An Algorithm for Translating Boolean Expressions
Journal of the ACM (JACM)
Report on the algorithmic language ALGOL 60
Communications of the ACM
A programming language
Index ranges for matrix calculi
Communications of the ACM
On the time required for a sequence of matrix products
Communications of the ACM
Some complete calculi for matrices
Communications of the ACM
A proposal for definitions in ALGOL
Communications of the ACM
EQL: A language for numerical computation
Computer Languages
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It is unfortunate that almost all of the presently used algebraic languages do not provide the capability of linear algebra. Operations such as the inner product of vectors, the product of two matrices, and the multiplication of a matrix by a scaler must inevitably be written out in detail in terms of the individual components. The reasons usually given for avoiding linear algebra in these languages are (1) the difficulties which would arise in scanning linear algebraic expressions, and (2) the uncertainty involved as to the amount of temporary storage needed during the evaluation of linear algebraic expressions when the program is executed. The purpose of this paper is to show how these two types of difficulties can be overcome. Although suggestions have been made for even further increasing the general capability of ALGOL such as including the ability to form a matrix from a collection of vectors, we shall be content here to consider the ordinary operations of linear algebra. Even if this much becomes available in algebraic languages, considerable progress will have been made. The following remarks constitute a suggestion for the addition to ALGOL of linear algebraic expressions.