Eigenvalue inequalities for positive block matrices with the inradius of the numerical range

We prove the operator norm inequality, for a positive matrix partitioned into four blocks in [Formula: see text], [Formula: see text] where [Formula: see text] is the diameter of the largest possible disc in the numerical range of [Formula: see text]. This shows that the inradius [Formula: see text] satisfies [Formula: see text] Several eigenvalue inequalities are derived. In particular, if [Formula: see text] is a normal matrix whose spectrum lies in a disc of radius [Formula: see text], the third eigenvalue of the full matrix is bounded by the second eigenvalue of the sum of the diagonal block, [Formula: see text] We think that [Formula: see text] is optimal and we propose a conjecture related to a norm inequality of Hayashi.

Generic differential operators on Siegel modular forms and special polynomials

Holomorphic vector valued differential operators acting on Siegel modular forms and preserving automorphy under the restriction to diagonal blocks are important in many respects, including application to critical values of L functions. Such differential operators are associated with vectors of new special polynomials of several variables defined by certain harmonic conditions. They include the classical Gegenbauer polynomial as a prototype, and are interesting as themselves independently of Siegel modular forms. We will give formulas for all such polynomials in two different ways. One is to describe them using polynomials characterized by monomials in off-diagonal block variables. We will give an explicit and practical algorithm to give the vectors of polynomials through these. The other one is rather theoretical but seems much deeper. We construct an explicit generating series of polynomials mutually related under certain mixed Laplacians. Here substituting the variables of the polynomials to partial derivatives, we obtain the generic differential operator from which any other differential operators of this sort are obtained by certain projections. This process exhausts all the differential operators in question. This is also generic in the sense that for any number of variables and block partitions, it is given by a recursive unified expression. As an application, we prove that the Taylor coefficients of Siegel modular forms with respect to off-diagonal block variables, or of corresponding expansion of Jacobi forms, are essentially vector valued Siegel modular forms of lower degrees, which are obtained as images of the differential operators given above. We also show that the original forms are recovered by the images of our operators. This is an ultimate generalization of Eichler–Zagier’s results on Jacobi forms of degree one. Several more explicit results and practical construction are also given.

NUMERICAL RANGE AND POSITIVE BLOCK MATRICES

We obtain several norm and eigenvalue inequalities for positive matrices partitioned into four blocks. The results involve the numerical range $W(X)$ of the off-diagonal block $X$ , especially the distance $d$ from $0$ to $W(X)$ . A special consequence is an estimate, $$\begin{eqnarray}\text{diam}\,W\left(\left[\begin{array}{@{}cc@{}}A & X\\ X^{\ast } & B\end{array}\right]\right)-\text{diam}\,W\biggl(\frac{A+B}{2}\biggr)\geq 2d,\end{eqnarray}$$ between the diameters of the numerical ranges for the full matrix and its partial trace.

The Tropical Matrix Groups with Symmetric Idempotents

In this paper we study the semigroup Mn(T) of all n×n tropical matrices under multiplication. We give a description of the tropical matrix groups containing a diagonal block idempotent matrix in which the main diagonal blocks are real matrices and other blocks are zero matrices. We show that each nonsingular symmetric idempotent matrix is equivalent to this type of block diagonal matrix. Based upon this result, we give some decompositions of the maximal subgroups of Mn(T) which contain symmetric idempotents.

Diagonal Block Method for Solving Two-Point Boundary Value Problems with Robin Boundary Conditions

This numerical study presents the diagonal block method of order four for solving the second-order boundary value problems (BVPs) with Robin boundary conditions at two-point concurrently using constant step size. The solution is obtained directly without reducing to a system of first-order differential equations using a combination of predictor-corrector mode via shooting technique. The shooting method was adapted with the Newton divided difference interpolation approach as the strategy of seeking for the new initial estimate. Five numerical examples are included to examine and illustrate the practical usefulness of the proposed method. Numerical tested problem is also highlighted on the diffusion of heat generated application that imposed the Robin boundary conditions. The present findings revealed that the proposed method gives an efficient performance in terms of accuracy, total function calls, and execution time as compared with the existing method.

HETEROGENIOUS BLOCKED ALL-PAIRS SHORTEST PATHS ALGORITHM

The problem of finding the shortest paths between all pairs of vertices in a weighted directed graph is considered. The algorithms of Dijkstra and Floyd-Warshall, homogeneous block and parallel algorithms and other algorithms of solving this problem are known. A new heterogeneous block algorithm is proposed which considers various types of blocks and takes into account the shared hierarchical memory organization and multi-core processors for calculating each type of block. The proposed heterogeneous block computing algorithms are compared with the generally accepted homogeneous universal block calculation algorithm at theoretical and experimental levels. The main emphasis is on using the nature of the heterogeneity, the interaction of blocks during computation and the variation in block size, the size of the block matrix and the total number of blocks in order to identify the possibility of reducing the amount of computation performed during the calculation of the block, reducing the activity of the processor’s cache memory and determining the influence of the calculation time of each block type on the total execution time of the heterogeneous block algorithm. A recurrent resynchronized algorithm for calculating the diagonal block (D0) is proposed, which improves the use of the processor’s cache and reduces the number of iterations up to 3 times that are necessary to calculate the diagonal block, which implies the acceleration in calculating the diagonal block up to 60%. For more efficient work with the cache memory, variants of permutation of the basic loops k-i-j in the algorithms of calculating the blocks of the cross (C1 and C2) and the updated blocks (U3) are proposed. These permutations in combination with the proposed algorithm for calculating the diagonal block reduce the total runtime of the heterogeneous block algorithm to 13% on average against the homogeneous block algorithm.