Vasilios N. Katsikis

An improved method for the computation of the Moore―Penrose inverse matrix

Abstract In this article we provide a fast computational method in order to calculate the Moore–Penrose inverse of singular square matrices and of rectangular matrices. The proposed method proves to be much faster and has significantly better accuracy than the already proposed methods, while works for full and sparse matrices.

Applications of the Moore-Penrose Inverse in Digital Image Restoration

This paper presents a fast computational method that finds application in a broad scientific field such as digital image restoration. The proposed method provides a new approach to the problem of image reconstruction by using the Moore-Penrose inverse. The resolution of the reconstructed image remains at a very high level but the main advantage of the method was found on the computational load that has been decreased considerably compared to the classic techniques.

Fast computing of the Moore-Penrose inverse matrix

In this article a fast computational method is provided in order to calculate the Moore-Penrose inverse of full rank m ×n matrices and of square matrices with at least one zero row or column. Sufficient conditions are also given for special type products of square matrices so that the reverse order law for the Moore-Penrose inverse is satisfied. is defined. In the case when T is a real m × n matrix, Penrose showed that there is a unique matrix satisfying the four Penrose equations, called the generalized inverse of T. A lot of work concerning generalized inverses has been carried out, in finite and infinite dimension (e.g., (2, 11)). In this article, we provide a method for the fast computation of the generalized inverse of full rank matrices and of square matrices with at least one zero row or column. In order to reach our goal, we use a special type of tensor product of two vectors, that is usually used in infinite dimensional Hilbert spaces. Using this type of tensor product, we also give sufficient conditions for products of square matrices so that the reverse order law for the Moore-Penrose inverse ((1, 4, 5)) is satisfied. There are several methods for computing the Moore-Penrose inverse matrix (cf. (2)). One of the most commonly used methods is the Singular Value Decomposition (SVD) method. This method is very accurate but also time-intensive since it requires a large amount of computational resources, especially in the case of large matrices. In the recent work of P. Courrieu (3), an algorithm for fast computation of Moore- Penrose inverse matrices is presented based on a known reverse order law (eq. 3.2

Full-rank representations of outer inverses based on the QR decomposition

Abstract An efficient algorithm for computing A T , S ( 2 ) inverses of a given constant matrix A, based on the QR decomposition of an appropriate matrix W, is presented. Correlations between the derived representation of outer inverses and corresponding general representation based on arbitrary full-rank factorization are derived. In particular cases we derive representations of { 2 , 4 } and { 2 , 3 } -inverses. Numerical examples on different test matrices (dense or sparse) are presented as well as the comparison with several well-known methods for computing the Moore–Penrose inverse and the Drazin inverse.

Digital Image Reconstruction in the Spectral Domain Utilizing the Moore-Penrose Inverse

The field of image restoration has seen a tremendous growth in interest over the last two decades. The recovery of an original image from degraded observations is a crucial method and finds application in several scientific areas including medical imaging and diagnosis, military surveillance, satellite and astronomical imaging, and remote sensing. The proposed approach presented in this work employs Fourier coefficients for moment-based image analysis. The main contributions of the presented technique, are that the image is first analyzed in orthogonal basis matrix formulation increasing the selectivity on image components, and then transmitted in the spectral domain. After the transmission has taken place, at the receiving end the image is transformed back and reconstructed from a set of its geometrical moments. The calculation of the Moore-Penrose inverse of 𝑟×𝑚 matrices provides the computation framework of the method. The method has been tested by reconstructing an image represented by an 𝑟×𝑚 matrix after the removal of blur caused by uniform linear motions. The noise during the transmission process is another issue that is considered in the current work.

Computational methods in portfolio insurance

Abstract In this article we develop a computational method for an algorithmic process first posed by Abramovich–Aliprantis–Polyrakis in 1994 in order to check whether a finite collection of linearly independent positive vectors in R m forms a lattice-subspace. Lattice-subspaces are closely related to a cost minimization problem in the theory of finance that ensures the minimum-cost insured portfolio and this connection is further investigated here. Finally, we propose a computational method in order to solve the minimization problem and to calculate the minimum-cost insured portfolio. All of the numerical work is performed using the Matlab high-level language.

Computational methods in lattice-subspaces of C[a, b] with applications in portfolio insurance

In this article, we develop a computational method for an algorithmic process first posed by Polyrakis in 1996 in order to check whether a finite collection of linearly independent positive functions in C[a,b] forms a lattice-subspace. Lattice-subspaces are closely related to a cost minimization problem in the theory of finance that ensures the minimum-cost insured portfolio and this connection is further investigated here. Finally, we propose a computational method in order to solve the minimization problem and to calculate the minimum-cost insured portfolio. All of the numerical work is performed using the Matlab high-level language.

A Matlab-based rapid method for computing lattice-subspaces and vector sublattices of R n: Applications in portfolio insurance

This paper provides the construction of a powerful and efficient computational method, that translates Polyrakis algorithm [I.A. Polyrakis, Minimal lattice-subspaces, Trans. Am. Math. Soc. 351 (1999) 4183-4203, Theorem 3.19] for the calculation of lattice-subspaces and vector sublattices in R^n. In the theory of finance, lattice-subspaces have been extensively used in order to provide a characterization of market structures in which the cost-minimizing portfolio is price-independent. Specifically, we apply our computational method in order to solve a cost minimization problem that ensures the minimum-cost insured portfolio.

Computational methods for option replication

A computational method is described for option replication. In particular, a procedure is provided for computing the projection basis that corresponds to a positive basis of ℝ m . Application of this procedure in order to compute maximal submarkets that replicate any option is demonstrated. Specifically, we provide a computational study for the replication of options in security markets with a finite number of states and a finite number of primitive assets with payoffs given by linearly independent vectors of ℝ m . The theoretical background of this work follows the results in Polyrakis and Xanthos [Maximal submarkets that replicate any option, Ann. Finance, DOI: 10.1007/s10436-009-0143-9]. Our goal is to make option replication computationally tractable and hence more viable as a financial tool.

Representations and properties of the W-Weighted Drazin inverse

Several new representations of the W-weighted Drazin inverse are introduced. These representations are expressed in terms of various matrix powers as well as in terms of matrix products involving the Moore–Penrose inverse and the usual matrix inverse. Also, the properties of various generalized inverses which arise from derived representations are investigated. The computational complexity and efficiency of the proposed representations are considered. Representations are tested and compared among themselves in a substantial number of randomly generated test examples.