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.An alternative to performing the singular value decomposition is to factor a matrixVT' where U and V are orthogonal matrices and C is a lower triangular matrix which indicates a separation between two subspaces by the size of its columns. These subspaces are denoted by V = (V1 , V2 ), where the columns of C are partitioned conformally into C = (Cl, C2) with II C2 IIF< E . Here e is some tolerance . In recent years, this has been called the ULV decomposition (ULVD) .If the matrix A results from statistical observations, it is often desired to remove old observations, thus deleting a row from A and its ULVD . In matrix terms, this is called a downdate . A downdating algorithm is proposed that preserves the structure in the downdated matrix C to the extent possible . Strong stability results are proven for these algorithms based upon a new perturbation theory. AMS subject classifications : (Primary) 65F20, 65G05 (Secondary) 65F15 .

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The problem of interconnecting nets with multi-port terminals in VLSI circuits is a direct generalization of the Group Steiner Problem (GSP). The GSP is a combinatorial optimization problem which arises in the routing phase of VLSI circuit design. This problem has been intractable, making it impractical to be used in real-world VLSI applications. This paper presents our work on designing and implementing a parallel approximation algorithm for the GSP based off an existing heuristic on a distributed architecture. Our implementation uses the CUDA-aware MPI approach to compute the approximate minimum-cost Group Steiner tree for several industry-standard VLSI graphs. Our implementation achieves up to 103x speedup compared to the best known serial work for the same graph. We present the speedup results for graphs up to 3k vertices. We also investigate some performance bottleneck issues by analyzing and interpreting the program performance data.