This Article 
 Bibliographic References 
 Add to: 
The Real Two-Zero Algorithm: A Parallel Algorithm to Reduce a Real Matrix to a Real Schur Form
January 1995 (vol. 6 no. 1)
pp. 48-62

Abstract—In this paper, we introduce a new method to reduce a real matrix to a real Schur form by a sequence of similarity transformations that are 3-D orthogonal transformations. Two significant features of this method are that (1) all the transformed matrices and all the computations are done in the real field, and (2) it can be easily parallelized. We call the algorithm that uses this method the real two-zero (RTZ) algorithm. We describe in this paper both serial and parallel implementations of the RTZ algorithm. Our tests indicate that the rate of convergence to a real Schur form is quadratic for real near-normal matrices with real distinct Eigenvalues. Suppose $n$ is the order of a real matrix $A$. In order to choose a sequence of 3-D orthogonal transformations on $A$, we need to determine some ordering on triples in ${\cal T} = \{(k,l,m)\mid 1 \leq k \char'74 l \char'74 m \leq n\}$, where $(k,l,m)$ defines the three coordinates under the 3-D transformation. We show how the ordering of the triples used in our implementations can be generated cyclically in an algorithm.

Index Terms—Real two-zero algorithm, 3-D orthogonal transformations, ordering on triples, real Schur form, algorithm to generate triples, householder matrix, modified Gao–Thomas algorithm, quadratic convergence, real field

[1] C. Bischof,“A parallel ordering for the block Jacobi method on a hypercube architecture for parallel computation,”in M. Heath, Ed.,SIAM Hypercube II, 1987.
[2] A. W. Bojanczyk and A. Lutoborski,“Computation of the Euler angles of a symmetric matrix,”SIAM J. Matrix Analysis, to appear.
[3] R. Brent and F. Luk,“The solution of singular-value and symmetric Eigenvalue problems on multiprocessor arrays,”SIAM SISC, vol. 6, pp. 69–84, 1985.
[4] P. J. Eberlein,“A Jacobi-like method for the automatic computation of Eigenvalues and Eigenvectors of an arbitrary matrix,”J. Soc. Indus. Appl. Math., vol. 10, no. 1,PP.——, Mar. 1962.
[5] ——,“Solution to the complex Eigenproblem by a norm reducing Jacobi-type method,”Math., vol. 14, pp. 232–245, 1970.
[6] ——,“On the diagonalization of complex symmetric matrices,”J. Inst. Math. Applics., vol. 7, pp. 337–383, 1971.
[7] P. J. Eberlein and J. Boothroyd,“Solution to the Eigenproblem by a norm reducing Jacobi-type method,”Numer. Math., vol. 11 pp. 1–12, 1968.
[8] P. J. Eberlein,“On using the Jacobi method for parallel computation,”SIAM J. Alg. Disc. Math., vol. 8, pp. 790–796, 1987.
[9] ——,“On using the Jacobi method on the hypercube,”in M. T. Heath, Ed.,Proc. 2nd Conf. Hypercube Multiprocessors, 1987, pp. 605–611.
[10] P. J. Eberlein and M. Mantharam,“Jacobi-sets for the Eigenproblem and their effect on convergence studied by graphic representations,”ch. 3,Proc. IV SIAM Conf. on Parallel Process for Sci. Comput., Chicago IL, USA, Dec. 1989.
[11] P. J. Eberlein and H. Park,“Efficient implementation of Jacobi algorithms and Jacobi sets on distributed memory architectures,”J. Parallel Disrib. Computing, vol. 8, pp. 358–366, 1990.
[12] G. E. Forsythe and P. Henrici,“The cyclic Jacobi method for computing the principal values of a complex matrix,”Trans. Amer. Math. Soc., vol. 94, pp. 1–23, 1960.
[13] G. R. Gao and S. J. Thomas,“An optimal parallel Jacobi-like algorithm for the singular value decomposition,”Tech. Rep. TR-SOCS-87.16, McGill Univ., Montréal, PQ, Canada, 1988.
[14] F. T. Luk and H. T. Park,“A proof of convergence for two parallel Jacobi SVD algorithms,”IEEE Trans. Comput., vol. 38, pp. 806–811, June 1989.
[15] M. Mantharam and P. J. Eberlein,“Two new Jacobi-sets for parallel Jacobi methods,”Parallel Computing, vol. 1, pp. 437–454, 1993.
[16] ——,“Block recursive algorithm to generate Jacobi-sets,”Parallel Computing, vol. 19, pp. 481–496, 1993.
[17] G. W. Stewart,“A Jacobi-like algorithm for computing the Schur decomposition of a non-hermitian matrix,”SIAM J. Sci. Statis. Comput., vol. 6, pp. 853–864, 1985.
[18] K. Veseli\' c,“On a class of Jacobi-like procedures for diagonalizing arbitrary real matrices,”Numer. Math., vol. 33, pp. 157–172, 1979.
[19] ——,“A quadratically convergent Jacobi-like method for real matrices with complex Eigenvalues,”Numer. Math., vol. 33, pp. 425–435, 1979.

Mythili Mantharam, P. J. Eberlein, "The Real Two-Zero Algorithm: A Parallel Algorithm to Reduce a Real Matrix to a Real Schur Form," IEEE Transactions on Parallel and Distributed Systems, vol. 6, no. 1, pp. 48-62, Jan. 1995, doi:10.1109/71.363411
Usage of this product signifies your acceptance of the Terms of Use.