By Arieh Iserles

ISBN-10: 0521443563

ISBN-13: 9780521443562

This is often the second one quantity within the annual sequence, Acta Numerica, a set of invited survey papers encompassing all elements of numerical research. Following the good fortune of the 1st quantity, this year's papers conceal some of the middle issues in numerical equipment at the present time. Papers on annealing difficulties, parallel equipment, area decomposition, and multigrid tools will permit somebody with an curiosity in computational and mathematical numerics to fast take hold of the newest advancements and rising tendencies during this varied topic.

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**Extra resources for Acta Numerica 1993 (Volume 2)**

**Sample text**

13) We make the following standard assumption. 10 The rank of A equals the number of its rows, and the interior feasible set of the primaldual problem T° := {(x, z) : x, z > 0, Ax = b, A*y + z = c for some y} is not empty. It is well established that the linear programming problem has a unique CONTINUATION AND PATH FOLLOWING 33 solution under these assumptions. 9 is lin < c*x u > Y Inx, : Ax = o, x min s c x — u XX { ^* where // > 0 is the barrier penalty parameter. 10, the logarithmic barrier function is strictly convex and has a unique minimal point x(n) for all /z > 0.

A typical example is the Toda flow which has been studied as a continuous analogue of the QR algorithm. A survey of these ideas has been given by Watkins (1984). According to Watkins, although it seems that the Toda flow and related flows yield insight into the workings of algorithms, they do not necessarily directly offer algorithms which are competitive with standard library algo rithms that have been developed and polished over numerous years. Surprisingly, Li and Li (1992), Li and Rhee (1989), Li, Zeng and Cong CONTINUATION AND PATH FOLLOWING 29 (1992) and Li, Zhang and Sun (1991) have been able to construct special implementations of homotopy methods which are now at least competitive with the library routines of EISPACK and IMSL for linear eigenvalue prob lems.

Further computational experience comparing an interior point method OBI and a simplex method CPLEX is reported in technical reports Bixby, Gregory, Lustig, Marsten and Shanno (1991), Carpenter and Shanno (1991) and Lustig, Marsten and Shanno (1991). Polak, Higgins and Mayne (1992) have given an algorithm for solving semiinfinite minimax problems which bears a resemblance to the interior penalty function methods. They report numerical results which show that the algorithm is extremely robust and its performance is at least comparable to that of current firstorder minimax algorithms.

### Acta Numerica 1993 (Volume 2) by Arieh Iserles

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