ ││A│ │CD│ │EFG│ │H││ │└─┘ └──┘ └───┘ └─┘│ └∊─────────────────┘
Disclose	⊃X	Remove one level of nesting from X		Pick	A⊃B	Indices A select an item from within nested array B
Depth	≡X	Deepest level of nesting in X		Match (see Boolean Functions)
Ravel	,X	Change X to a vector, preserving depth		Catenate	A,B	Connect two arrays along their last axis
Enlist	∈X	Change X to a vector of depth 1		Membership (see Boolean Functions)

Operators

Each	f¨X	Apply function f to each item of X
Reduction	f/X	Apply function f along the last axis (rows) of X
Reduction	f⌿X	Apply function f along the first axis of X
N-Wise Reduction	A f/B	Apply function f along the last axis (rows) of B with a moving window of size A
N-Wise Reduction	A f⌿B	Apply function f along the first axis of B with a moving window of size A
Scan	f\X	Apply function f along the last axis (rows) of X, displaying intermediate results
Scan	f⍀X	Apply function f along the first axis of X, displaying intermediate results
Inner Product	A f.g B	Apply function g between rows of A and corresponding columns of B, then apply f along each result
Outer Product	A ∘.f B	Apply function f between every combination of one element from A and one element from B

Note: A+.×B is the standard matrix multiplication operation for matrices A and B.

Scans of boolean vectors are often useful:

∧\X	Make everything 0 after the first 0
<\X	Make everything 0 after the first 1
≤\X	Make everything 1 after the first 0
∨\X	Make everything 1 after the first 1
X∨≠\X	Make everything 1 between odd and even 1s