Inverses Of Tridiagonal Matrices Research Articles

On Inverses of Tridiagonal Matrices Arising From Markov Chain-Random Walk I

Explicit expressions for the entries of the inverse of a general diagonal matrix are derived via the decomposition approach. In particular, three-term recurrence relations with variable coefficients are given, and with their solutions closed form expressions for each entry of the fundamental matrix of a Markov chain-random walk (MC-RW) model are established. We also obtain explicit formulas for the absorption probabilities and the mean time to absorption as well as the mean number of steps before absorption of the MC-RW model in the presence of two semiabsorbing boundaries.

Block tridiagonal matrix inversion and fast transmission calculations

A method for the inversion of block tridiagonal matrices encountered in electronic structure calculations is developed, with the goal of efficiently determining the matrices involved in the Fisher–Lee relation for the calculation of electron transmission coefficients. The new method leads to faster transmission calculations compared to traditional methods, as well as freedom in choosing alternate Green’s function matrix blocks for transmission calculations. The new method also lends itself to calculation of the tridiagonal part of the Green’s function matrix. The effect of inaccuracies in the electrode self-energies on the transmission coefficient is analyzed and reveals that the new algorithm is potentially more stable towards such inaccuracies.

The inverses of block tridiagonal matrices

Firstly, the twisted block decompositions of the block tridiagonal matrices are presented. According to the special structure of the decomposition, the formulae of computing the block elements of each column of the inverse matrices are obtained. Then an algorithm of inverting the block tridiagonal matrices has been established. The explicit expressions of the block elements of the inverse matrices are also presented. At last, for the algorithm in this paper and some existed algorithms for the inverse matrices, the calculating complexity and the calculating time have been compared.

A small note on the scaling of symmetric positive definite semiseparable matrices

In this paper we will adapt a known method for diagonal scaling of symmetric positive definite tridiagonal matrices towards the semiseparable case. Based on the fact that a symmetric, positive definite tridiagonal matrix $$T$$ satisfies property A, one can easily construct a diagonal matrix $$\hat{D}$$ such that $$\hat{D}T\hat{D}$$ has the lowest condition number over all matrices $$DTD$$ , for any choice of diagonal matrix $$D$$ . Knowing that semiseparable matrices are the inverses of tridiagonal matrices, one can derive similar properties for semiseparable matrices. Here, we will construct the optimal diagonal scaling of a semiseparable matrix, based on a new inversion formula for semiseparable matrices. Some numerical experiments are performed. In a first experiment we compare the condition numbers of the semiseparable matrices before and after the scaling. In a second numerical experiment we compare the scalability of matrices coming from the reduction to semiseparable form and matrices coming from the reduction to tridiagonal form.

Inversion of general tridiagonal matrices

In the current work, the authors present a symbolic algorithm for finding the inverse of any general nonsingular tridiagonal matrix. The algorithm is mainly based on the work presented in [Y. Huang, W.F. McColl, Analytic inversion of general tridiagonal matrices, J. Phys. A 30 (1997) 7919–7933] and [M.E.A. El-Mikkawy, A fast algorithm for evaluating nth order tridiagonal determinants, J. Comput. Appl. Math. 166 (2004) 581–584]. It removes all cases where the numeric algorithm in [Y. Huang, W.F. McColl, Analytic inversion of general tridiagonal matrices, J. Phys. A 30 (1997) 7919–7933] fails. The symbolic algorithm is suited for implementation using Computer Algebra Systems (CAS) such as MACSYMA, MAPLE and MATHEMATICA. An illustrative example is given.

Multilevel Hierarchical Matrices

This paper deals with a multilevel construction of hierarchical matrix approximations to the inverses of finite element stiffness matrices. Given a sequence of discretizations $A_{\ell} x_{\ell} = f_{\ell}$, $\ell=0,\ldots,n$, where $A_0 x_0 = f_0$ denotes the coarse grid problem, we will compute $A^{-1}_0$ exactly and then use interpolation to obtain an $\mathcal{H}$‐matrix approximation $A^{-\cal H}_{\ell+1}$ from the approximate $\mathcal{H}$‐matrix inverse $A^{-\cal H}_{\ell}$ on the next coarser grid. We develop an exact interpolation scheme for the inverse of tridiagonal matrices as they appear in the finite element discretization of one‐dimensional differential equations. We then generalize this approach to two spatial dimensions where these efficiently computed approximations to the inverse may serve as preconditioners in iterative solution methods. We illustrate this approach with some numerical tests for convection‐dominated convection‐diffusion problems.

A bibliography on semiseparable matrices*

Currently there is a growing interest in semiseparable matrices and generalized semiseparable matrices. To gain an appreciation of the historical evolution of this concept, we present in this paper an extensive list of publications related to the field of semiseparable matrices. It is interesting to see that semiseparable matrices were investigated in different fields, e.g., integral equations, statistics, vibrational analysis, independently of each other. Also notable is the fact that leading statisticians at that time used semiseparable matrices without knowing their inverses to be tridiagonal. During this historical evolution the definition of semiseparable matrices has always been a difficult point leading to misunderstandings; sometimes they were defined as the inverses of irreducible tridiagonal matrices leading to generator representable matrices, while in other cases they were defined as matrices having low rank blocks below the diagonal. In this overview we present a list of interesting results which contributed to the evolution of the concept of semiseparable matrices as we now know them.

Third‐order‐accurate semi‐implicit Runge–Kutta scheme for incompressible Navier–Stokes equations

AbstractA semi‐implicit three‐step Runge–Kutta scheme for the unsteady incompressible Navier–Stokes equations with third‐order accuracy in time is presented. The higher order of accuracy as compared to the existing semi‐implicit Runge–Kutta schemes is achieved due to one additional inversion of the implicit operator I‐τγL, which requires inversion of tridiagonal matrices when using approximate factorization method. No additional solution of the pressure‐Poisson equation or evaluation of Navier–Stokes operator is needed. The scheme is supplied with a local error estimation and time‐step control algorithm. The temporal third‐order accuracy of the scheme is proved analytically and ascertained by analysing both local and global errors in a numerical example. Copyright © 2005 John Wiley & Sons, Ltd.

An Implicit Q Theorem for Hessenberg-like Matrices

The implicit Q theorem for Hessenberg matrices is a widespread and powerful theorem. It is used in the development of, for example, implicit QR algorithms to compute the eigendecomposition of Hessenberg matrices. Moreover it can also be used to prove the essential uniqueness of orthogonal similarity transformations of matrices to Hessenberg form. The theorem is also valid for symmetric tridiagonal matrices, proving thereby also in the symmetric case its power. Currently there is a growing interest to so-called semiseparable matrices. These matrices can be considered as the inverses of tridiagonal matrices. In a similar way, one can consider Hessenberg-like matrices as the inverses of Hessenberg matrices. In this paper, we formulate and prove an implicit Q theorem for the class of Hessenberg-like matrices. We introduce the notion of strongly unreduced Hessenberg-like matrices and also a method for transforming matrices via orthogonal transformations to this form is proposed. Moreover, as the theorem is valid for Hessenberg-like matrices it is also valid for symmetric semiseparable matrices.

On inverses of tridiagonal matrices

An algorithm for computing the inverse of a general tridiagonal matrix is introduced. This algorithm is obtained by factoring this matrix into the product of two bidiagonal matrices using Crout’s LU factorization, one upper and one lower bidiagonal. A simple recurrence relation is used to generate a sequence of numbers, this sequence is then used to fill in the matrices L, U, L −1, U −1 and consequently the required inverse.

Right‐angled triangle property in inversion of general tridiagonal matrices

In this study a further relationship is extended to the elements in the lower triangle of the inverse of a general tridiagonal matrix for a non‐block case. Once the upper triangle of the inverse is determined based on Huang and McColl's analytical inversion formula, the corresponding lower triangle can be calculated efficiently using two proposed theorems. Each element in the lower triangle is decomposed into two parts: one is the coefficient; the other the counterpart element in the upper triangle. The coefficient is a cross product function of the elements in the tridiagonal matrix and can be easily obtained by using the right‐angled triangle property among the coefficients. This results in a faster computation of the lower triangle of the inverse of a general tridiagonal matrix. Several examples are given to demonstrate the superiority of two theorems developed by the author to Huang and McColl's algorithm. It is shown that the algorithm based on the right‐angled triangle property outperforms Huang and McColl's with regard to the speed of computing the inverse of the tridiagonal matrix. Empirically it is shown that the improvement rates are about 20% in calculating the inverse of Hermite matrices of sizes ranging from 7 by 7 to 20 by 20 for both the clamped and natural cubic spline.

Further study on the high-order double-Fourier-series spectral filtering on a sphere

A high-order harmonic spectral filter (HSF) is further studied using double Fourier series (DFS), which performs filtering in terms of successive inversion of tridiagonal matrices with complex-valued elements. The high-order harmonics filter equation is split into multiple Helmholtz equations. It is found that the filter provides the same order of accuracy as the spectral filter in [J. Comput. Phys. 177 (2002) 313] that consists of the pentadiagonal matrices with real-valued elements. The advantage of the filter over the previous one lies on the simplicity and easiness of numerical implementation or computer coding, just requiring the same complexity as Poisson’s equation solver. However, the operation count associated with the filter increases by a factor of about 2. To circumvent the inefficiency while preserving the simplicity, an easy way to construct pentadiagonal matrices associated with the biharmonic equation is presented in which the tridiagonal matrices related with Poisson’s equation are manipulated. Computational efficiency of the spectral filter is discussed in terms of the relative computing time to the spectral transform. It is revealed that the computing cost (requiring O( N 2) operations with N being the truncation) for the spectral filtering, even with the complex-valued matrices, is not significant in the DFS spectral model that is characterized by O( N 2log 2 N) operations. Filtering with different DFS expansions is discussed with a focus on the accuracy and pole condition. It is shown that the DFS violating the pole conditions produces a discontinuity at poles in case of wave truncation, and its influence spreads over the globe. The spectral filter is applied to two kinds of uniform-grid data, the regular and the shifted grids, and the results are compared with each other. The operator splitting (or spherical harmonics factorization) makes it feasible to apply the finite difference method to the high-order harmonics filter with ease because only the five-point stencil computations are required. The application could also be extended to other numerical methods only if the Helmholtz equation solver is available.

INVERSION FORMULAS FOR TRIDIAGONAL MATRICES WITH APPLICATIONS TO BOUNDARY VALUE PROBLEMS1*

This paper first reviews some results on the structure of inverses of nonsingular and irreducible tridiagonal matrices. Next, explicit inversion formulas are given for certain, not necessarily symmetric, tridiagonal matrices. The results are then applied to matrices arising from discretization of two-point boundary value problems of the Sturm-Liouville type. The results show harmonic relations between the Green functions and the discrete Green functions for the problems. Finally the results are extended to block tridiagonal matrices. This work was supported by Scientific Research Grant-in-Aid from JSPS.

Explicit inverses of some tridiagonal matrices

We give explicit inverses of tridiagonal 2-Toeplitz and 3-Toeplitz matrices which generalize some well-known results concerning the inverse of a tridiagonal Toeplitz matrix.

A Sparse Matrix Arithmetic Based on $\Cal H$ -Matrices. Part I: Introduction to ${\Cal H}$ -Matrices

A class of matrices ( $\Cal H$ -matrices) is introduced which have the following properties. (i) They are sparse in the sense that only few data are needed for their representation. (ii) The matrix-vector multiplication is of almost linear complexity. (iii) In general, sums and products of these matrices are no longer in the same set, but their truncations to the $\Cal H$ -matrix format are again of almost linear complexity. (iv) The same statement holds for the inverse of an $\Cal H$ -matrix. This paper is the first of a series and is devoted to the first introduction of the $\Cal H$ -matrix concept. Two concret formats are described. The first one is the simplest possible. Nevertheless, it allows the exact inversion of tridiagonal matrices. The second one is able to approximate discrete integral operators.

Decay Rates of the Inverse of Nonsymmetric Tridiagonal and Band Matrices

It is well known that the inverse C = [ci,j] of an irreducible nonsingular symmetric tridiagonal matrix is given by two sequences of real numbers, {ui} and {vi}, such that ci,j = u i vj for $i \leq j$. A similar result holds for nonsymmetric matrices A. There the inverse can be described by four sequences {ui},{vi}, {xi},$ and {vi} with u ivi = xiyi. Here we characterize certain properties of A, i.e., being an M-matrix or positive definite, in terms of the ui, vi,xi, and yi. We also establish a relation of zero row sums and zero column sums of A and pairwise constant ui,vi, xi, and yi. Moreover, we consider decay rates for the entries of the inverse of tridiagonal and block tridiagonal (banded) matrices. For diagonally dominant matrices we show that the entries of the inverse strictly decay along a row or column. We give a sharp decay result for tridiagonal irreducible M-matrices and tridiagonal positive definite matrices. We also give a decay rate for arbitrary banded M-matrices.

Analytical inversion of general tridiagonal matrices

In this paper we give a complete analysis for general tridiagonal matrix inversion for both non-block and block cases, and provide some very simple analytical formulae which immediately lead to closed forms for some special cases such as symmetric or Toeplitz tridiagonal matrices.

Analytical inversion of symmetric tridiagonal matrices

In this paper we present an analytical formula for the inversion of symmetrical tridiagonal matrices. The result is of relevance to the solution of a variety of problems in mathematics and physics. As an example, the formula is used to derive an exact analytical solution for the one-dimensional discrete Poisson equation with Dirichlet boundary conditions.

A Review on the Inverse of Symmetric Tridiagonal and Block Tridiagonal Matrices

In this paper some results are reviewed concerning the characterization of inverses of symmetric tridiagonal and block tridiagonal matrices as well as results concerning the decay of the elements of the inverses. These results are obtained by relating the elements of inverses to elements of the Cholesky decompositions of these matrices. This gives explicit formulas for the elements of the inverse and gives rise to stable algorithms to compute them. These expressions also lead to bounds for the decay of the elements of the inverse for problems arising from discretization schemes.

Matrix homographic iterations and bounds for the inverses of certain band matrices

We establish a convergence result for the homographic iteration U i+1 = A−BU −1 iC, where A, B, C, U i are m × m matrices. This enables us to prove the exponential decay for the inverse of certain tridiagonal block matrices.