Hyperpower Method Research Articles

Iterative Matrix Inversion Methods for Precoding in Cell-Free Massive MIMO Systems

Cell-free massive multiple-input multiple-output (mMIMO) systems are an alternative topology for mMIMO deployment, wherein a large number of access points are distributed over the coverage area to jointly serve users. Linear precoding methods such as zero-forcing are sufficient to achieve near-optimal performance in mMIMO systems. However, a key challenge in implementing these precoders can be a channel matrix inversion operation, which results in significant computational complexity in systems with large-scale antenna arrays. Hence, instead of direct matrix inversion, we examine several iterative methods to calculate the precoding matrix in a cell-free mMIMO system. We investigate their computational complexity and convergence rate in the presence of small- and large-scale fading and spatial correlation between antennas. Notably, we demonstrate that some iterative methods previously proposed for conventional (co-located) mMIMO do not always converge for cell-free mMIMO. Our main focus is the hyper-power iterative inversion method, which can be applied to both matrix inverses and pseudoinverses with guaranteed convergence; its factorized version also reduces its computational complexity. Although the hyper-power method does not reduce the complexity compared to direct matrix inversion, it converges quickly with high accuracy and strong numerical stability, and is conducive to parallel computation. These qualities make it a good candidate for matrix inversion in cell-free mMIMO system precoders.

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.

Exploiting higher computational efficiency index for computing outer generalized inverses

In this work, we study a new iterative method for computing outer generalized inverses that minimizes the required number of matrix-by-matrix (MM) products per loop for a specific high order. We consider the hyperpower method of order nineteen and improve it by using only seven MM products per iteration. Hence, the proposed scheme is efficient by reaching a higher computational efficiency index. Its theoretical convergence is investigated in detail. Several tests are worked out to validate the new iterative scheme as well as its computational aspects.

Calculating the Weighted Moore–Penrose Inverse by a High Order Iteration Scheme

The goal of this research is to extend and investigate an improved approach for calculating the weighted Moore–Penrose (WMP) inverses of singular or rectangular matrices. The scheme is constructed based on a hyperpower method of order ten. It is shown that the improved scheme converges with this rate using only six matrix products per cycle. Several tests are conducted to reveal the applicability and efficiency of the discussed method, in contrast with its well-known competitors.

Hyper-power zero forcing detector for massive MIMO systems

In massive multiple-input multiple-output (MIMO) systems when the number of base station antennas is much higher than the number of users, linear detectors, such as zero forcing (ZF) and minimum mean-square error (MMSE), are able to achieve the near-optimal performance due to the favorable massive MIMO channel propagation. But, these detectors employ, in general, exact matrix inversion which is computationally complex for such systems. In this paper, we affirm that computing the exact matrix inversion by direct methods is not necessary to find ZF or MMSE solution. An iterative matrix inversion procedure would yield similar performance. Thus, an efficient iterative matrix inversion based on the hyper-power (HP) method is proposed for massive MIMO detection. The computing efficiency of the iterative matrix inversion is further improved by optimizing the number of terms from the infinite series used in the HP method. Analytical results show that the optimum choice for the number of terms is three from the HP method. Simulation results show that the HP method with the optimum number of terms achieves the near-optimal ZF performance in a small number of iterations. Finally, the Coppersmith–Winograd algorithm for matrix multiplication is employed in order to reduce the computational complexity from \(O(K^{3})\) to \(O(K^{2.373})\), where K represents the number of users.

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.

Approximate Schur-Block ILU Preconditioners for Regularized Solution of Discrete Ill-Posed Problems

High order iterative methods with a recurrence formula for approximate matrix inversion are proposed such that the matrix multiplications and additions in the calculation of matrix polynomials for the hyperpower methods of orders of convergence p=4k+3, where k≥1 is integer, are reduced through factorizations and nested loops in which the iterations are defined using a recurrence formula. Therefore, the computational cost is lowered from κ=4k+3 to κ=k+4 matrix multiplications per step. An algorithm is proposed to obtain regularized solution of ill-posed discrete problems with noisy data by constructing approximate Schur-Block Incomplete LU (Schur-BILU) preconditioner and by preconditioning the one step stationary iterative method. From the proposed methods of approximate matrix inversion, the methods of orders p=7,11,15,19 are applied for approximating the Schur complement matrices. This algorithm is applied to solve two problems of Fredholm integral equation of first kind. The first example is the harmonic continuation problem and the second example is Phillip’s problem. Furthermore, experimental study on some nonsymmetric linear systems of coefficient matrices with strong indefinite symmetric components from Harwell-Boeing collection is also given. Numerical analysis for the regularized solutions of the considered problems is given and numerical comparisons with methods from the literature are provided through tables and figures.

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.

On the two classes of high‐order convergent methods of approximate inverse preconditioners for solving linear systems

SummaryTwo classes of methods for approximate matrix inversion with convergence orders p=3∗2k+1 (Class 1) and p=5∗2k−1 (Class 2), k≥1 an integer, are given based on matrix multiplication and matrix addition. These methods perform less number of matrix multiplications compared to the known hyperpower method or pth‐order method for the same orders and can be used to construct approximate inverse preconditioners for solving linear systems. Convergence, error, and stability analyses of the proposed classes of methods are provided. Theoretical results are justified with numerical results obtained by using the proposed methods of orders p=7,13 from Class 1 and the methods with orders p=9,19 from Class 2 to obtain polynomial preconditioners for preconditioning the biconjugate gradient (BICG) method for solving well‐ and ill‐posed problems. From the literature, methods with orders p=8,16 belonging to a family developed by the effective representation of the pth‐order method for orders p=2k, k is integer k≥1, and other recently given high‐order convergent methods of orders p=6,7,8,12 for approximate matrix inversion are also used to construct polynomial preconditioners for preconditioning the BICG method to solve the considered problems. Numerical comparisons are given to show the applicability, stability, and computational complexity of the proposed methods by paying attention to the asymptotic convergence rates. It is shown that the BICG method converges very quickly when applied to solve the preconditioned system. Therefore, the cost of constructing these preconditioners is amortized if the preconditioner is to be reused over several systems of same coefficient matrix with different right sides.

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.

Hyper-power methods for the computation of outer inverses

In this paper, we propose new iterative schemes for the computation of outer inverse which reduce the total number of matrix multiplications per iteration. In particular, we consider how the hyper-power method of orders 5 and 9 can be accelerated such that they require 4 and 5 matrix multiplications per iteration, respectively. These improvements are tested against quadratically convergent Schultz’ method and fastest Horner scheme hyper-power method of order three. Numerical results show the superiority and practical applicability of the proposed methods. Finally, it is shown that a possibly more efficient method should have the order at least r≥14, making it useless for practical applications.

The semi-iterative method applied to the hyper-power iteration

The semi-iterative method (SIM) is applied to the hyper-power (HP) iteration, and necessary and sufficient conditions are given for the convergence of the semi-iterative–hyper-power (SIM–HP) iteration. The root convergence rate is computed for both the HP and SIM–HP methods, and the quotient convergence rate is given for the HP iteration. Copyright © 2005 John Wiley & Sons, Ltd.

Applications of the hyper-power method for computing matrix products

In this paper the solutions some practical problems of cointegration analysis are discussed. We mainly focus on the following phases of the Johansen procedure: the baseline model specification, t determination of the number of cointegrating vectors, the inclusion of the determines components and design of partial mode This is the most commonly used approach analyzing nonstationary macroeconomic time series, due to its complexity and simple modeling based on the computer packages. However, as the application of statistic packages may be useless when methodology is partly known, its full understanding makes computer software a reliable source. The purpose of this paper is to popularize (correct) application of cointegration methods. .

A geometrical approach on generalized inverses by Neumann-type series

The convergence of the Neumann-type series to {1,2}-inverses has been shown by K. Tanabe [Linear Algebra Appl. 10 (1975) 163]. In this paper, these results indicating conditions characterizing the convergence of this series to different generalized inverses are extended. In addition, these results for obtaining different generalized inverses from the hyperpower method are applied. Finally, generalized involutory matrices are introduced and characterized using the obtained results.

The hyperpower iteration revisited

The hyperpower method is generalized by inseting an idempotent matrix P. It is shown that most of the iterations that can be used for computing generalized inverses can be recast in this form by suitable choice of the idempotent, the limit, or the initial values. The convergence behavior of B q k is analyzed and the results are applied to give a detailed investigation of this iteration.

Efficient algorithms for the inclusion of the inverse matrix using error-bounds for hyperpower methods

By exploiting generalized error-bounds for the well-known hyperpower methods for approximating the inverse of a matrix we derive inclusion methods for the inverse matrix. These methods make use of interval operations in order to give guaranteed inclusions whenever the convergence of the applied hyperpower method can be shown. The efficiency index of some of the new methods is greater than that of the optimal methods in [2] or [5]. A numerical example is given.

Using error-bounds for hyperpower methods to calculate inclusions for the inverse of a matrix

By means of error-bounds for the well-known hyperpower methods for approximating the inverse of a matrix we define inclusion methods for the inverse matrix. These methods are using machine interval operations and are giving guaranteed inclusions for the inverse matrix whenever the convergence of the applied hyperpower methods can be shown. In comparison with the very efficient interval Schulz's method in the literature, our methods are more efficient in terms of the efficiency index. Some numerical examples are given.

A class of optimal iterative methods of inverting a linear bounded operator

A family of higher-order iterative methods for inverting a linear bounded operator in a Banach space is presented. This family depending on two integer parameters can be considered as hyperpower methods of [1]. A convergence Theorem is proved and an optimal class with respect to the efficiency index is determined . Several error bounds are derived and compared with those in the literature.

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.