We derive iterative methods for computing the Fréchet derivative of the map which 6 sends a full-rank matrix A to the factor U in its polar decomposition A = UH, where U has 7 orthonormal columns and H is Hermitian positive definite. The methods apply to square matrices 8 as well as rectangular matrices having more rows than columns. Our derivation relies on a novel 9 identity that relates the ...

We derive iterative methods for computing the Fréchet derivative of the map which sends a full-rank matrix A to the factor U in its polar decomposition A = UH, where U has orthonormal columns and H is Hermitian positive definite. The methods apply to square matrices as well as rectangular matrices having more rows than columns. Our derivation relies on a novel identity that relates the Fréchet ...

A novel boundary integral equation (BIE) is developed for eddy-current nondestructive evaluation problems with surface crack under a uniform applied magnetic field. Once the field and its normal derivative are obtained for the structure in the absence of cracks, normal derivative of scattered field on the conductor surface can be calculated by solving this equation with the aid of method of mom...

We develop a theory of quasi-Newton and least-change update methods for solving systems of nonlinear equations F (x) = 0. In this theory, no diierentiability conditions are necessary. Instead, we assume that F can be approximated, in a weak sense, by an aane function in a neighborhood of a solution. Using this assumption, we prove local and ideal convergence. Our theory can be applied to B-diie...

We derive iterative methods for computing the Fréchet derivative of the map which sends a full-rank matrix A to the factor U in its polar decomposition A = UH, where U has orthonormal columns and H is Hermitian positive definite. The methods apply to square matrices as well as rectangular matrices having more rows than columns. Our derivation relies on a novel identity that relates the Fréchet ...

In this paper, the well-known nonconforming Morley element for biharmonic equations in two spatial dimensions is extended to any higher dimensions in a canonical fashion. The general -dimensional Morley element consists of all quadratic polynomials defined on each -simplex with degrees of freedom given by the integral average of the normal derivative on each -subsimplex and the integral average...

Recently Nolan constructed a spherically-symmetric spacetime admitting a spacelike curvature singularity with a regular C metric. We show here that this singularity is in fact weak. In a recent paper Nolan [1] constructed a simple spherically-symmetric spacetime which includes a spacelike curvature singularity with a continuous (C) metric. The goal was to use this example to demonstrate that a ...

We prove Kantorovich’s theorem on Newton’s method using a convergence analysis which makes clear, with respect to Newton’s Method, the relationship of the majorant function and the non-linear operator under consideration. This approach enable us to drop out the assumption of existence of a second root for the majorant function, still guaranteeing Q-quadratic convergence rate and to obtain a new...

1 A Model Diffusion Problem . . . . . . . . . . . . . . . . . . . . 1 x.1 Domain . . . . . . . . . . . . . . . . . . . . . . . . . . 1 x.2 Continuous Function . . . . . . . . . . . . . . . . . . . 2 x.3 Normed Vector Space . . . . . . . . . . . . . . . . . . 2 x.4 Square Integrable Function . . . . . . . . . . . . . . . 3 x.5 Inner Product Space . . . . . . . . . . . . . . . . . . . 4 x.6 Cauch...

We provide a comprehensive treatment of oscillation theory for Jacobi operators with separated boundary conditions. Our main results are as follows: If u solves the Jacobi equation (Hu)(n) = a(n)u(n + 1) + a(n − 1)u(n − 1) − b(n)u(n) = λu(n), λ ∈ R (in the weak sense) on an arbitrary interval and satisfies the boundary condition on the left or right, then the dimension of the spectral projectio...