Hyperpower Iterative Method Research Articles

Towards Higher-Order Zeroing Neural Network Dynamics for Solving Time-Varying Algebraic Riccati Equations

One of the most often used approaches for approximating various matrix equation problems is the hyperpower family of iterative methods with arbitrary convergence order, whereas the zeroing neural network (ZNN) is a type of neural dynamics intended for handling time-varying problems. A family of ZNN models that correlate with the hyperpower iterative methods is defined on the basis of the analogy that was discovered. These models, known as higher-order ZNN models (HOZNN), can be used to find real symmetric solutions of time-varying algebraic Riccati equations. Furthermore, a noise-handling HOZNN (NHOZNN) class of dynamical systems is introduced. The traditional ZNN and HOZNN dynamic flows are compared theoretically and numerically.

Hyperpower least squares progressive iterative approximation

Geometric iterative methods (GIMs) generate curves or surfaces that fit (interpolate or approximate) given sets of data points. Standard GIMs use the Richardson iteration or gradient descent method, so they converge relatively slowly. This paper investigates the GIMs acceleration method, which is based on the use of more efficient methods for the matrix inverse approximation. That is, the possibility to use hyperpower iterative methods is being explored. As a result, a novel GIM named hyperpower least squares progressive iterative approximation (HPLSPIA) is proposed. The convergence of the HPLSPIA method is analyzed, and the computational complexity is briefly discussed. The method is then applied to tensor product B-spline surface fitting. Several HPLSPIA methods utilizing different hyperpower methods for the matrix inverse approximation are experimentally compared. It is shown that polynomial factorizations used in the hyperpower iterative methods significantly affect the efficiency of the HPLSPIA methods.

An efficient hyperpower iterative method for computing weighted MoorePenrose inverse

In this paper, we propose a new hyperpower iterative method for approximating the weighted Moore-Penrose inverse of a given matrix. The main objective of the current work is to minimize the computational complexity of the hyperpower iterative method using some transformations. The proposed method attains the fifth-order of convergence using four matrix multiplications per iteration step. The theoretical convergence analysis of the method is discussed in detail. A wide range of numerical problems is considered from scientific literature, which demonstrates the applicability and superiority of the proposed method.

Computing the Moore-Penrose inverse using its error bounds

A new iterative scheme for the computation of the Moore-Penrose generalized inverse of an arbitrary rectangular or singular complex matrix is proposed. The method uses appropriate error bounds and is applicable without restrictions on the rank of the matrix. But, it requires that the rank of the matrix is known in advance or computed beforehand. The method computes a sequence of monotonic inclusion interval matrices which contain the Moore-Penrose generalized inverse and converge to it. Successive interval matrices are constructed by using previous approximations generated from the hyperpower iterative method of an arbitrary order and appropriate error bounds of the Moore-Penrose inverse. A convergence theorem of the introduced method is established. Numerical examples involving randomly generated matrices are presented to demonstrate the efficacy of the proposed approach. The main property of our method is that the successive interval matrices are not defined using principles of interval arithmetic, but using accurately defined error bounds of the Moore-Penrose inverse.

Higher-Order ZNN Dynamics

Several improvements of the Zhang neural network (ZNN) dynamics for solving the time-varying matrix inversion problem are presented. Introduced ZNN dynamical design is termed as ZNN models of the order p, $$p\ge 2$$, and it is based on the analogy between the proposed continuous-time dynamical systems and underlying discrete-time pth order hyperpower iterative methods for computing the constant matrix inverse. Such ZNN design is denoted by $$\hbox {ZNN}_H^p$$. Particularly, the $$\hbox {ZNN}_H^2$$ design coincides with the standard ZNN design. Moreover, $$\hbox {ZNN}_H^3$$ design represents a time-varying generalization of the previously defined ZNNCM model. In addition, an integration-enhanced noise-handling $$\hbox {ZNN}_H^p$$ model, termed as $$\hbox {IENHZNN}_H^p$$, is introduced. In the time-invariant case, we present a hybrid enhancement of the $$\hbox {ZNN}_H^p$$ model, shortly termed as $$\hbox {HZNN}_H^p$$, and investigate it theoretically and numerically. Theoretical and numerical comparisons between the improved and standard ZNN dynamics are considered.

Further efficient hyperpower iterative methods for the computation of generalized inverses $$A_{T,S}^{(2)}$$

In this paper, we shall investigate new efficient variants of the hyperpower iterative methods for computing the generalized inverses \(A_{T,S}^{(2)}\). Our main intention is to find efficient transformations of the hyperpower iterative method in order to minimize the number of required matrix by matrix products per cycle. In particular, we will transform the normalized hyperpower method of order 11 and the non-normalized hyperpower method of order 13 such that each of them requires six matrix by matrix products per iterative step. This will make the proposed methods quite computationally economic in comparison to the existing Schulz-type iterative methods. A variety of numerical experiments are also executed to manifest the capability and the practical applicability of our proposed iterative schemes. Additionally, it is also shown that the scalar by matrix multiplications involved in a non-normalized iterative method can reduce its computational efficiency in finding generalized inverses.

Modified discrete iterations for computing the inverse and pseudoinverse of the time-varying matrix

Abstract The general discretization scheme for transforming continuous-time ZNN models for matrix inversion and pseudoinversion into corresponding discrete-time iterative methods is developed and investigated. The proposed discrete-time ZNN models incorporate scaled Hyperpower iterative methods as well as the Newton iteration in certain cases. The general linear Multi-step method is applied in order to obtain the proposed discretization rule which comprises all previously proposed discretization schemes. Both the Euler difference rule and the Taylor-type difference rules are included in the general scheme. In particular, the iterative scheme based on the 4th order Adams–Bashforth method is proposed and numerically compared with other known iterative schemes. In addition, the ZNN model for computing the time-varying matrix inverse is extended to the singular or rectangular case for the pseudoinverse computation. Convergence properties of the continuous-time ZNN model in the case of the Moore–Penrose inverse and its discretization are also considered.

From Zhang Neural Network to scaled hyperpower iterations

A class of scaled hyperpower iterative methods for computing outer inverses is considered. This class appears during the construction of the discrete-time Zhang neural network for computing the usual matrix inverse. The usual hyperpower iterative methods belong to this class. Additionally, a more general class of scaled iterative methods, which includes the scaled hyperpower method, is defined and studied. Different values of the real scaling parameter are investigated both theoretically and numerically.

A hyperpower iterative method for computing the generalized Drazin inverse of Banach algebra element

A quadratically convergent Newton-type iterative scheme is proposed for approximating the generalized Drazin inverse \(b{^\mathrm d}\) of the Banach algebra element b. Further, its extension into the form of the hyperpower iterative method of arbitrary order \(p\ge 2\) is presented. Convergence criteria along with the estimation of error bounds in the computation of \(b{^\mathrm d}\) are discussed. Convergence results confirms the high order convergence rate of the proposed iterative scheme.

ZNN models for computing matrix inverse based on hyperpower iterative methods

Our goal is to investigate and exploit an analogy between the scaled hyperpower family (SHPI family) of iterative methods for computing the matrix inverse and the discretization of Zhang Neural Network (ZNN) models. A class of ZNN models corresponding to the family of hyperpower iterative methods for computing generalized inverses is defined on the basis of the discovered analogy. The Simulink implementation in Matlab of the introduced ZNN models is described in the case of scaled hyperpower methods of the order 2 and 3. Convergence properties of the proposed ZNN models are investigated as well as their numerical behavior.

Universal Iterative Methods for Computing Generalized Inverses

In this paper we construct a few iterative processes for computing {1,2} inverses of a linear bounded operator, based on the hyper-power iterative method or the Neumann-type expansion. Under suitable conditions these methods converge to the {1,2,3} or {1,2,4} inverses. Also, we specify conditions when the iterative processes converge to the Moore-Penrose inverse, the weighted Moore-Penrose inverse or to the group inverse. A few error estimates are derived. The advantages of the introduced methods over Tanabe's method [16] for computing reflexive generalized inverses are also investigated.

Neumann-type expansion of reflexive generalized inverses of a matrix and the hyperpower iterative method

Neumann-type series expansion of reflexive generalized inverses of a matrix is investigated, and [dota] necessary and sufficient condition is given in relation to the expansion for the convergence of the hyperpower iterative method for generating reflexive generalized inverses.

An optimum hyperpower iterative method for computing generalized inverses and associated projections

In an earlier paper (Gupta 1971), an iterative hyperpower method of any finite- order for computing generalized inverse A + and the associated projections AA + and A + A for any complex matrix A was developed. In this paper, hyperpower methods of different orders are compared and it is shown that there exists an optimum hyperpower method of order three.

A Hyperpower Iterative Method for Computing Matrix Products Involving the Generalized Inverse

Apth order iterative method for computing $A^\dag B$ is studied, where $p \geqq 2$, A and B are arbitrary complex matrices with equal number of rows and $A^\dag $ is the Moore–Penrose generalized inverse of A. The rate of convergence and the computational effort in the pth order method are studied, and the optimum p is given in terms of the numbers of columns of A and B.