RANK ADDITIVITY AND MATRIX POLYNOMIALS

Historically, work on multiple roots for the internal rate of return (PW = 0) equation focuses on tests for the root's uniqueness and interpreting the meaningfulness of computed roots. Instead, this article starts with problems that consist of P, A, and F cash flows. Then for all possible values of these cash flows, this article develops the boundary conditions for no, single, and multiple real roots of either positive or negative sign. The results of characterizing these roots are applied to two examples as a foundation for conclusions about those multiple root problems and hypotheses about other problems with multiple roots. The first example is an important class of resource development projects that may require significant expenditures for environmental remediation at the conclusion of operation. The second example of buying versus leasing a home is often used to demonstrate the value of economic analysis to students.

Asymptotic distributions of the latent roots of the covariance matrix with multiple population roots

Spectral break theory, as presented in this article, offers an algebraic-geometric framework unifying the concepts of renormalization and decoherence in the context of complex polynomials with multiplicity at least 2. The study focuses on transforming an unstable complex polynomial P , possessing at least one multiple root, into a simple polynomial Q , devoid of multiple or complex roots, through the composite operator . The renormalization operator R reduces root multiplicities to 1, while the decoherence operator D projects non-real roots onto their real parts. A spectral metric is defined to quantify the distance between the roots of P and Q, proving that ( d(P,Q) > 0 ) for any non-trivial P , establishing qualitative spectral non-proximity. Local continuity of the operators R and D is demonstrated on strata with fixed multiplicities in a non-compact space, leveraging the Weierstrass preparation theorem. Two indicators are introduced: e, measuring the impact of decoherence, and t, combining the spectral distance and e to assess the net effect of the spectral break. Examples illustrate that t reflects the dominance of either R or D . The study concludes with perspectives for extending the theory to non-linear polynomials and enhancing numerical algorithms for practical applications

Prevalence and morphology of multiple roots, root canals and C-shaped canals in mandibular premolars from cone-beam computed tomography images in a Thai population

A full integral of a polynomial is defined as its integral with the property that any multiple root of the polynomial is a root of this integral. The paper investigates relationships between the existence of a full integral and the form of a polynomial. In particular, it is proved that a full integral exists if the polynomial has no more than one multiple root. On the other hand, if the number of multiple roots of a polynomial strictly exceeds the number of its simple roots increased by one, then the polynomial has no full integral. Bibliography: 7 titles.

The features of generating the n th sequence of refined Fibonacci polynomial matrices of the m th order, the elements of which are Fibonacci polynomials of no higher than (m+n–2)th number, are considered. The obtained Fibonacci matrices allow finding both their determinants and inverse matrices suitable for matrix encryption of block data. It was found that even over the last decade a significant number of publications have been published, each of which substantiates different approaches to generating sequences of Fibonacci polynomial matrices and proves the feasibility of their use for encrypting block data. It was found that the use of such matrices as a separate procedure for protecting block data in the theory and practice of cryptography is extremely rare. The Fibonacci polynomial matrix sequences from the 2nd to the 5th order are analyzed, according to which shortcomings in the traditional approach to forming the structure of the elements of such matrices are revealed, first of all, the number (k) of its different elements, which are Fibonacci polynomials of the not higher than (n–1) th power. It is established that for Fibonacci matrices of any order (m) of such different Fibonacci polynomials there will be only k=3 , the structure of which will depend only on the number (n) of the sequence of the Fibonacci polynomial matrix. Such a small number is not only uninformative and easily analyzable by a cryptanalyst but also vulnerable to cryptanalysis. The structure of the elements of Fibonacci polynomial matrices is specified, the number of which already depends on the order of the matrix(m) and is k=m+1 . A method for generating a sequence of refined Fibonacci polynomial matrices has been developed, which consists in using a recurrent matrix relation, according to which the next polynomial matrix is formed by multiplying the variable x sequentially by the elements of the current matrix, adding the elements of the newly generated matrix to those of the previous matrix, after which all similar terms are grouped in the formed elements. The mechanisms for forming a sequence of 8 polynomial refined Fibonacci matrices from the 2nd to the 4th order, the elements of which are Fibonacci polynomials of no higher than (m+n–2) th power, have been presented, which made it possible to analyze not only the features of their construction, but also to understand the corresponding procedures for finding their determinants and inverse matrices. It was found that the proposed structure of the elements of the nth sequence of Fibonacci polynomial matrices of the mth order has an interesting property, according to which it is possible to avoid the use of the recurrent matrix relation, and to generate the corresponding Fibonacci polynomial matrices only by the numbers (m+n–2–j) th sequence of Fibonacci polynomials, the specific values of which depend on their location in the matrix and the number of its column, namely ∀j∈[0÷(m−1)] . Software has been developed that allows generating both sequences of refined Fibonacci polynomial matrices of the mth order, and finding their determinants and inverse polynomial matrices of a similar order. An example of the application of the matrix method of encrypting block data with a refined Fibonacci polynomial matrix is given, which allows the interested reader to understand the basic principle of encrypting both the initial message and decrypting the encrypted message.

Durand-Kerner's method for simultaneous rootfinding of a polynomial is locally second order convergent if all the zeros are simple. If this condition is violated numerical experiences still show linear convergence. For this case of multiple roots, Fraigniaud [4] proves that the means of clustering approximants for a multiple root is a better approximant for the zero and called this Quadratic-Like-Convergence of the Means. This note gives a new proof and a refinement of this property. The proof is based on the related Grau's method for simultaneous factoring of a polynomial. A similar property of some coefficients of the third order method due to Borsch-Supan, Maehly, Ehrlich, Aberth and others is proved.

A superlinear procedure for finding a multiple root is presented. In it the secant method is applied to the given function divided by a divided difference whose increment shrinks toward zero as the root is approached. Two function evaluations per step are required, but no derivatives need be calculated.

We have a good number of eighth-order iterative methods for simple zeros of nonlinear equations in the available literature. But, unfortunately, we don’t have a single iterative method of eighth-order for multiple zeros with known or unknown multiplicity. Some scholars from the worldwide have tried to present optimal or non-optimal multipoint eighth-order iteration functions. But, unfortunately, none of them get success in this direction and attained maximum sixth-order convergence in the case of multiple zeros with known multiplicity m. Motivated and inspired by this fact, we propose an optimal scheme with eighth-order convergence based on weight function approach. In addition, an extensive convergence study is discussed in order to demonstrate the eighth-order convergence of the present scheme. Moreover, we also show the applicability of our scheme on some real life and academic problems. These problems illustrate that our methods are more efficient among the available multiple root finding techniques.

Spaces of multiparameter rational vectors, i.e., of vectors whose components are rational functions in several variables, and polynomial bases of their subspaces are considered. The conjecture that any subspace in the space of multiparameter rational vectors possesses a “free” polynomial basis, i.e., a basis such that the associated basis multiparameter polynomial matrix has no finite regular spectrum, is disproved by an example. Some consequences of this fact are indicated. Simpler proofs of some properties of the singular spectra of basis polynomial matrices corresponding to the null-spaces of a singular polynomial matrix are presented. Bibliography: 5 titles.

Let P be an arbitrary field and let P(x) be a ring of polynomials over P. Denote by Pn and Pn (x) the rings of n xn matrices over P and P(x), respectively. Consider a nonsingular polynomial matrix A(x) e P~ (x) (detA(x) $ 0) and represent it as a matrix polynomial over the field P: is a unital matrix polynomial. In this paper, we establish conditions under which a unital divisor D (x) extracted from the matrix polynomial A (x) is determined uniquely by its characteristic polynomial detA (x) = d(x). The re- sult obtained enables one to indicate the class of matrix polynomials for which this unital divisor is unique for a given characteristic polynomial. We also show how this result can be used when solving matrix polynomial equations. The uniqueness of a tmital divisor for a matrix polynomial over an algebraically closed field with char- acteristic zero was studied in ( 1 ). It is known (2) that, for a matrix A (x) e Pn (x), there exist matrices U(x), V(x) ~ GL n (Pn (x)) such that U (x)A (x) V(x) = F A (X) = diag (al(x), a2(x) ..... an(x)), where aj(x) e P(x), j= 1, 2 ..... n, are unital polynomials and ai(x ) (ai+l(x), i = 1, 2 ..... n - 1. The matrix FA(X ) is called a canonical diagonal form of the matrix A(x). It is also known that if a matrix is nonsingular, then it can be reduced to the normal Hermite form by right elementary transformations, i.e., for the matrix A (x), there exists a matrix W(x) e GLn (P(x)) such that

This chapter studies, in the setting of vector spaces over a field, the basics concerning multilinear functions, tensor products, spaces of linear functions, and algebras related to tensor products. Sections 1–5 concern special properties of bilinear forms, all vector spaces being assumed to be finite-dimensional. Section 1 associates a matrix to each bilinear form in the presence of an ordered basis, and the section shows the effect on the matrix of changing the ordered basis. It then addresses the extent to which the notion of “orthogonal complement” in the theory of inner-product spaces applies to nondegenerate bilinear forms. Sections 2–3 treat symmetric and alternating bilinear forms, producing bases for which the matrix of such a form is particularly simple. Section 4 treats a related subject, Hermitian forms when the field is the complex numbers. Section 5 discusses the groups that leave some particular bilinear and Hermitian forms invariant. Section 6 introduces the tensor product of two vector spaces, working with it in a way that does not depend on a choice of basis. The tensor product has a universal mapping property—that bilinear functions on the product of the two vector spaces extend uniquely to linear functions on the tensor product. The tensor product turns out to be a vector space whose dual is the vector space of all bilinear forms. One particular application is that tensor products provide a basis-independent way of extending scalars for a vector space from a field to a larger field. The section includes a number of results about the vector space of linear mappings from one vector space to another that go hand in hand with results about tensor products. These have convenient formulations in the language of category theory as “natural isomorphisms.” Section 7 begins with the tensor product of three and then $n$ vector spaces, carefully considering the universal mapping property and the question of associativity. The section defines an algebra over a field as a vector space with a bilinear multiplication, not necessarily associative. If $E$ is a vector space, the tensor algebra $T(E)$ of $E$ is the direct sum over $n\geq0$ of the $n$-fold tensor product of $E$ with itself. This is an associative algebra with a universal mapping property relative to any linear mapping of $E$ into an associative algebra $A$ with identity: the linear map extends to an algebra homomorphism of $T(E)$ into $A$ carrying 1 into 1. Sections 8–9 define the symmetric and exterior algebras of a vector space $E$. The symmetric algebra $S(E)$ is a quotient of $T(E)$ with the following universal mapping property: any linear mapping of $E$ into a commutative associative algebra $A$ with identity extends to an algebra homomorphism of $S(E)$ into $A$ carrying 1 into 1. The symmetric algebra is commutative. Similarly the exterior algebra $\bigwedge(E)$ is a quotient of $T(E)$ with this universal mapping property: any linear mapping $l$ of $E$ into an associative algebra $A$ with identity such that $l(v)^2=0$ for all $v\in E$ extends to an algebra homomorphism of $\bigwedge(E)$ into $A$ carrying 1 into 1. The problems at the end of the chapter introduce some other algebras that are of importance in applications, and the problems relate some of these algebras to tensor, symmetric, and exterior algebras. Among the objects studied are Lie algebras, universal enveloping algebras, Clifford algebras, Weyl algebras, Jordan algebras, and the division algebra of octonions.

COUNTING AND COMPUTING THE EIGENVALUES OF A COMPLEX TRIDIAGONAL MATRIX, LYING IN A GIVEN REGION OF THE COMPLEX PLANE

1. Linear systems: Introduction to Linear Systems. Gauss Elimination. Numerical Solutions. 2. Vectors: Vector Operations. Dot Product. Span. Linear Independence. The Product [Ax]. Cross Product. Lines, Planes, and Hyperplanes. 3. Matrices: Matrix Operations. Matrix Inverse. Elementary and Invertible Matrices. LU Factorization. 4. Vector spaces: Subspaces of [R to the n Power]. Vector Spaces. Linear Independence Bases. Dimension. Coordinate Vectors and Change of Basis. Rank and Nullity. Applications to Coding Theory. 5. Linear transformations: Matrix Transformations. Linear Transformations. Kernel and Range. The Matrix of a Linear Transformation. The Algebra of Linear Transformations. 6. Determinants: Determinants Cofactor Expansion. Properties of Determinants. The Adjoint Cramer's Rule. Determinants with Permutations. 7. Eigenvalues and eigenvectors: Eigenvalues and Eigenvectors. Diagonalization. Approximations of Eigenvalues and Eigenvectors. Applications to Dynamical Systems. Applications to Markov Chains. 8. Dot and inner products: Orthogonal Sets and Matrices. Orthogonal Projections Gram-Schmidt Process. The QR Factorization. Least Squares. Orthogonalization of Symmetric Matrices. Quadratic Forms and Conic Sections. The Singular Value Decomposition (SVD). Inner Products. Appendices: Answers to selected exercises Each chapter concludes with Applications, Mini-Projects, and Computer Exercises.

Polynomial Equations Giving a Proper Feedback Compensator for a Strictly Proper Plant