Acta Numerica 2002: Volume 11 (Acta Numerica) by Arieh Iserles PDF

By Arieh Iserles

Acta Numerica anually surveys an important advancements in numerical arithmetic and medical computing. the topics and authors of the substantial articles are selected by means of a distinct foreign editorial board, in an effort to file an important and well timed advancements in a way available to the broader neighborhood of execs with an curiosity in medical computing. Acta Numerica volumes are a necessary software not just for researchers and execs wishing to improve their realizing of numerical suggestions and algorithms and stick to new advancements. also they are used as complicated educating aids at faculties and universities (many of the unique articles are used because the top source for graduate courses).

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Regarding the ToIEP as a nonlinear system of n equations in n unknowns, a natural tactic would be a Newton-type iteration. The schemes in Friedland, Nocedal and Overton (1987), originally proposed for the more general class of PIEPs, are in this class and certainly applicable to the ToIEP. The inverse Rayleigh quotient algorithm in Laurie (1988, 1991) is also equivalent to a Newton-type variation. These methods do not exploit the Toeplitz structure and can suffer from local convergence. The iterative scheme proposed by Trench (1997) seems to have more robust performance, but still no global convergence can be proved.

Mass-spring system Suppose that the ith. particle has mass mj, that the springs satisfy Hooke's law, and that the ith. spring has spring constant fcj. Let Ui(t) denote the 24 MOODY T. CHU AND GENE H. GOLUB horizontal displacement of the ith particle at time t. Then it is easy to see that the equation of motion is given by mi + k2(u2 - at - «*)> at at « = 2,... 11). 12). The direct problem calculates the natural frequencies and modes of the mass-sprint system from given values of rrii and kk- The inverse problem requires calculating quantities such as *+ i+1 and , i+1 from the spectral data.

An}, Let the cardinality £ of the index set C be denoted by |£|. The PEIEP is to determine (complete) values for the n 2 — \C\ positions that do not belong to £ so as to satisfy the spectral constraint. The Jacobi structure can be considered as a special case of PEIEP where, in addition to the desired symmetry of the band, elements outside the tridiagonal band are required to be zero. Another interesting variation of the PEIEP is the completion problem, where only a one-to-one correspondence between the £ positions in C and the £ prescribed values { a i , .

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