Abstract
The quadtrees matrix representation has been recently proposed as an alternative to the conventional linear storage of matrices. If all elements of a matrix are zero, then the matrix is represented by an empty tree; otherwise it is represented by a tree consisting of four subtrees, each representing, recursively, a quadrant of the matrix. Using four-way block decomposition, algorithms on quadtrees accelerate on blocks entirely of zeroes, and thereby offer improved performance on sparse matrices. This paper reports the results of experiments done with a quadtree matrix package implemented in REDUCE to compare the performance of quadtree representation with REDUCE's built-in sequential representation of matrices. Tests on addition, multiplication, and inversion of dense, triangular, tridiagonal, and diagonal matrices (both symbolic and numeric) of sizes up to 100×100 show that the quadtree algorithms perform well in a broad range of circumstances, sometimes running orders of magnitude faster than their sequential counterparts.
isiting scientist at Tektronix Labs during the summer of 1987 when this work was done.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
S. K. Abdali & D. D. Saunders. Transitive closure and related semiring properties via eliminants. Theoretical Computer Science 40, 2,3 (1985), 257–274.
E. H. Bareiss. Sylvester's identity and multistep integer-preserving Gaussian elimination. Math. Comp. 22}, 103 (July, 1968), 565–578.
V. N. Faddeeva. Computational Methods of Linear Algebra, Dover, New York (1959).
F. R. Gantmacher. The Theory of Matrices 1, Chelsea, New York (1960).
A. C. Hearn. REDUCE User's Manual, Version 3.3. Rand Publication CP78, The Rand Corp., Santa Monica, CA (July, 1987).
D. E. Knuth. The Art of Computer Programming, I, Fundamental Algorithms, 2nd Ed., Addison-Wesley, Reading, MA (1975).
A. C. McKellar & E. G. Coffman, Jr. Organizing matrices and matrix operations for paged memory systems. Comm. ACM 12, 3 (March, 1969), 153–165.
T. Sasaki & H. Murao. Efficient Gaussian elimination method for symbolic determinants and linear systems. In P. S. Wang (Ed.), Proc. 1981 ACM Symp. on Symbolic and Algebraic Computation, ACM Order No. 505810 (August, 1981), 155–159.
M. K. Sridhar. A new algorithm for parallel solutions of linear equations. Inf. Proc. Lett. 24, (April, 1987), 407–412.
V. Strassen. Gaussian elimination is not optimal. Numer. Math. 13, 4 (August, 1969), 354–356.
D. S. Wise. Representing matrices as quadtrees for parallel processors (extended abstract). ACM SIGSAM Bulletin 18, 3 (August, 1984), 24–25.
D. S. Wise. Parallel decomposition of matrix inversion using quadtrees. In Hwang, K., Jacobs, S. J., and Swartzlander E. E. (Eds.), Proc. 1986 International Conference on Parallel Processing, IEEE Computer Society Press, Washington, 1986, pp. 92–99.
D. S. Wise. Matrix algebra and applicative programming. In G. Kahn (Ed.), Functional Programming Languages and Computer Architecture, Lecture Notes in Computer Science 274, Springer, Berlin (1987), pp. 134–153.
D. S. Wise & J. Franco. Costs of quadtree representation of non-dense matrices. Technical Report No. 229, Computer Science Department, Indiana University (October, 1987).
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1989 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Abdali, S.K., Wise, D.S. (1989). Experiments with quadtree representation of matrices. In: Gianni, P. (eds) Symbolic and Algebraic Computation. ISSAC 1988. Lecture Notes in Computer Science, vol 358. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-51084-2_9
Download citation
DOI: https://doi.org/10.1007/3-540-51084-2_9
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-51084-0
Online ISBN: 978-3-540-46153-1
eBook Packages: Springer Book Archive