Arrowhead Matrices Research Articles

On the arrowhead-Fibonacci numbers

AbstractIn this paper, we define the arrowhead-Fibonacci numbers by using the arrowhead matrix of the characteristic polynomial of thek-step Fibonacci sequence and then we give some of their properties. Also, we study the arrowhead-Fibonacci sequence modulomand we obtain the cyclic groups from the generating matrix of the arrowhead-Fibonacci numbers when read modulom. Then we derive the relationships between the orders of the cyclic groups obtained and the periods of the arrowhead-Fibonacci sequence modulom.

Forward stable eigenvalue decomposition of rank-one modifications of diagonal matrices

We present a new algorithm for solving an eigenvalue problem for a real symmetric matrix which is a rank-one modification of a diagonal matrix. The algorithm computes each eigenvalue and all components of the corresponding eigenvector with high relative accuracy in O(n) operations. The algorithm is based on a shift-and-invert approach. Only a single element of the inverse of the shifted matrix eventually needs to be computed with double the working precision. Each eigenvalue and the corresponding eigenvector can be computed separately, which makes the algorithm adaptable for parallel computing. Our results extend to the complex Hermitian case. The algorithm is similar to the algorithm for solving the eigenvalue problem for real symmetric arrowhead matrices from N. Jakovčević Stor et al. (2015) [16].

AN ITERATIVE ALGORITHM FOR THE LEAST SQUARES SOLUTIONS OF MATRIX EQUATIONS OVER SYMMETRIC ARROWHEAD MATRICES

This paper concerns with exploiting an oblique projection technique to solve a general class of large and sparse least squares problem over symmetric arrowhead matrices. As a matter of fact, we develop the conjugate gradient least squares (CGLS) algorithm to obtain the minimum norm symmetric arrowhead least squares solution of the general coupled matrix equations. Furthermore, an approach is offered for computing the optimal approximate symmetric arrowhead solution of the mentioned least squares problem corresponding to a given arbitrary matrix group. In addition, the minimization property of the proposed algorithm is established by utilizing the feature of approximate solutions derived by the projection method. Finally, some numerical experiments are examined which reveal the applicability and feasibility of the handled algorithm.

A Fast Method for Computing the Inverse of Symmetric Block Arrowhead Matrices

Abstract: We propose an effective method to find the inverse of symmetric block arrowhead matrices which often appear in areas of applied science and engineering such as head-positioning systems of hard disk drives or kinematic chains of industrial robots. Block arrowhead matrices can be considered as generalisation of arrowhead matrices occurring in physical problems and engineering. The proposed method is based on LDLT decomposition and we show that the inversion of the large block arrowhead matrices can be more effective when one uses our method. Numerical results are presented in the examples.

An efficient method for computing the inverse of arrowhead matrices

Abstract In this paper we propose a simple and effective method to find the inverse of arrowhead matrices which often appear in wide areas of applied science and engineering such as wireless communications systems, molecular physics, oscillators vibrationally coupled with Fermi liquid, and eigenvalue problems. A modified Sherman–Morrison inverse matrix method is proposed for computing the inverse of an arrowhead matrix. The effectiveness of the proposed method is illustrated and numerical results are presented along with comparative results.

Least-Squares Solutions of the Matrix EquationsAXB+CYD=HandAXB+CXD=Hfor Symmetric Arrowhead Matrices and Associated Approximation Problems

The least-squares solutions of the matrix equationsAXB+CYD=HandAXB+CXD=Hfor symmetric arrowhead matrices are discussed. By using the Kronecker product and stretching function of matrices, the explicit representations of the general solution are given. Also, it is shown that the best approximation solution is unique and an explicit expression of the solution is derived.

Least squares solutions of the matrix equation [formula omitted] with the least norm for symmetric arrowhead matrices

In this paper, the least squares solution of the matrix equation AXB+CYD=E for symmetric arrowhead matrices with the least norm is discussed. By using Moore–Penrose inverse and the Kronecker product, the general expression of the solution to this problem is derived. A corresponding numerical algorithm and an example are also given.

Accurate eigenvalue decomposition of real symmetric arrowhead matrices and applications

We present a new algorithm for solving the eigenvalue problem for an n×n real symmetric arrowhead matrix. The algorithm computes all eigenvalues and all components of the corresponding eigenvectors with high relative accuracy in O(n2) operations under certain circumstances. The algorithm is based on a shift-and-invert approach. Only a single element of the inverse of the shifted matrix eventually needs to be computed with double the working precision. Each eigenvalue and the corresponding eigenvector can be computed separately, which makes the algorithm adaptable for parallel computing. Our results extend to Hermitian arrowhead matrices, real symmetric diagonal-plus-rank-one matrices and singular value decomposition of real triangular arrowhead matrices.

An Inverse Eigenvalue Problem for Star Spring-Mass System

The mass-normalized stiffness matrix of the star spring-mass system is an arrowhead matrix. Given two eigenpairs of the arrowhead matrix. It is assumed that the total mass of a star spring-mass system is known. The problem of constructing the physical elements of the system from the known data is considered. The necessary and sufficient conditions for the construction of a physical realizable system with positive mass and spring stiffness are established. If these conditions are satisfied, the system may be constructed uniquely.

Bounds for Eigenvalues of Arrowhead Matrices and Their Applications to Hub Matrices and Wireless Communications

This paper considers the lower and upper bounds of eigenvalues of arrow-head matrices. We propose a parameterized decomposition of an arrowhead matrix which is a sum of a diagonal matrix and a special kind of arrowhead matrix whose eigenvalues can be computed explicitly. The eigenvalues of the arrowhead matrix are then estimated in terms of eigenvalues of the diagonal matrix and the special arrowhead matrix by using Weyl's theorem. Improved bounds of the eigenvalues are obtained by choosing a decomposition of the arrowhead matrix which can provide best bounds. Some applications of these results to hub matrices and wireless communications are discussed.

Linking the TPR1, DPR1 and Arrow-Head Matrix Structures

Some recent polynomial root-finders rely on effective solution of the eigenproblem for special matrices such as DPR1 (that is, diagonal plus rank-one) and arrow-head matrices. We examine the correlation between these two classes and their links to the Frobenius companion matrix, and we show a Gauss similarity transform of a TPR1 (that is, triangular plus rank-one) matrix into DPR1 and arrow-head matrices. Theoretically, the known unitary similarity transforms of a general matrix into a block triangular matrix with TPR1 diagonal blocks enable the extension of the cited effective eigen-solvers from DPR1 and arrow-head matrices to general matrices. Practically, however, the numerical stability problems with these transforms may limit their value to some special classes of input matrices.

Fast and stable QR eigenvalue algorithms for generalized companion matrices and secular equations

We introduce a class ** of n×n structured matrices which includes three well-known classes of generalized companion matrices: tridiagonal plus rank-one matrices (comrade matrices), diagonal plus rank-one matrices and arrowhead matrices. Relying on the structure properties of **, we show that if A ∈ ** then A′=RQ ∈ **, where A=QR is the QR decomposition of A. This allows one to implement the QR iteration for computing the eigenvalues and the eigenvectors of any A ∈ ** with O(n) arithmetic operations per iteration and with O(n) memory storage. This iteration, applied to generalized companion matrices, provides new O(n2) flops algorithms for computing polynomial zeros and for solving the associated (rational) secular equations. Numerical experiments confirm the effectiveness and the robustness of our approach.

Processor-efficient sparse matrix-vector multiplication

The matrix-vector multiplication operation is the kernel of most numerical algorithms.Typically, matrix-vector multiplication algorithms exploit the sparsity of a matrix, either to reduce the time taken or the memory used. In the case of parallel implementations of numerical algorithms, the sparsity of a matrix can also be used to reduce the number of processing elements (PEs) required. For example, in Leiserson's systolic array algorithm for matrix-vector multiplication the nonzeros in the matrix are partitioned into bands and each band is fed into a different PE. Similarly, in Melhem's algorithm, the nonzeros are partitioned into stripes and each stripe is fed into a distinct PE. However, in many practical applications, such as the numerical solution of partial differential equations, a matrix is much sparser than is indicated by its bandwidth or by its stripe width. A new measure of sparsity of a matrix called staircase width is introduced. A staircase in a matrix M = (m i,j)n×n is a set of matrix positions S = {(i, j) | m i,j ≠ 0, such that no position in S is strictly to the northeast of another position in S. The staircase cover of a matrix is a partition of the positions of nonzero entries in the matrix into staircases and the staircase width of a matrix is the size of a smallest staircase cover of the matrix. The staircase width of a matrix is never larger than the bandwidth or the stripe width of a matrix. Moreover, it is significantly smaller than either for some commonly occurring types of matrices such as arrowhead matrices. Experiments with sparse matrices derived from a variety of engineering problems suggest that, in practice, the staircase width of a matrix is about half the stripe width of the matrix. An efficient algorithm to compute a minimum width staircase cover of a matrix is presented. Staircase covers are used as the basis of a new parallel algorithm to compute a matrix-vector product on a linear systolic array of PEs. This algorithm uses as many PEs as the staircase width of the matrix, and therefore, saves on the number of PEs relative to algorithms of Leiserson and Melhem. It is shown that despite using fewer PEs, the algorithm uses no more time steps than Leiserson's or Melhem's algorithm.

The Feasibility of the Interval Gaussian Algorithm for Arrowhead Matrices

We give a new class of interval matrices for which the interval Gaussian algorithm is feasible.

A Parallel Davidson-Type Algorithm for Several Eigenvalues

In this paper we propose a new parallelization of the Davidson algorithm adapted for many eigenvalues. In our parallelization we use a relationship between two consecutive subspaces which allows us to calculate eigenvalues in the subspace through an arrowhead matrix. Theoretical timing estimates for the parallel algorithm are developed and compared against our numerical results on the Paragon. Finally our algorithm is compared against another recent parallel algorithm for multiple eigenvalues, but based on Arnoldi: PARPACK.

An approximate inverse matrix technique for arrowhead matrices

A new class of approximate inverse matrix techniques based on the concept of sparse LU-type factorization procedures is introduced for computing explicitly inverses of arrowhead matrices without inverting the decomposition factors. Explicit preconditioned iterative schemes in conjunction with AIM techniques are presented for the efficient solution of linear systems. Applications of the method on a linear system are discussed and numerical results are given.

A new parallel chasing algorithm for transforming arrowhead matrices to tridiagonal form

Rutishauser, Gragg and Harrod and finally H.Y. Zha used the same class of chasing algorithms for transforming arrowhead matrices to tridiagonal form. Using a graphical theoretical approach, we propose a new chasing algorithm. Although this algorithm has the same sequential computational complexity and backward error properties as the old algorithms, it is better suited for a pipelined approach. The parallel algorithm for this new chasing method is described, with performance results on the Paragon and nCUBE. Comparison results between the old and the new algorithms are also presented.

A graph-theoretic model of symmetric givens operations and its implications

Symmetric Givens operations ( A′ ← GAG T , G a Givens rotation matrix) are a basic tool in many matrix computations, especially for eigenvalue-eigenvector computations. A graph-theoretic model of these operations is given for symmetric matrices, analogous to the graph-theoretic model of Cholesky factorization. Using this model, it is shown that unless there is “accidental cancellation,” it is impossible to reduce a range of different matrix classes to tridiagonal form in o( n 2) Givens operations; these classes include arrowhead matrices, pentadiagonal matrices, and cyclic tridiagonal matrices.

Efficient reduction algorithms for bordered band matrices

AbstractVarious plane rotation patterns are presented, which provide stable algorithms for reducing a b‐band matrix bordered by p rows and/or columns to (b + p)‐band form. These schemes generalize previously presented O(N2) reduction algorithms for matrices of order N, b = 1, and p = 1 to the reduction of more general b‐band, p‐bordered matrices where b ≥ 1 and p ≥ 1. Moreover, by splitting the matrix into two similarly structured submatrices and chasing nonzeros to the corners in two directions, the newly proposed patterns reduce the number of required rotations and hence the computational cost by one half compared to the other existing one‐way chasing algorithms. Symmetric, as well as more general matrices, are considered. An example of the first type is the symmetric arrowhead matrix that arises in solving inverse eigenvalue problems. Examples of the second type are found in updating the singular value decomposition (SVD) and the partial SVD.

A Divide-and-Conquer Algorithm for the Symmetric Tridiagonal Eigenproblem

The authors present a stable and efficient divide-and-conquer algorithm for computing the spectral decomposition of an $N \times N$ symmetric tridiagonal matrix. The key elements are a new, stable method for finding the spectral decomposition of a symmetric arrowhead matrix and a new implementation of deflation. Numerical results show that this algorithm is competitive with bisection with inverse iteration, Cuppen's divide-and-conquer algorithm, and the QR algorithm for solving the symmetric tridiagonal eigenproblem.