Operand space

Iterative algorithms are recommended for some linear systems Ax = b as an alternative to direct algorithms. An iteration usually amounts to one or two multiplications of the matrix A by a vector and to a few linear operations with vectors. If A is sparse, small storage space suffices. This is a major advantage of iterative methods where the direct methods have large fill-in. Furthermore, with appropriate data structures, arithmetic operations are actually performed only where both operands are nonzeros then, D A) or 2D A) flops per iteration and D(A) + 2n units of storage space suffice, where D(A) denotes the number of nonzeros in A. Finally, iterative methods allow implicit symmetrization, when the iteration applies to the symmetrized system A Ax = A b without explicit evaluation of A A, which would have replaced A by less sparse matrix A A. 

Modem microprocessors distinguish different operand bit sizes p as early as in the decode stage. Usually these have sizes in apoweroftwo. Hence n-p bits ofann-bit operation unit are not used at all. This paper shows how to exploit this space to execute additions in parallel or redundantly. 

By definition, the dimension of the DG of an algorithm is identical to the index space dimension. In particular, the bit-level algorithm of equation (14) should be described by a W-D DG. However, the construction of the multidimensional DG is avoided by using combinatorial logic to compute the values of Zk -)j thus reducing the problem to multiple-operand binary addition, which can be represented by a 2-D DG. Moreover, in this DG, the properties of the target architecture may be embodied. 

It is important to understand Mendeleevs modus operand regarding the placement of elements in the periodic system if we are to appreciate the motivation for many of his corrections of atomic weights and his predictions of unknown elements. Mendeleev considered a number of criteria in addition to atomic weight ordering, such as family resemblance among elements and the concept of the single occupancy of elements in any space in the periodic table. However, all these criteria could be, and often were, overridden as individual cases presented themselves to him. 

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