Real Symmetric Positive Definite Matrix Research Articles

The behavior of the Gauss-Radau upper bound of the error norm in CG

Consider the problem of solving systems of linear algebraic equations varvec{A}varvec{x}=varvec{b} with a real symmetric positive definite matrix varvec{A} using the conjugate gradient (CG) method. To stop the algorithm at the appropriate moment, it is important to monitor the quality of the approximate solution. One of the most relevant quantities for measuring the quality of the approximate solution is the varvec{A}-norm of the error. This quantity cannot be easily computed; however, it can be estimated. In this paper we discuss and analyze the behavior of the Gauss-Radau upper bound on the varvec{A}-norm of the error, based on viewing CG as a procedure for approximating a certain Riemann-Stieltjes integral. This upper bound depends on a prescribed underestimate varvec{mu } to the smallest eigenvalue of varvec{A}. We concentrate on explaining a phenomenon observed during computations showing that, in later CG iterations, the upper bound loses its accuracy, and is almost independent of varvec{mu }. We construct a model problem that is used to demonstrate and study the behavior of the upper bound in dependence of varvec{mu }, and developed formulas that are helpful in understanding this behavior. We show that the above-mentioned phenomenon is closely related to the convergence of the smallest Ritz value to the smallest eigenvalue of varvec{A}. It occurs when the smallest Ritz value is a better approximation to the smallest eigenvalue than the prescribed underestimate varvec{mu }. We also suggest an adaptive strategy for improving the accuracy of the upper bounds in the previous iterations.

From symplectic eigenvalues of positive definite matrices to their pseudo-orthogonal eigenvalues

Williamson's theorem states that every real symmetric positive definite matrix $A$ of even order can be brought to diagonal form via a symplectic $T$-congruence transformation. The diagonal entries of the resulting diagonal form are called the symplectic eigenvalues of $A$. We point at an analog of this classical result related to Hermitian positive definite matrices, *-congruences, and another class of transformation matrices, namely, pseudo-unitary matrices. This leads to the concept of pseudo-unitary (or pseudo-orthogonal, in the real case) eigenvalues of positive definite matrices.

Accurate error estimation in CG

In practical computations, the (preconditioned) conjugate gradient (P)CG method is the iterative method of choice for solving systems of linear algebraic equations Ax = b with a real symmetric positive definite matrix A. During the iterations, it is important to monitor the quality of the approximate solution xk so that the process could be stopped whenever xk is accurate enough. One of the most relevant quantities for monitoring the quality of xk is the squared A-norm of the error vector x − xk. This quantity cannot be easily evaluated; however, it can be estimated. Many of the existing estimation techniques are inspired by the view of CG as a procedure for approximating a certain Riemann–Stieltjes integral. The most natural technique is based on the Gauss quadrature approximation and provides a lower bound on the quantity of interest. The bound can be cheaply evaluated using terms that have to be computed anyway in the forthcoming CG iterations. If the squared A-norm of the error vector decreases rapidly, then the lower bound represents a tight estimate. In this paper, we suggest a heuristic strategy aiming to answer the question of how many forthcoming CG iterations are needed to get an estimate with the prescribed accuracy. Numerical experiments demonstrate that the suggested strategy is efficient and robust.

Design of Time–Frequency-Localized Two-Band Orthogonal Wavelet Filter Banks

In this paper, we design time–frequency-localized two-band orthogonal wavelet filter banks using convex semidefinite programming (SDP). The sum of the time variance and frequency variance of the filter is used to formulate a real symmetric positive definite matrix for joint time–frequency localization of filters. Time–frequency-localized orthogonal low-pass filter with specified length and regularity order is designed. For nonmaximally regular two-band filter banks of length twenty, it is found that, as we increase the regularity order, the solution of the SDP converges to the filters with time–frequency product (TFP) almost same as the Daubechies maximally regular filter of length twenty. Unlike the class of Daubechies maximally regular minimum phase wavelet filter banks, a rank minimization algorithm in a SDP is employed to obtain mixed-phase low-pass filters with TFP of the filters as well as the scaling and wavelet function better than the equivalent two-band Daubechies filter bank.

On the Symplectic Eigenvalues of Positive Definite Matrices

A simple proof of Williamson’s theorem is given. This theorem states that a real symmetric positive definite matrix A of even order can be brought to diagonal form Λ by a symplectic congruence transformation. The diagonal entries of Λ are called symplectic eigenvalues of A. The problem of calculating these values is also discussed.

Modified quasi-Chebyshev acceleration to nonoverlapping parallel multisplitting method

In this study, we propose a modified quasi-Chebyshev acceleration to the nonoverlopping multisplitting iteration method for solving the linear systems Ax = b where A is a real symmetric positive definite matrix or an H-matrix. In the process of the parallel multisplitting method, the distributive tasks are parallelly computed by each processor, then a global modified acceleration is used to obtain the solution of the system Ax = b for every τ steps, such that the efficiency of the computation can be improved. The convergence theory of the new algorithm is given under some reasonable conditions. Finally, numerical experiments show that the method is efficient and effective.

A limited memory BFGS algorithm for non-convex minimization with applications in matrix largest eigenvalue problem

This study aims to present a limited memory BFGS algorithm to solve non-convex minimization problems, and then use it to find the largest eigenvalue of a real symmetric positive definite matrix. The proposed algorithm is based on the modified secant equation, which is used to the limited memory BFGS method without more storage or arithmetic operations. The proposed method uses an Armijo line search and converges to a critical point without convexity assumption on the objective function. More importantly, we do extensive experiments to compute the largest eigenvalue of the symmetric positive definite matrix of order up to 54,929 from the UF sparse matrix collection, and do performance comparisons with EIGS (a Matlab implementation for computing the first finite number of eigenvalues with largest magnitude). Although the proposed algorithm converges to a critical point, not a global minimum theoretically, the compared results demonstrate that it works well, and usually finds the largest eigenvalue of medium accuracy.

Infinitely many homoclinic solutions for second-order discrete Hamiltonian systems

In this paper, we study the second-order self-adjoint discrete Hamiltonian systemwhere is an real symmetric positive definite matrix for all , is unnecessarily positive definite for all and is indefinite sign. By using fountain theorem, we establish some new criteria to guarantee that the above system has infinitely many homoclinic solutions under the assumption that is superquadratic as . Our results generalize and improve some existing results in the literature.

Annulus criteria for mixed nonlinear elliptic differential equations

New oscillation criteria are obtained for forced second order elliptic partial differential equations with damping and mixed nonlinearities of the form ∇ ⋅ ( A ( x ) | ∇ y | α − 1 ∇ y ) + 〈 b ( x ) , | ∇ y | α − 1 ∇ y 〉 + f ( x , y ) = e ( x ) , x ∈ Ω where Ω is an exterior domain in R N , f ( x , y ) = c ( x ) | y | α − 1 y + c 1 ( x ) | y | β − 1 y + c 2 ( x ) | y | γ − 1 y and β > α > γ > 0 . It is assumed that A = ( a i j ) N × N is a real symmetric positive definite matrix function, b = ( b i ) N × 1 is a real vector function, a i j ∈ C loc 1 + μ ( Ω , R ) , and b i , c , c 1 , c 2 , e ∈ C loc μ ( Ω , R ) for all i , j , for some μ ∈ ( 0 , 1 ) . Examples are given to illustrate the results.

Nonstationary multisplittings with general weighting matrices for mildly nonlinear systems

In this paper, we present a new nonstationary multisplitting iterative method for solving the mildly nonlinear equations Ax = F( x), where A ∈ R n× n is an n-by- n sparse real symmetric positive definite matrix and F : R n → R n is a general nonlinear mapping, and establish the convergence theories with general weighting matrices by more than one inner iteration. Meanwhile, we also give the lower bound of the inner iteration number in this nonstationary multisplitting iterative method.

Geometric interpretation of the three-dimensional coherence matrix for nonparaxial polarization

The 3-dimensional coherence matrix is interpreted by emphasising its invariance with respect to spatial rotations. Under these transformations, it naturally decomposes into a real symmetric positive definite matrix, interpreted as the moment of inertia of the ensemble (and the corresponding ellipsoid), and a real axial vector, corresponding to the mean angular momentum of the ensemble. This vector and tensor are related by several inequalities, and the interpretation is compared to those in which unitary invariants of the coherence matrix are studied.

Scaling symmetric positive definite matrices to prescribed row sums

We give a constructive proof of a theorem of Marshall and Olkin that any real symmetric positive definite matrix can be symmetrically scaled by a positive diagonal matrix to have arbitrary positive row sums. We give a slight extension of the result, showing that given a sign pattern, there is a unique diagonal scaling with that sign pattern, and we give upper and lower bounds on the entries of the scaling matrix.

On Generalized Antieigenvalue and Antieigenmatrix of Order r

SYNOPTIC ABSTRACTThe concept of the smallest antieigenvalue of a real symmetric positive definite matrix is extended to the generalized antieigenvalue of order r, and thus instead of a corresponding antieigenvector, one confronts the problem of finding an antieigenmatrix of order r. We provide a closed form expression for the generalized antieigenvalue and generalized antieigenmatrix. These quantities have found certain applications in statistics, and these applications will be illustrated in a separate communication.

On the Calculation of Antieigenvalues and Antieigenvectors

We provide a simple approach for computing antieigenvalues and antieigenvectors of a real symmetric positive definite matrix. Our approach closely follows that used for solving a statistical problem considered by Rao and Rao (1987). The approach is illustrated using certain examples.

Euclidean Norm Minimization of the SOR Operators

Because the spectral radius is only an asymptotic measure of the rate of convergence of a linear iterative method, Golub and dePillis [ Toward an effective two-parameter method, in Iterative Methods for Large Linear Systems, Academic Press, New York, 1990] have raised in a recent paper the question of determining, for each $k\geq 1$, a relaxation parameter $\omega\in (0,2)$ and a pair of relaxation parameters $\omega_1$ and $\omega_2$ which minimize the Euclidean norm of the kth power of the SOR and MSOR iteration matrices, respectively, associated with a real symmetric positive definite matrix with Property A. Here we use a reduction of these operators which they derived from the SVD of the associated block Jacobi matrix to obtain the minimizing relaxation parameters for the case k = 1 for both operators. We conclude the paper with two brief sections in which we assess what our results imply.

Estimates in quadratic formulas

LetA be a real symmetric positive definite matrix. We consider three particular questions, namely estimates for the error in linear systemsAx=b, minimizing quadratic functional minx(x T Ax−2b T x) subject to the constraint ‖x‖=α, α<‖A −1 b‖, and estimates for the entries of the matrix inverseA −1. All of these questions can be formulated as a problem of finding an estimate or an upper and lower bound onu T F(A)u, whereF(A)=A −1 resp.F(A)=A −2,u is a real vector. This problem can be considered in terms of estimates in the Gauss-type quadrature formulas which can be effectively computed exploiting the underlying Lanczos process. Using this approach, we first recall the exact arithmetic solution of the questions formulated above and then analyze the effect of rounding errors in the quadrature calculations. It is proved that the basic relation between the accuracy of Gauss quadrature forf(λ)=λ−1 and the rate of convergence of the corresponding conjugate gradient process holds true even for finite precision computation. This allows us to explain experimental results observed in quadrature calculations and in physical chemistry and solid state physics computations which are based on continued fraction recurrences.

Noniterative and fast iterative methods for interpolation and extrapolation

In this correspondence we study the band-limited interpolation and extrapolation problems for finite-dimensional signals. We show that these problems can be easily reduced to the solution of a set of linear equations with a real symmetric positive-definite matrix S with spectral radius /spl rho/(S) >

On the convergence of the ordered roots of a sequence of determinantal equations

Let λ 1( n)⩾λ 2( n)⩾⋯⩾λ p ( n) be the ordered roots of | A( n)−λ B( n)| = 0, where A( n) is a p × p real symmetric matrix and B( n) is a p p real symmetric positive definite matrix. Assume that A( n) → A and B( n) → B as n → ∞, where the rank of B may be less than p. Using a result on the continuity of the zeros of a polynomial, the limiting behavior of p sequences {λ j ( n)} ∞ n = 1 , j = 1, 2,…, p, is investigated.

Stationary values of the ratio of quadratic forms subject to linear constraints

Let A be a real symmetric matrix of order n, B a real symmetric positive definite matrix of order n, and C an n$\times$p matrix of rank r with r $\leq$ p > n. We wish to determine vectors $\underset ~\to x$ for which ${\underset ~\to x}^T\ A\underset ~\to x\ / {\underset ~\to x}^T\ B\underset ~\to x$ is stationary and $C^T \underset ~\to x\ = \underset ~\to \Theta$, the null vector. An algorithm is given for generating a symmetric eigensystem whose eigenvalues are the stationary values and for determining the vectors $\underset ~\to x$. Several Algol procedures are included.