Inverse eigenvalue problem

As was already mentioned in Section 3.4, we can calculate the vibration—inversion-rotation energy levels of ammonia by solving the Schrodinger equation [Eq. (3.46)]. We are of course primarily interested in the determination of the potential function of ammonia from the experimental frequencies of transitions between these levels (Fig. 11), Le. we must solve the inverse eigenvalue problem [Eq. (3.46)]. 

In the non-rigid bender approximation, we solved the inverse eigenvalue problem described by Eq. (5.4), i.e. we determined the potential function parameters given in Table 3 for NX3 (X = H, D, T). We have used the experimental infrared frequencies of transitions from the ground state to the i>2,2 2 > 2. and 41 2 inversion states and the zero-order frequencies of vibrations (Table 4). The zero-order frequencies have been obtained from the observed fundamental frequencies of NH3 [Ref. >], ND3 [Ref. °>], NTg [Refs." and [Ref.- 3)] corrected for... 

Fig. 1. Input and output for solving the direct and the inverse eigenvalue problem...

For the numerical study of the whole spectrum (for g R fixed), [79] uses a spectral tau-Chebychev discretization in y and the Arnold method (see [88]) to solve the generalized eigenvalue problem (see [89]). This numerical method is based on the orthonormalization of the Krylov space of the iterates of the inverse of the matrix A B. This method has been used more recently in [90]. It has been proven efficient in the stiff problems arising in the study of spectral stability of viscoelastic fluids. 

The algebraic problem (10a) can be numerically solved to provide results for the eigenvalues and eigenvectors vji from this matrix eigenvalue problem analysis [31], which will be combined within the inverse formula (9a) to provide the desired eigenfunctions of the original eigenvalue problem. 

So far we have obtained parametrisations of the logarithmic derivative and potential functions which are appropriate when the Schrodinger equation is regarded as a differential equation, and which allow us to find and whenever E is given. In the ASA, however, Schrodinger s equation is treated as an eigenvalue problem subject to boundary conditions in the form of specified logarithmic derivatives at the sphere. Therefore, we need to find a parametrisation of the function E (D) inverse to D (E), valid around E. ... 

Unlike the case with the perturbation matrix, this is an eigenvalue problem not a matrix inversion problem. That is, the above equation only has solutions for certain values of w oi is not input to the equations but is determined by them. Since the matrices A, B etc. are of dimension n(m - n), there are therefore n(m — n) values of and corresponding sets of values of X, Y. 

The method established by iterating (11.5.19) is known as the inverse-iteration method with the Rayleigh quotient or simply as the Rayleigh method [7,9]. As should be clear from our discussion, the Rayleigh method is just Newton s method applied to the eigenvalue problem (11.4.28). 

Multiplying the generalized eigenvalue equations (12.4.12) from the left by the inverse square root of Tfa), we arrive at a standard eigenvalue problem... 

In practice, the solution of Equation 3.16 for the estimation of the parameters is not done by computing the inverse of matrix A. Instead, any good linear equation solver should be employed. Our preference is to perform first an eigenvalue decomposition of the real symmetric matrix A which provides significant additional information about potential ill-conditioning of the parameter estimation problem (see Chapter 8). 

A step taken with this approximate inverse Hessian led to an increase in energy and a succession of right line search failures. The eighth evaluation is finally taken with a very small step, a = 0.007. Most optimization, of course, do not work with the normal modes and do not diagonalize the Hessian as we have done here to examine the problem. If this were done, then this particular negative eigenvalue could have been set back to its previous value, or even back to unity. Working in normal... 

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