8 Concluding Remarks

\Numerical inverting of matrices of high order", Bull. Amer. 18 Rounding errors do grow rapidly with the problem size n. The nal columns in the table give averages and maxima of`relative errors' jb RBT i ? b i j j j b RBT j j 1 where the b i values are either the computed solution from LINPACK, or the exact solution. For poorly-conditioned problems the RBT has large roundoo errors, which are somewhat worse than those of LINPACK. The ill-conditioned problems C(i+j,j) and Turing have respective condition numbers 2 4n (spectral norm) and n 2 n?1 (row sum norm). Edelman 3] has shown that normally distributed random matrices are well-behaved, with expected spectral norm essentially p 2n (actually this has been known for quite some time 26, p.299]). Although the errors were not as bad as we feared, they do suggest the RBT approach as implemented here should be limited to well-behaved problems. In spite of the fact that the code is not optimized or reened, the results show that the approach is already usable, and for the test matrices used often gives numerical results comparable in quality to that of LINPACK's dgefa/dgesl. The Random Butterry Transformation (RBT) pre-and post-multiplies a given matrix with non-singular random matrices. In this paper we have studied RBTs based on butterry and recursive butterry matrices that resemble the Fast Fourier Transform (FFT). We noted that RBTs have three important beneets. First, RBTs have provable randomization properties. Second, they can be performed eeciently, and in particular should be useful on many parallel and vector machines with architectures that support FFT-like computations. Third, RBTs have useful numerical properties , if restricted to unitary matrices: many important matrix norms are invariant under unitary transformations, and therefore condition numbers with respect to these norms are also invariant. Non-unitary RBTs can be constructed that also have similar numerical properties. The emphasis of the paper has been on describing how RBTs can be used to transform nonsin-gular matrices into nondegenerate matrices (with probability 1). This transformation has various applications in computational linear algebra. A noteworthy possibility is the development of new Gaussian elimination algorithms that operate without pivoting. The RBT appears to ooer new strategies for linear systems solution that are well-suited to high-performance computers. Putting the results here in perspective, we have shown that by paying a bit in preprocessing with certain random matrices, we can buy out …