New zeroing neural dynamics models for diagonalization of symmetric matrix stream

Abstract

In this paper, the problem of diagonalizing a symmetric matrix stream (or say, time-varying matrix) is investigated. To fulfill our goal of diagonalization, two error functions are constructed. By making the error functions converge to zero with zeroing neural dynamics (ZND) design formulas, a continuous ZND model is established and its effectiveness is then substantiated by simulative results. Furthermore, a Zhang et al. discretization (ZeaD) formula with high precision is developed to discretize the continuous ZND model. Thus, a new 5-point discrete ZND (DZND) model is further proposed for diagonalization of matrix stream. Theoretical analyses prove the stability and convergence of the 5-point DZND model. In addition, simulative experiments are carried out, of which the results substantiate not only the efficacy of the proposed 5-point DZND model but also its higher computational precision as compared with the conventional Euler-type and 4-point DZND models for diagonalization of symmetric matrix stream.

Appendices

Appendix 1

Suppose that

$$ D = \left[ {\begin{array}{*{20}{c}} {{d_{11}}}&0& {\cdots} &0\\ 0&{{d_{22}}}& {\cdots} &0\\ {\vdots} & {\vdots} & {\ddots} & {\vdots} \\ 0&0& {\cdots} &{{d_{nn}}} \end{array}} \right] \in {\mathbb{R}^{{n} \times n}} $$

and

$$ Y = \left[ {\begin{array}{*{20}{c}} {{y_{11}}}&{{y_{12}}}& {\cdots} &{{y_{1n}}}\\ {{y_{21}}}&{{y_{22}}}& {\cdots} &{{y_{2n}}}\\ {\vdots} & {\vdots} & {\ddots} & {\vdots} \\ {{y_{n1}}}&{{y_{n2}}}& {\cdots} &{{y_{nn}}} \end{array}} \right] \in {\mathbb{R}^{{n} \times n}}. $$

Let d_ij and y_ij denote the ij th elements of D and Y, respectively. We have

$$ \begin{array}{@{}rcl@{}} E &=& DY - YD \\ &=& \left[ {\begin{array}{*{20}{c}} 0&{{y_{12}}({d_{11}} - {d_{22}})}& {\cdots} &{{y_{1n}}({d_{11}} - {d_{nn}})}\\ {{y_{21}}({d_{22}} - {d_{11}})}&0& {\cdots} &{{y_{2n}}({d_{22}} - {d_{nn}})}\\ {\vdots} & {\vdots} & {\ddots} & {\vdots} \\ {{y_{n1}}({d_{nn}} - {d_{11}})}&{{y_{n2}}({d_{nn}} - {d_{22}})}& {\cdots} &0 \end{array}} \right]. \end{array} $$

Then, the ij th element of matrix E equals y_ij(d_ii − d_jj) with i≠j and i,j = 1, 2,…,n. Since the diagonal elements of D are distinct, y_ij (with i≠j and i,j = 1, 2,…,n) must equal zero so that y_ij(d_ii − d_jj) equals zero. Thus, matrix Y is diagonal. The proof is completed.

Appendix 2

Given an N-step method $\sum \nolimits _{i= 0}^{N} {\alpha _{i} x_{k +i}}=\tau \sum \nolimits _{i = 0}^{N} \beta _{i} \psi _{k +i}$ with its first and second characteristic polynomials being $P_{N}(\varsigma ) = \sum \nolimits _{i = 0}^{N} \alpha _{i} \varsigma ^{i}$ and $p_{N}(\varsigma ) = \sum \nolimits _{i = 0}^{N} \beta _{i} \varsigma ^{i}$, we have the following definition and results [32, 33] as the basis of DZND research.

Definition of root condition: :

A polynomial ρ(ς) satisfies the root condition if all its roots satisfy |ς|≤ 1 and any roots that satisfy |ς| = 1 are simple.

Result 1: :

An N-step method is said to be zero-stable if the first characteristic polynomial satisfies the root condition.

Result 2: :

An N-step method is consistent if its first and second characteristic polynomials satisfy P_N(1) = 0 and $P^{\prime }_{N}(1)=p_{N}(1)\neq 0$. The N-step method is consistent of order q if the truncation error for the exact solution is of order O(τ^q).

Result 3: :

An N-step method is convergent if and only if the method is zero-stable and consistent. That is, zero-stability plus consistency means convergence, which is also known as Dahlquist equivalence theorem.

Result 4: :

A zero-stable consistent N-step method converges with the order of its truncation error.