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A Proposal for Toeplitz Matrix Calculations
Author(s) -
Strang Gilbert
Publication year - 1986
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1002/sapm1986742171
Subject(s) - toeplitz matrix , circulant matrix , preconditioner , conjugate gradient method , eigenvalues and eigenvectors , mathematics , diagonal , matrix (chemical analysis) , positive definite matrix , computation , levinson recursion , algebra over a field , linear system , algorithm , pure mathematics , mathematical analysis , physics , geometry , materials science , quantum mechanics , composite material
In contrast to the usual (and successful) direct methods for Toeplitz systems Ax = b , we propose an algorithm based on the conjugate gradient method. The preconditioner is a circulant, so that all matrices have constant diagonals and all matrix‐vector multiplications use the Fast Fourier Transform. We also suggest a technique for the eigenvalue problem, where current methods are less satisfactory. If the first indications are supported by further experiment, this new approach may have useful applications—including nearly Toeplitz systems, and parallel computations.