Matrix diagonalization behavior

Figure 4 Typical timing behavior of the quadratic Fock matrix formation versus the cubically scaling diagonalization step (small prefactor) in SCF energy calculations. The timings for a conventional Fock matrix formation, the linear-scaling CFMM/LinK schemes (as explained later in this review), and Fock matrix diagonalization for a series of DNA molecules (A-T), = 1 — 16 are depicted. Integral threshold is 10 , basis set 6-31G. ...

So far, lumping has been defined, but nothing has been said concerning the dynamic behavior of the" system. Now we come to the definition of exact lumping A system is said to be exactly lumpable by the matrix L if there exists an N X N matrix K, enjoying the same properties as K does (i.e., off-diagonal elements of K are nonpositive, = 0, and there exists at least an m" =... 

This disagreement can be accounted for by noticing that the rate provided by the Mori theory is determined from the value of the area below the curve whereas the AEP is equivalent to fitting with an exponential decay the long-time decay behavior of (see Fig. 2). The exact time revolution of 9 (t) can be obtained by means of the diagonalization of the matrix A given... 

The exact exponents Xj are directly computed by diagonalizing the symmetric matrix MTM with high-precision computation to evade its roundoff error. Figure 2 plots the error in the smallest exponents, (t, 0) — ki(t, 0), against t. We can see that until t 30 the error rapidly decreases. After the initial dropping stage, it decreases slowly as 1/f (inset). This quite slow convergence shows that A,9R is not a sufficiently accurate approximation of the finite-time Lyapunov exponent. Note here that behavior of Aj is similar to that of iL-... 

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