Inverses Of Tridiagonal Matrices Research Articles

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.

Parallel solution of block tridiagonal linear systems

The explicit structure of the inverse of block tridiagonal matrices is presented in terms of blocks defined by linear recurrence relations. Parallel algorithms are shown which solve block second order linear recurrences without using commutativity. Moreover we investigate the parallel solution of the associated block tridiagonal linear system. Using this theoretical background, the implementation of the algorithms is analyzed both on a small number of processors and on a hypercube. The resulting complexity is given in terms of parallel steps, each consisting of block operations, and the cost due to interprocessor communications is taken into account, too.

The inversion of cyclic tridiagonal matrices

The inversion of cyclic tridiagonal matrices

Hermitian operator methods for reaction‐diffusion equations

AbstractA variety of time‐linearization, quasilinearization, operator‐splitting, and implicit techniques which use compact or Hermitian operators has been developed for and applied to one‐dimensional reaction‐diffusion equations. Compact operators are compared with second‐order accurate spatial approximations in order to assess the accuracy and efficiency of Hermitian techniques. It is shown that time‐linearization, quasilinearization, and implicit techniques which use compact operators are less accurate than second‐order accurate spatial discretizations if first‐order approximations are employed to evaluate the time derivatives. This is attributed to first‐order accurati temporal truncation errors. Compact operator techniques which use second‐order temporal approximations are found to be more accurate and efficient than second‐order accurate, in both space and time, algorithms. Quasilinearization methods are found to be more accurate than time‐linearization schemes. However, quasilinearization techniques are less efficient because they require the inversion of block tridiagonal matrices at each iteration. Some improvements in accuracy can be obtained by using partial quasilinearization and linearizing each equation with respect to the variable whose equation is being solved. Operator‐splitting methods which use compact differences to evaluate the diffusion operator were found to be less accurate than operator‐splitting procedures employ second‐order accurate spatial approximations. Comparisons among the methods presented in this paper are shown in terms of the L2‐norm errors and computed wave speeds for a variety of time steps and grid spacings: The numerical efficiency is assessed in terms of the CPU time required to achieve the same accuracy.

On the additive structure of the inverses of banded matrices

The additive structure of the inverses of banded matrices is investigated. Under certain conditions, the inverse of a (2 k+1)-diagonal symmetric banded matrix can be expressed as a sum of k symmetric matrices belonging to the class of inverses of symmetric irreducible tridiagonal matrices. In the nonsymmetric case, a more complicated structure is obtained. Applications are mentioned for the resolution of constant coefficient banded linear systems in VLSI models.

The explicit inverses of general tridiagonal matrices and some applications

Tridiagonal or Jacobi matrices arise in several problems in electrical engineering as well as in different areas of mathematics. — However, there is little known on the inverses of such matrices. The explicit inverses of general tridiagonal matrices are presented in this paper. The method is based on an efficient implementation of Cramer's rule or Coate's flow graph technique and the theory of continuants. The technique is well suited to computer programming and is extremely efficient in symbolical analysis of electrical networks. It could also be efficiently implemented on a parallel computer. — Sensitivity studies or modern optimization techniques used in iterative network synthesis often require the computation of partial derivatives. The proposed technique permits these to be calculated analytically with great case.

Estimates for the inverse of tridiagonal matrices arising in boundary-value problems

By considering tridiagonal matrices as three-term recurrence relations with Dirichlet boundary conditions, one can formulate their inverses in terms of Green's functions. This analysis is applied to three-point difference schemes for 1-D problems, and five-point difference schemes for 2-D problems. We derive either an explicit inverse of the Jacobian or a sharp estimate for both uniform and nonuniform grids.

Analysis of Economic Time Series: A Synthesis

A History of the Idea of Unobserved Components in the Analysis of Economic Time Series. Introduction to the Theory of Stationary Time Series. The Spectral Representation and Its Estimation. Formulation and Analysis of Unobserved-Components Models. Elements of the Theory of Prediction and Extraction. Formulation of Unobserved-Components Models and Canonical Forms. Estimation of Unobserved-Components and Canonical Models. Appraisal of Seasonal Adjustment Techniques. On the Comparative Structure of Serial Dependence in Some U.S. Price Series. Formulation and Estimation of Mixed Moving-Average Autoregressive Models for Single Time Series: Examples. Formulation and Estimation of Multivariate Mixed Moving-Average Autoregressive Time-Series Models. Formulation and Estimation of Unobserved-Components Models: Examples. Application to the Formulation of Distributed-Lag Models. A Time-Series Model of the U.S. Cattle Industry. Appendices: The Work of Buys Ballot. Some Requisite Theory of Functions of a Complex Variable. Fourier Series and Analysis. Whittle's Theorem. Inversion of Tridiagonal Matrices and a Method for Inverting Toeplitz Matrices. Spectral Densities, Actual and Theoretical, Eight Series. Derivation of a Distributed-Lag Relation between Sales and Production: A Simple Example. References. Author Index. Subject Index.

A theorem on inverse of tridiagonal matrices

Tridiagonal or Jacobi matrices arise in many diverse branches of mathematics and have been studied extensively. However, there is little written about the inverses of such matrices. In this paper we characterize those matrices with nonzero diagonal elements whose inverses are tridiagonal. The arguments given are elementaryand show that matrices with tridiagonal inverses have an interesting structure.

Gaussian families and a theorem on patterned matrices

In this paper we use the properties of the covariance matrix of a Gaussian Markovian family to give a probabilistic proof of a theorem about inverses of tridiagonal matrices.

Gaussian families and a theorem on patterned matrices

In this paper we use the properties of the covariance matrix of a Gaussian Markovian family to give a probabilistic proof of a theorem about inverses of tridiagonal matrices.

Interpolation by quadratic splines

Convergence properties of quadratic spline interpolation of continuous functions that does not necessarily take place at the midpoints of mesh intervals are investigated. A theorem giving lower bounds on the elements of the inverse of certain tridiagonal matrices is proved. This result is used to precisely relate the norm of certain interpolating projections to the points of interpolation and local mesh ratios. It is shown, for example, that for Lipschitz continuous functions, any choice of interpolation points, one in each mesh interval, uniformly bounded away from the mesh points, yields convergence at the best possible rate with no mesh ratio restriction.

Methods of inverting tridiagonal matrices

EFFICIENT algorithms for the inversion of symmetric tridiagonal matrices are obtained. The results were published in [1]. Tridiagonal matrices are used not only in the application of finite difference methods to boundary value problems for second-order differential equations [2], but also in the solution of problems of nuclear physics [3]. Hence there is great interest in economical methods for the inversion of highorder band matrices by computer. In this paper efficient algorithms for the inversion of symmetric tridiagonal matrices are obtained. The methods of inversion obtained are compared with other methods. The theorem proved is useful for the solution in analytic form of the problem of processing physical information about the motion of charged particles in bubble chambers [4].

The Explicit Inverses of Two Commonly Occurring Matrices

Explicit formulae are given for the inverses of certain tridiagonal scalar and block matrices. During an investigation into the convergence properties of natural splines it was found convenient to have the explicit forms for the inverses of certain tridiagonal matrices. Special forms of these matrices arise in other branches of numerical analysis and so it may be useful to record them. The derivation is elementary but complicated and will only be indicated.

The explicit inverses of two commonly occurring matrices

Explicit formulae are given for the inverses of certain tridiagonal scalar and block matrices. During an investigation into the convergence properties of natural splines it was found convenient to have the explicit forms for the inverses of certain tridiagonal matrices. Special forms of these matrices arise in other branches of numerical analysis and so it may be useful to record them. The derivation is elementary but complicated and will only be indicated.

On the Propagation of Errors in the Inversion of Certain Tridiagonal Matrices

When the differential equation of heat conduction is replaced by the implicit difference analog, one is led to the solution of

On the propagation of errors in the inversion of certain tridiagonal matrices

When the differential equation of heat conduction is replaced by the implicit difference analog, one is led to the solution of Ay = b where A is a tridiagonal matrix whose elements on the principal diagonal are = 2 + 2r and whose elements off the principal diagonal are = -r. The system of equations may be solved by the following algorithm: \[ β k = u = r 2 β k − 1 − 1 , β 1 = u 1 ; γ k = − r β − 1 ; z k = ( b k + r z k − 1 ) β k − 1 , z 1 = b 1 u − 1 ; y k = z k − γ k y k + 1 , y M = z M . {\beta _k}\, = \,u\, = \,{r^2}\beta _{k - 1}^{ - 1},\,{\beta _1}\, = \,{u_1}\,;\qquad {\gamma _k}\, = \, - r{\beta ^{ - 1}};\qquad {z_k}\, = \,({b_k}\, + \,r{z_{k - 1}})\beta _k^{ - 1},\,{z_1}\, = \,{b_1}{u^{ - 1}};\qquad {y_k}\, = \,{z_k}\, - \,{\gamma _k}{y_{k + 1}},\qquad {y_M}\,=\,{z_M}. \] An upper bound of the round-off errors in the computed values of the y k {y_k} ’s is obtained. An actual test case showed that the theoretical upper bound is about four times larger than the true round-off error. Moreover, the theoretical upper bound does not seem to vary appreciably with r.