Operation count

So far the theory has been completely general. In order to apply perturbation theory to the calculation of correlation energy, the unperturbed Hamilton operator must be selected. The most common choice is to take this as a sum over Fock operators, leading to Mdller-Plesset (MP) perturbation theory. The sum of Fock operators counts the (average) electron-electron repulsion twice (eq. (3.43)), and the perturbation becomes... 

The operation count for an n x n matrix with m right-hand sides is... 

MATLAB s polynomial-roots finder roots does not handle repeated or clustered roots very well, but otherwise it is the best 0(n3) root finder available. Note that an operations count of 0(nP) for an algorithm signifies that the algorithm performs K n additions and multiplications (for some algorithm specific constants K and j, but depending on n) to obtain its output from n input data. Most of the polynomial-root finders of the last century unfortunately were even slower 0(n4) algorithms and all in all much too slow and inaccurate. 

Rewrite the code quadnonequicol. m to run in 0(N) time as detailed on p. 376. To achieve this fast operations count, you need to implement all matrix times matrix or matrix times vector products in O(N) time by recognizing the sparsity of the matrices Df x, Dfiy, D/o.x, and Df-i/j that make up Df(z0id)-... 

Truncated Newton methods can be competitive only with preconditioning. Thus, the operation count for obtaining p in TN reflects IT inner PCG iterations per Newton step. Each such inner iteration involves the following operations an Hd multiplication (an, for an additional gradient evaluation in this finite-difference approximation [58]) calculation of the PCG vectors and scalars (7 , Algorithm [A4]) and numerical solution of Mz = r by forward and backward substitution (0(1), see [61]). 

Each element in b must be multiplied by the relevant nonzero matrix elements, leading to an overall operation count on the order of... 

The Knowles-Handy approach was expected to be very efficient on vector supercomputers, and indeed it enabled many important full Cl benchmark calculations.15 Nevertheless, one can see that this algorithm does more work than is strictly necessary. Equation (106) demonstrates that the operation count for the time-consuming step is approximately j/Vde(n4, where n is the number of orbitals and JV is the number of interacting intermediate states,... 

In order to avoid the unnecessarily large operation counts in the full Cl algorithm of Knowles and Handy, Olsen et al. abandoned the explicit use of a complete set of intermediate states and returned to some of Handy s original (1980) ideas44 concerning string-driven full Cl approaches. 

When Na = Np, the overall operation count is thus approximately... 

Finally, we discuss queries with aggregate operators COUNT, SUM, AVG, MAX, and MIN based on results from Gal et al. (2009). We consider three common extensions to semantics with aggregates and probabilistic information the range semantics returns the range of the aggregate (i.e., the minimum and the maximum value) the expected-value semantics returns the expected value of the aggregate, and the distribution semantics returns all possible values with their probabilities. Note that the answer under the former two semantics can be derived from that under the last semantics in other words, the distribution semantics is the richest one. We next formally define the three semantics. 

The feasibility of continuous operation counts among the major advantages of the HiPCo-process over most alternative methods. The catalytic decomposition of carbon monoxide can thus be considered as a potential way of scaling up SWNT-production from a few grams to kilograms or even tons. 

This form of the Hamiltonian shows explicitly the couplings between wave functions with different Mk values and makes possible to factorize occupation vectors in alpha- and beta-strings like done in non-relativistic Cl theory. The difference with non-relativistic theory is that calculations are not restricted to one value of Mk. Applied without further approximation the formalism gives therefore no dramatic reduction in operation count over the symmetry-adapted unrestricted scheme described in the previous section. An advantage of the formalism is, however, that it facilitates incorporation of the relativistic scheme in non-relativistic Cl or MCSCF implementations [35] and that the scheme gives a natural subdivision of the full Cl matrix. 

This means that the operator V is diagonal in the grid representation. The application of local operators reduces the operation count of the operator mapping from 0(N2H) to 0 NS). 

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