Direct algorithm

Lengsfield B H III 1980 General second-order MC-SCF theory a density matrix directed algorithm J. Chem. Phys. 73 382... 

Here is our plot. It is a log-log plot, so we can estimate the exponent for N by computing the slope of each line. The actual scaling for the direct algorithm is more likeN - than N ... 

Evaluate the integrals. In a conventional algorithm, they are stored on disk and read in for each iteration. In a direct algorithm, integrals are computed a few at a time as the Fock matrix is formed. 

The above is an example of how direct algorithms may be formulated for methods involving electron correlation. It illustrates that it is not as straightforward to apply direct methods at the correlated level as at the SCF level. However, the steady increase in CPU performance, and especially the evolution of multiprocessor machines, favours direct (and semi-direct where some intermediate results are stored on disk) algorithms. Recently direct methods have also been implemented at the coupled cluster level. 

As a result, one may justifiably extrapolate the (T) contribution from smaller basis sets than its CCSD counterpart in W1 theory, we extrapolate from the small and medium basis sets, and in W2 theory from the medium and large basis sets. This means that the most extensive basis sets in the calculations, namely large in W1 theory and extra large in W2 theory only require CCSD calculations, which are both much less expensive than CCSD(T) and much more amenable to direct algorithms such as those described in Refs. 40-41. 

RI methods also can be applied to electron propagator calculations in the quasiparticle approximation [28], Savings in storage are dramatic. When compared to semi-direct algorithms [21, 22], the pre-factor for RI results in a considerable speed-up. The implementation is also much simpler. Test calculations with different approximations and basis sets show the reliability of this approach. 

M.J. Frisch et al., Semi-direct algorithms for the MP2 energy and gradient Chem. Phys. Lett. 

In test programs, where the numerical solution is compared with the analytical solution, the latter often involves the error function erf or the complementary error function erfc. The latter could be obtained simply by subtracting erf from unity but a better approximation is obtained by the direct algorithm. The two routines, ERF and ERFC, were given to the author by a colleague, who probably obtained them from an IBM collection. They have been adapted to Fortran 90/95 by the author, and coupled to the above module. The comments in capitals are the original comments. 

P PjV 2 (pp4>q)) [ qj / ) reading/writing to disk, it is desirable also to have direct algorithms for electron correlation method. Direct in this context means that the integrals are calculated as needed and then discarded. The need for integrals over MOs instead of AOs, however, makes the development of direct methods in electron correlation somewhat more complicated than at the HF level. Consider for example the MP2 energy expression." ... 

The above is an example of how direct algorithms may be formulated for methods... 

M. J. Frisch, M. Head-Gordon, and J. A. Pople, Chem. Phys. Lett., 166, 281 (1990). Semi-Direct Algorithms for the MP2 Energy and Gradient. 

Since we will, in general, not have a direct algorithm to create samples Xi according to the distribution P xi) we will use a Markov process in which starting from an initial configuration xq a Markov chain of configuration is generated ... 

A direct algorithm for contractions involving ab cd) integrals may be employed when storage and recovery of these integrals is not feasible. Here, components of Y are transformed into the atomic basis in their last two indices ... 

TURBOMOLE running on the DEC 5000/200 workstation. A full-direct algorithm, integral neglect threshold equal to 10 a.u. 

GAUSSIAN 90 running on CRAY Y-MP4/32. A full-direct algorithm, 14 MW of the core memory used, core orbitals not correlated. The numbers of integral evaluations are given in parentheses. 

VII. Direct Algorithms for Sparse and Well-Structured Linear Systems... 

VII. DIRECT ALGORITHMS FOR SPARSE AND WELL-STRUCTURED LINEAR SYSTEMS... 

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. 

Toplds, D.M. and Veinott, A.F. (1967) On the convergence of some feasible direction algorithms for nonlinear programming. 

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