Vectorized executions

The use of the space computing proved reliable and efficient. It was a straightforward process to add computers to the SORCER service Cloud as needed during the course of the two optimization studies. This flexibility proved valuable as the number of computers available varied from day-to-day. Parallelization of var-responses in parametric models with exertion space coordination resulted in significant savings. In this case it reduced the computational time to perform the optimization from 24 h to proximately 2 h [47]. The reduction is achieved mainly by the parallelization of SORCER parametric models for each parametric response (a vector of var values) and parallelization of var exertion evaluators (for each vector) executed using exertion space as illustrated in Fig. 4.7. 

Vectorization A specialized form of parallelization that tries to expose computations suitable for SIMD, or vector, execution. 

As shown from numbers reported above, all routines restructured for vectorizing (independently upon whether they are run in scalar or in vector mode) are more efficient than POTJKL. This means that a DO loop reduction and, in general, a reorganization of the code for vector execution on a supercomputer frequently pays off also in terms of scalar speed-up. Moreover, times measured for POTPOW on the IBM 3090 show that this may be true even... 

Here, since the measurements were done in an integral reactor, calculation must start with the Conversion vs. Temperature function. For an example see Appendix G. Calculation of kinetic constants starts with listed conversion values as vX and corresponding temperatures as vT in array forms. The Vectorize operator of Mathcad 6 tells the program to use the operators and functions with their scalar meanings, element by element. This way, operations that are usually illegal with vectors can be executed and a new vector formed. The v in these expressions indicates a vector. 

An unpaired electron executes a spin about its own axis. The mechanical spin momentum is related to a spin vector which specifies the direction of the rotation axis and the magnitude of the momentum. The spin vector s of an electron has an exactly defined magnitude ... 

Any finite interpretation is necessarily recursive. There are only a finite number of function letters and predicate letters in P and so for each finite domain D only a finite number of possible assignments of functions from iP to D or eP to 0,1. We can recursively enumerate all finite interpretations. A program must loop if it ever enters the sane statement twice with all values specified alike. If finite domain D of interpretation I has d objects and P has n statements and m variables of any kind, then any execution sequence under I with more than ncP steps must twice enter the same statement with the same specification of all variables and hence must represent an infinite loop. Hence for each input vector a computation (P,I,a) diverges if and only if it fails to halt within ndm steps. So for each finite interpretation we can decide whether P baits for some inputs or all inputs. Thus (5) and (6) are partially decidable. 

Interestingly, the spectral transform Lanczos algorithm can be made more efficient if the filtering is not executed to the fullest extent. This can be achieved by truncating the Chebyshev expansion of the filter,76,81 or by terminating the recursive linear equation solver prematurely.82 In doing so, the number of vector-matrix multiplications can be reduced substantially. 

Because of round-off errors, symmetry contamination is often present even when the initial vector is properly symmetrized. To circumvent this problem, an effective scheme to reinforce the symmetry at every Lanczos recursion step has been proposed independently by Chen and Guo100 and by Wang and Carrington.195 Specifically, the Lanczos recursion is executed with symmetry-adapted vectors, but the matrix-vector multiplication is performed at every Lanczos step with the unsymmetrized vector. In other words, the symmetrized vectors are combined just before the operation Hq, and the resultant vector is symmetrized using the projection operators ... 

For a polyatomic reactant with many degrees of freedom the numerical calculations required to execute the program outlined above can easily achieve a scale that is impossible to handle even with a vectorized parallel processor supercomputer. The simplest approximation that reduces the scale of the numerical calculations is the neglect of some subset of the internal molecular motions, but this approximation usually leads to considerable error. A more sophisticated and intuitively reasonable approximation [72, 73] is to reduce the system dimensionality by placing constraints on the values of the internal molecular coordinates (instead of omitting them from the analysis). 

The perceptron network is the simplest of these three methods, in that its execution typically involves the simple multiplication of class-specific weight vectors to the analytical profile, followed by a hard limit function that assigns either 1 or 0 to the output (to indicate membership, or no membership, to a specific class). Such networks are best suited for applications where the classes are linearly separable in the classification space. 

As we realise, the matrix (XTX) must not be singular, otherwise the inversion process is not executable. If the matrix is nearly singular we will get more or less biased results for the coefficient vector b. 

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