|
A Brief J Reference |
|
|
|
This brief reference gives informal descriptions of most of the J primitives. Not every |
|
primitive is included and some idioms, examples and other resources have been added |
|
where appropriate. Since the presentation is brief and informal, it is not a replacement for |
|
the main J references: the J Introduction and Dictionary, the J User manual and the J |
|
Primer. |
|
|
|
However, since the material is informally organized by topic, this reference may be useful |
|
when considering which J features might be relevant to a given problem. Some users may |
|
also find it helps locate gaps in knowledge that can then be filled in with the main references. |
|
|
|
Chris Burke |
|
Jsoftware Inc. |
|
[email protected] |
|
www.jsoftware.com |
|
|
|
Cliff Reiter |
|
Department of Mathematics |
|
Lafayette College |
|
webbox.lafayette.edu/~reiterc |
|
|
|
Last updated August 2014 for J802. |
|
|
|
2 |
|
|
|
Contents |
|
|
|
A Brief J Reference |
|
|
|
1 Language Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
|
|
|
2 Nouns |
|
|
|
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
|
|
|
3 Constants |
|
|
|
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
|
|
|
4 Basic Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
|
|
|
5 Boolean and Relational Verbs . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
|
|
|
6 Name Assignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
|
|
|
7 Data Information and Building . . . . . . . . . . . . . . . . . . . . . . . . . . . |
|
|
|
8 Data Copying . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
|
|
|
9 Data Indexing and Amendment . . . . . . . . . . . . . . . . . . . . . . . . . . . |
|
|
|
10 Boxed Arrays |
|
|
|
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
|
|
|
11 Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
|
|
|
12 Explicit Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
|
|
|
13 Tacit Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
|
|
|
14 Verb Application to Subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
|
|
|
15 Gerunds and Controlled Application of Verbs |
|
|
|
. . . . . . . . . . . . . . . . . . . |
|
|
|
16 Program Flow Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
|
|
|
17 Recursion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
|
|
|
18 Function Composition |
|
|
|
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
|
|
|
19 More Verbs from Verbs |
|
|
|
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
|
|
|
20 Conversion: Literal, Numeric, Base, Binary . . . . . . . . . . . . . . . . . . . . . |
|
|
|
21 Reading and Writing Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
|
|
|
22 Scripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
|
|
|
23 Sorting and Searching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
|
|
|
24 Efficiency, Error Trapping, and Debugging . . . . . . . . . . . . . . . . . . . . . |
|
|
|
25 Randomization and Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . |
|
|
|
26 Constant and Identity Verbs |
|
|
|
. . . . . . . . . . . . . . . . . . . . . . . . . . . . |
|
|
|
5 |
|
|
|
6 |
|
|
|
6 |
|
|
|
7 |
|
|
|
8 |
|
|
|
8 |
|
|
|
9 |
|
|
|
9 |
|
|
|
10 |
|
|
|
11 |
|
|
|
11 |
|
|
|
11 |
|
|
|
12 |
|
|
|
13 |
|
|
|
13 |
|
|
|
14 |
|
|
|
16 |
|
|
|
17 |
|
|
|
19 |
|
|
|
19 |
|
|
|
20 |
|
|
|
21 |
|
|
|
22 |
|
|
|
22 |
|
|
|
23 |
|
|
|
23 |
|
|
|
CONTENTS |
|
|
|
27 Exact Computations |
|
|
|
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
|
|
|
28 Number Theory and Combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . |
|
|
|
29 Circular and Numeric Verbs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
|
|
|
30 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
|
|
|
31 Matrix Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
|
|
|
32 Calculus, Roots and Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . |
|
|
|
33 Special Datatypes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
|
|
|
34 Graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
|
|
|
35 Qt Session Manager Short-Cut Keys |
|
|
|
. . . . . . . . . . . . . . . . . . . . . . . . |
|
|
|
36 Addons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
|
|
|
37 Parts of Speech and Grammar |
|
|
|
. . . . . . . . . . . . . . . . . . . . . . . . . . . |
|
|
|
38 Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
|
|
|
39 Vocabulary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
|
|
|
3 |
|
|
|
24 |
|
|
|
24 |
|
|
|
25 |
|
|
|
25 |
|
|
|
26 |
|
|
|
26 |
|
|
|
27 |
|
|
|
28 |
|
|
|
28 |
|
|
|
29 |
|
|
|
30 |
|
|
|
31 |
|
|
|
32 |
|
|
|
4 |
|
|
|
A Brief J Reference |
|
|
|
5 |
|
|
|
1 Language Basics |
|
|
|
Examples: |
|
|
|
fahrenheit =: 50 |
|
(fahrenheit - 32) * 5%9 |
|
|
|
10 |
|
|
|
prices=: 3 1 4 2 |
|
orders=: 2 0 2 1 |
|
orders * prices |
|
|
|
6 0 8 2 |
|
|
|
+/ orders * prices |
|
|
|
16 |
|
|
|
+/ \ 1 2 3 4 5 |
|
|
|
1 3 6 10 15 |
|
|
|
2 3 * / 1 2 3 4 5 |
|
|
|
2 4 6 8 10 |
|
3 6 9 12 15 |
|
|
|
cube=: ^&3 |
|
cube i. 9 |
|
|
|
0 1 8 27 64 125 216 343 512 |
|
|
|
Names |
|
|
|
50 fahrenheit |
|
|
|
+ - * % |
|
|
|
/ \ |
|
|
|
& |
|
|
|
( ) |
|
|
|
=: |
|
|
|
Nouns |
|
|
|
Verbs |
|
|
|
Adverbs |
|
|
|
Conjunction |
|
|
|
Punctuation |
|
|
|
Assignment |
|
|
|
• Verbs act upon nouns (their arguments) to produce noun results |
|
|
|
• A verb may have two distinct (usually related) meanings depending on whether it is |
|
|
|
applied to one argument (to its right), or to two arguments (left and right). |
|
|
|
• An adverb acts on a single noun or verb to its left, typically returning a verb. For |
|
|
|
example: +/ is a verb that sums its argument. |
|
|
|
• A conjunction applies to two arguments, either nouns or verbs, typically returning a |
|
|
|
verb. In the example above, ^&3 is the verb cube. |
|
|
|
A Brief J Reference |
|
|
|
6 |
|
|
|
2 Nouns |
|
|
|
Nouns are classified in three independent ways: |
|
|
|
• numeric or literal |
|
|
|
• open or boxed |
|
|
|
• arrays of various ranks |
|
|
|
The atoms of any array must belong to a single class: numeric, literal, or boxed. Arrays of |
|
ranks 0, 1, and 2 are also called atom, list, and table, or, in mathematics, scalar, vector, |
|
and matrix. |
|
|
|
A single entity such as 2.3 or ’A’ is called an atom. The number of atoms in the shape of |
|
a noun is called its rank. Each position of the shape is called an axis of the array, and axes |
|
are referred to by indices 0, 1, 2, etc. |
|
|
|
Boxed nouns are produced by the verb < (box). The result of box is an atom, and boxed |
|
nouns are displayed in boxes. Box allows one to treat any array (such as the list of letters |
|
that represent a word) as a single entity, or atom. |
|
|
|
3 Constants |
|
|
|
r |
|
b |
|
e |
|
p |
|
x |
|
x |
|
j |
|
ad |
|
ar |
|
|
|
a. |
|
a: |
|
|
|
_1 |
|
_ |
|
__ |
|
_. |
|
|
|
rationals; 5r4 is 5 |
|
4 |
|
base representations; 2b101 is 5 |
|
base 10 exponential notation (scientific notation); 1.2e14 is 1.2 × 1014 |
|
base π exponential notation; 3p6 is 3π6 |
|
base e exponential notation; 3x2 is 3e2 |
|
extended precision; 2^100x is the exact integer 2100 |
|
complex numbers; 3j4 is 3 + 4i |
|
angle in degrees; 1ad45 is approximately 0.707j0.707 |
|
angle in radians; 1ar1 is ^0j1 |
|
|
|
alphabet; list of all 256 ASCII characters |
|
boxed empty |
|
negative one (−1) |
|
infinity (∞) |
|
negative infinity (−∞) |
|
indeterminate |
|
|
|
7 |
|
|
|
4 Basic Arithmetic |
|
|
|
x + y |
|
+ y |
|
x - y |
|
- y |
|
x * y |
|
* y |
|
x % y |
|
% y |
|
+: y |
|
-: y |
|
*: y |
|
x %: y |
|
%: y |
|
|
|
x ^ y |
|
^ y |
|
x ^. y |
|
^. y |
|
|
|
x | y |
|
| y |
|
|
|
x plus y |
|
y; identity function for real y, conjugate for complex y |
|
x minus y |
|
negate y |
|
x times y |
|
signum of y is _1, 0 or 1 depending on the sign of real y |
|
x divide y |
|
reciprocal of y |
|
double y |
|
halve y |
|
square y |
|
xth root of y |
|
square root of y |
|
x to the power y |
|
exponential base e |
|
base x logarithm of y |
|
natural logarithm (base e) |
|
residue (remainder); y mod x |
|
absolute value of y |
|
|
|
x <. y minimum of x and y; (smaller of, lesser of ) |
|
|
|
<. y |
|
|
|
greatest integer less than or equal to y; called the floor |
|
|
|
x >. y maximum of x and y; (larger of, greater of ) |
|
|
|
>. y |
|
<: y |
|
>: y |
|
|
|
least integer greater than or equal to y; called the ceiling |
|
predecessor of y; y-1 (decrement) |
|
successor of y; y+1 (increment) |
|
|
|
8 |
|
|
|
A Brief J Reference |
|
|
|
5 Boolean and Relational Verbs |
|
|
|
Result of tests are 0 if false or 1 if true. |
|
|
|
x < y |
|
x <: y |
|
x = y |
|
x >: y |
|
x > y |
|
x ~: y |
|
x -: y |
|
|
|
test if x is less than y |
|
test if x is less than or equal to y |
|
test if x is equal to y |
|
test if x greater than or equal to y (larger than or equal ) |
|
test if x is greater than y (larger than) |
|
test if x is not equal to y |
|
test if x is identically same as y (match) |
|
|
|
-. y not y; 1-y for numeric y. |
|
|
|
x +. y |
|
x *. y |
|
x +: y |
|
x *: y |
|
x e. y |
|
e. y |
|
|
|
x or y; the greatest common divisor (gcd ) of x and y |
|
x and y; the least common multiple (lcm) of x and y |
|
x nor y (not-or ) |
|
x nand y (not-and ) |
|
test if x is an item in y (member of ) |
|
test if the raze is in each open |
|
|
|
x E. y mark beginnings of list x as a sublist in y (pattern occurrence) |
|
|
|
Boolean tests are subject to a default comparison tolerance of t=:2^_44. For example, x=y |
|
is 1 if the magnitude of the difference between x and y is less than t times the larger of |
|
the absolute values of x and y. The comparison tolerance may be modified with the fit |
|
conjunction, !., as in x=!.0 y, tests if x and y are the same to the last digit. |
|
|
|
6 Name Assignment |
|
|
|
abc=: 1 2 3 |
|
abc=. 1 2 3 |
|
|
|
'abc'=: 1 2 3 |
|
'a b c'=: 1 2 3 |
|
'a b'=: 1 2;3 |
|
cube=: ^ & 3 |
|
(exp)=: 1 2 3 |
|
|
|
names '' |
|
erase 'a b c' |
|
(4!:5) 1 |
|
|
|
global assignment of 1 2 3 to name abc |
|
local assignment of 1 2 3 to name abc. The value is only |
|
available inside the definition where it is made. |
|
indirect assignment of 1 2 3 to name abc |
|
parallel assignment of 1 to a, 2 to b and 3 to c. |
|
parallel unboxed assignment of 1 2 to a and 3 to b |
|
function assignment |
|
result of expression exp is assigned 1 2 3 |
|
|
|
list of names defined in current locale |
|
erases the names a, b, and c |
|
(snap) returns names changed since last execution of |
|
(4!:5) 1 |
|
|
|
Several foreign conjunctions of the form 4!:n deal with names. See also the Locales lab to |
|
learn about using locales to create different locations for global names. |
|
|
|
9 |
|
|
|
7 Data Information and Building |
|
|
|
# y |
|
$ y |
|
x $ y |
|
|
|
i. y |
|
|
|
i: y |
|
i: a j. b |
|
|
|
x F/ y |
|
x , y |
|
x ,. y |
|
x ,: y |
|
, y |
|
,. y |
|
,: y |
|
$. y |
|
|
|
number of items in y (tally) |
|
shape of array y |
|
shape x reshape of y (cyclically using/reusing data) |
|
list of indices filling an array of shape y (integer ); negative y reverses |
|
axis |
|
symmetric arithmetic sequence; for integer y, the integers from -y to y |
|
list of numbers from -a to a in b equal steps |
|
table of values of dyad F with arguments from x and y (outer product) |
|
append x to y where axis 0 is lengthened (catenate) |
|
stitch x beside y (append items) where axis 1 is lengthened; |
|
x laminated to y giving an array with 2 items |
|
ravel (string out) elements of y |
|
ravel items of y |
|
itemize, make y into a single item by adding a new length-1 leading axis |
|
sparse matrix representation of y |
|
|
|
8 Data Copying |
|
|
|
x # y |
|
|
|
x #^:_1 y |
|
|
|
(G # ])y |
|
|
|
x {. y |
|
|
|
{. y |
|
{: y |
|
x }. y |
|
}. y |
|
}: y |
|
|: y |
|
|
|
(<0 1) |: y |
|
|
|
replicate or copy items in y the number of times indicated by x; the |
|
imaginary part of x is used to specify the size of expansion with fill |
|
elements |
|
expand (inverse of #) selects items of y according to 1 in Boolean x, pads |
|
with fills where 0 in x |
|
selects elements of y according to Boolean test G; thus, (2&< # ]) y |
|
gives the elements of y greater than 2. |
|
shape x take of y; negative entries cause take from end of axes; entries |
|
larger than axis length cause padding with fill elements. |
|
item in 1 {. y for non-empty arrays; in general 0{y (called head ) |
|
item in _1{.y or _1{y (called tail ) |
|
drop shape x part of y; negative entries cause drop from end of axes. |
|
1 }. y (one drop) or behead |
|
_1 }. y (negative one drop) or curtail |
|
transpose of y |
|
trace (diagonal) of matrix y |
|
|
|
10 |
|
|
|
A Brief J Reference |
|
|
|
9 Data Indexing and Amendment |
|
|
|
I { y |
|
|
|
x I } y |
|
|
|
item at position I in y (index or from) |
|
Arrays of I give corresponding arrays of items. |
|
Boxed arrays I give positions on corresponding axes. An empty box |
|
gives all values along that axis. |
|
y amended at positions I by data x |
|
|
|
[A=: i.3 4 |
|
|
|
0 1 2 3 |
|
4 5 6 7 |
|
8 9 10 11 |
|
|
|
0 2 { A |
|
|
|
0 1 2 3 |
|
8 9 10 11 |
|
|
|
0 2 {"1 A |
|
|
|
0 2 |
|
4 6 |
|
8 10 |
|
|
|
(<0 2;3) { A |
|
|
|
3 11 |
|
|
|
1000 (<0 2;3) } A |
|
|
|
0 1 2 1000 |
|
4 5 6 7 |
|
8 9 10 1000 |
|
|
|
11 |
|
|
|
10 Boxed Arrays |
|
|
|
box y |
|
open (unbox) y one level |
|
link x and y; box x and append to y; if y is unboxed, then box y first |
|
raze y; remove one level of boxing |
|
apply F inside of each boxed element of y |
|
apply F to inside of each boxed element of y and adjoin the results. |
|
boxed empty (noun called ace) |
|
boxed list of J words in string y; (word formation) |
|
depth or deepest level of boxing in y |
|
apply F at level n and maintain boxing. May be used dyadically and |
|
left and right level specified. If boxing is thought of as creating a tree |
|
structure, then adverb L: 0 may be called leaf |
|
apply F at level n and list the result (spread ) |
|
map has the same boxing as y and gives the paths to each leaf |
|
fetch the data from y specified by the path x |
|
|
|
< y |
|
> y |
|
x ; y |
|
; y |
|
F &.> y |
|
F &> y |
|
a: |
|
;: y |
|
L. y |
|
F L: n y |
|
|
|
F S: n y |
|
{:: y |
|
x {:: y |
|
|
|
11 Rank |
|
|
|
Rank can be specified by one, two or three numbers. If the rank r contains three numbers, |
|
the first is the monadic rank, the second the left dyadic rank and last the right dyadic rank. |
|
If it contains two numbers, the first gives the left dyadic rank and the second gives the |
|
monadic and right dyadic rank. All the ranks are the same when a single number is given. |
|
|
|
F"r y |
|
|
|
x F"(lr,rr) y |
|
x F"0 _ y |
|
|
|
N"r |
|
F b. 0 |
|
|
|
apply F on rank r cells of the data y |
|
apply F on rank lr cells from x and rank rr cells from y |
|
table builder when F is a scalar function |
|
constant function of rank r and result N |
|
gives the monadic, left and right ranks of the verb F |
|
|
|
12 Explicit Definition |
|
|
|
Explicit definitions can be made with m : n where m is a number that specifies whether the |
|
result is a noun, verb, adverb or conjunction. When n is 0, successive lines of input give the |
|
defining expressions until an isolated, closing right parenthesis is reached. Noun arguments |
|
to adverbs and conjunctions may be specified by m on the left and n on the right. Verb |
|
arguments are u and v and the derived functions use x and y to denote their arguments. |
|
|
|
4 : 0 |
|
3 : 0 |
|
|
|
2 : 0 |
|
1 : 0 |
|
0 : 0 |
|
|
|
input mode for a dyadic verb |
|
input mode for general verb. This is the monadic definition, optionally |
|
followed by an isolated colon and the dyadic definition. |
|
input mode for conjunction |
|
input mode for an adverb |
|
input mode for a noun |
|
|
|
12 |
|
|
|
A Brief J Reference |
|
|
|
The right argument n as in m : n, may be a string, a CRLF delimited string, a matrix, or a |
|
boxed list of strings. For example: |
|
|
|
f=: 3 : '(*:y) + ^y' |
|
|
|
defines f (y) = y2 + ey |
|
|
|
13 : n converts to tacit form of a verb (if possible). For example: |
|
|
|
13 : 'x , 2 * x + y' |
|
|
|
[ , 2 * + |
|
|
|
13 Tacit Definition |
|
|
|
In a tacit definition the arguments are not named and do not appear in the definition. |
|
|
|
In many cases the tacit form of definition is much simpler and more obvious than the |
|
equivalent explicit definition. |
|
|
|
For example: |
|
|
|
plus=: + |
|
sum=: +/ |
|
max=: >./ |
|
mean=: +/ % # |
|
|
|
assigns name plus to + |
|
sum of numeric list |
|
maximum of numeric list |
|
average of numeric list |
|
|
|
Compare the last definition with an equivalent explicit definition: |
|
|
|
mean=: 3 : '(+/y) % #y' |
|
|
|
13 |
|
|
|
14 Verb Application to Subsets |
|
|
|
F/ y |
|
|
|
G\ y |
|
F/ \ y |
|
x G\ y |
|
|
|
G\. y |
|
x G\. y |
|
|
|
x G;._3 y |
|
|
|
G;.3 y |
|
G;._2 y |
|
|
|
x F/. y |
|
|
|
F/. y |
|
|
|
insert verb F between items of y, also called F-reduction; thus +/2 3 4 |
|
is 2+3+4 |
|
apply G to prefixes of y; generalized scan |
|
F scan of y |
|
apply G to sublists of length x in y (the lists are infixes); negative x gives |
|
non-overlapping sublists. For example x avg y gives length x moving |
|
averages of data in y (where avg=: +/ % #). |
|
apply G to suffixes of y (order of execution makes this fast!) |
|
apply G to lists where sublists of length x in y are excluded (the sublists |
|
are outfixes) |
|
cut; apply G to shape x tessellations of y. |
|
In general, the rows of x |
|
give the shape and offset used for the tessellation. Include shards by |
|
specifying 3 instead of _3. |
|
cut; generalized suffix |
|
cut; apply G to sublists marked by ending with the last item in y. |
|
So }:<@}:;._2 y,CRLF gives the boxed lines of CRLF delimited text y. |
|
G;.2 y includes marked positions in sublists. G;._1 y and G;.1 y use |
|
first item to mark beginnings of sublists. G;.0 y and dyads and gerunds |
|
G are also defined. |
|
function F is applied to parts of x selected by distinct items (keys) in y. |
|
For example #/.~ y gives frequency of occurrence of items of y |
|
apply F to oblique lists from y. |
|
|
|
15 Gerunds and Controlled Application of Verbs |
|
|
|
^: |
|
F^:n y |
|
F^:_ y |
|
F^:(i.n)y |
|
F^:G^:_ y |
|
|
|
F`G |
|
F/. y |
|
|
|
F`:0 y |
|
|
|
[email protected] |
|
F::G |
|
|
|
iterate function (power ) |
|
iterate F n times on y; see the Dictionary for gerund n |
|
iterate F until convergence (limit) |
|
result of F iterated 0 to n-1 times on y |
|
iterate F on y until G gives false |
|
tie verbs F and G together forming a gerund |
|
evaluate each verb in gerund F taken cyclically on data y (evoke |
|
gerund ) |
|
alternative form of evoke gerund, returning all combinations of |
|
functions from F on y |
|
agenda: use G to select verb from gerund F to apply |
|
adverse: apply F, if an error occurs, apply G instead |
|
|
|
Many adverbs and conjunctions have gerund meanings that give generalizations; e.g. gerund |
|
insert cyclically inserts verbs from the gerund. Thus, the following are the same: |
|
|
|
+`% / 1 2 3 4 |
|
1 + 2 % 3 + 4 |
|
|
|
14 |
|
|
|
A Brief J Reference |
|
|
|
16 Program Flow Control |
|
|
|
Execution control is provided by words such as: if. else. while. etc. These control |
|
words: |
|
|
|
• can occur anywhere in a line of code |
|
|
|
• group the code into blocks |
|
|
|
Here, a block is zero or more sentences, which may themselves contain control words. A J |
|
block is true if the first element of the result is not zero. In particular, an empty block is |
|
true. |
|
|
|
In all cases, the result of the last expression executed that was not a test, is returned as |
|
the verb result. |
|
|
|
if. elseif. |
|
|
|
signum=: 3 : 0 |
|
if. y < 0 do. _1 |
|
elseif. y=0 do. 0 |
|
elseif. do. 1 |
|
end. |
|
) |
|
|
|
signum &> _5 7 8 0 |
|
|
|
_1 1 1 0 |
|
|
|
Compare: |
|
|
|
* _5 7 8 0 |
|
|
|
_1 1 1 0 |
|
|
|
select. |
|
|
|
The select. control word allows execution of expressions when a value matches those in |
|
a given case or cases. Evaluation of the select. control structure then terminates for a |
|
case. statement, or continues with the next block for an fcase. statement. An empty |
|
case matches all. |
|
|
|
15 |
|
|
|
atype=: 3 : 0 |
|
select. #$y |
|
case. 0 do. 'scalar' |
|
case. 1 do. 'vector' |
|
case. do. 'array of dimension greater than 1' |
|
end. |
|
) |
|
|
|
atype <i.3 3 |
|
|
|
scalar |
|
|
|
atype i.3 3 3 |
|
|
|
array of dimension greater than 1 |
|
|
|
while. whilst. |
|
|
|
The control word while. executes the loop while the control condition is true. |
|
|
|
whilst. is the same as while, except the steps of the loop are executed once before the |
|
control condition is tested. |
|
|
|
sumint=: 3 : 0 |
|
k=.0 |
|
s=.0 |
|
while. k<:y do. |
|
|
|
s=.s+k |
|
k=.k+1 |
|
|
|
end. |
|
s |
|
) |
|
|
|
sumint 10 |
|
|
|
55 |
|
|
|
for name. |
|
|
|
Sumint=: 3 : 0 |
|
s=.0 |
|
for_k. 1+i.y |
|
|
|
do. s=.s+k |
|
|
|
end. |
|
s |
|
) |
|
|
|
Sumint 10 |
|
|
|
55 |
|
|
|
16 |
|
|
|
break. |
|
|
|
A Brief J Reference |
|
|
|
loop, |
|
The control word break. |
|
and continue. returns to the top of the loop. The control word return. can be used to |
|
exit from function execution. |
|
|
|
is used to step out of a while. or whilst. or for_name. |
|
|
|
try. catch. |
|
|
|
The following line runs expression2 if running expression1 causes an error. |
|
|
|
try. expression1 catch. expression2 end. |
|
|
|
There are also control words for labeling lines and going to those lines: label_name. and |
|
goto_name. . |
|
|
|
17 Recursion |
|
|
|
One can use self reference of verbs that are named. For example, the factorial can be |
|
computed recursively as follows. |
|
|
|
fac=: 3 : 'if. y <: 1 do. 1 else. y * fac y - 1 end.' |
|
fac 3 |
|
|
|
6 |
|
|
|
Also: |
|
|
|
fac=: 1:`(*fac@<:)@.* |
|
fac 3 |
|
|
|
6 |
|
|
|
fac"0 i. 6 |
|
|
|
1 1 2 6 24 120 |
|
|
|
One can also create a recursive function without naming the function by using $: for self- |
|
reference. The factorial function can be defined recursively without name as follows. |
|
|
|
(1:`(*$:@<:)@.*) 3 |
|
|
|
6 |
|
|
|
(1:`(*$:@<:)@.*)"0 i.6 |
|
|
|
1 1 2 6 24 120 |
|
|
|
18 Function Composition |
|
|
|
17 |
|
|
|
Atop |
|
|
|
F@G y |
|
|
|
Compose |
|
|
|
F&G y |
|
|
|
Under |
|
|
|
F&.G y |
|
|
|
F |
|
| |
|
G |
|
| |
|
y |
|
|
|
F |
|
| |
|
G |
|
| |
|
y |
|
|
|
-1 |
|
|
|
G |
|
| |
|
F |
|
| |
|
G |
|
| |
|
y |
|
|
|
G |
|
/ \ |
|
|
|
y |
|
|
|
H |
|
| |
|
y |
|
|
|
Hook |
|
|
|
(G H) y |
|
|
|
x F@G y |
|
|
|
x F&G y |
|
|
|
F |
|
| |
|
G |
|
/ \ |
|
|
|
x |
|
|
|
y |
|
|
|
F |
|
/ \ |
|
|
|
G |
|
| |
|
x |
|
|
|
G |
|
| |
|
y |
|
|
|
x F&.G y |
|
|
|
-1 |
|
|
|
G |
|
| |
|
F |
|
/ \ |
|
|
|
G |
|
| |
|
x |
|
|
|
G |
|
| |
|
y |
|
|
|
x (G H) y |
|
|
|
G |
|
/ \ |
|
|
|
x |
|
|
|
H |
|
| |
|
y |
|
|
|
18 |
|
|
|
Fork |
|
|
|
A Brief J Reference |
|
|
|
(F G H) y |
|
|
|
x (F G H) y |
|
|
|
G |
|
/ \ |
|
|
|
F |
|
| |
|
y |
|
|
|
H |
|
| |
|
y |
|
|
|
G |
|
|
|
/ |
|
|
|
F |
|
/ \ |
|
|
|
\ |
|
|
|
H |
|
/ \ |
|
|
|
x |
|
|
|
y x |
|
|
|
y |
|
|
|
The rank of F@G and F&G is the rank of G. At is denoted @: and is the same as @ except the |
|
rank is infinite. Appose is denoted &: which is the same as & except the rank is infinite. |
|
The ranks of the hook and fork are infinite. |
|
|
|
Longer trains of verbs are interpreted by taking forks on the right. Thus F G H J is the |
|
hook F (G H J) where G H J is a fork, and F G H J K is the fork F G (H J K). |
|
|
|
Under |
|
|
|
The verb u &. v is equivalent to the composition u & v except that the verb obverse |
|
(inverse) to v is applied to the result for each cell. For example, multiplication is sum |
|
under log: |
|
|
|
3 + &. ^. 4 |
|
|
|
12 |
|
|
|
However, the rank of the result of u &. v is the monadic rank of v, which for many verbs is |
|
zero, whereas it is often the case that you want the rank to be infinite. An alternate form |
|
of under is &.:, which is equivalent to u &. (v"_). For example: |
|
|
|
+/ &. ^. 3 4 5 |
|
|
|
3 4 5 |
|
|
|
+/ &.: ^. 3 4 5 |
|
|
|
60 |
|
|
|
Cap |
|
|
|
[: F G has the effect of passing no left argument to F as part of the fork - the left branch |
|
of the fork is capped - thus F is applied monadically. |
|
|
|
19 |
|
|
|
19 More Verbs from Verbs |
|
|
|
N&G |
|
G&N |
|
|
|
F~y |
|
x F~ y |
|
|
|
F : G |
|
F :. G |
|
G f. |
|
|
|
F b. _1 |
|
F b. 1 |
|
|
|
monad derived from dyad G with N as the fixed left argument |
|
monad derived from dyad G with N as the fixed right argument (known |
|
as bond or curry) |
|
reflects y to both arguments; i.e. y F y (reflex ) |
|
pass interchanges arguments; i.e. y F x (commute) |
|
function with monad F and dyad G (monad/dyad definition) |
|
function F with obverse (restricted inverse) G |
|
function G with names appearing in its definition recursively replaced by |
|
their meaning. This fix es (makes permanent) the function meaning |
|
obverse (inverse) of F |
|
identity function for F |
|
|
|
20 Conversion: Literal, Numeric, Base, Binary |
|
|
|
": y |
|
a.b ": y |
|
ajb ": y |
|
|
|
":!. c y |
|
". y |
|
x ". y |
|
|
|
".@}: ;._2 y |
|
|
|
'm'~ |
|
#: y |
|
x #: y |
|
#. y |
|
x #. y |
|
3 !: n |
|
|
|
8 !: n |
|
|
|
format array y as a literal array |
|
format data in y with field width a and b decimal digits |
|
format data in y with field width a and b decimal digits, for |
|
example: 15j10 ": o.i.3 4 |
|
format data showing c significant digits |
|
execute or do string y |
|
convert y to numeric using x for illegal numbers. J syntax |
|
is relaxed so appearances of - in y are treated like _ |
|
execute expressions in CRLF delimited substrings appearing |
|
in y (that ends with CRLF) and adjoin the results |
|
value of name m is evoked |
|
binary representation of y (antibase-two) |
|
representation of y in base x (antibase) |
|
value of binary rank-1 cells of y (base-two) |
|
value of base x rank-1 cells of y (base) |
|
various binary conversions; for example, 1 (3 !:4) y con- |
|
verts J floats to binary short floats while _1 (3!:4) y con- |
|
verts binary short floats to J floats. |
|
format y according to format phrase x. For example, format |
|
to width 11, decimal places 2, comma-separated, with zeros |
|
formatted to nil, and infinities to n/a: |
|
|
|
'b<nil>d<n/a>c11.2' (8!:2) 1.23 12345 0.123,__ 0 _1234.5,:_44 0.5 0.1 |
|
|
|
1.23 12,345.00 |
|
|
|
0.12 |
|
nil -1,234.50 |
|
0.10 |
|
|
|
0.50 |
|
|
|
n/a |
|
-44.00 |
|
|
|
See also the Foreign Conjunction help. |
|
|
|
20 |
|
|
|
A Brief J Reference |
|
|
|
21 Reading and Writing Files |
|
|
|
The following verbs are based on foreign conjunctions in the form 1!:n. These provide for |
|
file reading/writing including indexed reads and writes and creating directories, reading |
|
and setting attributes and permissions. Chopping file data in appropriate places can be |
|
accomplished with ;._2 (cut). Simple substitution (e.g., _ for -) may be accomplished |
|
with charsub from strings. See regex for more complex processing. Memory mapped |
|
files should be considered for huge data sets. |
|
|
|
1!:0 y |
|
|
|
1!:1 y |
|
x 1!:2 y |
|
|
|
x 1!:3 y |
|
|
|
1!:11 y |
|
|
|
x 1!:12 y |
|
|
|
For example: |
|
|
|
directory information matching path and pattern in y (see |
|
fdir) |
|
read file y specified by a boxed name (see fread and freads) |
|
write file y with raw, (a. or text) data x (see fwrite and |
|
fwrites) |
|
append file y with raw, a. or text data x (see fappend and |
|
fappends) |
|
indexed read. y is a pair: file name; index and length. The |
|
index may be negative. If the length is elided, the read goes |
|
to the end. |
|
indexed write. y is a pair: file name; index. |
|
|
|
'abcdefgh' 1!:2 <F=: 't1.dat' |
|
1!:1 <F |
|
|
|
abcdefgh |
|
|
|
1!:11 F;2 5 |
|
|
|
cdefg |
|
|
|
'XYZ' 1!:12 F;3 |
|
1!:11 F;2 5 |
|
|
|
cXYZg |
|
|
|
Files may also be referenced by number; keyboard and screen input/output are supported, |
|
and other facilities give other useful file access including indexed i/o, permissions, erasure, |
|
locking, attributes. |
|
|
|
Convenient utilities are defined in the standard library. For example, read in a file, returning |
|
the result in a matrix: |
|
|
|
'm' fread 'mydata.dat' |
|
|
|
See also the Files lab. |
|
|
|
21 |
|
|
|
22 Scripts |
|
|
|
Scripts are plain text files containing J expressions. Typically the file extension is .ijs. |
|
Loading the scripts runs the J expressions. |
|
|
|
Ctrl+N |
|
0!:0 <'filename.ijs' |
|
0!:1 <'filename.ijs' |
|
|
|
open a new script window |
|
run the script filename.ijs; note boxing of filename |
|
run the script with display |
|
|
|
0!:0 y |
|
0!:1 y |
|
0!:10 y |
|
|
|
load 'filename.ijs' |
|
|
|
load 'scriptname' |
|
|
|
load '~addons/../filename.ijs' |
|
|
|
require 'filename' |
|
|
|
run the J noun y as a script |
|
run the J noun y displaying the result |
|
run the J noun y and continue on errors |
|
|
|
similar to 0!:0 except that the script is loaded within an |
|
explicit definitions, and hence local definitions in the script |
|
do not exist upon completion. This allows a script to have |
|
definitions that are local to the script. |
|
loads a library script, for example pacman is the script |
|
system/util/pacman.ijs. Enter scripts'v' to see the |
|
list of script names. |
|
loads a script from the addons directory, |
|
load '~addons/stats/r/rserve.ijs' |
|
similar to load except that if the script has already been |
|
loaded, it is not loaded again. |
|
|
|
for example |
|
|
|
Typically, applications are built from several scripts. |
|
|
|
The Project system helps you access and manage your J script files. |
|
|
|
For example, it lets build you applications from several scripts. Scripts are maintained |
|
individually during development, and can be compiled into a single output script for distri- |
|
bution/runtime/installation purposes. You can customize the build to suit your application. |
|
|
|
For more information, see the wiki pages Guides/Folders and Projects and Guides/J8 Stan- |
|
dalone. |
|
|
|
22 |
|
|
|
A Brief J Reference |
|
|
|
23 Sorting and Searching |
|
|
|
/: y |
|
|
|
x /: y |
|
|
|
/:~ y |
|
/:/: y |
|
\: y |
|
|
|
x i. y |
|
x e. y |
|
e. y |
|
x E. y |
|
I. y |
|
|
|
x I. y |
|
~. y |
|
|
|
grade up; indices of items of y ordered so that the corresponding items |
|
of y would be in nondecreasing order |
|
sort x according to indices in /:y |
|
sorts items of y into nondecreasing order |
|
rank order of items in y |
|
grade down; indices of items of y ordered so that the corresponding items |
|
of y are in nonincreasing order |
|
indices of items of y in the reference list x |
|
test if x is an item in y (member of ) |
|
test if the raze is in each open |
|
mark beginnings of list x as a sublist in y (pattern occurrence) |
|
indices of 1 in boolean list y; thus I.y<4 gives indices where y is less |
|
than 4 |
|
indices of y in the intervals defined by x |
|
nub of y; that is, items of y with duplicates removed (unique) |
|
|
|
~: y |
|
= y |
|
(G # ])y |
|
|
|
({.,#)/.~y may use key to get nub and frequencies appearing in y |
|
nubsieve: Boolean vector v so v#y is ~.y |
|
self-classify y according to ~. y |
|
selects items of y according to Boolean test G; thus, (2&< # ])y gives |
|
items of y greater than 2. |
|
items of x less those in y |
|
|
|
x -. y |
|
|
|
24 Efficiency, Error Trapping, and Debugging |
|
|
|
6!:2 y |
|
|
|
7!:2 y |
|
u :: v |
|
|
|
try. e1 catch. e2 end. |
|
|
|
time (seconds) required to execute string y. Optional left |
|
argument specifies number of repetitions used to obtain av- |
|
erage run time |
|
space (bytes) required to execute string y |
|
result of applying verb u unless that results in an error in |
|
which case v is applied (adverse) |
|
is similar except expressions in explicit definition mode are |
|
executed instead of verbs being applied |
|
|
|
The try/catch control structure may contain one or more distinct occurrences of catch. |
|
catchd. catcht. (in any order). For example: |
|
|
|
try. B0 catch. B1 end. |
|
try. B0 catcht. B1 catchd. B2 end. |
|
try. B0 catcht. B1 catch. B2 catchd. B3 end. |
|
|
|
The B0 block is executed and: |
|
|
|
23 |
|
|
|
catch. |
|
catchd. |
|
catcht. |
|
|
|
catches errors, whatever the setting of the debug flag 13!:0 |
|
catches errors, but only if the debug flag is 0 |
|
catches a throw. expression |
|
|
|
The foreign conjunctions 13!:n provide the underlying debugging facilities, while the Debug |
|
application provides interactive debugging, see the Debug lab. |
|
|
|
The Performance Monitor provides detailed execution time and space used when running |
|
an application, see the Performance Monitor lab. |
|
|
|
25 Randomization and Simulation |
|
|
|
? y |
|
|
|
?. y |
|
x ? y |
|
? 0 |
|
9!:0 '' |
|
9!:1 y |
|
randomize '' |
|
|
|
example, |
|
|
|
for |
|
|
|
roll ; |
|
|
|
called |
|
|
|
from i.y; |
|
|
|
random index |
|
+/\(?100#2){_1 1 is a 100 step random _1 1 walk |
|
default random index from i.y using 16807 as the random seed |
|
x random indices dealt from i.y without duplication |
|
random number in range [0,1) |
|
query random seed |
|
set random seed to y |
|
randomize random seed; randomize is defined in numeric.ijs |
|
|
|
See also ~addons/stats/base/random.ijs for various random number utilities, and |
|
~addons/stats/base/distribution.ijs for generating various distributions. |
|
|
|
For example: |
|
|
|
3 deal ;: 'anne henry mary susan tom' |
|
|
|
+-----+----+---+ |
|
|susan|mary|tom| |
|
+-----+----+---+ |
|
|
|
normalrand 5 NB. mean 0, sd 1 |
|
|
|
0.719033 _0.512529 0.801304 0.436659 _0.0496758 |
|
|
|
26 Constant and Identity Verbs |
|
|
|
] y |
|
x ] y |
|
[ y |
|
x [ y |
|
+ y |
|
0: y |
|
1: y |
|
_: y |
|
N"r |
|
|
|
result is y, the identity function on y |
|
result is y (right) |
|
result is y, the identity function on y |
|
result is x (left) |
|
result is y if y is a real number |
|
result is the scalar 0 |
|
result is 1; likewise, there are constant functions _9: to 9: |
|
result is the infinite scalar _ |
|
constant function with value N for each rank r cell |
|
|
|
24 |
|
|
|
A Brief J Reference |
|
|
|
27 Exact Computations |
|
|
|
2x or 2r1 |
|
2x^100 |
|
2r3 |
|
x: y |
|
x:^:_1 y |
|
|
|
2 x: y |
|
|
|
exact integer 2 (extended precision) |
|
exact integer 2100 |
|
exact rational number 2 |
|
convert y to extended precision rational |
|
convert y to fixed precision numeric |
|
numerator and denominator of extended precision rationals |
|
|
|
3 (extended precision) |
|
|
|
There is special code to avoid exponentiation for extended precision arguments when using |
|
residue, for example: |
|
|
|
m&|@(2x&^) y |
|
|
|
computes 2y mod m efficiently (without computing 2y) |
|
|
|
28 Number Theory and Combinatorics |
|
|
|
p: y |
|
p:^:_1 y number of primes less than y |
|
|
|
y-th prime number (in origin 0) |
|
|
|
x p: y |
|
q: y |
|
x q: y |
|
x +. y |
|
x *. y |
|
gcd y |
|
|
|
x | y |
|
! y |
|
x ! y |
|
|
|
A. y |
|
x A. y |
|
|
|
C. y |
|
|
|
x C. y |
|
|
|
{ y |
|
|
|
various number theoretic functions: next prime, totient, etc. |
|
prime factors of y |
|
prime factors of y with limited factor base |
|
greatest common divisor (gcd) |
|
least common multiple (lcm) |
|
function gcd defined in ~addons/math/misc/gcd.ijs re- |
|
sults in the gcd of the elements of y along with the coef- |
|
ficients whose dot product with y gives the gcd. Also useful |
|
for finding inverses modulo m. |
|
residue (remainder) y modulo x |
|
factorial of y for integer y and Γ(y + 1) in general |
|
number of combinations of x things from y things (general- |
|
ized) |
|
atomic representation (position) of permutation y |
|
applies permutation with atomic representation x to y |
|
(atomic permute, anagram). For example, (i.!n) A. i.n |
|
is all permutations of order n |
|
cycle representation of numeric permutation y as a boxed |
|
list; visa versa when y is boxed |
|
permutes y according to permutation x (either in numeric |
|
or boxed cyclic representation) |
|
Cartesian product: all selections of one item from each box |
|
in y. |
|
|
|
29 Circular and Numeric Verbs |
|
|
|
Many trigonometric functions and other functions associated with circles are obtained using |
|
o. with various numeric left arguments. |
|
|
|
25 |
|
|
|
o. y |
|
|
|
πy (pi times) |
|
|
|
sin(y) |
|
cos(y) |
|
tan(y) |
|
|
|
0 o. y (cid:112)1 − y2 (circle functions) |
|
1 o. y |
|
2 o. y |
|
3 o. y |
|
4 o. y (cid:112)1 + y2 |
|
sinh(y) |
|
5 o. y |
|
cosh(y) |
|
6 o. y |
|
tanh(y) |
|
7 o. y |
|
8 o. y (cid:112)−(1 + y2) |
|
real part(y) |
|
9 o. y |
|
abs(y) which is |y |
|
10 o. y |
|
imaginary part(y) |
|
11 o. y |
|
arg(y) |
|
12 o. y |
|
|
|
sin−1(y) |
|
_1 o. y |
|
cos−1(y) |
|
_2 o. y |
|
tan−1(y) |
|
_3 o. y |
|
_4 o. y (cid:112)y2 − 1 |
|
sinh−1(y) |
|
_5 o. y |
|
cosh−1(y) |
|
_6 o. y |
|
tanh−1(y) |
|
_7 o. y |
|
−(cid:112)−(1 + y2) |
|
_8 o. y |
|
y |
|
_9 o. y |
|
conjugate(y) |
|
_10 o. y |
|
yi where i is |
|
_11 o. y |
|
eiy |
|
_12 o. y |
|
|
|
√ |
|
|
|
−1 |
|
|
|
m H. n y |
|
x m H. n y hypergeometric function using x terms |
|
|
|
hypergeometric function; sometimes denoted F (m; n, y) |
|
|
|
30 Complex Numbers |
|
|
|
Complex numbers are denoted with a j separating the real and imaginary parts. Thus, the |
|
complex number commonly written 3.1 + 4i is denoted 3.1j4. |
|
|
|
complex conjugate of y |
|
|
|
+ y |
|
| y magnitude of y |
|
* y |
|
j. y |
|
x j. y |
|
|
|
generalized signum; complex number in y direction |
|
complex number 0jy; that is, iy (imaginary) |
|
complex number xjy; that is, x + iy (complex ) |
|
|
|
+. y pair containing real(y) and imaginary(y) |
|
*. y polar pair (r, θ) where y = reiθ, (length, angle) |
|
r. y |
|
x r. y |
|
|
|
is eiy (angle to complex ) |
|
is xeiy (polar to complex ) |
|
|
|
26 |
|
|
|
A Brief J Reference |
|
|
|
31 Matrix Arithmetic |
|
|
|
x +/ . * y matrix product of x and y (dot product for vectors) |
|
x +/ . = y number of places where vector arguments match |
|
x F/ . G y |
|
|
|
inner product; F-insert applied to pairwise G’s applied row by column; |
|
the last axis of x and first axis of y need to be compatible (same or 1) |
|
and that axis collapses in the product. |
|
inner product; H applied to cells of G applied rank _1 _ |
|
determinant of y |
|
generalized determinant; +/ . * gives the permanent. |
|
matrix divide; solution z to the linear matrix system x = y +/ . * z; |
|
least squares solution is given when appropriate |
|
matrix inverse or pseudo-inverse of matrix y |
|
transpose of y |
|
generalized transpose of y. Axes listed in x are successively moved to |
|
the end. |
|
reverse items in y |
|
rotate items in y by x positions downward along the last axis |
|
y by y identity matrix; multiply by diagonal to get a diagonal matrix |
|
QR decomposition of y |
|
|
|
x H . G y |
|
|
|
-/ . * y |
|
F . G y |
|
|
|
x %. y |
|
|
|
%. y |
|
|: y |
|
x |: y |
|
|
|
|. y |
|
x |. y |
|
|
|
=@i. y |
|
128!:0 y |
|
|
|
The J Addons Lapack and FFTW give extensive linear algebra and fast Fourier transform |
|
utilities, respectively. |
|
|
|
32 Calculus, Roots and Polynomials |
|
|
|
F D. n y |
|
F d. n y |
|
x F D: n y |
|
F t. n |
|
F t: n |
|
F T. n y |
|
p. y |
|
|
|
x p. y |
|
|
|
p.. y |
|
x p.. y |
|
|
|
n-th derivative of F at y |
|
n-th derivative rank zero: compare to (F D. 1)"0 y |
|
slope of secant of F at y and x+y |
|
n-th Taylor series coefficient of F about 0 |
|
n-th Taylor series coefficient of F about 0 weighted by !n |
|
n-th degree Taylor polynomial for F about 0 evaluated at y |
|
polynomial ; |
|
coefficient-with-root boxed representation of polynomials. |
|
polynomial specified by x evaluated at points y. The coefficients x are in |
|
ascending powers, for example 2 1 3 p. is the polynomial 3x2 + x + 2. |
|
coefficients of derivative of polynomial y |
|
integral of polynomial y with a constant term x |
|
|
|
toggles between coefficient representation and leading- |
|
|
|
27 |
|
|
|
33 Special Datatypes |
|
|
|
Sparse arrays provide a compact and efficient storage form for very large arrays where most |
|
elements are zero or some other sparse element. |
|
|
|
The verb $. converts a dense array to sparse, and $.^:_1 y ($. inverse) converts a sparse |
|
array to dense. |
|
|
|
A sparse array has a single sparse element, plus an array of other values and a matrix of |
|
their corresponding indices. |
|
|
|
The sparse attribute can be assigned to axes individually. Non-sparse axes are known as |
|
dense axes. |
|
|
|
J primitives work directly on sparse arrays, and operations give the same results when |
|
In other words, the following |
|
applied to dense and sparse versions of the same arrays. |
|
identities hold for any function f , with the exception only of those (like overtake {.) which |
|
use the sparse element as the fill. |
|
|
|
f -: f &. $. |
|
f -: f &. ($.^:_1) |
|
|
|
All primitives accept sparse or dense arrays as arguments (e.g. sparse+dense or sparse$sparse). |
|
|
|
Symbols are a mechanism for searching, sorting, and comparisons on data that is much |
|
more efficient than alternatives such as boxed strings. Structural, selection, and relational |
|
verbs work on symbols. |
|
|
|
The monad verb s: converts arrays into symbols. Several types of arguments are acceptable: |
|
|
|
• string with the leading character as the separator |
|
|
|
• literal array where each row, excluding trailing blanks, is the name of a symbol |
|
|
|
• array of boxed strings |
|
|
|
Unicode is a 2-byte (16-bit) character datatype. |
|
|
|
The verb u: creates unicode arrays. The monad applies as follows: |
|
|
|
Argument |
|
1-byte characters |
|
2-byte characters |
|
integers |
|
|
|
Result |
|
same as 2&u: |
|
copy of argument |
|
same as 4&u: |
|
|
|
The inverse of the monad u: is 3&u: |
|
|
|
The dyad u: takes a scalar integer left argument and applies to several kinds of arguments: |
|
|
|
28 |
|
|
|
A Brief J Reference |
|
|
|
Left Right |
|
1 |
|
2 |
|
3 |
|
4 |
|
5 |
|
|
|
2-byte characters |
|
1-byte characters |
|
2-byte characters |
|
integers |
|
2-byte characters |
|
|
|
6 |
|
|
|
1-byte characters |
|
|
|
Result |
|
1-byte characters; high order bytes are discarded |
|
2-byte characters; high order bytes are 0 |
|
integers |
|
2-byte characters; integers must be from 0 to 65535 |
|
1-byte characters; high order bytes must be 0 (and are dis- |
|
carded) |
|
2-byte characters; pairs of 1-byte characters are converted |
|
to 2-byte characters |
|
|
|
1&u: and 2&u: is an inverse pair, as are 3&u: and 4&u: . |
|
|
|
34 Graphics |
|
|
|
J offers a great number of graphics facilities. Running the Graph Utilities, Plot and View- |
|
mat labs is recommended. |
|
|
|
Plot provides a powerful high level set of standard plotting functions, while Viewmat gives |
|
a visual display of a table. |
|
|
|
The underlying functions are in the gl2.ijs script. |
|
|
|
35 Qt Session Manager Short-Cut Keys |
|
|
|
Many J menu short-cut keys are defined, for example: |
|
|
|
Enter |
|
F1 |
|
Ctrl+F1 |
|
Ctrl+Shift+F1 |
|
Ctrl+Shift-up-arrow |
|
Ctrl+D |
|
Ctrl+E |
|
|
|
captures current line for editing on the execution input line |
|
Vocabulary |
|
context sensitive Vocabulary |
|
context sensitive NuVoc |
|
scroll up in execution log history |
|
window with execution history |
|
load selection |
|
|
|
36 Addons |
|
|
|
There are several addon packages available from the J wiki, see the page JAL, such as: |
|
|
|
29 |
|
|
|
fast fourier transform package. |
|
fftw |
|
J object database |
|
jod |
|
linear algebra package. |
|
lapack |
|
publish builds pdf reports from markup. |
|
SQLite |
|
stats |
|
tabula |
|
tara |
|
|
|
provides J bindings to SQLite embedded engine |
|
various statistical functions |
|
scientific calculator. |
|
reads and writes files in Excel format. |
|
|
|
Install from menu Tools|Package Manager. |
|
|
|
30 |
|
|
|
A Brief J Reference |
|
|
|
37 Parts of Speech and Grammar |
|
|
|
Most words are denoted with an ASCII symbol found on standard keyboards, or such a |
|
symbol followed by a period or colon. For example, we may think of % as denoting a J word |
|
meaning reciprocal, and %. as an inflection of that word meaning matrix inverse. |
|
|
|
Basic data objects in the language are nouns. These include scalars, such as 3.14, as well as |
|
lists (vectors) such as 2 3 5 7, matrices which are a rectangular arrangement of atoms and |
|
higher dimensional arrays of atoms. In general, arrays contain atoms that are organized |
|
along axes. These arrays may be literal, numeric or boxed. Any array may be boxed and, |
|
thereby, be declared to be a scalar. Nested boxing allows for rich data structures. |
|
|
|
The number of axes of an array gives its dimension. Thus, a scalar is 0-dimensional, a |
|
vector is 1-dimensional, a matrix is 2-dimensional and so on. The shape of an array is a list |
|
of the lengths of its axes. Often, the shape can be imagined as being split into two portions, |
|
giving an array of arrays. The leading portion of the split gives the frame (the shape of the |
|
outer array) and the other portion corresponds to the shape of the arrays, giving what are |
|
called cells. The items are the cells that occur by thinking of an n-dimensional array as a |
|
list of (n-1)-dimensional arrays. That is, items are rank _1 cells. |
|
|
|
Functions are known as verbs. For example, + denotes plus, %: denotes root, and (+/ % #) |
|
denotes average. Adverbs take one argument (often a verb) and typically result in a verb. |
|
For example, insert, denoted by / is an adverb. It takes a verb argument such as + and |
|
results in a derived verb +/ that sums items. Notice that adverbs take arguments on the |
|
left. The derived verb may itself take one noun argument (where it is a monad) or two |
|
noun arguments (where it is a dyad). It is sometimes helpful to be able to view a function |
|
as an object that can be formally manipulated. This facility is inherent in the J gerund. |
|
Gerunds are verbs playing the role of a noun. |
|
|
|
Conjunctions take two arguments and typically result in a verb. For example, . (dot) is a |
|
conjunction (be careful to distinguish this from a dot that is the last character in a name). |
|
For example, with left argument sum and right argument times, we get the matrix product |
|
+/ . * as the derived verb. |
|
|
|
The application of verbs to arguments proceeds from right to left. Thus 3*5+2 is 21 since |
|
the 5+2 is evaluated first. However, it is possible to think of the expression as being read |
|
left to right: 3 times the result of 5 plus 2. Therefore, verbs have long right scope and |
|
short left scope. Of course, one can use parentheses to order computations however desired: |
|
(3*5)+2 is 17. |
|
|
|
In contrast to verbs, adverbs and conjunctions bond to their arguments before verbs do. |
|
Also in contrast, they have long left scope and short right scope. Thus, we do not need the |
|
parentheses in (+/) . * to denote the matrix product since the left argument of the dot |
|
is the entire expression on its left, namely, +/ which gives the sum. Thus +/ . * denotes |
|
the matrix product. |
|
|
|
31 |
|
|
|
38 Glossary |
|
|
|
Adverb |
|
|
|
Atom |
|
|
|
Axis |
|
|
|
Cell |
|
|
|
A part of speech that takes an argument on the left and |
|
typically results in a verb. For example, insert / is an ad- |
|
verb such that with argument plus as in +/ the result is the |
|
derived verb sum. |
|
A 0-dimensional element of an array; it may be numeric, |
|
literal or boxed. |
|
An organizational direction of an array. The shape of an |
|
array gives the lengths of the axes of the array. |
|
A subarray of an array that consists of all the entries from |
|
the array with some fixed leading set of indices. |
|
|
|
Fork |
|
|
|
Dimension |
|
|
|
Gerund |
|
Hook |
|
|
|
Dyad |
|
Explicit |
|
|
|
Conjunction A part of speech that takes two arguments and typically |
|
results in a verb. For example, *:^:3 is a function that |
|
iterates squaring three times (^: is a conjunction). |
|
The dimension of an array is the number of axes given by |
|
the array’s shape. |
|
A verb with two arguments. |
|
Describes a definition which uses named arguments; for ex- |
|
ample, a verb defined using x and y.. |
|
A list of three verbs isolated in a train so that composition |
|
of functions occurs (see Section 18) |
|
A verb playing the role of a noun. |
|
A list of two verbs isolated in a train so that composition of |
|
functions occurs (see Section 18) |
|
The use of a period or colon suffix to change the meaning of |
|
a J word. |
|
A cell of rank _1. Thus, an array may be thought of as a |
|
list of its items. |
|
A verb with one argument. |
|
A data object that is numeric, literal or boxed. |
|
The dimension of cells upon which a verb operates; addi- |
|
tional leading axes are handled uniformly. |
|
Function definition without explicit (named) reference to the |
|
arguments |
|
Lists of conjunctions, adverbs, verbs and nouns; for example, |
|
a train of three verbs is a fork. |
|
A function; when it uses two arguments, it is a dyad; and |
|
when it uses one argument, it is a monad. |
|
|
|
Monad |
|
Noun |
|
Rank |
|
|
|
Inflection |
|
|
|
Trains |
|
|
|
Tacit |
|
|
|
Verb |
|
|
|
Item |
|
|
|
32 |
|
|
|
A Brief J Reference |
|
|
|
39 Vocabulary |
|
|
|
= Self-classify • Equal |
|
< Box • Less Than |
|
> Open • Larger Than |
|
_ Negative Sign, Infinity |
|
|
|
+ Conjugate • Plus |
|
* Signum • Times |
|
- Negate • Minus |
|
% Reciprocal • Divide |
|
|
|
^ Exponential • Power |
|
$ Shape Of • Shape |
|
~ Reflex • Pass, Evoke |
|
| Magnitude • Residue |
|
|
|
. Det • Dot Product |
|
: Explicit (monad, dyad) |
|
, Ravel • Append |
|
; Raze • Link |
|
|
|
# Tally • Copy |
|
! Factorial • Out Of |
|
/ Insert • Table, Insert |
|
\ Prefix • Infix, Train |
|
|
|
[ Same • Left |
|
] Same • Right |
|
{ Catalogue • From |
|
} Item Amend • Amend |
|
|
|
" Rank • Constant |
|
` Tie (gerund) |
|
@ Atop |
|
& Bond, Compose |
|
? Roll • Deal |
|
|
|
a. Alphabet |
|
b. Boolean, Basic |
|
D. Derivative |
|
E. • Member Of Interval |
|
|
|
i. Integers • Index Of |
|
j. Imaginary • Complex |
|
M. Memo |
|
p. Roots • Polynomial |
|
|
|
=. Is (local) |
|
<. Floor • Lesser Of |
|
>. Ceiling • Larger Of |
|
_. Indeterminate |
|
|
|
=: Is (global) |
|
<: Decrem • Less Or Equal |
|
>: Increm • Larger Or Equal |
|
_: Infinity |
|
|
|
+. Real/imaginary • GCD (Or) |
|
*. Length/angle • LCM (And) |
|
-. Not (1-) • Less |
|
%. Mat Inv • Mat Divide |
|
|
|
+: Double • Not Or |
|
*: Square • Not And |
|
-: Halve • Match |
|
%: Square Root • Root |
|
|
|
^. Natural Log • Logarithm |
|
$. Sparse |
|
~. Nub |
|
|. Reverse • Rotate |
|
|
|
^: Power |
|
$: Self Reference |
|
~: Nub Sieve • Not Equal |
|
|: Transpose |
|
|
|
.. Even |
|
:. Obverse |
|
,. Ravel Items • Stitch |
|
;. Cut |
|
|
|
.: Odd |
|
:: Adverse |
|
,: Itemize • Laminate |
|
;: Words • Sequential Machine |
|
|
|
#. Base 2 • Base |
|
!. Fit |
|
/. Oblique • Key, Append |
|
\. Suffix • Outfix |
|
|
|
#: Antibase 2 • Antibase |
|
!: Foreign |
|
/: Grade Up • Sort Up |
|
\: Grade Down • Sort Down |
|
|
|
{. Head • Take |
|
}. Behead • Drop |
|
|
|
". Do • Numbers |
|
|
|
@. Agenda |
|
&. &.: Under (Dual) |
|
?. Roll • Deal (fixed seed) |
|
|
|
a: Ace (Boxed Empty) |
|
C. Cycle Direct • Permute |
|
D: Secant Slope |
|
f. Fix |
|
|
|
[: Cap |
|
|
|
{: Tail, {:: Map, Fetch |
|
}: Curtail |
|
|
|
": Default Format • Format |
|
`: Evoke Gerund |
|
@: At |
|
&: Appose |
|
|
|
A. Anagram Index • Anagram |
|
d. Derivative |
|
e. Raze In • Member In |
|
H. Hypergeometric |
|
|
|
i: Axis Integers • Index Of Last |
|
L. Level Verb |
|
NB. Comment |
|
p.. Poly Deriv • Poly Integral |
|
|
|
I. Indices • Interval Index |
|
L: Level At |
|
o. Pi Times • Circle Function |
|
p: Primes |
|
|
|
q: Prime Factors • Prime Exponents |
|
S: Spread |
|
T. Taylor Approximationu: Unicode |
|
_9: to 9: Constant Verbs |
|
|
|
r. Angle • Polar |
|
t. Taylor Coeff. (m t. u t.) |
|
x: Extended Precision |
|
|
|
s: Symbol |
|
t: Weighted Taylor |
|
|
|
Font styles in the Vocabulary: noun, verb, adverb, conjunction. |
|
|
|
|