Curvature matrix

Here m is the number of points in y and np the number of fitted parameters. The difference m-np is the number of degrees of freedom, df The elements djj in equation (4.32) are the diagonal elements of the inverse of the so-called curvature matrix, Curv, that contains the second derivatives of the sum of squares with respect to the parameters. The definition of the element Curvjk is... 

The implementation into nglm. m is straightforward. The curvature matrix needs to be passed back to the main program, ... 

The routine below is essentially the same as Ma in chrom. m, which we have used earlier (p.J5S). The additions are, that nglm. m returns the Curvature matrix Curv and the few lines at the end for the actual computation and output of the standard deviations. 

JlJ is an approximation for the curvature matrix. It is approximately 0.5 times the Hessian matrix of second derivatives of ssq with respect to the... 

What is the effect on the iterative refinement of the parameters The minimum is defined by Jtr=0. The curvature matrix is only required to guide the iterative process towards the minimum and thus the approximation, J J, for the curvature matrix does not compromise the exact location of the minimum. The approximation only results in a slightly different path taken by the algorithm towards the minimum. Ignoring the terms... 

The critical concepts encompassed by the Levenberg-Marquardt Method are the selection of the scaling factor for the Method of Steepest Descent and an approach for making a smooth transition from one method to the other. The curvature matrix a is replaced by cc such that... 

In the author s experience, the curvature matrix will always become singular, at least at some points in the parameter refinement. 

Th c Newton-Raph son block dingotial method is a second order optim izer. It calculates both the first and second derivatives of potential energy with respect to Cartesian coordinates. I hese derivatives provide information ahont both the slope and curvature of lh e poten tial en ergy surface, Un like a full Newton -Raph son method, the block diagonal algorilh m calculates the second derivative matrix for one atom at a lime, avoiding the second derivatives with respect to two atoms. 

The second energy derivatives with respect to the x, y, and z directions of centers a and b (for example, the x, y component for centers a and b is Hax,by = (3 E/dxa3yb)o) form the Hessian matrix H. The elements of H give the local curvatures of the energy surface along the 3N cartesian directions. 

The individual laminae used by Tsai [4-6] consist of unidirectional glass fibers in a resin matrix (U.S. Polymeric Co. E-787-NUF) with moduli given in Table 2-3. A series of special cross-ply laminates was constructed with M = 1,2,3,10 for two-layered laminates and M = 1,2,5,10 for three-layered laminates. The laminates were subjected to axial loads and bending moments whereupon surface strains were measured. Accordingly, the stiffness relations as strains and curvatures in terms of forces and moments, that is. 

The effect of the specific values of the B j can be readily calculated for some simple laminates and can be calculated without significant difficulty for many more complex laminates. The influence of bending-extension coupling can be evaluated by use of the reduced bending stiffness approximation suggested by Ashton [7-20]. If you examine the matrix manipulations for the inversion of the force-strain-curvature and moment-strain-curvature relations (see Section 4.4), you will find a definition that relates to the reduced bending stiffness approximation. You will find that you could use as the bending stiffness of the entire structure,... 

Most optimization algorithms also estimate or compute the value of the second derivative of the energy with respect to the molecular coordinates, updating the matrix of force constants (known as the Hessian). These force constants specify the curvature of the surface at that point, which provides additional information useful for determining the next step. 

H is a real symmetric matrix, and its eigenvalues are therefore real. Its eigenvectors are referred to as the principal axes of curvature . 

FegNi. Frozen phonon calculations combined with the determination of the electron-phonon matrix in the framework of the theory of Varma and Weber have been carried out for the ferrous alloy. The resulting phonon dispersion for the bet phase was already presented elsewhere . As expected, no softening or anomalous curvatures have been detected. This confirms the existence of a bet ground state for FesNi. 

The classification of critical points in one dimension is based on the curvature or second derivative of the function evaluated at the critical point. The concept of local curvature can be extended to more than one dimension by considering partial second derivatives. d2f/dqidqj, where qt and qj are x or y in two dimensions, or x, y, or z in three dimensions. These partial curvatures are dependent on the choice of the local axis system. There is a mathematical procedure called matrix diagonalization that enables us to extract local intrinsic curvatures independent of the axis system (Popelier 1999). These local intrinsic curvatures are called eigenvalues. In three dimensions we have three eigenvalues, conventionally ranked as A < A2 < A3. Each eigenvalue corresponds to an eigenvector, which yields the direction in which the curvature is measured. 

Instead of a formal development of conditions that define a local optimum, we present a more intuitive kinematic illustration. Consider the contour plot of the objective function fix), given in Fig. 3-54, as a smooth valley in space of the variables X and x2. For the contour plot of this unconstrained problem Min/(x), consider a ball rolling in this valley to the lowest point offix), denoted by x. This point is at least a local minimum and is defined by a point with a zero gradient and at least nonnegative curvature in all (nonzero) directions p. We use the first-derivative (gradient) vector Vf(x) and second-derivative (Hessian) matrix V /(x) to state the necessary first- and second-order conditions for unconstrained optimality ... 

These necessary conditions for local optimality can be strengthened to sufficient conditions by making the inequality in (3-87) strict (i.e., positive curvature in all directions). Equivalently, the sufficient (necessary) curvature conditions can be stated as follows V /(x ) has all positive (nonnegative) eigenvalues and is therefore defined as a positive (semidefinite) definite matrix. 

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