Orthomin, an Iterative Method for Solving Sparse Sets of Simultaneous Linear Equations

An international conference on large sparse sets of simultaneous linear equations was held at St. Catherine's College, Oxford, from 5th to 8th April, 1970. It was organized by the Institute of Mathematics and its Applications, and it brought together many specialists from widely differing fields, all dealing with the problem of solving large sets of linear equations.

In cases where successive integration of F ( x ) is not practical, a solution of the equation d 2 y/dx 2 =F(x) is obtained by approximation. The solution attempted is to find the value of y at selected points in the range of integration. A common procedure is to reduce the problem to the solution of a set of linear simultaneous equations, and these equations are then solved by “relaxation”. It does not appear to be generally known that an explicit and simple solution of these simultaneous equations exists. In this note the detail of such a solution is carried out for functions F ( x ) as occur in the theory of thin beams.

the timeliness of information received during the implementation of any production process, as well as its qualitative composition, significantly affect the efficiency of this process. This is particularly important for agricultural production, as field operations are highly time-sensitive, both in terms of start-up and duration. The real possibility of a significant increase in the efficiency of production processes in crop production appeared at the current level of information technology and the spread of digitalization of various fields of activity. On the basis of existing methodological approaches to the system representation of objects of reality, we have developed an information system of the production process in crop production. The elements of the system are the amount of knowledge of specialists in the cultivation of crops, information about the composition of cultivation technologies, information about the objects processed in the implementation of production processes, recorded in any form replenished special knowledge, knowledge about the goals achieved in the implementation of production processes. Elements of the system are interconnected, developing in a certain way and, at the same time, limiting each other. The formalization of the relationships between elements of the information system allowed us to obtain ten equations with fifteen unknowns. To solve this system of equations, a set of simultaneous linear equations is presented. Five sets of simultaneous equations can be defined, depending on which of the relationships of the elements of the information system of the type fi will be broken. The obtained solutions will allow to quantify the volume of elements of the information system, which makes it very promising to use information technology management of production processes in crop production.

The singular value problem, in which it is required to solve a set of simultaneous linear equations, when there are more variables than equations, is considered, and a simple geometric interpretation described. This geometric interpretation complements the more abstract matrix algebra approach.

This paper compares the efficiency of line successive overrelaxation (LSOR) with a two‐dimensional correction procedure (2DC), the iterative, alternating direction implicit procedure (ADI), and the strongly implicit procedure (SIP) to solve finite‐difference equations used to simulate several groundwater reservoirs. Three of the reservoirs are linear, two are isotropic areal problems, and the third is an anisotropic cross‐section simulation. The fourth is a nonlinear water table aquifer with areas of thin saturation. SIP is generally the best method for the linear simulations and with the addition of another iteration parameter is the only method that gives an adequate rate of convergence for the water table problem. LSOR with 2DC is competitive with SIP on isotropic and anisotropic linear problems that are dominated by no‐flow boundaries. ADI is generally more efficient than LSOR if a good set of iteration parameters are used, but this advantage is offset by the relative ease of finding the best acceleration parameter for LSOR.

A procedure for solving the differential groundwater flow equation is presented herein. Using a finite difference or finite element discretization scheme, a set of simultaneous linear equations is obtained. The eigenvalues and eigenvectors of a matrix, which is a function of the coefficients of the set, are the key to the solution. A vector L is obtained straightforwardly by combining the eigenvector matrix A, the eigenvalue vector α, the pumping vector P, and the initial head vector H. Vector L, which depends on time, can be expressed simply and explicitly as a function of the eigenvalues. Piezometric heads can be obtained by combining A and L. L is the only vector that needs to be computed as P changes with time. In this way, influence functions of a piezometric head, flow velocity and flow depletion of a stream connected with the aquifer under a unit stress, can be obtained explicitly and continuously in time. The method can be applied to confined as well as to leaky aquifers and to one‐, two‐, or three‐ dimensional linear models. Its main advantage lies in the fact that it is unnecessary to repeatedly solve a matrix for every time increment. The method is particularly useful for groundwater management problems in which a large number of alternatives have to be evaluated.

Examination of tables 1 and 3 illustrate clearly that not only is the procedure outlined in section II biased, but that bias of an order of magnitude shown in tables 1 and 3 entries will occur with extremely high probability. For example, for n = 10, r = 10, fW2 = .617 with sample variance of only .0172 even when the population values of lpiI i = 2, . . , r are controlled to be 0! It can easily be seen that if the data are time series and contain a common trend or serial correlations, it is even more dangerous to interpret high multiple sample correlation coefficients as representing meaningful structural relationships.3 One effective way of dealing with the problem is to reformulate it in Bayesian terms so that we may systematically incorporate information that the researcher possesses from sources other than the sample into the analysis. In [2] we indicate how Bayesian estimation of the mean vector and variance-covariance matrix of a multinormal process may be done even when the dimensionality of the process is r and there are only n < r sample observations available. In [3] we show how Bayesian estimation can be done when the data generating process is one frequently occurring in econometric analysis a set of simultaneous linear equations with stochastic components and there are fewer vector observations than parameters.

A coupled electron pair approximation is derived and illustrative calculations are presented. The present approximation, which we refer to as the unitary coupled electron pair approximation (UCEPA), provides multiconfigurational (MC) reference capability and is as computationally tractable as multireference configuration interaction (MRCI). The method is capable of yielding size-consistent energies if the MC reference function is of the complete active space (CAS) variety. The coefficient matrix of the resultant set of simultaneous linear equations is evaluated using internal/external orbital space partitioning within a unitary group approach (UGA) treatment of the state space. We demonstrate the accuracy of the method on several small benchmark molecules for which full CI results are known, and on nontrivial studies on the singlet–triplet splitting in methylene and the electron affinity of the oxygen atom.

PurposeThe purpose of this paper is to compute the values of induced voltages and currents in an underground gas pipeline, which shares the same right of way with a high voltage power line (HVPL) in unbalanced phase currents regime, in order to detect the possibility of the AC corrosion occurring in the pipeline.Design/methodology/approachThe analysis of the induced voltages in metallic structures (gas pipeline) was done using a professional analysis and modeling software which uses the electromagnetic field method.FindingsIt was shown that during fault conditions, large currents and voltages are induced on the pipelines, which may pose a threat to operating personnel and equipment. Induced voltages also occur in cases of unbalanced phase currents of the HVPL, which may pose a threat voltage that could lead to damages of the underground gas pipeline due to corrosion phenomena.Practical implicationsBecause the electromagnetic field approach involves complicated analytical expression and introduces serious computational difficulties, among which matrix of coefficients representing the set of simultaneous linear equations, the use of professional analysis and modeling software is justified.Originality/valueThe paper reveals that even in unbalanced phase currents regime of the HVPL, the nearby metallic structures that share the same right of way with the transmission line are in danger of degradation because of the induced voltages, whose values exceed the values imposed by regulation.

This algorithm for the calculation of the induced multipole tensors in a set of charge distributions includes the contributions of partial derivatives of arbitrary order of the potentials defined by both induced and permanent multipoles as well as non-uniform external fields of arbitrary strength. Specific equations are given for both systems with and without translational order. The algorithm, which is based on the Maxwell invariant form, uses direct extensions of algorithms previously developed and tested for the calculation of permanent multipole energies and induced dipole vectors (when the non-uniformity of the field is neglected). The induced tensor components are calculated iteratively. The first approximation, which gives the components as the solution of a set of simultaneous linear equations, includes all nonlinear, non-uniform contributions of permanent multipoles and external fields as well as all contributions linear in derivatives of the induced potentials. The induced tensor components are then used to calculate the net induction energy. The general relations between the polarizability tensors with respect to a centre and the moments of the polarizability densities about the centre are derived.

System identification is important in science and engineering. This contribution proposes a simple yet elegant scheme for identifying a fractional order dynamic system based on its observed response to a standard excitation. Our approach consists in obtaining and solving a set of simultaneous linear equations connecting the parameters to yield the desired estimates. To minimise the effect of noise, we have performed an in-depth analysis of the Grunwald-Letnikov definition and, as far as we know, are the first ones to do so. Results show that the proposed method offers a very high degree of accuracy even for erroneous data.

This paper describes a method to find accurate mass and stiffness matrices in system identification. A set of simultaneous linear equations is solved to identify the mass and stiffness matrix of structures. An exact solution cannot necessarily be found because the coefficientmatrix in the equation is sometimes ill-conditioned. The singularvalue decomposition is used to find an approximate solution. The proposedmethod introduces the projected norm as an indicator to determine howmany singular values should be considered for identifying accuratemass and stiffness matrices. Anumerical example is given to demonstrate that the proposedmethod is effective for system identification. The system identification method was previously developed by the author and the contribution of the paper is a method to select singular values. It is found that the proposed method identifies mass and stiffness matrices that can accurately predict the dynamic characteristics and the mass properties of structures.

The analog to the Bloch equation for the case of thermal conduction in a superconductor limited by phonon scattering is derived by introducing an appropriate general form for the nonequilibrium part of the distribution function into the corresponding Boltzmann equation. This integral equation for the deviation function is solved numerically for different temperatures $T$ by replacing it by sets of simultaneous linear equations with dimensions up to 39. The limiting curve for the deviation function when $T$ approaches the transition temperature ${T}_{c}$ from below turns out to be identical to the curve which has been reported by Klemens for the normal state. With $T$ decreasing below ${T}_{c}$ the maximum of the deviation function rises and shifts to higher energies. The ratio of the thermal conductivity in the superconducting state to that in the normal state, $\frac{{\ensuremath{\kappa}}_{s}}{{\ensuremath{\kappa}}_{n}}$, plotted against $\frac{T}{{T}_{c}}$ is found to increase monotonically and to have a limiting slope of about 1.62 at ${T}_{c}$. Consideration of the energy dependence of the energy gap in the case of lead yields a sizable effect on the plot of $\frac{{\ensuremath{\kappa}}_{s}}{{\ensuremath{\kappa}}_{n}}$ vs $\frac{T}{{T}_{c}}$.

The problem of calculating the reflected and transmitted fields as a function of scan angle in an infinite phased array of parallel-plate waveguides is studied on an analytical basis. Two infinite sets of simultaneous linear equations are derived for the scattered fields internal and external to the waveguides. These sets of equations are attacked by a newly developed modified residue calculus method. Application of the results to the problem of designing rectangular waveguide arrays with improved impedance performance over a wide range of H -plane scan angles is discussed. It is indicated that the present analysis may serve as a first step toward investigating the more general problem of phased arrays scanning at an arbitrary angle.

A general approach for eliminating higher order aliasings is developed by using multiple interlaced samplings. The M-fold interlaced samplings can be used to eliminate aliasing up to the M'th order by solving a set of simultaneous linear equations involving an MxM Vandermonde matrix, thereby increasing the sampling rate by M times. The concept is demonstrated by considering the case of Fresnel zone sampling and reconstruction.