In this paper, we study the eigenvalues of real Tridiagonal 3-Toeplitz matrices of different order. When the order of a Tridiagonal 3-Toeplitz matrix is n = 3k+ 2, the eigenvalues were found explicitly. Here, we consider the distribution of eigenvalues for a Tridiagonal 3-Toeplitz matrix of orders n = 3k and n = 3k + 1. We explain our method by finding roots of a combination of Chebyshev polynomials of the second kind. This distribution solves the eigenproblem for integer powers of such matrices.

In this paper,  rstly we derive a general expression for the mth power (m 2 N) for one type of Tridiagonal matrices of even order. Secondly we present a method for computing integer powers of the antiTridiagonal matrices that is corresponding with these matrices. Then, we present some examples to illustrate our results and give Maple 18 procedure in order to verify our calculations.

A large group of structures hold tri-diagonal stiffness matrices. The eigenpairs and inverse of these matrices are found simpler than the ones of common matrices. In addition, using the householder transformation, symmetric matrices can be converted to the similar tri-diagonal matrices. Therefore, since stiffness matrices are symmetric, they can be changed to the similar tri-diagonal ones. In other words, all symmetric matrices can be converted to the Tridiagonal ones and the simpler solution of tri-diagonal matrices can be used for all stiffness matrices. Such a comparison is also true for block tri-diagonal matrices and block symmetric matrices. Although block matrices are a specific kind of common matrices, we want to study them independently because working with blocks can be more time-saving and efficient in many cases. In this paper, efficient solutions are presented for tri-diagonal and block Tridiagonal matrices. Besides, using the features of symmetric and block symmetric matrices they are converted to the tri-diagonal and block tri-diagonal ones.

In this paper, at first for a given set of real or complex numbers with nonnegative summation, we introduce some special conditions that with them there is no nonnegative Tridiagonal matrix in which s is its spectrum. In continue we present some conditions for existence such nonnegative Tridiagonal matrices.

In this paper, a class of matrices, namely, Drazin-dagger matrices, in which the Drazin inverse andthe Moore-Penrose inverse commute, is introduced. Also, some properties of the generalized inverses of these matrices, are investigated. Moreover, some results about the Moore-Penrose inverse, the Drazin inverse and the numerical range of some reciprocal matrices are obtained. In particular, the relations between reciprocal matrices, Drazin-Dagger matrices and star order are established. Also, some properties of the generalized inverses of the conjugate EP matrices are studied. To illustrate the results, some numerical examples are also given.

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General Purpose Graphics Processing Unite (GPGPU) allows the user to utilize GPU for general computing purposes. Using these processors can cause a great speedup in numerical calculations.Several studies have been performed to investigate the advantages of using the GPGPU in numerical calculations including solving Tridiagonal set of equations. The main focus of the mentioned studies was on improving parallel methods, for example, CR and PCR algorithms. Although these algorithms are consistent with GPU architecture, they have higher arithmetic complexity compared with serial Thomas algorithm, they also have limitations in dimensions of the equations’ set. Therefore, in the present study, according to the advantages of Thomas algorithm compared with the parallel algorithms, a novel method entitled checkerboard Thomas has been developed to accommodate Thomas algorithm for running on GPU. This method has been used for solving 2D steady heat conduction problem and the results show an increase in the solution precision compared to Thomas and PCR algorithms. Also, the results indicate that the new algorithm can cause computing to increase in speedup between 5.7 to 22.2x, compared with Thomas algorithm. Furthermore, results show that the new method is about 2x faster than PCR algorithm. It has also been seen that speed decrement for uncoalesced access to global memory is 42.7% minimum and 81.9% maximum for128×128 and 1024×1024 grid size, respectively.

In this paper using the eigenvalues and eigenvectors of symmetric and asymmetric of Tridiagonal matrices with infinite dimension, inverse of these matrices are found. Moreover numerical method of finite difference is used for numerical modeling of engineering problems. Many engineering problems modeled by numerical methods result in solution of linear systems. Many of these linear systems are in tri-diagonal form. The inverse of this kinds of matrices have many applications. In this paper, response of linear and nonlinear structures resulting from different types of arbitrary dynamics and statics analyses are presented. The method of this paper provides closed form solutions for the analysis of structures under different static and seismic loadings.