Parallel solution of certain toeplitz linear systems
SIAM Journal on Computing
An improved Newton interaction for the generalized inverse of a Matrix, with applications
SIAM Journal on Scientific and Statistical Computing
Stability results for collocation methods for Volterra integral equations
Applied Mathematics and Computation
Fast inversion of triangular Toeplitz matrices
Theoretical Computer Science - Algebraic and numerical algorithm
Approximate iterations for structured matrices
Numerische Mathematik
Linear algebra for tensor problems
Computing
Approximation of $2^d\times2^d$ Matrices Using Tensor Decomposition
SIAM Journal on Matrix Analysis and Applications
Acceleration of the inversion of triangular Toeplitz matrices and polynomial division
CASC'11 Proceedings of the 13th international conference on Computer algebra in scientific computing
SIAM Journal on Scientific Computing
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We address a linear fractional differential equation and develop effective solution methods using algorithms for the inversion of triangular Toeplitz matrices and the recently proposed QTT format. The inverses of such matrices can be computed by the divide and conquer and modified Bini's algorithms, for which we present the versions with the QTT approximation. We also present an efficient formula for the shift of vectors given in QTT format, which is used in the divide and conquer algorithm. As a result, we reduce the complexity of inversion from the fast Fourier level O(nlogn) to the speed of superfast Fourier transform, i.e., O(log^2n). The results of the paper are illustrated by numerical examples.