Ball and beam, Navier-Stokes, stability of an airplane wrt to pilot commands. nonlinear systems, number of variables = number of equations The solution means determining some values of variables/parameters that strictly satisfy the given relationships (equations). Impossible to solve them by hand even if they are homogeneous.
Direct solve – reuse the factorization (quasi-implicit Newton, chord method)
Computational fluid dynamics – millions of variables (chesapeake bay)
Examples of cost functions Talk about differentiability
Convergence rates - LINEAR We want Bk to mimic the behavior of newton => symmetric positive definite (at least in a neighborhood)
Trust regions determine a region around the current iterate where the approximate model of the objective function is accurate. The step is taken to be the minimizer on the trust region.
Newton’s method is a very powerful method with many applications and uses (solving nonlinear systems, finding minima of cost functions). Newton’s method can be used together with many other numerical algorithms (factorizations, linear solvers)
The optimization and parallelization of matrix-vector, matrix-matrix products, decompositions and other numerical methods can have a significant impact in overall performance