Solution Of Simultaneous Linear Equations Research Articles

Torsion problem for elastic cylinder with holes

An integral equation method is given to solve the classical torsion problem in elasticity theory for a multiply connected region. As is well known, the solution depends upon finding the solution of the two-dimensional Laplace's equation which takes the value 1 2 (x 2+y 2)+ c i on the boundary, where x,y are the usual Cartesian coordinates, and c i are unknown constants. The usual approaches are extremely cumbersome. In this paper a new approach is suggested to solve such problems. This method is simple and straightforward and requires the solution of simultaneous linear equations. An example is given which substantiates the theory. The method is very general and can take into account the discontinuity in the displacement component w. The result can therefore be applied to dislocation theory.

Engineering Status of Computerized Tomographic Scanning

With the advances in computer technology during the 1960s several authors, including Cormack, Kuhl and Edwards, began to discuss the possibility of improving the quality of ordinary x-ray tomography using exact mathematical reconstruction techniques. The basic method is to measure x-ray attenuation along sets of parallel lines oriented at multiple angles, all contained within a single transverse cross-sectional slice of the body. These measurements can be represented mathematically as linear combinations of the unknown mass attenuation coefficients in the slice. Standard mathematical techniques developed for the solution of simultaneous linear equations can be adopted to reconstruct the unknown cross-section distribution from such data. The result can be displayed as a two-dimensional array using computer generated display techniques, and closely resembles the image that could have been obtained if the cross-sectional slice were removed from the body and imaged by conventional projection radiography.

Determination of Available Power from Resistive Multiports

The concept of available power (P_{A}) is extended to multiports. A theorem is proved, which gives a necessary condition for P_{A} to be achieved with all positive load resistances for the case of resistive-reciprocal multiports. Based on the theorem, a procedure is outlined for determining P_{A} for resistive multiports. The method requires the solution of simultaneous linear equations only. Two examples are given.

On computing generalized inverses

A method for computing the generalized inverse of a matrix is described, which makes use of elementary orthogonal matrices and theGaussian elimination. The method also yields orthonormal bases for the ranges and the null spaces of the matrix and the generalized inverse. Modifications of the method for the solution of simultaneous linear equations are given. Compact storage schemes, in the case of sparse matrices, are also described.

Projection Methods for Solving Sparse Linear Systems

Some methods of successive approximation for the solution of simultaneous linear equations are discussed. The coefficient matrix A of the linear system is assumed to be sparse. It is shown that savings in the computer storage and the computing time are possible, if there exists a subset of the rows (columns) of A, consisting of only orthogonal rows (columns). Such savings are also possible, if for some permutation matrices P and Q, PAQ has a particular structure, viz., singly bordered block diagonal form. It is shown that the set of orthogonal rows (columns) of A, as well as P and Q can be determined by using some results from graph theory (e.g., incidence matrices, row and column graphs, points of attachment). Geometrical interpretations of the methods and their inter-relatiohip are given.

AN ANTHOLOGY OF NUMERICAL SOLUTIONS OF SIMULTANEOUS LINEAR EQUATIONS

"AN ANTHOLOGY OF NUMERICAL SOLUTIONS OF SIMULTANEOUS LINEAR EQUATIONS." Survey Review, 19(149), p. 334

AN ANTHOLOGY OF NUMERICAL SOLUTIONS OF SIMULTANEOUS LINEAR EQUATIONS

"AN ANTHOLOGY OF NUMERICAL SOLUTIONS OF SIMULTANEOUS LINEAR EQUATIONS." Survey Review, 19(149), p. 334

Solution of linear equations with coefficient matrix in band form

An algorithm is given for the solution of simultaneous linear equations having a band type coefficient matrix. The algorithm utilizes a suitable partitioning of the coefficient matrix. Some particular cases are described in which the use of the algorithm is advisable.

AN ANTHOLOGY OF NUMERICAL SOLUTIONS OF SIMULTANEOUS LINEAR EQUATIONS

AbstractThe advent of the electronic computer has brought the problem of the solution of simultaneous linear equations to the fore, particularly in regard to the solution of symmetrical equations such as normal equations in least squares solutions. The object of this paper will be to remind readers of some of the usual ways of solving simultaneous equations, and then to discuss in greater detail the solution of normal equations by the Gauss–Doolittle, Cholesky and other methods with modifications and explanations.

AN ANTHOLOGY OF NUMERICAL SOLUTIONS OF SIMULTANEOUS LINEAR EQUATIONS

The advent of the electronic computer has brought the problem of the solution of simultaneous linear equations to the fore, particularly in regard to the solution of symmetrical equations such as normal equations in least squares solutions. The object of this paper will be to remind readers of some of the usual ways of solving simultaneous equations, and then to discuss in greater detail the solution of normal equations by the Gauss–Doolittle, Cholesky and other methods with modifications and explanations.

Automatic Controlled Precision Calculations

Recent developments in computer design and error analysis have made feasible the use of variable precision arithmetic and the preparation of programs that automatically determine their own precision requirements. Such programs enable the user to specify the accuracy he wants, and yield answers guaranteed to lie within the bounds prescribed. A class of such programs, called “contracting error programs,” is defined in which the precision is determined by prescribing error bounds on the data. A variant of interval arithmetic is defined which enables a limited class of algorithms to be programmed as contracting error programs. A contracting error program for the solution of simultaneous linear equations is described, demonstrating the application of the idea to a wider class of problems.

Magnetic field problem with axial symmetry and its solution in terms of Fourier-Bessel expansions

A method for reaching implicit analytical solutions is explained, demonstrated and verified for certain problems of potential theory, characterized by having boundaries which are parts of co-ordinate surfaces. Different Fourier expansions for the potential are matched over an interface of the problem by the solution of linear simultaneous equations in their coefficients. Two problems are completed by numerical computations and the construction of equipotentials. Comparisons are made with known results for the easier problem, and with experimental results for the other. The method is considered as a computational alternative to relaxation and finite differences, and the axisymmetric problem is shown to be aimed at applications in the design of electrical machines.

A Vector Solution of Simultaneous Linear Equations

A Vector Solution of Simultaneous Linear Equations

A Quantitative Determination of several Short-Lived Iodine, Barium and Strontium Fission Products in Gas-Cooled Reactor Effluents

Gamma scintillation spectrometry is used as a basis for the quantitative determination of absolute disintegration rate values for I131, I132, I133, I134, I135, Ba139, Ba140, Sr91 and Sr92 collected from reactor effluent air on activated coconut charcoal. After chemical separations for the various elements have been made, the complex γ-spectra resulting are treated mathematically, resulting in the solution of simultaneous linear equations to yield d/m values for the specific isotope. Corrections are made for any absorber between the source and NaI(Tl) crystal, for the total absolute detection efficiency for the counting geometry used, for the peak to total ratios for the particular photopeak, for the decay scheme, and for any internal conversion, thus obviating the need for standard plates of the isotope in question. The method is applicable to γ-emitters in general, providing values for corrections to be applied are available.

Simple electrical analogue for the solution of linear simultaneous equations

A simple resistance analogue provides a rapid solution for simultaneous equations of the type yi = aijxj, where aij = 0, i > j.

A routine to find the solution of simultaneous linear equations with polynomials coefficients

A routine to find the solution of simultaneous linear equations with polynomials coefficients

A Note on the Solution of Simultaneous Linear Equations

A Note on the Solution of Simultaneous Linear Equations

Further Contributions to the Solution of Simultaneous Linear Equations and the Determination of Eigenvalues.

Further Contributions to the Solution of Simultaneous Linear Equations and the Determination of Eigenvalues.

Further Contributions to the Solution of Simultaneous Linear Equations and the Determination of Eigenvalues

Further Contributions to the Solution of Simultaneous Linear Equations and the Determination of Eigenvalues

Monte Carlo method applied to the solution of simultaneous linear equations

Monte Carlo method applied to the solution of simultaneous linear equations