Vector operand

This example shows that vector operands can be used in expressions. The four bits of A are and ed with the four bits of B, the result of which is or ed with the four bits of C. The result is assigned (starting with the rightmost bit) to the target net RFile. 

In these equations = (1 — R RR ) is a projection operator that removes from its vector operand the component parallel to R. (The symbol is used because Q is the conventional notation for this type of projection operator it should not be confused with the multipole moment operators.)... 

Banked Memory. Another characteristic of many vector supercomputers is banked memory. The main memory is usually divided into a small number of electronically separate banks. A given memory bank can absorb or supply operands at a much slower rate than the rate at which the central processing unit (CPU) can produce or use data. If the data can be spread across multiple memory banks, the effective memory bandwidth, or rate at which memory can absorb or supply data, is increased. For example, if a single memory bank can supply one operand every 16 clock cycles, then 16 memory banks would enable the entire memory subsystem to deflver one operand per clock cycle, assuming that the data come sequentially from different memory banks. 

The pipelines achieve their highest sustained flow rate in vector mode. In this situation, a single instruction is interpreted and a single operation performed on many pairs of operands. For example, if A, B, and C are arrays, only one vector instruction is required for computing the sum C(I) = A(I) + B(I), 1=1,100. The A and B values stream continuously into the pipes, additions are performed in discrete steps and results flow back to CM at the rate of one per CPU clock cycle per pipe. This is in contrast to scalar execution which requires five instructions to be executed 100 times 2 fetches, 1 add, 1 store, and 1 counter incremention. 

The vector instruction is useful since it allows a CPU to do one operation on a large set of data previously each separate set of data (two operands) required decoding an instruction and then fetching the data finally performing the required operation. Even in the pipeline mode, it has not been possible to achieve the theoretical efficiency of a computer, namely, one result per machine cycle, without the vector capability. Basically, the vector instruction tells the CPU where itwill have to go to get data, and how much data will be required. 

Here is another example where the operands of a logical operator are vectors. In such a case, a series of logic gates to cover the range of the vector are produced. 

It may be worth while to review the different kinds of multiplicity involved in the symbols appearing in Eqs. (3-6) and (3-15). Equation (3-6) is merely a shorthand way of writing the material balance for each of the key components, each term being a row matrix having as many elements as there are independent reactions. The equation asserts that when these matrices are combined as indicated, each element in the resulting matrix will be zero. The elements in the first two terms are obtained by vector differential operation, but the elements are scalars. Equation (3-15), on the other hand, is a scalar equation, from the point of view of both vector analysis and matrix algebra, although some of its terms involve vector operations and matrix products. No account need be taken of the interrelation of the vectors and matrices in these equations, but the order of vector differential operators and their operands as well as of all matrix products must be observed. 

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. 

FIFOs as operand memories are well suited for vector/matrix operations. 

Although vector processors have existed that loaded their operands directly from memory and stored the results... 

Because of the generic nature of Fig. 1, no details of the interconnection between the VPU and the memory are shown. Still, these details are very important for the effective speed of a vector operation when the bandwidth between the memory and the VPU is too small, it is not possible to take full advantage of the VPU because it has to wait for operands and/or has to wait before it can store results. When the ratio of arithmetic-to-load/store operations is not high enough to compensate for such situations, severe performance losses may be incurred. The influence of the number of load/store paths for the dyadic vector operation c = a- -b (a, b, and c vectors) is depicted in Fig. 2. 

VHDL defines logical operators for the basic types of BIT and BOOLEAN or any one-dimensional array of either type, such as BIT VECTOR. The result generated by each operator is of the same type and has the same length as the operands supplied. Hence, the operands must be identical in type and length to allow each matching element to be compared. The synthesizer should detect any errors of this t)rpe, but they may go undetected until run time when simulating the description. The operators are listed in Table 4.1. 

Whenever the + operator is encountered in a description and the operands are both of type BIT VECTOR, the overloaded function will be used. Note that all operators require two operands except the sign operators, which take one. An operator cannot be overloaded witfi a procedure. When using an overloaded operator it can be referenced in two ways - with operator or function notation as shown below. 

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