On the Calculation of the Moore–Penrose and Drazin Inverses: Application to Fractional Calculus

Sayevand, Khosro; Pourdarvish, A.; Machado, J. A. Tenreiro; Erfanifar, Raziye

doi:10.3390/math9192501

Cited by 12 publications

(3 citation statements)

References 42 publications

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“…Therefore, numerical methods and especially iterative methods have been important to find the tensor inverse. There exist some iterative methods to compute the Moore-Penrose inverse matrix [12,16,24,36]. The most famous method to approximate the matrix A −1 is the Newton method, and this method was developed for finding the tensor A −1 as follows [25]: 25]).…”

Section: Lemma 12 ([23]) Suppose Thatmentioning

confidence: 99%

A Novel Iterative Method to Find the Moore-Penrose Inverse of a Tensor with Einstein Product

Raziyeh Erfanifar,

Masoud Hajarian,

Khosro Sayevand

2024

NMTMA

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In this study, based on an iterative method to solve nonlinear equations, a third-order convergent iterative method to compute the Moore-Penrose inverse of a tensor with the Einstein product is presented and analyzed. Numerical comparisons of the proposed method with other methods show that the average number of iterations, number of the Einstein products, and CPU time of our method are considerably less than other methods. In some applications, partial and fractional differential equations that lead to sparse matrices are considered as prototypes. We use the iterates obtained by the method as a preconditioner, based on tensor form to solve the multilinear system A * N X = B. Finally, several practical numerical examples are also given to display the accuracy and efficiency of the new method. The presented results show that the proposed method is very robust for obtaining the Moore-Penrose inverse of tensors.

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Section: Lemma 12 ([23]) Suppose Thatmentioning

confidence: 99%

A Novel Iterative Method to Find the Moore-Penrose Inverse of a Tensor with Einstein Product

Raziyeh Erfanifar,

Masoud Hajarian,

Khosro Sayevand

2024

NMTMA

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“…In general, most existing methods for their obtaining are iterative algorithms for approximating generalized inverses of complex matrices (some recent papers, see, e.g. [38][39][40]). There are only several direct methods for finding MP-inverse for an arbitrary complex matrix A ∈  mÂn .…”

Section: The General Solution Of the Homogeneous Equationmentioning

confidence: 99%

Solving and Algorithm for Least-Norm General Solution to Constrained Sylvester Matrix Equation

Rehman¹,

Kyrchei²

2023

Inverse Problems - Recent Advances and Applications

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Keeping in view that a lot of physical systems with inverse problems can be written by matrix equations, the least-norm of the solution to a general Sylvester matrix equation with restrictions A1X1=C1,X1B1=C2,A2X2=C3,X2B2=C4,A3X1B3+A4X2B4=Cc, is researched in this chapter. A novel expression of the general solution to this system is established and necessary and sufficient conditions for its existence are constituted. The novelty of the proposed results is not only obtaining a formal representation of the solution in terms of generalized inverses but the construction of an algorithm to find its explicit expression as well. To conduct an algorithm and numerical example, it is used the determinantal representations of the Moore–Penrose inverse previously obtained by one of the authors.

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“…In this part, we propose a scheme to solve the nonlinear matrix equation (1.1), by using Newton's method (Pan and Schreiber, 1991; Sayevand et al ., 2021). In fact, for finding the root of scriptX-1-scriptA=0, we can getwhere η > 0.…”

Section: Iterative Schemes To Solve the Nonlinear Matrix Equation (11)mentioning

confidence: 99%

Efficient iterative schemes based on Newton's method and fixed-point iteration for solving nonlinear matrix equation X^p = Q±A(X⁻¹+B)⁻¹A^T

Erfanifar,

Hajarian

2023

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PurposeIn this paper, the authors study the nonlinear matrix equation Xp=Q±A(X-1+B)-1AT, that occurs in many applications such as in filtering, network systems, optimal control and control theory.Design/methodology/approachThe authors present some theoretical results for the existence of the solution of this nonlinear matrix equation. Then the authors propose two iterative schemes without inversion to find the solution to the nonlinear matrix equation based on Newton's method and fixed-point iteration. Also the authors show that the proposed iterative schemes converge to the solution of the nonlinear matrix equation, under situations.Findings The efficiency indices of the proposed schemes are presented, and since the initial guesses of the proposed iterative schemes have a high cost, the authors reduce their cost by changing them. Therefore, compared to the previous scheme, the proposed schemes have superior efficiency indices.Originality/value Finally, the accuracy and effectiveness of the proposed schemes in comparison to an existing scheme are demonstrated by various numerical examples. Moreover, as an application, by using the proposed schemes, the authors can get the optimal controller state feedback of $x(t+1) = A x(t) + C v(t)$.

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On the Calculation of the Moore–Penrose and Drazin Inverses: Application to Fractional Calculus

Cited by 12 publications

References 42 publications

A Novel Iterative Method to Find the Moore-Penrose Inverse of a Tensor with Einstein Product

A Novel Iterative Method to Find the Moore-Penrose Inverse of a Tensor with Einstein Product

Solving and Algorithm for Least-Norm General Solution to Constrained Sylvester Matrix Equation

Efficient iterative schemes based on Newton's method and fixed-point iteration for solving nonlinear matrix equation X^p = Q±A(X⁻¹+B)⁻¹A^T

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On the Calculation of the Moore–Penrose and Drazin Inverses: Application to Fractional Calculus

Cited by 12 publications

References 42 publications

A Novel Iterative Method to Find the Moore-Penrose Inverse of a Tensor with Einstein Product

A Novel Iterative Method to Find the Moore-Penrose Inverse of a Tensor with Einstein Product

Solving and Algorithm for Least-Norm General Solution to Constrained Sylvester Matrix Equation

Efficient iterative schemes based on Newton's method and fixed-point iteration for solving nonlinear matrix equation Xp = Q±A(X−1+B)−1AT

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Efficient iterative schemes based on Newton's method and fixed-point iteration for solving nonlinear matrix equation X^p = Q±A(X⁻¹+B)⁻¹A^T