Abstract
A symmetrizer of a nonsymmetric matrix A is the symmetric matrixX that satisfies the equationXA =A tX, wheret indicates the transpose. A symmetrizer is useful in converting a nonsymmetric eigenvalue problem into a symmetric one which is relatively easy to solve and finds applications in stability problems in control theory and in the study of general matrices. Three designs based on VLSI parallel processor arrays are presented to compute a symmetrizer of a lower Hessenberg matrix. Their scope is discussed. The first one is the Leiserson systolic design while the remaining two, viz., the double pipe design and the fitted diagonal design are the derived versions of the first design with improved performance.
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References
Anderson P and Loizou G, On the quadratic convergence of an algorithm which diagonalizes a complex symmetric matrix,J. Inst. Math. Its Appl. 12 (1973) 261–271
Anderson P and Loizou G, A Jacobi-type method for complex symmetric matrices (Handbook),Numer. Math. 25 (1976) 347–363
B N Datta, An algorithm for computing a symmetrizer of a Hessenberg matrix, (unpublished)
Dew P M, VLSI architectures for problems in numerical computation, (ed.) D J Paddon, Super-computers and Parallel Computation, New Series No. 1 (ed.), The Institute of Mathematics and Its Application Series, 1984
Eberlein P J, On the diagonalization of complex symmetric matrices,J. Inst. Math. Its Appl. 7 (1971) 377–383
Evans D J, Designing efficient systolic algorithms for VLSI parallel processor arrays, Parallel Architecture and Computer Vision, 1988
Krishnamurthy E V and Sen S K, Numerical algorithms: Computations in science and engineering (1993) (New Delhi: Affiliated East-West press)
Kung H T, Why systolic architectures?,IEEE Comput. 16 (1982) 37–46
Kung H T and Leiserson C E, Systolic arrays (for VLSI), (eds) I S Duffand and G W StewartSparse Matrix Proceedings 1978, 256–82;SIAM (1979)
Kung S Y, VLSI Array processors (1988) (New Jersey: Prentice-Hall, Englewood Cliffs)
Kung S Y, Arun K S, Gal-Ezer R J and Bhaskar Rao D, Wavefront array processor: language, architecture, and applications,IEEE Trans. Comput. C31 (1982) 1054–1066
Mead C and Conway L, Introduction to VLSI systems (1980) (Reading, Massachusetts: Addison-Wesley)
Seaton J J, Diagonalization of complex symmetric matrices using a modified Jacobi method,Comput. J. 12 (1969) 156–157
Sen S K and Venkaiah V Ch, On computing an equivalent symmetric matrix for a nonsymmetric matrix,Int. J. Comput. Math. 24 (1988) 169–80
Suros R and Montagne E, Optimizing systolic networks by fitted diagonals,Parallel Computing 4 (1987) 167–174
Taussky O, The role of symmetric matrices in the study of general matrices,Linear Algebra Appl. 51 (1972) 147–154
Ullman D J, Computational Aspects of VLSI, (1984) (Standford Univ.: Computer Science Press)
Uwe S and Lother T, Linear systolic arrays for matrix computations,J. Parallel and Distributed Computing 7 (1989) 28–39
Venkaiah V Ch and Sen S K, Computing a matrix symmetrizer exactly using modified multiple modulus residue arithmetic,J. Comput. Appl. Math. 21 (1988) 27–40
Venkaiah V Ch and Sen S K, Error-free symmetrizers and equivalent symmetric matrices,Acta Applicande Mathematicae 21 (1990) 291–313
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Kumar, F.R.K., Sen, S.K. Symmetrizing a Hessenberg matrix: Designs for VLSI parallel processor arrays. Proc. Indian Acad. Sci. (Math. Sci.) 105, 59–71 (1995). https://doi.org/10.1007/BF02840591
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DOI: https://doi.org/10.1007/BF02840591