Compound matrix block diagonalization for efficient solution of eigenproblems in structural mechanics

SummaryRecently, methods were developed for the decomposition of special matrices, leading to canonical forms I-IV. In this paper, the conditions required for the decomposability of such matrices are studied. It is shown that these canonical forms are obtainable by special block diagonalization of matrices, having certain properties. Here, tri- to penta-block diagonal matrices are studied and methods are developed for their decomposition. These methods are incorporated in the efficient solution of numerous eigenproblems of structural mechanics.

[1]  A. Kaveh,et al.  Graph symmetry and dynamic systems , 2004 .

[2]  G. Stewart,et al.  A block QR algorithm and the singular value decomposition , 1993 .

[3]  I. Hargittai,et al.  Symmetry 2 : unifying human understanding , 1990 .

[4]  Haesun Park.  Efficient diagonalization of oversized matrices on a distributed-memory multiprocessor , 1990 .

[5]  Ali Kaveh,et al.  Structural Mechanics: Graph and Matrix Methods , 1995 .

[6]  Ali Kaveh,et al.  Eigensolutions for factorable matrices of special patterns , 2003 .

[7]  EIGENVALUES OF FACTORABLE MATRICES WITH FORM IV SYMMETRY , 2005 .

[8]  Peter G. Glockner,et al.  SYMMETRY IN STRUCTURAL MECHANICS , 1973 .

[9]  Ali Kaveh,et al.  Eigensolution of symmetric frames using graph factorization , 2004 .

[10]  D. Schuch,et al.  Symmetries in Science XIV , 2010 .

[11]  Ali Kaveh,et al.  Block diagonalization of adjacency and Laplacian matrices for graph product; applications in structural mechanics , 2006 .

[12]  A. Kaveh,et al.  Eigensolutions for matrices of special structures , 2002 .

[13]  R. Livesley.  Mathematical Methods for Engineers , 1989 .

[14]  Milija N. Pavlović,et al.  A SYMMETRY-ADAPTED FLEXIBILITY APPROACH FOR MULTI-STOREY SPACE FRAMES: GENERAL OUTLINE AND SYMMETRY-ADAPTED REDUNDANTS , 1995 .

[15]  M. Sadkane,et al.  Convergence Analysis of the Block Householder Block Diagonalization Algorithm , 2005 .

[16]  Ali Kaveh,et al.  EIGENSOLUTIONS FOR MATRICES OF SPECIAL PATTERNS , 2003 .

[17]  Gene H. Golub,et al.  Matrix computations , 1983 .

[18]  M.A. Hasan,et al.  Block eigenvalue decomposition using nth roots of the identity matrix , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[19]  James Demmel,et al.  Applied Numerical Linear Algebra , 1997 .