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. 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.
Indeed, the very suggestiveness of a notation may make it seem harder to learn because of the many properties it suggests for explorations. 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. it is important to distinguish the difficulty of describing and of learning a piece of notation from the difficulty of mastering its implications. 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. 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. Iverson described the rationale behind array programming (actually referring to APL) as follows:
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. The fundamental idea behind array programming is that operations apply at once to an entire set of values. 4 Mathematical reasoning and language notation.The level of concision can be dramatic in certain cases: it is not uncommon to find array programming language one-liners that require several pages of object-oriented code. Array programming primitives concisely express broad ideas about data manipulation.
In these languages, an operation that operates on entire arrays can be called a vectorized operation, regardless of whether it is executed on a vector processor, which implements vector instructions. These include APL, J, Fortran 90, MATLAB, Analytica, TK Solver (as lists), Octave, R, Cilk Plus, Julia, Perl Data Language (PDL), and the NumPy extension to Python. Modern programming languages that support array programming (also known as vector or multidimensional languages) have been engineered specifically to generalize operations on scalars to apply transparently to vectors, matrices, and higher-dimensional arrays. Such solutions are commonly used in scientific and engineering settings. In computer science, array programming refers to solutions which allow the application of operations to an entire set of values at once.
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