Direct-diagonalization algorithm

In order to generate eigenvectors of the Hamiltonian matrix, we follow the two-step procedure (1) Compute the matrix elements of the Hamiltonian operator in the basis set, Hy = (i H j). (2) Using standard direct diagonalization algorithms, compute the eigenvalues and eigenvectors of the N X N real symmetric matrix H,... 

The equation matrix is tridiagonal, that is, all its elements are zero except the three middle diagonals. This matrix lends itself to a direct solution algorithm consisting of forward elimination followed by backward substitution. For simplicity the component subscript i is dropped since the matrix is solved for one component at a time. 

The density-matrix-spectral-moments algorithm (DSMA) °° is an approximate scheme for solving the TDHF equations that allows us to calculate from the source ( W) by solving eq A18 without a direct diagonalization of L. This is accomplished by computing the set of electronic oscillators that dominate the expansion of Without loss of generality, we can take rj itj to be real and express it in terms of our momentum variables as - °°... 

Should A be singular, this method still gives a basis inverse, namely, the inverse of the submatrix of A in which the pivots were found. A version of this algorithm using strictly positive, diagonal pivots is used for constrained parameter estimation in Subroutine GREGPLUS, as described in Sections 6.4 and 7.3 there the determinant of the basis submatrix is obtained directly as the product of the pivotal divisors. Basis inverse matrices also prove useful in Sections 6.6 and 7.5 for interval estimation of parameters and related functions. 

Enzyme activity is assumed to decay exponentially over the experiment. Fast controller response in both directions can be observed. Compared with the uncontrolled case, the controller controls the product purity and compensates the drift in the enzyme activity. The evolution of the results of the optimization algorithm during each cycle is plotted as a dashed line, shifted by one cycle to the right in order to vitalize the convergence. This shows that a feasible solution is found rapidly and that the controller can be implemented under real-time conditions. In this example, the control horizon was set to two cycles and the prediction horizon was set to ten cycles. A diagonal matrix i j = 0.02 I (3,3) was chosen for regularization. 

This is a system of equations of the form Ax = B. There are several numeral algorithms to solve this equation including Gauss elimination, Gauss-Jacobi method, Cholesky method, and the LU decomposition method, which are direct methods to solve equations of this type. For a general matrix A, with no special properties such as symmetric, band diagonal, and the like, the LU decomposition is a well-established and frequently used algorithm. 

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