Journal of research of the National Bureau of Standards 

About: Journal of research of the National Bureau of Standards is an academic journal. The journal publishes majorly in the area(s): Natural rubber & Combustion. It has an ISSN identifier of 0091-0635. It is also open access. Over the lifetime, 2480 publications have been published receiving 61378 citations.

Methods of Conjugate Gradients for Solving Linear Systems

Abstract: An iterative algorithm is given for solving a system Ax=k of n linear equations in n unknowns. The solution is given in n steps. It is shown that this method is a special case of a very general method which also includes Gaussian elimination. These general algorithms are essentially algorithms for finding an n dimensional ellipsoid. Connections are made with the theory of orthogonal polynomials and continued fractions.

An iteration method for the solution of the eigenvalue problem of linear differential and integral operators

Abstract: The present investigation designs a systematic method for finding the latent roots and the principal axes of a matrix, without reducing the order of the matrix. It is characterized by a wide field of applicability and great accuracy, since the accumulation of rounding errors is avoided, through the process of \"minimized iterations\". Moreover, the method leads to a well convergent successive approximation procedure by which the solution of integral equations of the Fredholm type and the solution of the eigenvalue problem of linear differential and integral operators may be accomplished.

Variation of Peak Temperature With Heating Rate in Differential Thermal Analysis

Abstract: In differential thermal analysis, the temperature at which the maximum deflection is observed varies with heating rate for certain types of reactions. An expression can be derived relating this variation with the kinetics of the reaction. By making a number of differential thermal patterns at different heating rates, the kinetic constants can be obtained directly from the differential thermal data. Measurements of the variation of peak temperature with heating rate have been made for several minerals of the kaolin group, the values of the kinetic constants determined, and these values compared with corresponding values obtained for both the same samples and similar material by conventional isothermal techniques. Some factors affecting the results are discussed. The method of differential thermal analysis (DTA) has been universally accepted by mineralogical laboratories as a rapid and convenient means for recording the thermal effects that occur as a sample is heated. Changes in heat content of the active sample are indicated by deflections shown by a line representing the differential temperature. It is conventional to represent an endothermic effect by a negative deflection and an exothermic effect by a positive deflection. The deflections, whether positive or negative, are called peaks.

Solution of Systems of Linear Equations by Minimized Iterations1

Abstract: In an earlier publication [14] a method was described which generated the eigenvalues and eigenvectors of a matrix by a successive algorithm based on minimizations by least squares. The advantage of this method consists in the fact that the successive iterations are constantly employed with maximum efficiency which guarantees fastest convergence for a given number of iterations. Moreover, with the proper care the accumulation of rounding errors can be avoided. The resulting high precision is of great advantage if the separation of closely bunched eigenvalues and eigenvectors is demanded [16]. It was pointed out in [14, p. 256] that the inversion of a matrix, and thus the solution of simultaneous systems of linear equations, is contained in the general procedure as a special case. However, in view of the great importance associated with the solution of large systems of linear equations, this problem deserved more than passing attention. It is the purpose of the present discussion to adopt the general principles of the previous investigation to the specific demands that arise if we are not interested in the complete analysis of a matrix but only in the more special problem of obtaining the solution of a given set of linear equations

On approximate solutions of systems of linear inequalities

Abstract: (briefly, -4x^6), one arrives at a vector JC that "almost" satisfies (1). It is almost obvious geometrically that, if (1) is consistent, one can infer that there is a solution x0 of (1) "close" to x. The purpose of this report is to justify and formulate precisely this assertion. We shall use fairly general definitions of functions that measure the size of vectors, since it may be possible to obtain better estimates of the constant c (whose important role is described in the statement of the theorem) for some measuring functions than for others. We shall make a few remarks on the estimation of c after completing the proof of the main theorem.