An inverse eigenvalue problem, where a matrix is to be constructed from some or all of its eigenvalues, may not have a real-valued solution at all. An approximate solution in the sense of least squares is sometimes desirable. Two types of least squares problems are formulated and explored in this paper. In spite of their different appearance, the two problems are shown to be equivalent. Thus one new numerical method, modified from the conventional alternating projection method, is proposed. The method converges linearly and globally and can be used to generate good starting values for other faster but more expensive and locally convergent methods. The idea can be applied to multiplicative inverse eigenvalue problems for the purpose of preconditioning. Numerical examples are presented.

This paper presents a recursive procedure to compute the Moore-Penrose inverse of a matrix A. The method is based on the expression for the Moore-Penrose inverse of rank-one modified matrix. The computational complexity of the method is analyzed and a numerical example is included. A variant of the algorithm with lower computational complexity is also proposed. Both algorithms are tested on randomly generated matrices. Numerical performance confirms our theoretic results.

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How to efficiently map the nodes and links in a given virtual network (VN) to those in the substrate network (SN) so that the residual substrate network (RSN) can host as many VN requests as possible is a major challenge in virtual network embedding. Most research has developed heuristic algorithms with interactive or two-stage methods. These methods, however, could cause the RSN fragmented into several disconnected components that are insufficient to host a large number of given VN requests.
Without loss of generality, we assume that SNs, as the Internet, are small world and scale free, meaning that the average number of hops between any two nodes is a small constant and that most nodes have small degrees while only a small number of nodes have large degrees. Taking advantage of these two properties, we devise in this paper a new twostage VN embedding approach to improve the connectivity of the residual substrate network so that it has the capacity to host more VN requests. Our algorithm uses a greedy strategy that maps neighboring VN nodes to substrate nodes whose distances are bounded by a small constant. We also map the edge nodes of given VN, i.e., nodes with small degrees, to nodes with large degrees in the SN. We then map the links by solving shortest path problems. Our experimental evaluations show that our algorithms offer better mappings and results in significantly fewer rejections for VN requests

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