We performed evaluations on the following set of sizes for matrix A, ranging from small to large: {100×100, 250×250, 500×500, 1000×1000} . Three matrix types were used: (i) ban- ded (with 5 diagonals), (ii) lower triangular, (iii) **symmetric** **positive** **definite** (sympd). For each matrix size and type, 1000 unique random systems were solved, with each random **system** solved using the standard LU-based solver (as per Section II-D) and the adaptive solver. For each solver, the average wall-clock time (in seconds) across the 1000 runs is reported. The results are presented in Tables I, II and III.

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where A is a large sparse non-Hermitian **positive** deﬁnite matrix, that is, its Hermitian part H = (A + A ∗ )/ is Hermitian **positive** deﬁnite, where A ∗ denotes the conjugate transpose of a matrix A. In order to solve **system** () by iterative methods, usually, eﬃcient splittings of the coeﬃcient matrix A are required. For example, the classic Jacobi and Gauss-Seidel iterations [–] split the matrix A into its diagonal and oﬀ-diagonal parts. Recently, consid- erable interest appears in the work on the Hermitian and skew-Hermitian splitting (HSS) method for this **system** introduced by Bai et al. [] and some generalized HSS methods such as the NSS method [], PSS method [], PHSS method [, ], and LHSS method [], and several signiﬁcant theoretical results are proposed. Meanwhile, these methods and theoretical results are applied to this **linear** **system** directly or indirectly (as a precondi- tioner); see [–]. It is shown in [, , ] that the successive over-relaxed (SOR) itera- tive method and **symmetric** SOR (SSOR) iterative method for Hermitian **positive** deﬁnite **linear** systems are convergent. But, is the same true for these iterative methods for non- Hermitian **positive** deﬁnite **linear** systems? In this paper, we mainly study the convergence of the SOR iterative and SSOR iterative method for non-Hermitian **positive** deﬁnite **linear** systems and propose some convergence conditions.

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prove that the monic Jacobi matrix associated with x 2 U can be obtained from the monic Jacobi matrix associated with U applying two consecutive Darboux transfor- mations without parameter with shifts − and , and taking limits when tends to zero. This problem was considered by Buhmann and Iserles [3] in a more general context. They considered a **positive** **definite** **linear** functional L and the **symmetric** Jacobi matrix associated with the orthonormal sequence of polynomials associated with L, and proved that one step of the QR method applied to the Jacobi matrix corre- sponds to finding the Jacobi matrix of the orthonormal polynomial **system** associated with x 2 L.

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Central difference formula, Finite difference methods, Generalized symmetric eigenvalue problem, Positive definite matrix, Two-point boundary value problem.. INTRODUCT ION We shall consi[r]

When W and T are both **symmetric** **positive** semi- **definite** satisfying that null {T } ∩ null {W } ̸= { 0 } , the co- efficient matrix of (I.1) is singular. For solving the non- Hermitian singular **linear** equations (I.1) efficiently, Bai et al. [ 8 ] investigated the semi-convergence property of the HSS iteration method. Recently, Chen et al. [ 16 ] , Yang et al. [ 30 ] and Wu et al. [ 29 ] proposed the semi-convergence properties of the MHSS iteration method for solving sin- gular complex **linear** systems. There are also some recent studies on iterative methods for singular **linear** systems in [ 25 ] , [ 19 ] , [ 24 ] . However, from the numerical results we can see that those iterative methods and the corresponding preconditioned Krylov subspace methods converge very slowly.

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In the case of one variable, this problem first arose from the work of A. M. Krall [1] when he studied the orthogonal polynomials that are eigenfunctions of a fourth order differential operator considered in [2,3], and showed that the polynomials are orthogonal with respect to a measure that is obtained from a continuous measure on an interval by adding masses at the end points of the interval. In [4], Koornwinder studied the case that the measure is the Jacobi weight function together with additional mass points at 1 and − 1; he constructed the corresponding orthogonal polynomials and studied their properties. Uvarov [5] considered the problem of orthogonal polynomials with respect to a measure obtained by adding a finite discrete part to another measure; his main result expresses the polynomials orthogonal with respect to the new measure in terms of the polynomials orthogonal with respect to the old one. More generally, one can consider perturbations of quasi-**definite** **linear** functionals via the addition of Dirac delta functionals, orthogonal polynomials in such a general setting has been studied extensively in recent years (see, for instance [6,7], [8] and the references therein).

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at each grid point of the skeleton (Smith et al., 2006). A model-based framework was developed to construct the medial manifolds of fiber tracts and then to test pointwise hypotheses based on diffusion properties along the medial manifolds (Yushkevich et al., 2008). However, since these two methods ignore the functional nature of diffusion properties along fiber tracts, they can suffer from low statistical power in detecting interesting features and in exploring variability in tract-based dif- fusion properties. A functional data analysis framework was used to compare a univariate diffusion property along fiber tracts across two (or more) populations for a single hypothesis test per tract by using functional principal component analysis and the Hotelling T 2 statistic (Goodlett et al., 2009). Their method has two major limitations including only consideration of a univariate diffusion property and the lack of control for other covariates of interest, such as age. To address these two limitations, a functional regression framework was proposed to analyze multiple diffusion properties along fiber tracts as functional responses with a set of covariates of interest, such as age, diagnostic status and gender (Zhu et al., 2010b). An alternative approach, called the generalized functional **linear** model, was developed with a scalar outcome (e.g., diagnostic group) as responses and fiber bundle diffusion properties as varying covariate functions (or functional predictors) (Goldsmith et al., 2010).

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by means of cosimilarity which plays an important role in modern quantum theory [11-15]. B.Zhou and his coauthors showed that (2) had totally different solutions from the solution of X +A ∗ X −1 A = Q. They gave some sufficient conditions and necessary conditions for the existence of **positive** **definite** solutions of (2) in [13]. Moreover, several iterative methods such as basic fixed-point iter- ations [14], an inversion-free algorithm [14] and a structure-preserving-doubling algorithm [15] for the maximal **positive** **definite** solution of (2) were proposed.

Abstract. Although Selberg-type single **positive** **definite** **symmetric** matrices gamma and beta integrals have been evaluated by several authors, see e.g., Askey and Richards [1], Gupta and Kabe [2, 4], Mathai [8], and elsewhere in the vast multivariate statistical analysis literature. However, several other types of Selberg-type integrals appear to have been neglected in the literature. Thus e.g., Selberg-type integrals associated with inverse Wishart densities, inverse multivariate beta densities, their noncentral counterparts, etc, have not been explored as yet. The present paper records Selberg-type generalized quadratic forms gamma and beta integrals. Our methodology is based on hypercomplex (HC) multivariate normal distribution theory, Kabe [6].

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Background: Multiple **system** atrophy (MSA) is an adult-onset, rare, and progressive neurodegenerative disorder characterized by a varying combination of autonomic failure, cerebellar ataxia, and parkinsonism. MSA is categorized as MSA-P with predominant parkinsonism, and as MSA-C with predominant cerebellar features. The prevalence of MSA has been reported to be between 1.86 and 4.9 cases per 100,000 individuals. In contrast, approximately 1% of the population is affected by schizophrenia during their lifetime; therefore, MSA-P comorbidity is very rare in schizophrenic patients. However, when the exacerbation or progression of parkinsonism occurs in patients with schizophrenia treated with antipsychotics, it is necessary to consider rare neurodegenerative disorders, including MSA-P, in the differential diagnosis of parkinsonism.

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is called the condition number for matrix inversion with respect to the matrix norm ·. Notice that κA A −1 A ≥ A −1 A I ≥ 1 for any matrix norm see, e.g., 1, page 336. The condition number κA AA −1 of a nonsingular matrix A plays an important role in the numerical solution of **linear** systems since it measures the sensitivity of the solution of **linear** systems Ax b to the perturbations on A and b. There are several methods that allow to find good approximations of the condition number of a general square matrix.

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via lasso penalized D-trace loss by an efficient accelerated gradient method. The **positive**-definiteness and sparsity are the most important property of large covariance matrices, our method not only efficiently achieves these property, but also shows an better convergence rate. Numerical results have show that our estimator also have a better performance, comparing to Zhang et al. ’s method and the Graphical lasso method.

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The matrix variate beta type 3 distribution can be derived by using independent gamma matrices. An m × m random **symmetric** **positive** **definite** matrix Y is said to have a matrix variate gamma distribution with parameters Ψ > 0, and κ > m − 1/2, denoted by Y ∼ Gam, κ, Ψ, if its p.d.f. is given by

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The role of **positive** **definite** functions in obtaining sharp inequalities for trigonometric polyno- mials and entire functions is well known (see, for instance, Boas [6, Ch. 11], Timan [22, Sect. 4.8], Lizorkin [13], Gorin [9], and Trigub and Belinsky [23]). For instance, the classical Bernstein in- equality max |f ′ (x)| ≤ n max |f(x)| for trigonometric polynomials of degree at most n is related to the **positive** definiteness of the function (1 − |x|) + . A historical survey of such inequalities and the

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OF HADAMARD'S INEQUALITY The inequality of Hadamard [24] holds for a matrix in MnC when the absolute value of its determinant is dominated ~ by the absolute value of the product of its [r]

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This problem has been intensely studied since the 1960s. Many alternative formulations have been investigated, and efficient algorithms have been proposed in Cottle et al. (1992)[1]. Much attention has recently been paid on a class of iterative methods called the matrix-splitting method [2-4]. Matrix splitting method for **linear** complementarity problem exploits particular features of matrices such as the sparsity and the block structure. Such an approach is motivated by matrix splitting methods for the **linear** complementarity problem.

for any u , v ∈ B s . The equation (3) is called the Möbius addition, known as Möbius translation on the open s-ball (see formula (4.5.5) of [13]). The binary **system** ( B s , ⊕ M ) forms also a gyrocommutative gyrogroup, called the nonstandard real relativistic gyrogroup or Möbius gyrogroup.

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While the convergence rates shown in the above theorems do not depend on the dimensionality of original spaces, the rates may not be optimal. In fact, in the case of kernel ridge regression, the optimal rates are known under additional information on the spectrum of covariance opera- tors (Caponnetto and De Vito, 2007). It is also known (Eberts and Steinwart, 2011) that, given the target function is in the Sobolev space of order α, the convergence rates is arbitrary close to O p (n − 2α/(2α+d) ), the best rate for any **linear** estimator (Stone, 1982), where d is the dimensionality

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In this work, we have presented a unified study to explain the relation between universal kernels, characteristic kernels and RKHS embedding of measures: while characteristic kernels are related to the injective RKHS embedding of Borel probability measures, the universal kernels are related to the injective RKHS embedding of finite signed Borel measures. We showed that for all practical purposes (e.g., Gaussian kernel, Laplacian kernel, etc.), the notions of characteristic and universal kernels are equivalent. In addition, we also explored their relation to various other notions of **positive** **definite** (pd) kernels: strictly pd, integrally strictly pd and conditionally strictly pd. As an example, we showed all these notions to be equivalent (except for conditionally strictly pd) in the case of radial kernels on R d . We would like to note that while this study assumes the kernel to be real- valued, all the results extend verbatim to the case of complex-valued kernels as well.

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Since A is **positive** **definite**, then any eigenvalue of A is **positive** real number. Thus, the eigenvalues of H are all **positive** real numbers. That is to say, all eigenvalues of M and N are **positive** real numbers. Thus, M and N are both **positive** **definite**. It is obvious that M -1 and N -1 are **positive** **definite** matrices.