# Systolic Algorithms for the Parallel Solution of Dense Symmetric Positive-Definite Toeplitz Systems

## Abstract

The most popular method for the solution of linear systems of equations with Toeplitz coefficient matrix on a single processor is Levinson’s algorithm, whose intermediate vectors form the Cholesky factor of the inverse of the Toeplitz matrix. However, Levinson’s method is not amenable to efficient parallel implementation. In contrast, use of the Schur algorithm, whose intermediate vectors form the Cholesky factor of the Toeplitz matrix proper, makes it possible to perform the entire solution procedure on one processor array in time linear in the order of the matrix.

By means of the Levinson recursions we will show that all three phases of the Toeplitz system solution process: factorisation, forward elimination and backsubstitution, can be based on Schur recursions. This increased exploitation of the Toeplitz structure then leads to more efficient parallel implementations on systolic arrays.

## Keywords

Systolic Array Toeplitz Matrix Cholesky Factor Very Large Scale Integrate Toeplitz Matrice## Preview

Unable to display preview. Download preview PDF.

## References

- [1]Bareiss, E.H.,
*Numerical Solution of Linear Equations with Toeplitz and Vector Toeplitz Matrices*, Numer. Math., 13 (1969), pp. 404–24.Google Scholar - [2]Brent, R.P., Gustavson, F.G. and Yun, D.Y.Y.,
*Fast Solution of Toeplitz Systems of Equations and Computation of Pade Approximants*, J. Algorithms, 1 (1980), pp. 159–95.MathSciNetCrossRefGoogle Scholar - [3]Brent, R.P., Kung, H.T. and Luk, F.T., Some Linear-Time Algorithms for Systolic Arrays,
*Proc. IFIP 9th World Computer Congress*, North Holland, Amsterdam, 1983, pp. 865–76.Google Scholar - [4]Brent, R.P. and Luk, F.T.,
*A Systolic Array for the Linear-Time Solution of Toeplitz Systemsof Equations*, J. VLSI and Computer Systems, 1 (1983), pp. 1–22.MathSciNetMATHGoogle Scholar - [5]Bunch, J.R.,
*Stability of Methods for Solving Toeplitz Systems of Equations*, SIAM J. Sci. Stat. Comput., 6(1985), pp. 349–64.Google Scholar - [6]Delosme, J.-M.,
*Algorithms for Finite Shift-Rank Processes*, Ph.D. Thesis, Dept of Electrical Engineering, Stanford University, 1982.Google Scholar - [7]Delosme, J.-M. and Ipsen, I.C.F.,
*Parallel Solution of Symmetric Positive Definite Systems with Hyperbolic Rotations*, Linear Algebra and its Applications, 77 (1986), pp. 75–111.MathSciNetMATHCrossRefGoogle Scholar - [8]Delosme, J.-M. and Ipsen, I.C.F., Efficient Systolic Arrays for the Solution of Toeplitz Systems: An Illustration of a Methodology for the Construction of Systolic Architectures in VLSI,
*Systolic Arrays*, Adam Hilger, 1987, pp. 37–46.Google Scholar - [9]Delosme, J.-M. and Morf, M., Normalized Doubling Algorithms for Finite Shift-Rank Processes,
*Proc. 20th IEEE Conference on Decision and Control*, 1981, pp. 246–8.Google Scholar - [10]Grenander, U. and Szego, G.,
*Toeplitz Forms and their Applications*, University of California Press, 1958.MATHGoogle Scholar - [11]Iohvidov, I.S.,
*Hankel and Toeplitz Matrices and Forms*, Birkhauser, 1982.Google Scholar - [12]Kung, S.-Y. and Hu, Y.H.,
*A Highly Concurrent Algorithm and Pipelined Architecture for Solving Toeplitz Systems*, IEEE Trans. Acoustics, Speech, and Signal Processing, ASSP-31 (1983), pp. 66–76.Google Scholar - [13]Kung, H.T. and Leiserson, C.E., Systolic Arrays (for VLSI),
*Sparse Matrix Proceedings*, SIAM, Philadelphia, PA, 1978, pp. 256–82.Google Scholar - [14]Levinson, N.,
*The Wiener RMS (Root-Mean-Square) Error Criterion in Filter Design and Prediction*, J. Math. Phys., 25 (1947), pp. 261–78.Google Scholar - [15]Morf, M.,
*Fast Algorithms for Multivariable Systems*, Ph.D. Thesis, Dept of Electrical Engineering, Stanford University, 1974.Google Scholar - [16]Nash, J.G., Hansen, S. and Nudd, G.R., VLSI Processor Array for Matrix Manipulation,
*CMU Conference on VLSI Systems and Computations*, Computer Science Press, 1981, pp. 367–73.Google Scholar - [17]Roebuck, P.A. and Barnett, S.,
*A Survey of Toeplitz and Related Matrices*, Int. J. Systems Sci., 9 (1978), pp. 921–34.MathSciNetMATHCrossRefGoogle Scholar - [18]Schur, I.,
*Ueber Potenzreihen die im Innern des Einheitskreises Beschraenkt Sind*, J. Reine Angewandte Mathematik, 147 (1917), pp. 205–32.Google Scholar - [19]Trench, W.F.,
*An Algorithm for The Inversion of Finite Toeplitz Matrices*, J. Soc. Indust. Appl. Math., 12(1964), pp. 515–22.MathSciNetMATHCrossRefGoogle Scholar