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

  • Ilse C. F. Ipsen
Conference paper
Part of the The IMA Volumes in Mathematics and Its Applications book series (IMA, volume 13)


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.


Systolic Array Toeplitz Matrix Cholesky Factor Very Large Scale Integrate Toeplitz Matrice 
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Copyright information

© Springer-Verlag New York Inc. 1988

Authors and Affiliations

  • Ilse C. F. Ipsen
    • 1
  1. 1.Department of Computer ScienceYale UniversityNew HavenUSA

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