Iterative Extrapolation Research Articles

Finite-difference solution of the coupled-channel Schrödinger equation using Richardson extrapolation

A method for numerical solution of the coupled-channel Schrödinger equation for bound states based on the finite-difference method of the second order with iterative extrapolation of net eigensolutions is described. It is shown that Richardson's extrapolation to the limit significantly increases the accuracy of the finite-difference solutions with substantial reduction of requirements in computer memory, disk storage and computational time. The same extrapolational procedure and error estimations are applied to the eigenvalues and eigenfunctions. Results of sample calculations for several quantum-mechanical problems are presented and discussed. One of the sample problems considered is the hydrogen atom in a uniform magnetic field.

Fast discrete extrapolation via the fast Hartley transform

A fast algorithm for A. Papoulis's (1975) and R.W. Gerchberg's (1974) iterative extrapolation based on the fast Hartley transform (FHT) is presented. The low-pass filtering in the iterative procedure can be implemented by the FHT directly instead of by the fast Fourier transform (FFT) and the inverse FFT. M.S. Sabri and W. Steenart's (1978) example demonstrates that the FHT approach is simple to use.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Schrödinger's radial equation: Solution by extrapolation

Combining an appropriate finite difference method with iterative extrapolation to the limit results in a simple, highly accurate, numerical method for solving a one-dimensional Schrödinger's equation appropriate for a diatomic molecule. This numerical procedure has several distinct advantages over the more conventional methods such as Numerov's method or the method of finite differences without extrapolation. The advantages are the following: (i) initial guesses for the term values are not needed; (ii) the algorithm is easy to implement, has a firm mathematical foundation and provides error estimates; (iii) the method is relatively less sensitive to round-off error since a small number of mesh points is used and, hence, can be implemented on small computers; (iv) the method is faster for equivalent accuracy. We demonstrate the advantages of the present algorithm by solving Schrödinger's equation for (a) a Morse potential function appropriate for HCl and (b) a numerically derived Rydberg-Klein-Rees potential function for the X 1 Σ + state of CO. A direct comparison of the results for the X 1 Σ + state of CO is made with results obtained using Numerov's method.

Theoretical analysis of collisional energy transfer in unimolecular reactions: Collision efficiencies in binary mixtures

The unimolecular decomposition of ethane and acetyl is studied by weak collision RRKM theory. The reactions take place in a binary mixture of weak and strong colliders represented in the master equation by exponential, shifted Gaussian or strong collision contributions to the energy transfer kernel. The pressure dependence of the rate coefficients is studied as a function of mixture composition over the full range from the low-pressure limit to the high-pressure limit. Attention is focused on the validity of linear mixture rules which predict mixture dependence in terms of single component rate coefficients. A recently proposed iterative extrapolation method for the pressure- and composition-dependent collision efficiencies of the different medium species is tested and found to reproduce the falloff of the rate coefficient to good accuracy. Deviations due to nonlinear composition dependence are seen in the low-pressure limit.

An extrapolative diagonalization of incomplete hamiltonians

We propose a new technique of constructing the low-lying bound states. It starts from an “available” non-square finite submatrix of the hamiltonian and its essence lies in a preliminary band-matrix iterative extrapolation of these matrix elements. The efficiency of the method is illustrated on the standard example — the anharmonic oscillator.

A method for improving the accuracy of finite‐element solutions using the extrapolation method

AbstractTo improve the accuracy of solutions by the finite‐element method (FEN), it is necessary to refine the element division or use higher‐order elements. However, in both approaches, large‐scale/long‐time computation is necessary because the dimension of the matrix equation to be solved increases. Especially, in vectorial or 3‐dimensional analysis, this is a serious problem with respect to computational cost. By the error analysis theory of FEM, the asymptotic property of errors is known when the interpolation polynomial of certain order is used. In light of this, we use the so‐called deferred approach to the limit and derive the extrapolation formula making use of the asymptotic property of errors. This leads to the more efficient use of FEM, which is one of our objectives. Also, an algorithm using the iterative extrapolation method is proposed to obtain more accurate solutions. The theory is tested numerically for the eigenvalue problem of a dielectric‐loaded rectangular waveguide. The estimate of the exact eigenvalue is obtained by using the extrapolation formula described above. Hence, it is shown that very accurate results are obtained by merely repeating small‐scale computations.

Optimality of certain iterative and non-iterative data extrapolation procedures

A fundamental aspect of many modern spectral analysis and image restoration procedures is the extrapolation of the measured data to achieve improved resolution. Iterative extrapolation poses some mathematical questions about convergence and about the optimality of the resulting limit. We consider these questions and discuss briefly the optimality of certain non-iterative extrapolation methods.

Limit of continuous and discrete finite-band Gerchberg iterative spectrum extrapolation

It is known that the Gerchberg method for iteratively extrapolating the spectrum of an object of finite support has a different limit when the infinite frequency band used in the iterations is replaced by a finite one. It is shown that, for both continuous and discrete data, this limit involves an optimal extrapolation within the finite band to minimize the component of the limit that lies beyond the object support.

Iterative time-limited signal restoration

The purpose of this paper is to show that time-limited restoration of shift-invariant blurred signals can be done by means of fixed-point solutions of contraction mappings, under rather general conditions for the distortion operator. All our results are valid for multidimensional signals. An application of these results to the iterative extrapolation of band-limited discrete images is shown.

Gerchberg’s extrapolation algorithm in two dimensions

Gerchberg's 1-D iterative extrapolation algorithm for bandlimited signals is generalized to two dimensions in two distinct ways. One generalization requires knowledge of the entire spectral pupil of the bandlimited image. The second requires only knowledge of two 1-D intervals formed by the vertical and horizontal projections of the pupil. For real bandlimited images of the low-pass type, this corresponds to knowing only the maximum x and y spatial frequencies of the image. The utilization of information of the known portion of the image in the extrapolation process is discussed for both algorithms. The second algorithm, reformulated discretely, is placed in closed form.

Coherent optical extrapolation of 2-D band-limited signals: processor theory

Given a truncated portion of a band-limited image with known x and y bandwidths, the extrapolation problem is to determine the signal over all x and y. An iterative extrapolation algorithm, recently proposed by Gerchberg, requires only the repeated operations of Fourier transformation and truncation. A coherent optical processor is presented that implements Gerchberg's iterative extrapolation algorithm in two dimensions. Iteration is performed by simple passive feedback.

Adaptive control system utilising sampling techniques and Aitken's approximation

A model-reference adaptive control system is described, where adaption is performed by means of sampled signals. Aitken's method of iterative interpolation and extrapolation is employed to predict the performance of the continuous system at the sampling instants. The predicted-performance information is used to compute the strength of adaption impulses by sampled-data analysis techniques.