On the Quadratic Convergence of an Algorithm which Diagonalizes a Complex Symmetric Matrix

Anderson, P. J.; Loizou, George

doi:10.1093/imamat/12.3.261

Cited by 5 publications

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“…An efficient Jacobi iteration for complex symmetric matrices was given by Eberlein (1971). Anderson & Loizou (1973) show that Eberlein's method converges rapidly when the off-diagonal elements are small; an ALGOL implementation of the algorithm can be found in Anderson & Loizou (1976).…”

Section: S -T S -L = Kmentioning

confidence: 99%

Useful properties of the system matrix for a homogeneous anisotropic visco-elastic solid

Frazer

Fryer

1989

Geophysical Journal International

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For the computation of synthetic seismograms in a generally anisotropic layered earth it is necessary to find the eigenvectors and eigenvalues of the first order elastic system matrix A for many values of wavenumber and frequency. The analytical formulas used to construct the eigenvectors of A in the isotropic case are not available in the general anisotropic case so one must use numerical methods whose speed often depends on an efficient use of the properties of A. First we review the symmetries of A and the conditions under which A is not semi-simple. Then we construct a perturbation theory for the eigenvectors of A. Finally we show how to make A symmetric so that special techniques for symmetric matrices, such as Jacobi iteration, can be used. All the results given here remain valid when the medium is attenuating, i.e. when the elastic coefficients are complex.Key words: elastic system matrix, perturbation theory 1 REVIEW As the elastic system matrix is discussed in many other papers and books we include here only a bare outline of its derivation and a few details that will be needed below. For general anisotropy it is convenient to use Cartesian coordinates {xl, x 2 , x 3 } with x j as the depth coordinate. The constitutive equation relating the stress tensor t and the displacement vector u is then (1) t.. = c . . * u tj rjkl k.1 in which '*' denotes time convolution and repeated subscripts imply summation. The momentum equation is in which p is mass density, t is time and f is body force per unit volume. To remove derivatives in xl, x 2 and t from these equations we take a triple Fourier transform: r m r mAfter some algebraic manipulations one then obtains the first order system in the formis the vector of motions, u = [u,, u 2 , u 3 ] 9 and scaled tractions, t = ( i / w ) [ t l , , 223, t33]T. One reason for scaling the tractions is that it makes the components of A real when the medium is lossless, which is useful in certains kinds of computations. The system matrix A has the form where T, S, and C are 3 x 3 submatrices and C and S are symmetric. Note that A has this same form and these same symmetries even if any or all of w, p , , p z , or the ciikl are complex. Thomson, Clarke & Garmany (1986) have summarized the symmetries of A that result from various definitions of b, including our equation (5), and from not factoring iw out of A. It is straightforward to translate the results of this paper into forms consistent with these other definitions. ISOTROPY A N D ANISOTROPYFor an isotropic medium one can calculate explicit formulas for the eigenvectors of A by the use of potentials (e.g. Aki & Richards 1980, p. 166). These formulas show that the eigenvectors of A have the following properties:(a) A is diagonable and the eigenvectors of A span the 173

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Section: S -T S -L = Kmentioning

confidence: 99%