In this paper we are interested in the higher-order derivatives of functions of the eigenvalues of symmetric matrices with respect to the matrix argument. We describe the formula for the k-th derivative of such functions in two general cases. The first case concerns the derivatives of the composition of an arbitrary (not necessarily symmetric) k-times differentiable function with the eigenvalue...

We describe a new extension to ScaLAPACK [2] for computing with symmetric (Hermi-tian) matrices stored in a packed form. The new code is built upon the ScaLAPACK routines for full dense storage for a high degree of software reuse. The original ScaLAPACK stores a symmetric matrix as a full matrix but accesses only the lower or upper triangular part. The new code enables more efficient use of mem...

Nowadays, observing and understanding the performance of a multithreaded application is untrivial, especially within a complex thread environment (multilevel scheduling). Thanks to our environment, the run of a multithreaded application can be precisely analyzed. In particular, our environment allows the programmer to measure the number of CPU cycles used by a given function or the precise sche...

In this review we discuss the relationship between random matrix theories and symmetric spaces. We show that the integration manifolds of random matrix theories, the eigenvalue distribution, and the Dyson and boundary indices characterizing the ensembles are in strict correspondence with symmetric spaces and the intrinsic characteristics of their restricted root lattices. Several important resu...

In this paper, static response and buckling analysis of functionally graded saturated porous beam resting on Winkler elastic foundation is investigated. The beam is modeled using higher-order shear deformation theory in conjunction with Biot constitutive law which has not been surveyed so far. Three different patterns are considered for porosity distribution along the thickness of the beam: 1) ...

The trace of A is Tr(A) ∑di 1 Aii ∑di 1 λi(A). The trace norm of A is ‖A‖∗ ∑di 1 |λi(A)|. A symmetric matrix is positive semide nite (PSD) if all its eigenvalues are nonnegative. Note that for a PSD matrix A, we have Tr(A) ‖A‖∗. We also recall the matrix exponential eA ∑∞k 0 Ak k! which is well-de ned for all real symmetric A and is itself also a real symmetric matrix. If A is symmetric, then e...

We describe a new extension to ScaLAPACK [2] for computing with symmetric (Hermi-tian) matrices stored in a packed form. The new code is built upon the ScaLAPACK routines for full dense storage for a high degree of software reuse. The original ScaLAPACK stores a symmetric matrix as a full matrix but accesses only the lower or upper triangular part. The new code enables more efficient use of mem...

A real symmetric matrix of order n has a full set of orthogonal eigenvectors. The most used approach to compute the spectrum of such matrices reduces first the dense symmetric matrix into a symmetric structured one, i.e., either a tridiagonal matrix [2, 3] or a semiseparable matrix [4]. This step is accomplished in O(n) operations. Once the latter symmetric structured matrix is available, its s...

This paper presents Local Log-Euclidean Covariance Matrix (LECM) to represent neighboring image properties by capturing correlation of various image cues. Our work is inspired by the structure tensor which computes the second-order moment of image gradients for representing local image properties, and the Diffusion Tensor Imaging which produces tensor-valued image characterizing the local tissu...