An implicit finite difference method is developed for a one-dimensional frac- tional percolation equation （FPE） with the Dirichlet and fractional boundary conditions. The stability and convergence ...are discussed for two special cases, i.e., a continued seep- age flow with a monotone percolation coefficient and a seepage flow with the fractional Neumann boundary condition. The accuracy and efficiency of the method are checked with two numerical examples.
The probability for two monic polynomials of a positive degree
n with coefficients in the finite field
F
q
to be relatively prime turns out to be identical with the probability for an
n
×
n
Hankel ...matrix over
F
q
to be nonsingular. Motivated by this, we give an explicit map from pairs of coprime polynomials to nonsingular Hankel matrices that explains this connection. A basic tool used here is the classical notion of Bezoutian of two polynomials. Moreover, we give simpler and direct proofs of the general formulae for the number of
m-tuples of relatively prime polynomials over
F
q
of given degrees and for the number of
n
×
n
Hankel matrices over
F
q
of a given rank.
In this paper we describe some properties of companion matrices and demonstrate some special
patterns that arisewhen a Toeplitz or a Hankel matrix is multiplied by a related companion matrix.We ...present
a necessary and sufficient condition, generalizing known results, for a matrix to be the transforming matrix
for a similarity between a pair of companion matrices. A special case of our main result shows that a Toeplitz
or a Hankel matrix can be extended using associated companion matrices, preserving the Toeplitz or Hankel
structure respectively.
In this paper we use a matrix approach to approximate solutions of variational inequalities in Hilbert spaces. The methods studied combine new or well-known iterative methods (as the original Mann ...method) with regularized processes that involve regular matrices in the sense of Toeplitz. We obtain ergodic type results and convergence.
In this letter, we first present explicit relations between block-oriented nonlinear representations and Volterra models. For an identification purpose, we show that the estimation of the diagonal ...coefficients of the Volterra kernels associated with the considered block-oriented nonlinear structures is sufficient to recover the overall model. An alternating least squares-type algorithm is provided to carry out this model identification.
In this paper, an implicit finite difference scheme with the shifted Grünwald formula, which is unconditionally stable, is used to discretize the fractional diffusion equations with constant ...diffusion coefficients. The coefficient matrix possesses the Toeplitz structure and the fast Toeplitz matrix-vector product can be utilized to reduce the computational complexity from
O
(
N
3
)
to
O
(
N
log
N
)
, where
N
is the number of grid points. Two preconditioned iterative methods, named bi-conjugate gradient method for Toeplitz matrix and bi-conjugate residual method for Toeplitz matrix, are proposed to solve the relevant discretized systems. Finally, numerical experiments are reported to show the effectiveness of our preconditioners.
On the eigenvectors of Toeplitz matrices Ikramov, Kh. D.
Moscow University computational mathematics and cybernetics,
04/2015, Volume:
39, Issue:
2
Journal Article
Peer reviewed
Although every nonzero vector
x
∈
C
n
can be an eigenvector of a nonscalar Toeplitz matrix
T
, this assertion is generally false if symmetry is additionally required of
T
. It is shown that every ...symmetric or skew-symmetric vector is an eigenvector of a symmetric Toeplitz (nonscalar) matrix. A problem in matrix analysis that results in the need to characterize such eigenvectors is described.
In this paper, we propose new schemes for subquadratic arithmetic complexity multiplication in binary fields using optimal normal bases. The schemes are based on a recently proposed method known as ...block recombination, which efficiently computes the sum of two products of Toeplitz matrices and vectors. Specifically, here we take advantage of some structural properties of the matrices and vectors involved in the formulation of field multiplication using optimal normal bases. This yields new space and time complexity results for corresponding bit parallel multipliers.
A field multiplication in the extended binary field is often expressed using Toeplitz matrix-vector products (TMVPs), whose matrices have special properties such as symmetric or triangular. We show ...that such TMVPs can be efficiently implemented by taking advantage of some properties of matrices. This yields an efficient multiplier when a field multiplication involves such TMVPs. For example, we propose an efficient multiplier based on the Dickson basis which requires the reduced number of XOR gates by an average of 34% compared with previously known results.
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