Identity matrix

About: Identity matrix is a research topic. Over the lifetime, 1253 publications have been published within this topic receiving 20575 citations. The topic is also known as: unit matrix.

Some numerical experiments with variable-storage quasi-Newton algorithms

Abstract: This paper describes some numerical experiments with variable-storage quasi-Newton methods for the optimization of some large-scale models (coming from fluid mechanics and molecular biology). In addition to assessing these kinds of methods in real-life situations, we compare an algorithm of A. Buckley with a proposal by J. Nocedal. The latter seems generally superior, provided that careful attention is given to some nontrivial implementation aspects, which concern the general question of properly initializing a quasi-Newton matrix. In this context, we find it appropriate to use a diagonal matrix, generated by an update of the identity matrix, so as to fit the Rayleigh ellipsoid of the local Hessian in the direction of the change in the gradient. Also, a variational derivation of some rank one and rank two updates in Hilbert spaces is given.

Iterative methods for solving partial difference equations of elliptic type

Abstract: Conditions (1.2) were formulated by Geiringer [4, p. 379](2). Evidently these conditions imply that ai,i 5#O (i=1, 2, -, 1N). It is easy to show by methods similar to those used in [4, pp. 379-381] that the determinant of the matrix A= (ai,j) does not vanish. Moreover, if the matrix A * = (a*j) is symmetric, where a*1=ai,iai,j/ ai,i (i, j= 1, 2, N * , N), then A * is positive definite. For if X is a nonpositive real number, then the matrix A * -XI, where I is the identity matrix, also satisfies (1.2) and hence its determinant cannot vanish. Therefore all eigenvalues of A * are positive, and A * is positive definite. On the other hand if A* is positive definite then ai,i5zQ (i=1, 2, , N). We shall be concerned with effective methods for obtaining numerical solu-

A Simple Way to Initialize Recurrent Networks of Rectified Linear Units

Abstract: Learning long term dependencies in recurrent networks is difficult due to vanishing and exploding gradients. To overcome this difficulty, researchers have developed sophisticated optimization techniques and network architectures. In this paper, we propose a simpler solution that use recurrent neural networks composed of rectified linear units. Key to our solution is the use of the identity matrix or its scaled version to initialize the recurrent weight matrix. We find that our solution is comparable to LSTM on our four benchmarks: two toy problems involving long-range temporal structures, a large language modeling problem and a benchmark speech recognition problem.

Improvement of a Large Analytical Model Using Test Data

Abstract: A method has been developed which uses measured normal modes and natural frequencies to improve an analytical mass and stiffness matrix model of a structure. The method directly identifies, without iteration, a set of minimum changes in the analytical matrices which force the eigensolutions to agree with the test measurements. An application is presented in which the analytical model had 508 degrees of freedom and 19 modes were measured at 101 locations on the structure. The resulting changes in the model are judged to be small compared to expectations of error in the analysis. Thus, the improved model is accepted as a reasonable model of the structure with improved dynamic response characteristics. In addition, it is shown that the procedure may be a useful tool in identifying apparent measured modes which are not true normal modes of the structure. Nomenclature - analytical matrix = matrix of changes = identity matrix = full improved stiffness and mass matrices (n x n) = full analytical K, M matrices (n x n) = partitions of KA,MA corresponding to test coordinates = partitions of KA,MA corresponding to coupling elements = partitions of KA,MA corresponding to unmea- sured coordinates = number of measured modes = number of degrees of freedom in the model = measures of changes, Eqs. (15-17) = matrix norm, sum of the squares of all the elements = rectangular modal matrix, normalized (n x m) = /th mode, /th column of $ = measured and unmeasured partitions of ,- = diagonal matrix of measured natural frequencies (m xm) = natural frequency of /th mode = 12 17 = sum of the squares of all elements of matrix ( )