Richard Steven Varga

• Ph.D.
• Professor at Kent State University

A numerical formulation of high order accuracy, based on variational methods, is proposed for the solution of multidimensional diffusion-convection-type equations. Accurate solutions are obtained without the difficulties that standard finite difference approximations present. In addition, tests show that accurate solutions of a one-dimensional prob...

This paper presents a new technique for solving some of the partial differential equations that are commonly used in simulating reservoir performance. The results of applying this technique to a simple problem show that one obtains accurate pressure problem show that one obtains accurate pressure values near wells, as well as accurate pressure grad...

We extend here the Perron–Frobenius theory of nonnegative matrices to certain complex matrices. Following the generalization of the Perron–Frobenius theory to matrices that have some negative entries, given by Noutsos [14], we introduce here two types of extensions of the Perron–Frobenius theory to complex matrices. We present and prove here some s...

In this paper, we consider the localization of generalized eigenvalues, and we discuss ways in which the Gersgorin set for generalized eigenvalues can be approximated. Earlier, Stewart proposed an approximation using a chordal metric. We will obtain here an improved approximation, and using the concept of generalized diagonal dominance, we prove th...

The following problem arose in a correspondence between the authors, concerning the exact location of the set of eigenvalues of special real matrices, arising from a problem of synchronization of chaotic oscillators.

Professor David M. Young, Jr was instrumental in the development of iterative algorithms and software for solving large sparse systems of linear algebraic equations of the type that typically arise in the numerical solution of partial differential equations. We present a brief overview of his life, accomplishments, and research. Copyright © 2010 Jo...

In this paper, we give new extensions of results by G. Szegő, in 1924, on the interrelationships between the zeros of the partial sums of ezez, and those of the partial sums of cos(z)cos(z) and sin(z)sin(z).We also include numerical results and figures which illustrate our new results.

Our investigations into solving systems of liner equations thus far have centered on the theoretical aspects of variants of the successive overrelaxation iterative method and the alternating-direction implicit methods. In all cases, the theoretical selection of associated optimum acceleration parameters was based on knowledge of certain key eigenva...

Many of the problems of physics and engineering that require numerical approximations are special cases of the following second-order linear parabolic differential equation: $$ \begin{array}{*{20}c} {\phi \left( {\rm x} \right)u_t \left( {{\rm x};t} \right)} \hfill & { = \sum\limits_{i = 1}^n {\left( {K_i \left( {\rm x} \right)u_{x_i } } \right)_{x...

The basis aims in this chapters are to derive finite difference approximations for certain elliptic differential equations, to study the properties of the associated matrix equations, and to describe rigorous iterative methods for the solution of such matrix equations, In certain cases, the derived properties of the associated matrix equations are...

Let A = [a i,j] be an n × n complex matrix, and let us seek the solution of the system of linear equations $$ \begin{array}{*{20}c} {\sum\limits_{j = 1}^n {a_{i,j} x_j = k_i ,} } \hfill & {1 \le i \le n,} \hfill \\ \end{array} $$ (3.1) which we write in matrix notation as $$ A{\rm x} = {\rm k}, $$ (2.3) where k is a given column vendor in Cn. The s...

The title of this book, Matrix Iterative Analysis, suggests that we might consider here all matrix numerical methods which are iterative in nature. However, such an ambitious goal is in fact replaced by the more practical one where we seek to consider in some detail that smaller branch of numerical analysis concerned with the efficient solution, by...

The point successice overrelaxation iterative method of Chap.3 was simultaneously introduced by Frankel (1950) and Young (1950). Whereas Frankel considered the special case of the numerical solutation of the Dirichlet problem for a rectangle and showed for this case that the point successive overrelaxation iterative method, with suitable chosen rel...

In our previous investigations concerning the acceleration of basic iterative methods, we would, in the final analysis, consider the convergence properties and rates of convergence of an iterative procedure of the form $$ \begin{array}{*{20}c} {{\rm x}^{\left( {m + 1} \right)} = M{\rm x}^{\left( m \right)} + {\rm g},} \hfill & {m \ge 0,} \hfill \\...

In the previous chapter, we considered various block iterative methods, such as the line and 2-line successive overrelaxation iterative methods, which as pratical methods involved the direct (or implicit) solution of particular lower-order matrix equations. Also, as a standard for comparison of the asymptotic rates of convergence of these methods,...

We introduce several localization techniques for the generalized eigenvalues of a matrix pair, obtained via the famous Geršgorin theorem and its generalizations. Specifically, we address the techniques of computing and graphing of the obtained localization sets of a matrix pair. The work that follows involves much about nonnegative matrices, strict...

Let α be a positive number, and let E n (x α; [0, 1]) denote the error of best uniform approximation to x α by polynomials of degree at most n on the interval [0, 1]. Russian mathematician S. N. Bernstein established the existence of a nonnegative constant β(α) such that β(α):=lim n→∞(2n)2α E n (x α;[0, 1]) (α>0). In addition, Bernstein showed t...

With s n (z):=∑ k=0 n z k /z! denoting the n-th partial sum of e z , let its zeros be denoted by {z k,n } k=1 n for any positive integer n. If θ 1 and θ 2 are any angles with 0<θ 1 <θ 2 <2π, let Z θ 1 ,θ 2 be the associated sector, in the z-plane, defined by Z θ 1 ,θ 2 :=z ∈ ℂ : θ 1 ≤ arg z ≤ θ 2 · If #{z k,n } k=1 n ∩ Z θ 1 ,θ 2 represents the num...

In this paper, we address the problem of finding a numerical approximation to the minimal Geršgorin set, Γ R (A), of an irreducible matrix A in C n,n . In particular, boundary points of Γ R (A) are related to a well-known result of Olga Taussky.

With E(n,n)(Absolute value of x; [-1, +1]) denoting the error of best uniform rational approximation from pi(n,n) to Absolute value of x on [-1, +1], we determine the numbers {E2n,2n(Absolute value of x; [-1, +1])}n=1(40), where each of these numbers was calculated with a precision of at least 200 significant digits. With these numbers, the Richard...

Let denote a nite-dimensional square complex matrix, and let denote a x ed singular value decomposition (SVD) of . In this note, we follow up work from Smithies and Varga (Linear Algebra Appl., 417 (2006), pp. 370ñ380), by dening the SV-normal estimator , (which satises ), and showing how it denes an upper bound on the norm, , of the commutant of a...

We continue the work of Szegő [18] on describing the angular distribution of the zeros of the normalized partial sum s n(nz) of e z, where $s _{n}(z):={\sum _{k=0} ^{n}}z ^k/k!$ . We imbed this problem in some inverse problem of potential theory and prove a so-called Erdős-Turán-type theorem, which is of interest in itself.

The behavior of the constants λn,n(e −x ), denoting the errors of best uniform approximation to e −z on the interval [0,+∞) by real rational functions having numerator and denominator polynomials of degree at most n, has generated much recent interest in the approximation theory literature. Based on high-precision calculations, we present here the...

In this note, we introduce the singular value decomposition Geršgorin set, ΓSV (A), of an N × N complex matrix A, where N ⩽ ∞. For N finite, the set ΓSV (A) is similar to the standard Geršgorin set, Γ (A), in that it is a union of N closed disks in the complex plane and it contains the spectrum, σ(A), of A. However, ΓSV (A) is constructed using col...

We give a generalization of a less well-known result of Dashnic and Zusmanovich [2] from 1970, and show how this generalization compares with related results in this area.

We give a generalization of a less well-known result of Dashnic and Zusmanovich [2] from 1970, and show how this generalization compares with related results in this area.

We study here in detail the location of the real and complex zeros of the partial sums of and , which extends results of Szegö (1924) and Kappert (1996).

We extend results of Szegő (1924) and Kappert (1996) on the location of the zeros of the normalized partial sums of cos (z) and sin (z), and their rates of convergence to the associated Szegő curves.

We construct polynomial approximations of Dzjadyk type (in terms of the k-th modulus of continuity, k⩾1) for analytic functions defined on a continuum E in the complex plane, which simultaneously interpolate at given points of E. Furthermore, the error in this approximation is decaying as e−cnα strictly inside E, where c and α are positive constant...

We establish a discrepancy theorem for signed measures, with a given positive part, which are supported on an arbitrary convex curve. As a main application, we obtain a result concerning the distribution of zeros of polynomials orthogonal on a convex domain.

For any p>1 and for any sequence $\{ a_j \}^\infty_{j=1}$ of nonnegative numbers, a classical inequality of Hardy gives that$$\sum^n_{k=1} \left({\sum \nolimits^k_{i=1} a_i\over k}\right)^p\les \left({p\over p-1}\right)^p\sum^n_{k=1}a^p_k \quad {\hbox{for each}}\;n \; \in \; {\open {N}},$$unless all $a_j=0$, where the constant $[p/(p-1)]^p$ is best...

The Riemann Hypothesis is equivalent to the conjecture that the de Bruijn-Newman constant satisfies 0. However, so far all the bounds that have been proved for go in the other direction, and provide support for the conjecture of Charles Newman that 0. This paper shows how to improve previous lower bounds and prove that Gamma2:7 Delta 10 Gamma9 ! :...

Given a triple (G, W, γ) of an open bounded set G in the complex plane, a weight function W(z) which is analytic and different from zero in G, and a number γ with 0 ≤ γ ≤ 1, we consider the problem of locally uniform rational approximation of any function ƒ(z), which is analytic in G, by weighted rational functions Wmi+ni(z)Rmi, ni(z)i = 0∞, where...

: Given an analytic Jordan curve Gamma, with interior G and exteriorOmegaGamma and given a sequence of complex numbers fa n g 1 n=0 satisfying lim sup n!1 ja n j 1=n = 1, we consider here three series of the form f(z) = 1 X n=0 a n P n (z); where the polynomials P n (z) are chosen to be (i) the Faber polynomials associated with G, (ii) the polynomi...

: It is shown here that a 1989 conjecture of Rigler, Trimble, and Varga in the theory of polynomials having concentration at low degrees, is not true in general, and counterexamples are explicitly derived here. What is intriguing is that these counterexamples produce only small deviations from the conjecture mentioned above. AMS 1980 Classification...

In 1924, Szeg} o showed that the zeros of the normalized partial sums, sn(nz), of ez tended to what is now called the Szeg} oc urve S ,w here S:= z2C :jze1 zj =1 andj zj 1 : Using modern methods of weighted potential theory, these zero distribution results of Szeg} o can be essentially recovered, along with an asymptotic formula for the weighted pa...

. Given an open bounded set G in the complex plane and a weight function W (z) which is analytic and different from zero in G, we consider the problem of locally uniform rational approximation of any function f(z), which is analytic in G, by particular ray sequences of weighted rational functions of the form W m+n (z)Rm;n (z), where Rm;n (z) = Pm (...

We establish a discrepancy theorem for signed measures, with a given positive part, which are supported on an arbitrary convex curve. As a main application, we obtain a result concerning the distribution of zeros of polynomials orthogonal on a convex domain. 1. Introduction and main results Let G ae C be a bounded Jordan domain, and let h(z) be a w...

Recently, two Gersgorin-type matrix questions were raised. These are answered here, using ovals of Cassini.

Given a pair (G,W) of an open bounded set G in the complex plane and a weight function W(z) which is analytic and different from zero in G , we consider the problem of the locally uniform approximation of any function f(z) , which is analytic in G , by weighted polynomials of the form {W n (z)P n (z) } $\infinity$ n=0 , where deg Pn\(\leq\)n. The m...

. The de Bruijn-Newman constant has been investigated extensively because the truth of the Riemann Hypothesis is equivalent to the assertion that 0. On the other hand, C. M. Newman conjectured that 0. This paper improves previous lower bounds by showing that Gamma5:895 Delta 10 Gamma9 ! : This is done with the help of a spectacularly close pair of...

Dedicated to Olga Taussky and John Todd, on the occasion of their important birthdays in 1996, for their inspiring work in matrix theory and numerical analysis. Abstract. Making use, from the preceding paper, of the affirmative solution of the Spectral Conjecture, it is shown here that the general boundaries, of the minimal Gerschgorin sets for par...

This study is concerned with k-step methods for the iterative solution of nonsymmetric systems of real linear equations. These are generalizations of the Chebyshev (2-step) iteration, with the potential for faster convergence in cases where the spectrum of the underlying coefficient matrix is not approximated well by an ellipse. We investigate the...

Recently, a classification of matrices of class Z was introduced by Fiedler and Markham. This classification contains the classes of M-matrices and the classes of N0- and F0-matrices studied by Fan, G. Johnson, and Smith. The problem of determining which nonsingular matrices have inverses which are Z-matrices is called the inverse Z-matrix problem....

Recently, Marti´nez, Michon, and San Marti´n introduced the new class of (symmetric)strictly ultrametric matrices. They proved that the inverse of a strictly ultrametric matrix is a strictly row and strictly column diagonally dominant Stieltjes matrix. Here, we generalize their result by introducing a class of nonsymmetric matrices, calledgeneraliz...

Recently, a classification of matrices of class Z was introduced by Fiedler and Markham. This classification contains the classes of M--matrices and the classes of N 0 -- and F 0 --matrices studied by K. Fan, G. Johnson, and R. Smith. The problem of determining which nonsingular matrices have inverses which are Z--matrices is called the inverse Z--...

We give here a rigorous formulation for a pair of consecutive simple positive zeros of the functionH 0 (which is closely related to the Riemann - 4.379 10 - 6 < L - 4.379 \cdot 10^{ - 6}< \Lambda

With s(n)(z) denoting the n-th partial sum of e(z), the exact rate of convergence of the zeros of the normalized partial sums, s(n)(nz), to the Szego curve D0,infinity was recently studied by Carpenter et al. (1991), where D0,infinity is defined by D0,infinity := {z is-an-element-of C : \ze1-z\ = 1 and Absolute value of z less-than-or-equal-to 1}....

It is well known that every $n \times n$ Stieltjes matrix has an inverse that is an $n \times n$ nonsingular symmetric matrix with nonnegative entries, and it is also easily seen that the converse of this statement fails in general to be true for $n > 2$. In the preceding paper by Martinez, Michon, and San Martin [SIAM J. Matrix Anal. Appl., 15 (19...

We present here a new hybrid method for the iterative solution of large sparse nonsymmetric systems of linear equations, say of the formAx=b, whereA N, N , withA nonsingular, andb N are given. This hybrid method begins with a limited number of steps of the Arnoldi method to obtain some information on the location of the spectrum ofA, and then swi...

It was recently shown that the inverse of a strictly ultrametric matrix is a strictly diagonally dominant Stieltjes matrix. On the other hand, as it is well-known that the inverse of a strictly diagonally dominant Stieltjes matrix is a real symmetric matrix with nonnegative entries, it is natural to ask, conversely, if every strictly diagonally dom...

Let , where ¦·¦ is the Euclidean norm, and for X ⊂ n, let X denote the closed convex hull of X in n. In 1990, Graham showed that if f is a normalized holomorphic map from Bn1 into n, and if f is either an open map or a polynomial map, then there is a sharp, uniform constant a, a given by ae1+a=1, such that f(Bn1) ⊃ Bna. Graham posed the question to...

The successive-overrelaxation (SOR) iterative method for linear systems is well understood if the associated Jacobi matrix B is consistently ordered and weakly cyclic of index 2. If, in addition, B2 has only nonnegative eigenvalues and if ϱ(B), the spectral radius of B, is strictly less than unity, then by D. M. Young's classical theorem, the optim...

The de Bruijn-Newman constant Λ has been investigated extensively because the truth of the Riemann hypothesis is equivalent to the assertion that Λ≤0. On the other hand, C. M. Newman conjectured that Λ≥0. This paper improves previous lower bounds by showing that -5·895·10 -9 <Λ. This is done with the help of a spectacularly close pair of consecutiv...

The level set structure of a real entire function f is investigated. The results establish a connection between the level sets of f the Laguerre expression for f and the distribution of zeros of f. An application to the Riemann Hypothesis is also given.

Let $A = I - B \in \mathbb{C}^{n,n} $, with diag$(B) = 0$, denote a nonsingular non-Hermitian matrix. To iteratively solve the linear system $A{\bf x} = {\bf b}$, two splittings of A, together with induced relaxation methods, have been recently investigated in [W. Niethammer and R. S. Varga, Results in Math., 16 (1989), pp. 308–320]. The Hermitian...

Let α be a positive number, and let En,n(xα;[0,1]) denote the error of best uniform rational approximation from πn,n to xα on the interval [0,1]. We rigorously determined the numbers {En,n(xα;[0,1])}n=1/30 for six values of α in the interval (0, 1), where these numbers were calculated with a precision of at least 200 significant digits. For each of...

We continue the work of Szego and others on describing the convergence of the zeros, {z(k,n)}k = 1n, of the normalized partial sum s(n)(nz) of e(z) where s(n)(z) := SIGMA-j = 0n z(j)/j!, to the Szego curve D infinity, where D infinity := {z an-element-of C:\ze1-z\ = 1 and \z\ less-than-or-equal-to 1}. It turns out that the convergence rate of these...

We investigate here a new numerical method, base on the Laguerre inequalities, for determining lower bounds for the de Bruijn-Newman constant L1 (Hl (x))*20c \text. \text. = (H¢l (x))2 - Hl (x) H"l (x)\text (x,\text l Î \mathbbR\text)L_1 (H_\lambda (x))\begin{array}{*{20}c} {\text{.}} \\ {\text{.}} \\ \end{array} = (H'_\lambda (x))^2 - H_\lambda...

In order to study the structure of defects in nematic liquid crystals, we have constructed a numerical procedure that minimizes the Landau-de Gennes free energy model. Using a new representation, a finite-element discretization, and a direct minimization scheme based on Newton's method and successive overrelaxation, this procedure determines the or...

In this paper, generalizations of some known results for Jensen polynomials, pertaining to (i)convexity, (ii) the Turán inequalities, and (iii) the Laguerre inequalities, are established. These results are then applied in general to real entire functions, which are representable by Fourier transforms, and in particular to the Riemann ξ-function.

RESUMEN RESUMEN It is shown that the kernel K_n  (z), n (even) ≥ 2, in the contour integral representation of the remainder term of the n-point Gauss formula for the Chebyshev weight function of the second kind, as z varies on the ellipse E _ρ =3D { z:z =3D ρ e ^iv + ρ^-1 e ^-iv ,...

Without Abstract

For the matrix equation Ax = b, we consider here two splittings A = M1 − N1 = M2 − N2 of the matrix A, where M 1 ≔ (A + A*)/2 is the Hermitian part of A, and M 2 ≔ I + (A − A*)/2 is the identity plus the skew-Hermitian part of A. To these two splittings of A, we apply an extrapolation, with extrapolation factor ω, and we find associated regions for...

A new necessary and sufficient condition for real entire functions, represented by Fourier transforms, to have only real zeros is proved. An application of this result to the Riemann ξ-function is also given.

The basic aim of the research proposed for this AFOSR Grant was to investigate, in depth, the optimization of iterative methods for solution of nonsymmetric nonpositive-definite matrix equations which arise in the numerical approximation (by finite difference or finite elements) of typical convection- diffusion problems. In the opinion of this Prin...

A unified approach is presented for determining all the constants λm, n (m ≥ 0, n ≥ 0) which occur in the study of real vs. complex rational Chebyshev approximation on an interval. In particular, it is shown that λm, m+2 = 1/3 (m ≥ 0), a problem which had remained open.

A new necessary and sufficient condition for real entire functions, represented by Fourier transforms, to have only real zeros is proved. An application of this result to the Riemann ξ-function is also given.

The purpose of this note is threefold:i) to derive the new functional equation, $\begin{gathered} \left[ {\lambda - \left( {1 - \omega } \right)\left( {1 - \hat \omega } \right)} \right]^p = \lambda ^k \left[ {\lambda \omega + \hat \omega - \omega \hat \omega } \right]^{\left| {\zeta _L } \right| - k} \left[ {\lambda \hat \omega + \omega - \omega \...

A unified approach is presented for determining all the constants $\gamma_{m,n} (m \geq 0, n \geq 0)$ which occur in the study of real vs. complex rational Chebyshev approximation on an interval. In particular, it is shown that $\gamma_{m,m+2} = 1/3 (m \geq 0)$, a problem which had remained open.

Given a large sparse system of linear algebraic equations in fixed point form ${\bf x} = T{\bf x} + {\bf c}$, one way to solve this system is to apply a semi-iterative method (SIM) to the basic iteration method ${\bf x}_m = T{\bf x}_{m - 1} + {\bf c}$. It is known that if the spectrum, $\sigma (T)$, of T is contained in some compact subset $\Omega...

It is known that the Riemann hypothesis is equivalent to the statement that all zeros of the Riemann [^(b)]m (l): = ò0¥ t2m elt2 F(t)dt\hat b_m (\lambda ): = \int_0^\infty {t^{2m} e^{\lambda t^2 } \Phi (t)dt} satisfy the Turn inequalities ([^(b)]m (l))2 > ( \frac2m - 12m + 1 )[^(b)]m - 1 (l)[^(b)]m + 1 (l)(m \geqslant 1,l\geqslant 0).(\hat b_m (\l...

A technique is developed whereby one can obtain asymptotic estimates of eigenvalues of first-order iteration matrices. The technique is applied to iteration matrices arising from the numerical solution of the 1- and 2-dimensional biharmonic equation. The eigenvalue estimates are computationally verified.

In this paper, we give a new proof, based on matrix theory, and sharpenings of a result of Fejér on the boundedness of partial sums of functions in H∞.

A measurement technique has been developed for noninvasive breast cancer detection. The process involves the use of close-range stereophotogrammetry as a data acquisition device for the determination of breast surface concavities. We report the methodology used to detect these surface depressions, the rationale for the study, and our preliminary fi...

A lower bound is constructively found for the de Bruijn-Newman constant , which is related to the Riemann Hypothesis. This lower bound is determined by explicitly exhibiting an associated jensen polynomial with nonreal zeros.

With E_{2n}(\vert x\vert) denoting the error of best uniform approximation to \vert x\vert by polynomials of degree at most 2n on the interval \lbrack-1,\ +1\rbrack, the famous Russian mathematician S. Bernstein in 1914 established the existence of a positive constant \beta for which \displaystyle \lim_{n\to\infty}2nE_{2n}(\vert x\vert)=:\beta.More...