Multi-dimensional array

From Rosetta Code
Multi-dimensional array is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

For the purposes of this task, the actual memory layout or access method of this data structure is not mandated. It is enough to:

  1. State the number and extent of each index to the array.
  2. Provide specific, ordered, integer indices for all dimensions of the array together with a new value to update the indexed value.
  3. Provide specific, ordered, numeric indices for all dimensions of the array to obtain the arrays value at that indexed position.
The task is to
  • State if the language supports multi-dimensional arrays in its syntax and usual implementation.
  • State whether the language uses row-major or column major order for multi-dimensional array storage, or any other relevant kind of storage.
  • Show how to create a four dimensional array in your language and set, access, set to another value; and access the new value of an integer-indexed item of the array.
    The idiomatic method for the language is preferred.
  • The array should allow a range of five, four, three and two (or two three four five if convenient), in each of the indices, in order. (For example, if indexing starts at zero for the first index then a range of 0..4 inclusive would suffice).
  • State if memory allocation is optimised for the array - especially if contiguous memory is likely to be allocated.
  • If the language has exceptional native multi-dimensional array support such as optional bounds checking, reshaping, or being able to state both the lower and upper bounds of index ranges, then this is the task to mention them.

Show all output here, (but you may judiciously use ellipses to shorten repetitive output text).

ALGOL 68[edit]

Works with: ALGOL 68G version Any - tested with release 2.8.win32

Algol 68 supports multi-dimensional arrays as standard. The maximum values for the indexes and the number of indexes is up to the implementation.

The upper and optionally the lower bounds of each index are specified when the array is created. If omitted the, the lower bound defaults to 1. The bounds can be arbitrary integer expressions.

The upper and lower bounds are separated by ":" and each pair of bounds is separated from the next by a ",". The array required by the task could be declared as follows: #

[ 1 : 5, 1 : 4, 1 : 3, 1 : 2 ]INT a;

As the lower bounds are all 1, this could be abbreviated to:

[ 5, 4, 3, 2 ]INT a;

Note that the lower and upper bounds can be 0 or negative, e.g. the following declares an array of the same size but different bounds:

[ -7 : -3. -3 : 0, -1 : 1, 0 : 1 ]INT x;

Individual array elements can be set and accessed by stating the index values separated by ",". The following sets the lowest element of the array to 0:

a[ 1, 1, 1, 1 ] := 0;

The following sets the highest element to the lowest:

a[ 5, 4, 3, 2 ] := a[ 1, 1, 1, 1 ];

Subsets of the array can be set and accessed by stating the index values of the subsets, e.g. the following sets a pair of elements near the start of the array to a pair of elements near the end: Selecting a subset of an array is known as slicing.

a[ 3 : 4, 1, 1, 1 ] := a[ 5, 4, 2 : 3, 1 ];

The whole array can also be assigned, e.g., the following sets the entire array to some values, using nested "literal arrays" (known as row-displays):

# Note this requires the lower bounds of each index be 1                #
a := ( ( ( ( 1111, 1112 ), ( 1121, 1122 ), ( 1131, 1132 ) )
, ( ( 1211, 1212 ), ( 1221, 1222 ), ( 1231, 1232 ) )
, ( ( 1311, 1312 ), ( 1321, 1322 ), ( 1331, 1332 ) )
, ( ( 1411, 1412 ), ( 1421, 1422 ), ( 1431, 1432 ) ) )
, ( ( ( 2111, 2112 ), ( 2121, 2122 ), ( 2131, 2132 ) )
, ( ( 2211, 2212 ), ( 2221, 2222 ), ( 2231, 2232 ) )
, ( ( 2311, 2312 ), ( 2321, 2322 ), ( 2331, 2332 ) )
, ( ( 2411, 2412 ), ( 2421, 2422 ), ( 2431, 2432 ) ) )
, ( ( ( 3111, 3112 ), ( 3121, 3122 ), ( 3131, 3132 ) )
, ( ( 3211, 3212 ), ( 3221, 3222 ), ( 3231, 3232 ) )
, ( ( 3311, 3312 ), ( 3321, 3322 ), ( 3331, 3332 ) )
, ( ( 3411, 3412 ), ( 3421, 3422 ), ( 3431, 3432 ) ) )
, ( ( ( 4111, 4112 ), ( 4121, 4122 ), ( 4131, 4132 ) )
, ( ( 4211, 4212 ), ( 4221, 4222 ), ( 4231, 4232 ) )
, ( ( 4311, 4312 ), ( 4321, 4322 ), ( 4331, 4332 ) )
, ( ( 4411, 4412 ), ( 4421, 4422 ), ( 4431, 4432 ) ) )
, ( ( ( 5111, 5112 ), ( 5121, 5122 ), ( 5131, 5132 ) )
, ( ( 5211, 5212 ), ( 5221, 5222 ), ( 5231, 5232 ) )
, ( ( 5311, 5312 ), ( 5321, 5322 ), ( 5331, 5332 ) )
, ( ( 5411, 5412 ), ( 5421, 5422 ), ( 5431, 5432 ) ) ) );

If the lower bounds are not 1, they can be temporarily be changed to 1 by using the "AT" construct, e.g. as in the following:

# declare an array the same size as a, but with all lower bounds equal to 2: #
[ 2 : 6, 2 : 5, 2 : 4, 2 : 3 ]INT b;
 
# set b to the same values as a above: #
b[ AT 1, AT 1, AT 1, AT 1 ] :=
( ( ( ( 1111, 1112 ), ( 1121, 1122 ), ( 1131, 1132 ) )
, ( ( 1211, 1212 ), ( 1221, 1222 ), ( 1231, 1232 ) )
, ( ( 1311, 1312 ), ( 1321, 1322 ), ( 1331, 1332 ) )
, ( ( 1411, 1412 ), ( 1421, 1422 ), ( 1431, 1432 ) ) )
, ( ( ( 2111, 2112 ), ( 2121, 2122 ), ( 2131, 2132 ) )
, ( ( 2211, 2212 ), ( 2221, 2222 ), ( 2231, 2232 ) )
, ( ( 2311, 2312 ), ( 2321, 2322 ), ( 2331, 2332 ) )
, ( ( 2411, 2412 ), ( 2421, 2422 ), ( 2431, 2432 ) ) )
, ( ( ( 3111, 3112 ), ( 3121, 3122 ), ( 3131, 3132 ) )
, ( ( 3211, 3212 ), ( 3221, 3222 ), ( 3231, 3232 ) )
, ( ( 3311, 3312 ), ( 3321, 3322 ), ( 3331, 3332 ) )
, ( ( 3411, 3412 ), ( 3421, 3422 ), ( 3431, 3432 ) ) )
, ( ( ( 4111, 4112 ), ( 4121, 4122 ), ( 4131, 4132 ) )
, ( ( 4211, 4212 ), ( 4221, 4222 ), ( 4231, 4232 ) )
, ( ( 4311, 4312 ), ( 4321, 4322 ), ( 4331, 4332 ) )
, ( ( 4411, 4412 ), ( 4421, 4422 ), ( 4431, 4432 ) ) )
, ( ( ( 5111, 5112 ), ( 5121, 5122 ), ( 5131, 5132 ) )
, ( ( 5211, 5212 ), ( 5221, 5222 ), ( 5231, 5232 ) )
, ( ( 5311, 5312 ), ( 5321, 5322 ), ( 5331, 5332 ) )
, ( ( 5411, 5412 ), ( 5421, 5422 ), ( 5431, 5432 ) ) ) );

Bounds checking is standard and there is no standard way to turn it off. There may be implementation-specific mechanisms ( probably using pragmatic comments ) to do so.

The memory layout of an array is determined by the implementation and is not visible to the programmer.

"Flexible" arrays whose size can change during the execution of the program can be created. These are used e.g. for the builtin STRING mode.

The size of a flexible array can be changed only be assigning a new array to it.

The bounds of the array can be determined using the LWB and UPB operators. For a 1 dimensional array, the unary LWB and UPB operators can be used.

# E.g. the following prints the lower and upper bounds of the first two      #
# indexes of a ( in this case, 1, 5, 1 and 4 ) #
print( ( 1 LWB a, 1 UPB a, 2 LWB a, 2 UPB a, newline ) );
Output:
         +1         +5         +1         +4

ALGOL W[edit]

Algol W supports multi-dimensional arrays as standard. The maximum values for the indexes and the number of indexes is up to the implementation. The upper and lower bounds of each index are specified when the array is declared. The bounds are evaluated on entry to the block containing the declaration and can be arbitrary integer expressions.

The upper and lower bounds are separated by "::" and each pair of bounds is separated from the next by a ",". The array required by the task could be declared as follows:

integer array a ( 1 :: 5, 1 :: 4, 1 :: 3, 1 :: 2 );

note that the lower and upper bounds can be 0 or negative, e.g. the following declares an array of the same size but different bounds:

integer array x ( -7 :: -3. -3 :: 0, -1 :: 1, 0 :: 1 );

individual array elements can be set and accessed by stating the index values separated by ",". The following sets the lowest element of the array to 0:

a( 1, 1, 1, 1 ) := 0;

The following sets the highest element to the lowest:

a( 5, 4, 3, 2 ) := a( 1, 1, 1, 1 );

Subsets of an array can be specified as parameters to procedures. E.g.:

% declare a procedure that has a three-dimensional array parameter       %
procedure p ( integer array a1 ( *, *, * ) ) ; a1( 1, 2, 1 ) := 3 ;
 
% call the procedure with a subset of the 4 dimensional array  %
p( a( *, 2, *, * ) );

The elements of the array can only be set individually, it is not possible to assign the whole array or multiple elements other than with element-by-element assignments.

Bounds checking is standard and there is no standard way to turn it off. There may be implementation-specific mechanisms to do so.

The memory layout of an array is determined by the implementation and is not visible to the programmer.

EchoLisp[edit]

EchoLisp natively supports 1 and 2-dimensions arrays : lib matrix. The following shows an implementation of multi-dimensional arrays, using contiguous memory and bound checking. Indices are vectors #(i j k ...) of integers.

 
(require 'math) ;; dot-product
 
;; dims = vector #(d1 d2 .....)
;; allocates a new m-array
(define (make-m-array dims (init 0))
;; allocate 2 + d1*d2*d3... consecutive cells
(define msize (apply * (vector->list dims)))
(define m-array (make-vector (+ 2 msize) init))
 
;; compute displacements vector once for all
;; m-array[0] = [1 d1 d1*d2 d1*d2*d3 ...]
(define disps (vector-rotate! (vector-dup dims) 1))
(vector-set! disps 0 1)
(for [(i(in-range 1 (vector-length disps)) )]
(vector-set! disps i (* [disps i] [disps (1- i)])))
(vector-set! m-array 0 disps)
 
(vector-set! m-array 1 dims) ;; remember dims
m-array)
 
;; from indices = #(i j k ...) to displacement
(define-syntax-rule (m-array-index ma indices)
(+ 2 (dot-product (ma 0) indices)))
 
;; check i < d1, j < d2, ...
(define (m-array-check ma indices)
(for [(dim [ma 1]) (idx indices)]
#:break (>= idx dim) => (error 'm-array:bad-index (list idx '>= dim))))
 
;; --------------------
;; A P I
;; --------------------
;; indices is a vector #[i j k ...]
;; (make-m-array (dims) [init])
 
(define (m-array-dims ma) [ma 1])
 
; return ma[indices]
(define (m-array-ref ma indices)
(m-array-check ma indices)
[ma (m-array-index ma indices)])
; sets ma[indices]
(define (m-array-set! ma indices value )
(m-array-check ma indices)
(vector-set! ma (m-array-index ma indices) value))
 
 
Output:
(define MA (make-m-array #(5 4 3 2) 1))
(m-array-dims MA)
    โ†’ #( 5 4 3 2)
(m-array-set! MA #(3 2 1 1 ) '๐Ÿ“ )
(m-array-ref MA #(3 2 1 1 ))
    โ†’ ๐Ÿ“
MA
    โ†’ #( #( 1 5 20 60) #( 5 4 3 2)  1 1 1 1 ...  1 1 ๐Ÿ“ 1 1 1 1 1 ... 1 1 1 1 1 1)

(m-array-ref MA #(1 1 42 2))
๐Ÿ˜ก error: m-array:bad-index (42 >= 3)

Fortran[edit]

Messing with multi-dimensional arrays has been central from the introduction of Fortran in 1954. Array indexing is with positive integers and always starts with one, so the DIMENSION statement merely specifies the upper bound of an index via integer constants and an array may have up to three dimensions. All storage for an array is contiguous. Access to an element of a one-dimensional array A is via A(integer expression) and early Fortran allowed only simple expressions, no more than c*v ยฑ b which is to say an integer constant times an integer variable plus (or minus) an integer constant. These restrictions simplified the thorough optimisation done by the first compiler. Later compilers abandoned those optimisations, relaxed the constraints on index expressions and allowed more dimensions.

 
DIMENSION A(5,4,3,2) !Declares a (real) array A of four dimensions, storage permitting.
X = 3*A(2,I,1,K) !Extracts a certain element, multiplies its value by three, result to X.
A(1,2,3,4) = X + 1 !Places a value (the result of the expression X + 1) ... somewhere...
 

Compilers typically perform no checking on the bounds of array indices, even when values are being written to some address. An immediate crash would be the best result. Array bound checking may be requested as a compiler option in some cases, resulting in larger code files, slower execution, and some bug detection.

With Fortran 90 came a large expansion of abilities. Lower bounds could also be specified for an array's dimension, arrays defined in subprograms would be sized according to values at execution time (not constants at compile time) as in A(-5:3,N + 7) where N was a parameter to the subroutine, and if that was insufficient, arrays of a desired size could be allocated (and deallocated) according to program logic within a routine.

As well, array data could be manipulated with array processing statements, rather than always having to prepare DO-loops or similar, in particular A = 0 would set all values of array A to zero, whatever its dimensionality. Thus, given an array B(0:2,0:2), a two-dimensional array, B(1:2,1:2) = 6*A(1,3:4,1,1:2) - 7 would select certain elements scattered about in A, perform the calculation, and store the results in a part of B. Hopefully, element-by element without using a temporary array. Even more complicated selections can be made via the RESHAPE intrinsic now available.

Data are stored in Fortran arrays in "column-major" order, with the result that successive elements are indexed with the left-most subscript varying most rapidly. Thus, given an array DIMENSION A(3,3) the elements are ordered as A(1,1), A(2,1), A(3,1), A(1,2), A(2,2), A(3,2), A(1,3), A(2,3), A(3,3). This doesn't matter when the array is referenced via something like A(row,column). However, should the array A be written (or read) or initialised via a DATA statement without specification of indices, that is the order of the values. Consider
      INTEGER A(3,3)	!An array, regarded as being a matrix.
DATA A/1,2,3, !Some initial values.
2 4,5,6, !Laid out as if a matrix.
3 7,8,9/ !But supplied as consecutive elements.
 
WRITE (6,*) "Writing A..."
WRITE (6,1) A !Using the array order.
1 FORMAT (3I2) !After three values, start a new line.
 
WRITE (6,*) "Writing A(row,col) down rows and across columns..."
DO I = 1,3 !Write each row, one after the other.
WRITE (6,1) (A(I,J), J = 1,3) !The columns along a row.
END DO !Onto the next row.
END

Output:

 Writing A...
 1 2 3
 4 5 6
 7 8 9
 Writing A(row,col) down rows and across columns...
 1 4 7
 2 5 8
 3 6 9

The same will happen if data were to be supplied with that layout and read by something like READ(in,*) A To read data with that layout but obtain the desired ordering, the read must proceed with explicit specification of the index ordering as with the two loops of the second attempt at WRITE. This done, programming proceeds and functions such as MATMUL will give results as expected.

This affects higher-dimension arrays also. A(3,3,100) will give a hundred sets of 3x3 arrays, which might be used in something like CALL PROCESS(A(1,1,IT)) where subroutine PROCESS expects a two-dimensional array and need only engage in two-level indexing rather than three. Conversely, A(100,3,3) will produce confusion.

For large arrays when memory is not truly random-access (on-chip memory versus main memory, main memory versus slower disc storage) a calculation that works along a row will be accessing widely-separated elements in storage, perhaps with only one "hit" per virtual memory page before leaping to another, while an equivalent calculation working down a column will enjoy multiple hits per page. The difference can be large enough to promote usages such as swapping the order of the DO-loops for portions of a calculation.

J[edit]

J supports multi-dimensional arrays, and provides a variety of ways of creating and working with them.

Perhaps the simplest mechanism to create a multidimensional array is to reshape a smaller dimensioned array with the desired dimensions:

A1=:5 4 3 2$0

This creates an array of 120 zeros arranged in contiguous memory.

Note that items along the leading dimension of the array being reshaped are repeated as necessary to fill in all the values of the array. Thus, this is equivalent:

A2=:5 4 $ 1 3 2$ 0

Another candidate for the simplest mechanism to create a multidimensional array is to ask for an array of indices with those dimensions:

   i.2 3 4 5
0 1 2 3 4
5 6 7 8 9
10 11 12 13 14
15 16 17 18 19
 
20 21 22 23 24
25 26 27 28 29
30 31 32 33 34
35 36 37 38 39
 
40 41 42 43 44
45 46 47 48 49
50 51 52 53 54
55 56 57 58 59
 
 
60 61 62 63 64
65 66 67 68 69
70 71 72 73 74
75 76 77 78 79
 
80 81 82 83 84
85 86 87 88 89
90 91 92 93 94
95 96 97 98 99
 
100 101 102 103 104
105 106 107 108 109
110 111 112 113 114
115 116 117 118 119

Note that these indices are not multi-dimensional indices but simple counting numbers. (Internally, values in the array are arranged contiguously and in this index order.) To obtain the corresponding index, you can use J's antibase verb. For example:

   2 3 4 5#:118
1 2 3 3

Normally, in J, you operate on "everything at once", which can be challenging if you have not worked in a language (such as sql or any of a variety of others) which encourages that kind of thinking. Nevertheless, here's some introductory "one at a time" operations:

Pulling a single value from an array:

   (<1 2 3 3) { i.2 3 4 5
118

Setting a single value in an array:

A3=: 987 (<1 2 3 3)} i. 2 3 4 5

And, to reduce vertical space used in this task, here's an example of extracting a sub-array:

   (<1 2){A3
100 101 102 103 104
105 106 107 108 109
110 111 112 113 114
115 116 117 987 119

Note that bounds are checked when indexing from the array, or when combining with another array.

   (<5 5 5 5){A3
|index error

and

   A1+A3
|length error

For more introductory material on multi-dimensional arrays, you might be interested in chapter 5 and chapter 7 of the Learning J book.

JavaScript[edit]

Javascript does not natively support multi-dimensional arrays.

But, as it does support one dimensional arrays, we can use them to construct multi-dimensional arrays.

One approach pairs an array of dimensions with an array of values.

 
function array() {
var dimensions= Array.prototype.slice.call(arguments);
var N=1, rank= dimensions.length;
for (var j= 0; j<rank; j++) N*= dimensions[j];
this.dimensions= dimensions;
this.values= new Array(N);
}
 

A routine to convert between single dimension indices and multi-dimensional indices can be useful here:

 
function tobase(base, vals) {
var r= 0, len= base.length;
for (j= 0; j < len; j++) {
r*= base[j];
r+= vals[j];
}
return r;
}
 
function frombase(base, val) {
var r= new Array(base.length);
for (j= base.length-1; j>= 0; j--) {
r[j]= val%base[j];
val= (val-r[j])/base[j];
}
return r;
}

For example, array indexing in this approach might be implemented something like:

 
array.prototype.index= function() {
var indices= Array.prototype.slice.call(arguments);
return this.values[tobase(this.dimensions, indices)];
}
 

This would allow things like:

 
a= new array(6,7,8,9);
a.index(2,3,5,6);
 

But a proper implementation of the index method should include array bounds checking.

That said, note that operating on the entire array at once avoids the need for any index calculations.

Another approach "nests" arrays inside other arrays:

 
function array(length) {
var rest= Array.prototype.slice.call(arguments);
var r= new Array(length);
if (0<rest.length) {
for (var j= 0; j<length; j++) {
r[j]= array.apply(rest);
}
}
}

This approach is superficially simpler but may turn out to be considerably slower.

That said, with this approach indexing can be performed directly - without implementing any index function. For example a[2][3][4][5], and this convenience may override any concerns about performance.

jq[edit]

jq only supports one-dimensional arrays natively, but these can be used to represent multi-dimensional arrays since a JSON array can contain elements of any type. Furthermore, jq has builtin functions which are well-suited for updating and accessing the elements in multi-dimensional arrays. As illustrated below, these are getpath(d) and setpath(d; value), where d is an array specifying the indices of a particular element.

Preliminaries

(1) The index origin for jq arrays is 0.

(2) All values in jq are immutable, but there are element-wise operators that in effect modify an array at a particular index; for example, if ary is an array, and if i is an index into ary, then the expression 'ary | .[i] = v' in effect sets ary[i] to v, though in fact it returns a copy of ary with the i-th element set to v.

(3) If ary is an array, then ary[i:j] is the array [a[i], ... a[j-1]], assuming 0<=i<j<=length. Other notational conveniences are supported, e.g. in jq 1.5, ary[-i] can be used to refer to ary[length-i].

(4) There is nothing in jq itself to force an array of arrays to be a "multi-dimensional array" in the sense that it is strictly "rectangular". A function for checking whether a JSON array is in fact a multi-dimensional array in the sense of this article is presented below.

Array Creation To create a one-dimensional array of n nulls, one may write [][n] = null.

To create a multi-dimensional array with dimensions specified by an array, d, we can define a recursive function as follows:

# The input is used to initialize the elements of the
# multi-dimensional array:
def multiarray(d):
. as $in
| if (d|length) == 1 then [range(0;d[0]) | $in]
else multiarray(d[1:]) | multiarray( d[0:1] )
end;

A four-dimensional array as specified by the task description can now be created as follows:

0 | multiarray( [5, 4, 3, 2] )

For convenience of reference, let us name this array by wrapping it in a function:

def ary: 0 | multiarray( [5, 4, 3, 2] );

Access and Update To access the [4,3,2,1] element of the previously defined multi-dimensional array, one can either write:

ary | .[4][3][2][1]

or

ary | getpath( [4,3,2,1])

To illustrate, let us define an array of indices:

def ix: [4,3,2,1];

To check that the two approaches for accessing an element are equivalent:

ary | setpath(ix; 100) | getpath(ix)
#=> 100
 
ary | setpath(ix; 100) | .[4][3][2][1]
#=> 100

Ascertaining the Dimensions of a Multi-dimensional Array

The following function returns the dimensions of the input if it is a regular multi-dimensional array; it returns [] for a scalar, and null for an irregularly shaped array:

def dimensions:
def same(f):
if length == 0 then true
else (.[0]|f) as $first | reduce .[] as $i (true; if . then ($i|f) == $first else . end)
end;
 
if type == "array"
then if length == 0 then [0]
elif same( dimensions ) then [length] + (.[0]|dimensions)
else null
end
else []
end;

For exampe:

ary | dimensions
#=> [5,4,3,2]

Lua[edit]

Lua does not support straightforward declaration of multidimensional arrrays but they can be created in the form of nested tables. Once such a structure has been created, elements can be read and written in a manner that will be familiar to users of C-like languages.

-- Variadic, first argument is the value with which to populate the array.  
function multiArray (initVal, ...)
local function copy (t)
local new = {}
for k, v in pairs(t) do
if type(v) == "table" then
new[k] = copy(v)
else
new[k] = v
end
end
return new
end
local dimensions, arr, newArr = {...}, {}
for i = 1, dimensions[#dimensions] do table.insert(arr, initVal) end
for d = #dimensions - 1, 1, -1 do
newArr = {}
for i = 1, dimensions[d] do table.insert(newArr, copy(arr)) end
arr = copy(newArr)
end
return arr
end
 
-- Function to print out the specific example created here
function show4dArray (a)
print("\nPrinting 4D array in 2D...")
for k, v in ipairs(a) do
print(k)
for l, w in ipairs(v) do
print("\t" .. l)
for m, x in ipairs(w) do
print("\t", m, unpack(x))
end
end
end
end
 
-- Main procedure
local t = multiArray("a", 2, 3, 4, 5)
show4dArray(t)
t[1][1][1][1] = true
show4dArray(t)
Output:
Printing 4D array in 2D...
1
        1
                1       a       a       a       a       a
                2       a       a       a       a       a
                3       a       a       a       a       a
                4       a       a       a       a       a
        2
                1       a       a       a       a       a
                2       a       a       a       a       a
                3       a       a       a       a       a
                4       a       a       a       a       a
        3
                1       a       a       a       a       a
                2       a       a       a       a       a
                3       a       a       a       a       a
                4       a       a       a       a       a
2
        1
                1       a       a       a       a       a
                2       a       a       a       a       a
                3       a       a       a       a       a
                4       a       a       a       a       a
        2
                1       a       a       a       a       a
                2       a       a       a       a       a
                3       a       a       a       a       a
                4       a       a       a       a       a
        3
                1       a       a       a       a       a
                2       a       a       a       a       a
                3       a       a       a       a       a
                4       a       a       a       a       a

Printing 4D array in 2D...
1
        1
                1       true    a       a       a       a
                2       a       a       a       a       a
                3       a       a       a       a       a
                4       a       a       a       a       a
        2
                1       a       a       a       a       a
                2       a       a       a       a       a
                3       a       a       a       a       a
                4       a       a       a       a       a
        3
                1       a       a       a       a       a
                2       a       a       a       a       a
                3       a       a       a       a       a
                4       a       a       a       a       a
2
        1
                1       a       a       a       a       a
                2       a       a       a       a       a
                3       a       a       a       a       a
                4       a       a       a       a       a
        2
                1       a       a       a       a       a
                2       a       a       a       a       a
                3       a       a       a       a       a
                4       a       a       a       a       a
        3
                1       a       a       a       a       a
                2       a       a       a       a       a
                3       a       a       a       a       a
                4       a       a       a       a       a

This is not memory-efficient and any advanced operations will have to be written if required. A much easier solution is to use an extension module such as densearray available from luarocks.org.

Perl 6[edit]

#Perl 6 supports multi dimension arrays natively. There are no arbitrary limits on the number of dimensions or maximum indicies. Theoretically, you could have an infinite number of dimensions of infinite length, though in practice more than a few dozen dimensions gets unwieldy. An infinite maximum index is a fairly common idiom though. You can assign an infinite lazy list to an array and it will only produce the values when they are accessed.
 
my @integers = 1 .. Inf; # an infinite array containing all positive integers
 
say @integers[100000]; #100001 (arrays are zero indexed.)
 
# Multi dimension arrays may be predeclared which constrains the indicies to the declared size:
 
my @dim5[3,3,3,3,3];
 
#Creates a preallocated 5 dimensional array where each branch has 3 storage slots and constrains the size to teh declared size.
#It can then be accessed like so:
 
@dim5[0;1;2;1;0] = 'Perl 6';
 
say @dim5[0;1;2;1;0]; # Perl 6
 
[email protected][0;1;2;1;4] = 'error'; # runtime error: Index 4 for dimension 5 out of range (must be 0..2)
 
# Note that the dimensions do not _need_ to be predeclared / allocated. Perl 6 will auto-vivify the necessary storage slots on first access.
 
my @a2;
 
@a2[0;1;2;1;0] = 'Perl 6';
 
@a2[0;1;2;1;4] = 'not an error';
 
# It is easy to access array "slices" in Perl 6.
 
my @b = map { [$_ X~ 1..5] }, <a b c d>;
 
.say for @b;
# [a1 a2 a3 a4 a5]
# [b1 b2 b3 b4 b5]
# [c1 c2 c3 c4 c5]
# [d1 d2 d3 d4 d5]
 
say @b[*;2]; # Get the all of the values in the third "column"
# (a3 b3 c3 d3)
 
# By default, Perl 6 can store any object in an array, and it is not very compact. You can constrain the type of values that may be stored which can allow the optimizer to store them much more efficiently.
 
my @c = 1 .. 10; # stores integers, but not very compact since there are no constraints on what the values _may_ be
 
my uint16 @d = 1 .. 10 # since there are and only can be unsigned 16 bit integers, the optimizer will use a much more compact memory layout.
 
# indicies must be a positive integer. negative indicies are not permitted, fractional indicies will be truncated to an integer.
 

Phix[edit]

Multi-dimensional arrays (sequences in Phix parlance) are not just supported, but the central and most important construct of the entire language.
(Quite often the phrase "nested sub-sequences" is specifically used to refer to multi-dimensional capabilities.)
Arrays are allocated piecemeal from the heap, so storage is closer to row-major order than column-major order, with integers allocated contiguously at the lowest level.
Arrays are not limited by type; they can contain any mixture of integers, floating point numbers, strings, or nested sub-sequences, and can easily be shrunk or expanded, at any level, at any time.

sequence xyz = repeat(repeat(repeat(0,x),y),z)

or if you prefer:

sequence line = repeat(0,x)
sequence plane = repeat(line,y)
sequence space = repeat(plane,z)

Internally Phix uses reference counting with copy-on-write semantics, so the initial space allocated is x+y+z, and as more and more individual points are populated the total space required gradually increases to x*y*z.
Following the above initialisation, a specific point would be referenced as space[z][y][x] (and obviously space[x][y][z] is perfectly possible with a different initialisation, as long as you accept that two logically adjacent x will be far from physically adjacent).
Note that all indexes are 1-based. You can also use negative indexes to count backwards from the end of the sequence.

sequence rqd = repeat(repeat(repeat(repeat(0,2),3),4),5),
row -- scratch var
 
rqd[5][4][3][2] := 5432
 ?rqd[5][4][3][2]
 
row = rqd[5][4][3] -- row is now [0,5432]
row[1] = 1 -- rqd remains unchanged
rqd[5][4][3] = row -- rqd[5][4][3][1] is now 1
rqd[5][4][2] = repeat(0,10) -- middle element of rqd[5][4] is now longer than the other two
 ?rqd[5][4]
Output:
5432
{{0,0},{0,0,0,0,0,0,0,0,0,0},{1,5432}}

Array bounds are automatically checked on all forms of subscripting. Slices can also be referenced, eg rqd[2..3] or rqd[5][4][1..3]. A slice cannot be followed by further subscripts. Use the following, experimenting with the nesting value, to quickly dump (smallish) nested sequences in an easily readable fashion:

?"==1=="
pp(rqd,{pp_Nest,1})
?"==2=="
pp(rqd,{pp_Nest,2})
?"==3=="
pp(rqd,{pp_Nest,3})
Output:
"==1=="
{{{{0,0}, {0,0}, {0,0}}, {{0,0}, {0,0}, {0,0}}, {{0,0}, {0,0}, {0,0}},
  {{0,0}, {0,0}, {0,0}}},
 ....
 {{{0,0}, {0,0}, {0,0}}, {{0,0}, {0,0}, {0,0}}, {{0,0}, {0,0}, {0,0}},
  {{0,0}, {0,0,0,0,0,0,0,0,0,0}, {1,5432}}}}
"==2=="
{{{{0,0}, {0,0}, {0,0}},
  {{0,0}, {0,0}, {0,0}},
  {{0,0}, {0,0}, {0,0}},
  {{0,0}, {0,0}, {0,0}}},
 ....
 {{{0,0}, {0,0}, {0,0}},
  {{0,0}, {0,0}, {0,0}},
  {{0,0}, {0,0}, {0,0}},
  {{0,0}, {0,0,0,0,0,0,0,0,0,0}, {1,5432}}}}
"==3=="
{{{{0,0},
   {0,0},
   {0,0}},
  ...
  {{0,0},
   {0,0},
   {0,0}}},
 ...
 {{{0,0},
   {0,0},
   {0,0}},
  ...
  {{0,0},
   {0,0,0,0,0,0,0,0,0,0},
   {1,5432}}}}

Python[edit]

Python: In-built[edit]

Python has syntax (and hidden) support for the access of multi-dimensional arrays, but no in-built datatype that supports it.

A common method of simulating multi-dimensional arrays is to use dicts with N-element tuples as keys.

Function dict_as_mdarray allows for the creation of an initialised multi-dimensional array of a given size. Note how indexing must use the round brackets of a tuple inside the square brackets of normal dict indexing:

>>> from pprint import pprint as pp   # Pretty printer
>>> from itertools import product
>>>
>>> def dict_as_mdarray(dimensions=(2, 3), init=0.0):
... return {indices: init for indices in product(*(range(i) for i in dimensions))}
...
>>>
>>> mdarray = dict_as_mdarray((2, 3, 4, 5))
>>> pp(mdarray)
{(0, 0, 0, 0): 0.0,
(0, 0, 0, 1): 0.0,
(0, 0, 0, 2): 0.0,
(0, 0, 0, 3): 0.0,
(0, 0, 0, 4): 0.0,
(0, 0, 1, 0): 0.0,
...
(1, 2, 3, 0): 0.0,
(1, 2, 3, 1): 0.0,
(1, 2, 3, 2): 0.0,
(1, 2, 3, 3): 0.0,
(1, 2, 3, 4): 0.0}
>>> mdarray[(0, 1, 2, 3)]
0.0
>>> mdarray[(0, 1, 2, 3)] = 6.78
>>> mdarray[(0, 1, 2, 3)]
6.78
>>> mdarray[(0, 1, 2, 3)] = 5.4321
>>> mdarray[(0, 1, 2, 3)]
5.4321
>>> pp(mdarray)
{(0, 0, 0, 0): 0.0,
(0, 0, 0, 1): 0.0,
(0, 0, 0, 2): 0.0,
...
(0, 1, 2, 2): 0.0,
(0, 1, 2, 3): 5.4321,
(0, 1, 2, 4): 0.0,
...
(1, 2, 3, 3): 0.0,
(1, 2, 3, 4): 0.0}
>>>

Python: numpy library[edit]

Python has the widely available numpy library for array specific operations. It creates numpy array types that take full advantage of Python's syntax support for multi-dimensional arrays.

Numpy arrays contain values of a single type arranged in a contiguous block of memory that can be further arranged to be compatible with C language or Fortran array layouts to aid the use of C and Fortran libraries.

The default is row-major order (compatible with C). When dealing with Fortran storage order, care must be taken in order to get the same behavious as in Fortran:

import numpy as np
a = np.array([[1, 2], [3, 4]], order="C")
b = np.array([[1, 2], [3, 4]], order="F")
np.reshape(a, (4,)) # [1, 2, 3, 4]
np.reshape(b, (4,)) # [1, 2, 3, 4]
np.reshape(b, (4,), order="A") # [1, 3, 2, 4]

In Fortran, one would expect the result [1, 3, 2, 4].

>>> from numpy import *
>>>
>>> mdarray = zeros((2, 3, 4, 5), dtype=int8, order='F')
 
>>> mdarray
array([[[[0, 0, 0, 0, 0],
[0, 0, 0, 0, 0],
[0, 0, 0, 0, 0],
[0, 0, 0, 0, 0]],
 
[[0, 0, 0, 0, 0],
[0, 0, 0, 0, 0],
[0, 0, 0, 0, 0],
[0, 0, 0, 0, 0]],
 
[[0, 0, 0, 0, 0],
[0, 0, 0, 0, 0],
[0, 0, 0, 0, 0],
[0, 0, 0, 0, 0]]],
 
 
[[[0, 0, 0, 0, 0],
[0, 0, 0, 0, 0],
[0, 0, 0, 0, 0],
[0, 0, 0, 0, 0]],
 
[[0, 0, 0, 0, 0],
[0, 0, 0, 0, 0],
[0, 0, 0, 0, 0],
[0, 0, 0, 0, 0]],
 
[[0, 0, 0, 0, 0],
[0, 0, 0, 0, 0],
[0, 0, 0, 0, 0],
[0, 0, 0, 0, 0]]]], dtype=int8)
>>> mdarray[0, 1, 2, 3]
0
>>> mdarray[0, 1, 2, 3] = 123
>>> mdarray[0, 1, 2, 3]
123
>>> mdarray[0, 1, 2, 3] = 666
>>> mdarray[0, 1, 2, 3]
-102
>>> mdarray[0, 1, 2, 3] = 255
>>> mdarray[0, 1, 2, 3]
-1
>>> mdarray[0, 1, 2, 3] = -128
>>> mdarray[0, 1, 2, 3]
-128
>>> mdarray
array([[[[ 0, 0, 0, 0, 0],
[ 0, 0, 0, 0, 0],
[ 0, 0, 0, 0, 0],
[ 0, 0, 0, 0, 0]],
 
[[ 0, 0, 0, 0, 0],
[ 0, 0, 0, 0, 0],
[ 0, 0, 0, -128, 0],
[ 0, 0, 0, 0, 0]],
 
[[ 0, 0, 0, 0, 0],
[ 0, 0, 0, 0, 0],
[ 0, 0, 0, 0, 0],
[ 0, 0, 0, 0, 0]]],
 
 
[[[ 0, 0, 0, 0, 0],
[ 0, 0, 0, 0, 0],
[ 0, 0, 0, 0, 0],
[ 0, 0, 0, 0, 0]],
 
[[ 0, 0, 0, 0, 0],
[ 0, 0, 0, 0, 0],
[ 0, 0, 0, 0, 0],
[ 0, 0, 0, 0, 0]],
 
[[ 0, 0, 0, 0, 0],
[ 0, 0, 0, 0, 0],
[ 0, 0, 0, 0, 0],
[ 0, 0, 0, 0, 0]]]], dtype=int8)
>>>

Racket[edit]

Racket has multi-dimensional arrays as part of the standard math library. Instead of repeating the whole thing here, see the quick start page of the documentation, which describes all of what's asked here.

REXX[edit]

REXX supports multi-dimension (stemmed) arrays and the limit of the number of dimension varies with individual REXX interpreters,   most (but not all) will probably be limited by the largest allowable clause length or source-line length, the smallest of which is 250 characters   (included the periods),   which would be around 125 dimensions.

The REXX language has something called   stemmed arrays.

For instance, for a stemmed array named   antenna   with two dimensions,   to reference the element   2, 0   (and set that value to the variable   g),   one could code   (note the use of periods after the variable name and also between the indices to the array dimensions:

g = antenna.2.0

Memory allocation for stemmed arrays is not optimized, and the array elements are not contiguous.

The index to the array dimensions may be any integer, and indeed, even non-numeric   (and possible non-viewable).

    a = '"'                    /*set variable  A  to a quote character  ["].  */
b = '~' /*set variable B to a tilde character [~]. */
 
h.a.b = '+++' /*set variable H.".~ to three pluses [+++]. */

The REXX language does not do bounds checking, but if     signal on novalue     is in effect, it can be used to trap (read) references of array elements that haven't been assigned a value (at the time of the read reference).

/*REXX program shows how to assign/show values of a  multidimensional  array. */
/*REXX arrays can start anywhere. */
y.=0 /*set all values of Y array to 0. */
/* [โ†‘] bounds need not be specified. */
#=0 /*the count for the number of SAYs. */
y.4.3.2.0= 3**7 /*set penultimate element to 2187 */
do i=0 for 5
do j=0 for 4
do k=0 for 3
do m=0 for 2; #=#+1 /*bump the SAY counter.*/
say 'y.'i"."j'.'k"."m '=' y.i.j.k.m
end /*m*/
end /*k*/
end /*j*/
end /*i*/
say
say '# of elements displayed = ' # /*should be 5 * 4 * 3 * 2 or 5! */
exit /*stick a fork in it, we're all done. */
 
/*other versions of the 1st (REXX) SAY */
say 'y.' || i || . || k || . || m '=' y.i.j.k.m
say 'y.'||i||.||k||.||m '=' y.i.j.k.m
say 'y.'i||.||k||.||m '=' y.i.j.k.m

output:

y.0.0.0.0 = 0
y.0.0.0.1 = 0
y.0.0.1.0 = 0
y.0.0.1.1 = 0
y.0.0.2.0 = 0
y.0.0.2.1 = 0
y.0.1.0.0 = 0
y.0.1.0.1 = 0
  ยท
  ยท
  ยท
y.4.2.2.1 = 0
y.4.3.0.0 = 0
y.4.3.0.1 = 0
y.4.3.1.0 = 0
y.4.3.1.1 = 0
y.4.3.2.0 = 2187
y.4.3.2.1 = 0

# of elements displayed =  120

Tcl[edit]

In Tcl, arrays are associative maps and lists are closer to what other languages name "arrays". Either can be used for multidimensional data, but the implementations (and implications!) are quite different.

It's worth briefly discussing both here. Since lists are closer to the theme of this page, they come first.

lists[edit]

Multi-dimensional lists are easily handled by nesting. Let's define a helper proc to construct such lists using lrepeat:

proc multilist {value args} {
set res $value
foreach dim [lreverse $args] {
set res [lrepeat $dim $res]
}
return $res
}
Output:
% multilist x 2
x x
% multilist x 2 3
{x x x} {x x x}
% multilist x 2 3 4
{{x x x x} {x x x x} {x x x x}} {{x x x x} {x x x x} {x x x x}}
 

Both lset and lindex know how to access multi-dimensional lists:

% set ml [multilist 0 2 3 4]
{{0 0 0 0} {0 0 0 0} {0 0 0 0}} {{0 0 0 0} {0 0 0 0} {0 0 0 0}}
% lset ml 1 2 3 11
{{0 0 0 0} {0 0 0 0} {0 0 0 0}} {{0 0 0 0} {0 0 0 0} {0 0 0 11}}
% lset ml 1 1 4 12
{{0 0 0 0} {0 0 0 0} {0 0 0 0}} {{0 0 0 0} {0 0 0 0 12} {0 0 0 11}}
% lindex $ml 1 2 3
11

lsort and lsearch are among other useful commands that support nested lists.

arrays[edit]

Tcl arrays are collections of variables, not collections of values: thus they cannot nest. But since keys are simply strings, multidimensional data can be kept like this:

% array set x {
0,0 a
0,1 b
1,0 c
1,1 d
}
% parray x
x(0,0) = a
x(0,1) = b
x(1,0) = c
x(1,1) = d
% puts $x(0,1)
b
% set a 0
1
% set b 1
% puts $x($b,$a)
c
% set $x($b,$a) "not c"
not c
% parray x $b,$a
x(1,0) = not c

Such an array can also be "sliced" with the array command:

% array get x 1,*
1,0 c 1,1 d
% array names x 0,*
0,0 0,1

Note however that the order in which elements are returned from these operations is undefined! The last command might return {0,0 0,1} or {0,1 0,0} depending on how its keys are hashed, which is not under the programmer's control.

Using arrays like this for ordered data is suboptimal for this reason, and because iterating over elements is cumbersome. But it's a common technique for records:

% array set players {
1,name Fido
1,score 0
1,colour green
 
2,name Scratchy
2,score 99
2,colour pink
}
% foreach player {1 2} {
puts "$players($player,name) is $players($player,colour) and has $players($player,score) points"
}
Fido is green and has 0 points
Scratchy is pink and has 99 points
%

The interested reader should also be aware of the difference between arrays and dictionaries, and know that the latter are often preferred for record-like structures.