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We are all familiar with the fact that if B is two miles from A, and if C is two miles from B, then C is not necessarily four miles from A. Only in very special circumstances are distances compounded according to the ordinary artimetical law of addition. Actually, there are many other entities which behave in this way as distances rather than as ordinary numbers; this motiviates the study of th...

We describe a subset of the level{1, level{2, and level{3 BLAS implemented for each node of the Connection Machine system CM{200. The routines, collectively called LBLAS, have interfaces consistent with languages with an array syntax such as Fortran 90. One novel feature, important for distributed memory architectures, is the capability of performing computations on multiple instances of object...

A linear process is a system of events and states related by an inner product, on which are defined the behaviorally motivated operations of tensor product or orthocurrence, sum or concurrence, sequence, and choice. Linear process algebra or LPA is the theory of this framework. LPA resembles Girard’s linear logic with the differences attributable to its focus on behavior instead of proof. As wi...

The main general reference for this and the next two lectures is a recent book [6] and its preliminary version [5] which is freely available on the web. 1 Why iterative methods? As we already know, there are two major classes of methods to solve linear systems: direct methods (like LU factorization, Cholesky factorization, etc.) and iterative methods. We underline the following properties of th...

Definition 1.1. Let K be a field. (V,+, ·) is a vector space over K, if (V,+) is an Abelian group, and · : K × V → V (called scalar multiplication) is distributive, associative and multiplication by 1 is the identity on V . This compact definition unwinds to give us the following ten(!) axioms: (1) + : V × V → V (2) u+ (v + w) = (u+ v) + w. (3) u+ v = v + u. (4) ∃0 ∈ V such that v + 0 = 0 + v =...

aij is called the (i, j)th element of A. Note that a special case of matrix is a vector where either m = 1 or n = 1. If m = 1 and n > 1, then it is called a row vector. If m > 1 and n = 1, then it is called a column vector. A vector has a nice geometric interpretation where the direction and length of the vector are determined by its elements. For example, the vector A = [1 3] has the opposite ...