Block Triangular Form Research Articles

An efficient estimation algorithm for the model parameters of robotic manipulators

An approach to identification of the model parameters of robotic manipulators is proposed in which neither prior knowledge of the geometric parameters nor restrictive robot motions are required. The number of the model parameters to be identified is minimized through a regrouping procedure for the Lagrangian functions of robotic manipulators. The identification model for the model parameters is formulated in an upper block triangular form. Based on these results, a computationally efficient estimation algorithm for the model parameters is obtained. To illustrate the practical use of the method, a four-degree-of-freedom SCARA robot is examined. >

The rank of powers of matrices in a block triangular form

A graph theoretic upper bound for the rank of powers of matrices in a block triangular form is given. The index theorem for general matrices is derived as a corollary

Sparsity structure and Gaussian elimination

In this paper we collate and discuss some results on the sparsity structure of a matrix. If a matrix is irreducible, Gaussian elimination yields an LU factorization in which L has at least one entry beneath the diagonal in every column except the last and U has at least one entry to the right of the diagonal in every row except the last. If this factorization is used to solve the equation Ax=b , the intermediate vector has an entry in its last component and the solution x is full. Furthermore, the inverse of A is full.Where the matrix is reducible, these results are applicable to the diagonal blocks of its block triangular form.

Undecomposable Hamiltonian matrices with pure imaginary spectrum

A subclass of Hamiltonian matrices with pure imaginary spectrum, not reducible to Hamiltonian block-triangular form, is described.

Basis factorization method for multistage linear programming problems with an application to optimal energy plant operation

An algorithm based on the basis factorization method for solving multistage linear programming problems is presented. In applying linear programming for optimal dynamic-plant operation, the constraint appears in the form of a staircase structure. When pivotings occur within each stage, the inversion of the basis matrix is easily obtained using inversions of small square diagonal blocks. In general cases, pivotings across stages occur. However, this basis matrix can be easily transformed into the former square block triangular form using the correction factor. The correction factor means multiple pivotings among the bases and contains structural columns corresponding to different columns among these bases. When these structural columns are maintained in computer storage, the relative cost factor for general bases is easily obtained using a set of elementary column operations. A program code using the above idea is developed and applied to in-plant energy operating systems. A computational example shows that the computing time of the proposed algorithm is less than 10 percent of the general purpose LP code MPS-II.

Combinatorial Canonical Form of Layered Mixed Matrices and Its Application to Block-Triangularization of Systems of Linear/Nonlinear Equations

With a view to obtaining an efficient procedure for solving large-scale systems of equations, canonical block-triangular forms are defined for layered mixed matrices and for mixed matrices, and some practical examples are presented. The canonical forms are obtained from a straightforward application of the decomposition principle for submodular functions. The relation to the existing decomposition techniques for electrical networks, as well as to the Dulmage–Mendelsohn decomposition, is also discussed.

Simultaneous reduction of sets of matrices under similarity

Let S be a set of n × n matrices over a field F , and A the algebra generated by S over F . The problem of deciding whether the elements of S can be simultaneously reduced (to block-triangular form with the diagonal blocks of some specified size) is considered, and an account is given of various methods used to attack the problem. Most of the techniques use representation theory to obtain information on A . The problems of simultaneous triangularization, existence of common eigenvectors, etc. are also considered. The aim of the paper is to survey the methods used to attack these problems and to give some typical results. The paper does not contain many new results.

Regularity of systems of linear differential equations of block-triangle form

wh~re x ~_ R n, i~ = dx/dt, t ~ R, A (t) is an n • n-dimensional matrix function, continuous and bounded on the entire axis R. We denote by C~ R ) the space of continuous vector or matrix functions bounded on R. We recall that the system (i) is said to be weakly regular on R if for every vector-function / (t) ~ C o (R) the nonhomogeneous system ,t = A (t) x qf (t) has at least one solution bounded on R and is said to be regular on R (e-dichotomous) if the nonhomogeneous system has only one bounded solution for each f (t)~ C o (R). It is well known (see [I, 2]) that weak regularity on R of system (I) is equivalent to the existence of a quadratic form V (t, x) --- with continuously differentiable matrix coefficients S(t) (S (t) ~ C i (R)) bounded on R such that the derivative V (t, x) along the solution of the adjoint system 2 = --A T (t) x is of fixed sign: ~--[I xll 2. The angle brackets here denote the ordinary scalar product in R n, ----]! x II z. In the case det S (t)~ 0, for Vt ~_ II the system (I) is regular on R (e-dichotomous). It is well known (see [3]) that one may always extend a weakly regular system (I) on R to a system of block-triangular form

A simple algorithm for computing canonical forms

It is well known that all linear time-invariant controllable systems can be transformed to Brunovsky canonical form by a transformation consisting only of coordinate changes and linear feedback. However, the actual procedures for doing this have tended to be overly complex. The technique intorduced here is envisioned as an on-line procedure and is inspired by George Meyer's tangent model for nonlinear systems. The process utilizes Meyer's block triangular form as an intermediate step in going to Brunovsky form. The method also involves orthogonal matrices, thus eliminating the need for the computation of matrix inverses. In addition, the Kronecker indices can be computed as a by-product of this transformation so it is not necessary to know them in advance.

Block Triangularization of Algebras of Matrices

This paper is concerned with the interdependence of the irreducible constituents of an algebra of n × n matrices over a field F. It is shown that there is a similarity transformation reducing the algebra to a block triangular form in which, ateach pair of diagonal places, the blocks either are always equal or may be occupied by any entries from the corresponding irreducible constituents. A recent theorem of Kaplan-sky is extended as an application of this result.

Application of sparse matrix techniques to inter-regional input-output analysis

Combining the concepts described in this paper makes it possible to solve inter-regional input-output systems (and other types of large, sparse, linear systems) with considerable efficiency in storage and computation. The exact number of operations and corresponding savings in computational time and storage depend on the particular zero-non zero structure of each matrix in the system, but in any case the savings can be enormous. A recommended procedure is summarized below. 1. Take the entire inter-regional IO system which includes the representation of each of the regions and of the links among them and express it in the formMx=Bz or, more simply,Mx=b (since bothB andz are known). It may be helpful for this purpose to draw a picture of the large matrixM. 2. PartitionM into blocks, exploiting the structure of the particular system. For example, at the very least, the matrix or matrices corresponding to each region will probably be separate blocks. The analysis required for this step may lead to reformulating the matrix representation of the given economic system by, for example, replacing a set of equations with linear combinations of these same equations, particularly for the equations representing the links among regions. 3. Identify an appropriate partition of the blocks ofM, and a corresponding partition of the vectorsx andb, for performing a block factorization. This solution algorithm, which solves forx inMx=b (whereM now represents the entire system) is not available as a packaged program and so for the foreseeable future must be written in ‘home-made’ computer code that assumes a sparse matrix storage scheme compatible with the package to be used (as in 4 below). The algorithm for block factorization in the 2×2 case relevant to our particular inter-regional IO system is given in Section 8 of this paper. 4. Some steps of the algorithm require solving smaller systems of the formHu=v foru, given some matrixH and some vectorv. Efficient packaged subroutines can be obtained at low cost in order to solve these systems by finding the block triangular form corresponding to a particularH, then performing theLU factorization of the diagonal blocks only. The home-made code will interact with these subroutines.

Algorithm 529: Permutations To Block Triangular Form [F1

article Free AccessArtifacts AvailableArtifacts Evaluated & Reusable Share on Algorithm 529: Permutations To Block Triangular Form [F1] Authors: I. S. Duff Computer Science and Systems Division, Building 8.9, AERE Harwell, Didcot, Oxfordshire, OX11 ORA, England Computer Science and Systems Division, Building 8.9, AERE Harwell, Didcot, Oxfordshire, OX11 ORA, EnglandView Profile , J. K. Reid Computer Science and Systems Division, Building 8.9, AERE Harwell, Didcot, Oxfordshire, OX11 ORA, England Computer Science and Systems Division, Building 8.9, AERE Harwell, Didcot, Oxfordshire, OX11 ORA, EnglandView Profile Authors Info & Claims ACM Transactions on Mathematical SoftwareVolume 4Issue 2June 1978 pp 189–192https://doi.org/10.1145/355780.355790Online:01 June 1978Publication History 40citation863DownloadsMetricsTotal Citations40Total Downloads863Last 12 Months30Last 6 weeks4 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

On the singular graph and the Weyr characteristic of anM-matrix

LetA be anM-matrix in standard lower block triangular form, with diagonal blocksAii irreducible. LetS be the set of indices α such that the diagonal blockAαα is singular. We define the singular graph ofA to be the setS with partial order defined by α > β if there exists a chain of non-zero blocksAαi, Aij, ⋯, Alβ.

A survey of sparse matrix research

This paper surveys the state of the art in sparse matrix research in January 1976. Much of the survey deals with the solution of sparse simultaneous linear equations, including the storage of such matrices and the effect of paging on sparse matrix algorithms. In the symmetric case, relevant terms from graph theory are defined. Band systems and matrices arising from the discretization of partial differential equations are treated as separate cases. Preordering techniques are surveyed with particular emphasis on partitioning (to block triangular form) and tearing (to bordered block triangular form). Methods for solving the least squares problem and for sparse linear programming are also reviewed. The sparse eigenproblem is discussed with particular reference to some fairly recent iterative methods. There is a short discussion of general iterative techniques, and reference is made to good standard texts in this field. Design considerations when implementing sparse matrix algorithms are examined and finally comments are made concerning the availability of codes in this area.

On Permutations to Block Triangular Form

On Permutations to Block Triangular Form

Use of sparse matrix techniques in numerical solution of differential equations for renal counterflow systems

Mammals concentrate urine by intricate counterflow systems in their kidneys. The mathematical models of such systems involve the numerical solution of a system of differential equations. (A preliminary report of this work was given in Notices of the American Mathematical Society 21, A498 (1974).) This leads to a system of nonlinear algebraic equations, that are solved by the Newton-Raphson method. The basic step in the Newton-Raphson method is the repeated solution of a system of linear equations with the Jacobian of the nonlinear equations as the coefficient matrix. It is shown that the zero-nonzero structure of the Jacobian can be predicted from the physical connectivity of the model. We exploit this zero-nonzero structure to permute the Jacobian to a bordered block triangular form, which is then used to compute the correction to the solution vector of the nonlinear equations. For this computation a modified form of partial pivoting is used. Results are given that show the high accuracy of the computed solutions, the optimum use of the internal computer storage, and the fast rate of computation.

A note on the decomposition of systems of sparse non-linear equations

In this paper we point out some advantages of decomposing a system of non-linear equations into block-triangular form directly instead of decomposing the linearized systems.

The logarithmic limit-set of an algebraic variety

Let C be the field of complex numbers and V a subvariety of (C{O})n. To study the exponential behavior of Vat infinity, we define V(a) as the set of limitpoints on the unit sphere Sn-1 of the set of real n-tuples (u, log I i u.,u log IXn ), where x e V and u, = (1 + (log lx,l)2) -2. More algebraically, in the case of arbitrary base-field k we can look places at infinity on V and use the values of the associated valuations on X1, Xn to construct an analogous set V(b). Thirdly, simply by studying the terms occurring in elements of the ideal I defining V, we define another closely related set, V',). These concepts are introduced to prove a conjecture of A. E. Zalessky on the action of GL(n, Z) on k[X1 1, . . ., Xn 1], then studied further. It is shown among other things that V(b) = V(c) (when defined) V(a. If a certain natural conjecture is true, then equality holds where we wrote - and the common set V. Sn1 is a finite union of convex spherical polytopes. 1. A conjecture of Zalessky. Let k be a field, and k[X ]=k[X11,..., Xn l] the ring obtained by adjoining n commuting indeterminates and their inverses to k. This is the group algebra on the free abelian group of rank n, Zn, so GL(n, Z) has a natural action on it. Call a subgroup of Zn nontrivial if it is of infinite order and infinite index in Zn; and call an ideal I( k[X'] nontrivial if it is of infinite dimension (i.e., nonzero) and infinite codimension in k[X+] as k-vector spaces. A. E. Zalessky conjectures in [1, Problem V.9], and we shall here prove: THEOREM 1. Let I be a nontrivial ideal in k[X ], and Hc GL(n, Z) the stabilizer subgroup of I. Then H has a subgroup Ho offinite index, which stabilizes a nontrivial subgroup of Zn (equivalently, which can be put into block-triangular form

The modifiedLR algorithm for complex Hessenberg matrices

The LR algorithm of Rutishauser [3] is based on the observation that if $$ A = LR $$ (1) where L is unit lower-triangular and R is upper-triangular then B defined by $$ B = {L^{ - 1}}AL = RL $$ (2) is similar to A. Hence if we write $$ \matrix{ {{A_s} = {L_s}{R_s},} & {{R_s}{L_s} = {A_{s + 1}},} \cr } $$ (3) a sequence of matrices is obtained each of which is similar to A 1. Rutishauser showed [3, 4, 5] if A 1 has roots of distinct moduli then, in general As tends to upper triangular form, the diagonal elements tending to the roots arranged in order of decreasing modulus. If A 1 has some roots of equal modulus then A s does not tend to strictly triangular form but rather to block-triangular form. Corresponding to a block of k roots of the same modulus there is a diagonal block of order k which does not tend to a limit, but its roots tend to the k eigenvalues.

A graph theoretic approach to matrix inversion by partitioning

LetM be a square matrix whose entries are in some field. Our object is to find a permutation matrixP such thatPM P ?1 is completely reduced, i.e., is partitioned in block triangular form, so that all submatrices below its diagonal are 0 and all diagonal submatrices are square and irreducible. LetA be the binary (0, 1) matrix obtained fromM by preserving the 0's ofM and replacing the nonzero entries ofM by 1's. ThenA may be regarded as the adjacency matrix of a directed graphD. CallD strongly connected orstrong if any two points ofD are mutually reachable by directed paths. Astrong component ofD is a maximal strong subgraph. Thecondensation D * ofD is that digraph whose points are the strong components ofD and whose lines are induced by those ofD. By known methods, we constructD * from the digraph,D whose adjacency matrixA was obtained from the original matrixM. LetA * be the adjacency matrix ofD *. It is easy to show that there exists a permutation matrixQ such thatQA * Q ?1 is an upper triangular matrix. The determination of an appropriate permutation matrixP from this matrixQ is straightforward.