Hermitian methods

In 1924, the Russian astronomer Numerov (transliterating his own name as Noumerov), published a paper [421] in which he described some improvements in approximations to derivatives, to help with numerical simulations of the movement of bodies in the solar system. His device has been adapted to the solution of pdes, and was introduced to electrochemistry by Bieniasz in 2003 [108]. The method described by Bieniasz is also called the Douglas equation in some texts such as that of Smith [514], where a rather clear description of the method is found. With the help of the Numerov method, it is possible to attain fourth order accuracy in the spatial second derivative, while using only the usual three points. The first paper by Bieniasz on this method treated equally spaced grids, and was followed by another on unequally spaced grids [107], The method makes it practical to use higher-order time derivative approximations without the complications of, say, the (6,5)-point scheme described above, which makes the solution of the system of equations a little complicated (and computer time consuming). 

The description in Smith [514, pp. 137-] is followed here. It starts with a statement that a second derivative can be approximated by 

Smith does not explain the origin of (9.29), but a derivation can be found in Lapidus and Pinder [350, pp. 19-], to which the reader is referred. The form seen in (9.29) is one of several equally valid forms, but is the one chosen in this context, as it allows the Numerov device. 

The method will be described as applied to BI, which is the basis for both extrapolation and BDF, both of which can be driven to fourth order accuracy, which is also achieved by the Numerov device applied to the right-hand side of the diffusion equation, 

Using the usual notation, C,[ denoting the next point in time after the present value Cj, i being the index along the X axis, we now discretise the left-hand 

We need not concern us with the implementation of the higher-order derivatives, as will shortly be clear. Now both sides are operated on by (1 -i- 8 ), which is the same as adding to each side the operation on that side. This gives 

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Wu and White [577] have described a new method that is reminiscent of the earher work of Kimble and White [338] but makes use of the Hermitian method (that is, using derivatives) to achieve higher-order solutions for several concentration rows at a time. They also suggest, but do not demonstrate, the use of their new scheme as a possible start-up for BDF. The reader is referred to their paper for details. 

Wu and White [98] have described a new method that is reminiscent of the earlier work of Kimble and White [14] but makes use of the Hermitian method (that is. 

A second method for achieving the same result will be mentioned briefly. If w is a unit vector, wHw = 1, then the matrix I — 2wwH is unitary and its own inverse (it is also hermitian). For any vector a, it is always possible to choose a unit vector w such that... 

The method of Jacobi applies not only to hermitian matrices, but, with some modification, to all normal matrices. It is not to be recommended, but it has some historical interest and is often mentioned in the literature. Hence, it will be described briefly as it is applied to the... 

A Hessenberg form H (the same form but not the same matrix) can also be obtained by a sequence of orthogonal transformations, either by plane rotations (the method of Givens), each rotation annihilating an individual element, or by using unitary hermitians, I — 2wiwf, wfwt = 1 (the method of Householder), each of which annihilates ill possible elements in a column. Thus, at the first step, if A = A, and... 

Hermitian operators for electric and magnetic field intensities, 561 Herzfeld, C. M., 768 Hessenberg form, 73 Hessenberg method, 75 Heteroperiodic oscillation, 372 Hilbert space abstract, 426... 

The aforementioned applications of recursive methods in reaction dynamics do not involve diagonalization explicitly. In some quantum mechanical formulations of reactive scattering problems, however, diagonalization of sub-Hamiltonian matrices is needed. Recursive diagonalizers for Hermitian and real-symmetric matrices described earlier in this chapter have been used by several authors.73,81... 

M. Rosina, (a) Direct variational calculation of the two-body density matrix (b) On the unique representation of the two-body density matrices corresponding to the AGP wave function (c) The characterization of the exposed points of a convex set bounded by matrix nonnegativity conditions (d) Hermitian operator method for calculations within the particle-hole space in Reduced Density Operators with Applications to Physical and Chemical Systems—II (R. M. Erdahl, ed.), Queen s Papers in Pure and Applied Mathematics No. 40, Queen s University, Kingston, Ontario, 1974, (a) p. 40, (b) p. 50, (c) p. 57, (d) p. 126. 

M. Bouten, P. Van Leuven, M. V. Mihailovic, and M. Rosina, Two exactly soluble models as a test of the Hermitian operator method. Nucl. Phys. A221, 173-182 (1974). 

Now, since the Hessian is the second derivative matrix, it is real and symmetric, and therefore hermitian. Thus, all its eigenvalues are real, and it is positive definite if all its eigenvalues are positive. We find that minimization amounts to finding a solution to g(x)=0 in a region where the Hessian is positive definite. Convergence properties of iterative methods to solve this equation have earlier been studied in terms of the Jacobian. We now find that for this type of problems the Jacobian is in fact a Hessian matrix. 

Equation (149) is valid provided that the matrix T and that within the brackets are non-singular. It has been shown that this is always true in the case considered. (47) Since Newmark and Sederholm have used the exchange superoperator in its non-Hermitian form, such as that given in equation (117), their method is rather time-consuming. Probably, this is the reason why the method has been largely forgotten. 

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