Diagonal Submatrix Research Articles

Identification of Generalized Memory Polynomials Using Two-Tone Signals

This paper shows that the coefficients of a generalized memory polynomial model of a nonlinear device can be estimated by examining the output when the input is a series of two-tone signals. There are several benefits to using two-tone signals rather than noise-like waveforms for system identification, including the relative ease of generating two-tone signals with a high signal-to-noise ratio and high dynamic range. The two-tone approach described here results in a block upper triangular linear system of equations that can be solved for the unknown coefficients. The block upper triangular structure provides a sufficient condition for the matrix to be nonsingular: if each diagonal submatrix has full column rank, so does the matrix. In addition to showing that certain sets of two-tone signals satisfy this sufficient condition, methods for controlling the condition number are also given. Simulations verify the two-tone estimation method and illustrate the effects of noise and matrix conditioning. Some useful results regarding certain Khatri–Rao products can be found in the proofs.

A parallel fast boundary element method using cyclic graph decompositions

We propose a method of a parallel distribution of densely populated matrices arising in boundary element discretizations of partial differential equations. In our method the underlying boundary element mesh consisting of n elements is decomposed into N submeshes. The related N×N submatrices are assigned to N concurrent processes to be assembled. Additionally we require each process to hold exactly one diagonal submatrix, since its assembling is typically most time consuming when applying fast boundary elements. We obtain a class of such optimal parallel distributions of the submeshes and corresponding submatrices by cyclic decompositions of undirected complete graphs. It results in a method the theoretical complexity of which is O((n/N)log(n/N))$O((n/\sqrt {N})\log (n/\sqrt {N}))$ in terms of time for the setup, assembling, matrix action, as well as memory consumption per process. Nevertheless, numerical experiments up to n=2744832 and N=273 on a real-world geometry document that the method exhibits superior parallel scalability O((n/N)logn)$O((n/N)\,\log n)$ of the overall time, while the memory consumption scales accordingly to the theoretical estimate.

Matrix Partitions with Finitely Many Obstructions

Each $m$ by $m$ symmetric matrix $M$ over $0, 1, *$, defines a partition problem, in which an input graph $G$ is to be partitioned into $m$ parts with adjacencies governed by $M$, in the sense that two distinct vertices in (possibly equal) parts $i$ and $j$ are adjacent if $M(i,j)=1$, and nonadjacent if $M(i,j)=0$. (The entry $*$ implies no restriction.) We ask which matrix partition problems admit a characterization by a finite set of forbidden induced subgraphs. We prove that matrices containing a certain two by two diagonal submatrix $S$ never have such characterizations. We then develop a recursive technique that allows us (with some extra effort) to verify that matrices without $S$ of size five or less always have a finite forbidden induced subgraph characterization. However, we exhibit a six by six matrix without $S$ which cannot be characterized by finitely many induced subgraphs. We also explore the connection between finite forbidden subgraph characterizations and related questions on the descriptive and computational complexity of matrix partition problems.

Stability of matrices with negative diagonal submatrices

This paper discusses stability conditions for matrices that determine the homogeneous dynamics of systems of linear second-order differential equations. In particular, we focus on situations in which these matrices have a negative diagonal submatrix. We present a number of theorems that provide conditions which are sufficient for either stability or instability of such matrices. In order to discuss the instability theorems and unify them with earlier results, we introduce the concept of the dominant diagonal number of a matrix.

An iterative solution strategy for boundary element equations from mixed boundary value problems

An iterative solution strategy is presented for mixed boundary value problems from the direct boundary element method, where both Dirichlet and Neumann constraints are prescribed on the boundary. In the mixed boundary value problem constraints are imposed upon the BEM system of equations Hu = Gq to Ax = b . The A matrix is fully-populated, unsymmetric, and made up of columns from H and G , corresponding to the Neumann and Dirichlet boundary conditions, respectively. In this work, the coefficient matrix A is rearranged to place the columns from H in the left-most columns of A and the columns from G in the right-most columns of A . The diagonal submatrix in A containing terms from G is then reduced by elimination while the diagonal submatrix containing terms from H is retained for iteration. Jacobi, Gauss-Seidel adn conjugate gradient normal iterative solvers are considered. Convergence of the proposed solution strategy is studied using four, two-dimensional potential and elasticity problems.

An iterative solution strategy for boundary element equations from mixed boundary value problems

An iterative solution strategy is presented for mixed boundary value problems from the direct boundary element method, where both Dirichlet and Neumann constraints are prescribed on the boundary. In the mixed boundary value problem constraints are imposed upon the BEM system of equations Hu = Gq to Ax = b . The A matrix is fully-populated, unsymmetric, and made up of columns from H and G, corresponding to the Neumann and Dirichlet boundary conditions, respectively. In this work, the coefficient matrix A is rearranged to place the columns from H in the left-most columns of A and the columns from G in the right-most columns of A. The diagonal submatrix in A containing terms from G is then reduced by elimination while the diagonal submatrix containing terms from H is retained for iteration. Jacobi, Gauss-Seidel adn conjugate gradient normal iterative solvers are considered. Convergence of the proposed solution strategy is studied using four, two-dimensional potential and elasticity problems.

Exact parametric analysis of stochastic Petri nets

An algorithm for exact parametric analysis of stochastic Petri nets is presented. The algorithm is derived from the theory of decomposition and aggregation of Markov chains. The transition rate of interest is confined into a diagonal submatrix of the associated Markov chain by row and column permutations. Every time a new value is assigned to the transition, a smaller Markov chain is analyzed. As a result, the computational cost is greatly reduced. >

State estimation using augmented blocked matrices

A novel method of blocking the Hachtel formulation for power system state estimation is proposed. The key idea is organization of the Hachtel formulation into a submatrix block structure that conforms to that of the incidence matrix of the network. This leads to the normal first-neighbor topological structure of the blocked matrix, which is processed in the same sparsity-preserving order as the nodal admittance matrix. Explicit inversion of diagonal submatrix blocks overcomes former difficulties with small or zero diagonals and further enhances numerical stability. Equality constraints are enforced exactly instead of as measurements with artificially large weighting factors. Another feature of the method is that it can accommodate a more realistic covariance matrix having nonzero off-diagonal weighting factors corresponding to correlations between the active and reactive power measurements at a bus. In fact, any covariance terms that fall within the submatrix blocks (in locations that are now zeros) can be accommodated with no effects on computational burden. As indicated by its large reduction in the numerical condition number of three representation test problems, the proposed method should be able to overcome the usual causes of ill-conditioning of power system state estimation. As shown by comparisons of its computational requirements with those of other methods, there is no sacrifice in speed to achieve robustness. >

Non‐iterative solution of a tridiagonal‐type matrix coupled by diagonal submatrices

AbstractA non‐iterative method is presented for solving a system of algebraic equations forming a banded matrix with each diagonal submatrix being tridiagonal and all other submatrices being diagonal. A recursion relation is assumed between successive elements of the solution vector. This relation is determined by equating a linearly combined form of it with the original set of equations also being linearly combined. The solution vector is determined by back‐substitution of elements in the recursion relation. Since the method takes advantage of the banded structure of the matrix, it is more efficient than the conventional methods. A FORTRAN program that incorporates the present method is included.

A problem by Jensen

Throughout, all matrices are real, S denotes a symmetric matrix, T a lower triangular matrix, and D a diagonal matrix. The following problem appeared in the May <u>Signum Newsletter</u> [1]. Given S, for what values of μ can T and D be found such that TDT t = (S - μI)? A sufficient condition is that μ not be an eigenvalue of any proper leading diagonal submatrix of S. A necessary and sufficient condition is this. Let m(ν) be the multiplicity of μ as an eigenvalue of the leading ν by ν diagonal submatrix of S, zero if μ is not an eigenvalue of that submatrix. The condition is that the sequence, m(ν), (including the element for S as an improper submatrix of itself) be nondecreasing. This follows from the lemma below.