We first recall some properties of infinite tridiagonal matrices considered as matrix transformations in sequence spaces of the forms sξ , sξ , s (c) ξ , or lp(ξ). Then, we give some results on the finite section method for approximating a solution of an infinite linear system. Finally, using a quasi-Newton method, we construct a sequence that converges fast to a solution of an infinite linear ...

As in Section 4.3, this objective is minimized using the Gauss-Newton method. Specifically, we start from an initial transformation T and perform optimization iteratively. In each iteration, we locally parameterize T with a 6-vector ξ, evaluate the residual r and Jacobian Jr at T, solve the linear system in (21) to compute ξ, and use ξ to update T. To compute the Jacobian, we need the partial d...

The single bootstrap already is popular in economics, though the double bootstrap has better convergence properties. We discuss the theory and implementation of the double bootstrap, both with and without the pivotal transformation, and give detailed examples of each. One example is a nonlinear double bootstrap of a Cobb-Douglas production function, and explains the use of Gauss-Newton Regressi...

Examples from Archimedes, Galileo, Newton, Einstein and others suggest that fundamental laws of physics were—or, at least, could have been—discovered by experiments performed not in the physical world but only in the mind. Though problematic for a strict empiricist, this mode of discovery may have become possible, in part, through the evolutionary emergence in humans of deeply internalized impl...

Nonrelativistic Newton and Schrödinger equations remain correct not only under holonomic but also under nonholonomic transformations of the spacetime coordinates. Here we study the properties of transformations which are holonomic in the space coordinates while additionally tranforming the time in a path-dependent way. This makes them nonholonomic in spacetime. The resulting transformation form...

We consider the formal reduction of a system of linear differential equations and show that, if the system can be block-diagonalised through transformation with a ramified Shearing-transformation and following application of the Splitting Lemma [9], and if the spectra of the leading block matrices of the ramified system satisfy a symmetry condition, this block-diagonalisation can also be achiev...

Solutions at infinity are solutions with z0 = 0 and not all other coordinates equal to zero. We see that only z 1z 2 2 = 0 remains, or equivalently we find [0 : 1 : 0] and [0 : 0 : 1] representing two distinct solutions at infinity, but each with multiplicity two. This is an unsatisfactory result, as we expected to find two, not four solutions at infinity. The projective transformation defined ...

We present a regularized Gauss-Newton method for solving the inverse problem of parameter reconstruction from boundary data in frequency-domain diffuse optical tomography. To avoid the explicit formation and inversion of the Hessian which is often prohibitively expensive in terms of memory resources and runtime for large-scale problems, we propose to solve the normal equation at each Newton ste...

In this letter, we present a computational framework based on the use of the Newton and level set methods and tailored for the modeling of bubbles with surface tension in a surrounding Newtonian fluid. We describe a fully implicit and monolithic finite element method that maintains stability for significantly larger time steps compared to the usual explicit method and features substantial compu...

The main di culty in the implementation of most standard implicit Runge-Kutta (IRK) methods applied to (sti ) ordinary di erential equations (ODE's) is to e ciently solve the nonlinear system of equations. In this article we propose the use of a preconditioner whose decomposition cost for a parallel implementation is equivalent to the cost for the implicit Euler method. The preconditioner is ba...