Efficient computer manipulation of tensor products with applications to multidimensional approximation

Pereyra, V.; Scherer, G.

doi:10.1090/S0025-5718-1973-0395196-6

Efficient computer manipulation of tensor products with applications to multidimensional approximation
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by V. Pereyra and G. Scherer PDF

Math. Comp. 27 (1973), 595-605 Request permission

Abstract:

The objective of this paper is twofold: (a) To make it possible to perform matrix-vector operations in tensor product spaces, using only the factors ($n \cdot {p^2}$ words of information for $\otimes _{i = 1}^n{A_i},{A_i} \in \mathcal {L}({E^p},{E^p})$) instead of the tensor-product operators themselves (${({p^2})^n}$ words of information). (b) To produce efficient algorithms for solving systems of linear equations with coefficient matrices being tensor products of nonsingular matrices, with special application to the approximation of multidimensional linear functionals.

References

Åke Björck and Victor Pereyra, Solution of Vandermonde systems of equations, Math. Comp. 24 (1970), 893–903. MR 290541, DOI 10.1090/S0025-5718-1970-0290541-1
Eugene Isaacson and Herbert Bishop Keller, Analysis of numerical methods, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR 0201039
Nathan Jacobson, Lectures in Abstract Algebra. Vol. I. Basic Concepts, D. Van Nostrand Co., Inc., Toronto-New York-London, 1951. MR 0041102
G. Birkhoff, M. H. Schultz, and R. S. Varga, Piecewise Hermite interpolation in one and two variables with applications to partial differential equations, Numer. Math. 11 (1968), 232–256. MR 226817, DOI 10.1007/BF02161845
G. Galimberti and V. Pereyra, Solving confluent Vandermonde systems of Hermite type, Numer. Math. 18 (1971/72), 44–60. MR 300417, DOI 10.1007/BF01398458
Philip J. Davis and Philip Rabinowitz, Numerical integration, Blaisdell Publishing Co. [Ginn and Co.], Waltham, Mass.-Toronto, Ont.-London, 1967. MR 0211604