Operand

An operator is a mathematical instruction. For example, the operator d/dx is the instruction to differentiate once with respect to a . Matrices in general, and the matrix R of Chapter 6 in particular, are operators. The matrix R is an instruction to rotate a part of the operand matrix through a certain angle, 0 as in Eq. (6-62). 

Thus the Jacobi procedure, by making many rotations of the elements of the operand matrix, ultimately arrives at the operator matrix that diagonalizes it. Mathematically, we can imagine one operator matr ix that would have diagonalized the operand matr ix R, all in one step... 

Tipping pan and horizontal filters are also used for leaching the modus operand of the Rotocel extractor resembles that of a tipping pan filter, although the details of its design differ slightly. 

A pipelined floating-point multiply unit might accomplish a floating-point multiply by performing four independent suboperations, labeled a, b, c, and d, on the operands. The suboperations can be envisioned as the four workers on a four-person assembly line. The floating-point multiply pipeline could accept a new set of operands every clock cycle. The pipeline occupancy of this code fragment would look like... 

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. 

Other Performance Considerations. Even if a program allows main memory to supply operands at peak rate, it may not be fast enough to keep the CPU operating at its peak rate. Consider the general SAXPY... 

Operator Description Operand(s) Type Result Type... 

It was shown in [ReaR68] that the operation N A B) which is required by the superposition theorem is particularly simple if the operands A and B are the symmetric functions known as -functions (or Schur functions). In fact, for any two -functions X and /z ... 

Since the integral is over time t, the resulting transform no longer depends on t, but instead is a function of the variable s which is introduced in the operand. Hence, the Laplace transform maps the function X(f) from the time domain into the s-domain. For this reason we will use the symbol when referring to Lap X t). To some extent, the variable s can be compared with the one which appears in the Fourier transform of periodic functions of time t (Section 40.3). While the Fourier domain can be associated with frequency, there is no obvious physical analogy for the Laplace domain. The Laplace transform plays an important role in the study of linear systems that often arise in mechanical, electrical and chemical kinetic systems. In particular, their interest lies in the transformation of linear differential equations with respect to time t into equations that only involve simple functions of s, such as polynomials, rational functions, etc. The latter are solved easily and the results can be transformed back to the original time domain. 

Equations 69-12, while derived from calculus, can be converted in matrix notation as follows (recall that in matrix multiplication, the rows of the left-hand operand are multiplied by the columns of the right-hand operand) ... 

Some expressions may have undefined values—for example, attributes of null, daft arithmetic expressions such as 0/0, or parameterized attributes whose precondition is false. Generally, an expression is undefined if any of its subexpressions is undefined. However, some operators do not depend on one of their inputs under certain circumstances 0 n is well defined even if you don t know n so is n 0. The same applies to (true b) and (false b), again no matter what the order of the operands. (This works no matter which way you write the operands—we re not writing a program.)... 

The model shows exactly what is expected of a spreadsheet that it maintain the arithmetic relationships between the cells no matter how they are altered. In particular, the invariant Sum ... says that the value of every Sum cell is always the addition of its two operands. 

User, ci Cell I addOperand(ci2 Cell I) post ci2 is appended to ci.sumpart.operands... 

No problem, I answer. All the information mentioned by the requirements is there— it s just that some of the names have changed. If you want to get the lef tand right operands of any Sum, look at the first and second items of my operands array. But because the requirements don t call for any operations that directly ask for the left or right, I haven t bothered to write them. Nevertheless, anyone using my code would see it behaving as expected from the spec. 

These abstractions happen to be particularly easy the correspondence to the spec model is not very far removed. Others are more complex. But it doesn t matter if an abstraction function is hopelessly inefficient It need only demonstrate that the information is in there somewhere. Nor does it matter if there is more information in the code than in the model. I can store more than two operands, although readers of the official spec won t use more than two. 

Sum impU.container.value = impl1.operands[1].abs.content.value... 

This is an invariant we could execute in the implementation in debug mode. We could make one further simplification to make more clear what is happening. Evaluating the expressions operands[n].abs will give a C qlland the retrievals say that C ell content.value = impll. value. So we could rewrite the invariant ... 

SumJ container.value = operands[1].abs.impl1.value... 

But that s still a bit long-winded. Something such as abs. impll is running one way up a one-to-one association only to come back down it again—the two cancel out. So we get SumJ container.value = operands[1].value + operands[2].value... 

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