Physics
Scientific paper
Aug 1987
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1987pepi...47..333o&link_type=abstract
Physics of the Earth and Planetary Interiors, Volume 47, p. 333-345.
Physics
12
Scientific paper
The Simultaneous Iterative Reconstruction Technique (SIRT) is a variation of Richardson's method for solving linear systems with positive definitive matrices, and can be used for solving any least squares problem. Previous SIRT methods used in tomography have suggested a constant normalization factor for the step size. With this normalization, the convergence rate of the eigencomponents decreases as the eigenvalue decreases, making these methods impractical for obtaining large bandwidth solutions. By allowing the normalization factor to change with each iteration, the error after k iterations is shown to be a k th order polynomial. The factors are then chosen to yield a Chebyshev polynomial so that the maximum error in the iterative method is minimized over a prescribed range of eigenvalues.
Compared with k iterations using a constant normalization, the Chebyshev method requires only sqrt(k) and has the property that all eigencomponents converge at the same rate. Simple expressions are given which permit the number of iterations to be determined in advanced based upon the desired accuracy and bandwidth. A stable ordering of the Chebyshev factors is also given which minimizes the effects of numerical roundoff. Since a good upper bound for the maximum eigenvalue of the normal matrix is essential to the calculations, the well known `power method with shift of origin' is combined with the Chebyshev method to estimate its value.
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