Array programming

In computer science, array programming refers to solutions that allow the application of operations to an entire set of values at once. Such solutions are commonly used in scientific and engineering settings.

Modern programming languages that support array programming (also known as

The fundamental idea behind array programming is that operations apply at once to an entire set of values. This makes it a high-level programming model as it allows the programmer to think and operate on whole aggregates of data, without having to resort to explicit loops of individual scalar operations.

Kenneth E. Iverson described the rationale behind array programming (actually referring to APL) as follows:^[2]

most programming languages are decidedly inferior to mathematical notation and are little used as tools of thought in ways that would be considered significant by, say, an applied mathematician.

The thesis is that the advantages of executability and universality found in programming languages can be effectively combined, in a single coherent language, with the advantages offered by mathematical notation. it is important to distinguish the difficulty of describing and of learning a piece of notation from the difficulty of mastering its implications. For example, learning the rules for computing a matrix product is easy, but a mastery of its implications (such as its associativity, its distributivity over addition, and its ability to represent linear functions and geometric operations) is a different and much more difficult matter.

Indeed, the very suggestiveness of a notation may make it seem harder to learn because of the many properties it suggests for explorations.

[...]

Users of computers and programming languages are often concerned primarily with the efficiency of execution of algorithms, and might, therefore, summarily dismiss many of the algorithms presented here. Such dismissal would be short-sighted since a clear statement of an algorithm can usually be used as a basis from which one may easily derive a more efficient algorithm.

The basis behind array programming and thinking is to find and exploit the properties of data where individual elements are similar or adjacent. Unlike object orientation which implicitly breaks down data to its constituent parts (or

Uses

Array programming is very well suited to

In scalar languages such as C and Pascal, operations apply only to single values, so a+b expresses the addition of two numbers. In such languages, adding one array to another requires indexing and looping, the coding of which is tedious.

for (i = 0; i < n; i++)
    for (j = 0; j < n; j++)
        a[i][j] += b[i][j];

In array-based languages, for example in Fortran, the nested for-loop above can be written in array-format in one line,

a = a + b

or alternatively, to emphasize the array nature of the objects,

a(:,:) = a(:,:) + b(:,:)

While scalar languages like C do not have native array programming elements as part of the language proper, this does not mean programs written in these languages never take advantage of the underlying techniques of vectorization (i.e., utilizing a CPU's vector-based instructions if it has them or by using multiple CPU cores). Some C compilers like GCC at some optimization levels detect and vectorize sections of code that its heuristics determine would benefit from it. Another approach is given by the OpenMP API, which allows one to parallelize applicable sections of code by taking advantage of multiple CPU cores.

Array languages

In array languages, operations are generalized to apply to both scalars and arrays. Thus, a+b expresses the sum of two scalars if a and b are scalars, or the sum of two arrays if they are arrays.

An array language simplifies programming but possibly at a cost known as the abstraction penalty.

Ada

The previous C code would become the following in the Ada language,^[6] which supports array-programming syntax.

A := A + B;

APL

APL uses single character Unicode symbols with no syntactic sugar.

A ← A + B

This operation works on arrays of any rank (including rank 0), and on a scalar and an array. Dyalog APL extends the original language with augmented assignments:

A +← B

Analytica

Analytica provides the same economy of expression as Ada.

A := A + B;

BASIC

Dartmouth BASIC had MAT statements for matrix and array manipulation in its third edition (1966).

DIM A(4),B(4),C(4)
MAT A = 1
MAT B = 2 * A
MAT C = A + B
MAT PRINT A,B,C

Mata

Stata's matrix programming language Mata supports array programming. Below, we illustrate addition, multiplication, addition of a matrix and a scalar, element by element multiplication, subscripting, and one of Mata's many inverse matrix functions.

. mata:

: A = (1,2,3) \(4,5,6)

: A
       1   2   3
    +-------------+
  1 |  1   2   3  |
  2 |  4   5   6  |
    +-------------+

: B = (2..4) \(1..3)

: B
       1   2   3
    +-------------+
  1 |  2   3   4  |
  2 |  1   2   3  |
    +-------------+

: C = J(3,2,1)           // A 3 by 2 matrix of ones

: C
       1   2
    +---------+
  1 |  1   1  |
  2 |  1   1  |
  3 |  1   1  |
    +---------+

: D = A + B

: D
       1   2   3
    +-------------+
  1 |  3   5   7  |
  2 |  5   7   9  |
    +-------------+

: E = A*C

: E
        1    2
    +-----------+
  1 |   6    6  |
  2 |  15   15  |
    +-----------+

: F = A:*B

: F
        1    2    3
    +----------------+
  1 |   2    6   12  |
  2 |   4   10   18  |
    +----------------+

: G = E :+ 3

: G
        1    2
    +-----------+
  1 |   9    9  |
  2 |  18   18  |
    +-----------+

: H = F[(2\1), (1, 2)]    // Subscripting to get a submatrix of F and

:                         // switch row 1 and 2
: H
        1    2
    +-----------+
  1 |   4   10  |
  2 |   2    6  |
    +-----------+

: I = invsym(F'*F)        // Generalized inverse (F*F^(-1)F=F) of a

:                         // symmetric positive semi-definite matrix
: I
[symmetric]
                 1             2             3
    +-------------------------------------------+
  1 |            0                              |
  2 |            0          3.25                |
  3 |            0         -1.75   .9444444444  |
    +-------------------------------------------+

: end

MATLAB

The implementation in MATLAB allows the same economy allowed by using the Fortran language.

A = A + B;

A variant of the MATLAB language is the GNU Octave language, which extends the original language with augmented assignments:

A += B;

Both MATLAB and GNU Octave natively support

The Nial example of the inner product of two arrays can be implemented using the native matrix multiplication operator. If a is a row vector of size [1 n] and b is a corresponding column vector of size [n 1].

a * b;

By contrast, the

a .* b;

The inner product between two matrices having the same number of elements can be implemented with the auxiliary operator (:), which reshapes a given matrix into a column vector, and the transpose operator ':

A(:)' * B(:);

rasql

The rasdaman query language is a database-oriented array-programming language. For example, two arrays could be added with the following query:

SELECT A + B
FROM   A, B

R

The R language supports array paradigm by default. The following example illustrates a process of multiplication of two matrices followed by an addition of a scalar (which is, in fact, a one-element vector) and a vector:

> A <- matrix(1:6, nrow=2)                             # !!this has nrow=2 ... and A has 2 rows
> A
     [,1] [,2] [,3]
[1,]    1    3    5
[2,]    2    4    6
> B <- t( matrix(6:1, nrow=2) )  # t() is a transpose operator                           !!this has nrow=2 ... and B has 3 rows --- a clear contradiction to the definition of A
> B
     [,1] [,2]
[1,]    6    5
[2,]    4    3
[3,]    2    1
> C <- A %*% B
> C
     [,1] [,2]
[1,]   28   19
[2,]   40   28
> D <- C + 1
> D
     [,1] [,2]
[1,]   29   20
[2,]   41   29
> D + c(1, 1)  # c() creates a vector
     [,1] [,2]
[1,]   30   21
[2,]   42   30

Mathematical reasoning and language notation

The matrix left-division operator concisely expresses some semantic properties of matrices. As in the scalar equivalent, if the (

where == is the equivalence relational operator. The previous statements are also valid MATLAB expressions if the third one is executed before the others (numerical comparisons may be false because of round-off errors).

If the system is overdetermined – so that A has more rows than columns – the pseudoinverse A+ (in MATLAB and GNU Octave languages: pinv(A)) can replace the inverse A−1, as follows:

pinv(A) *(A * x)==pinv(A) * (b)

(pinv(A) * A)* x ==pinv(A) * b       (matrix-multiplication associativity)

x = pinv(A) * b

However, these solutions are neither the most concise ones (e.g. still remains the need to notationally differentiate overdetermined systems) nor the most computationally efficient. The latter point is easy to understand when considering again the scalar equivalent a * x = b, for which the solution x = a^-1 * b would require two operations instead of the more efficient x = b / a. The problem is that generally matrix multiplications are not

(a * x)/ a ==b / a

(x * a)/ a ==b / a       (commutativity does not hold for matrices!)

x * (a / a)==b / a       (associativity also holds for matrices)

x = b / a

The MATLAB language introduces the left-division operator \ to maintain the essential part of the analogy with the scalar case, therefore simplifying the mathematical reasoning and preserving the conciseness:

A \ (A * x)==A \ b

(A \ A)* x ==A \ b       (associativity also holds for matrices, commutativity is no more required)

x = A \ b

This is not only an example of terse array programming from the coding point of view but also from the computational efficiency perspective, which in several array programming languages benefits from quite efficient linear algebra libraries such as ATLAS or LAPACK.^[9]

Returning to the previous quotation of Iverson, the rationale behind it should now be evident:

it is important to distinguish the difficulty of describing and of learning a piece of notation from the difficulty of mastering its implications. For example, learning the rules for computing a matrix product is easy, but a mastery of its implications (such as its associativity, its distributivity over addition, and its ability to represent linear functions and geometric operations) is a different and much more difficult matter. Indeed, the very suggestiveness of a notation may make it seem harder to learn because of the many properties it suggests for explorations.

Third-party libraries

The use of specialized and efficient libraries to provide more terse abstractions is also common in other programming languages. In C++ several linear algebra libraries exploit the language's ability to overload operators. In some cases a very terse abstraction in those languages is explicitly influenced by the array programming paradigm, as the NumPy extension library to Python, Armadillo and Blitz++ libraries do.^[10][11]

See also

• Array slicing
• List of array programming languages

References