126_MATH 685 CSI 700 OR 682 Lecture Notes

MATH 685/ CSI 700/ OR 682 Lecture Notes Lecture Nonlinear equations Nonlinear Equations  Given a function f, we are looking for a value x, s.t f(x)=0 (a root of the equation, or a zero of the function f) The problem is called root finding or zero finding Example Existence/uniqueness Examples in 1d Example of a system in 2d Multiplicity Sensitivity and conditioning Sensitivity and conditioning Sensitivity and conditioning Fixed-point iteration (FPI) Newton’s method Example Example Convergence of Newton’s method Cost of Newton’s method Secant updating methods Broyden’s method Broyden’s method Example Example Example Example (cont) Robust Newton-like methods Trust-region methods ...  Given a function f, we are looking for a value x, s.t f(x)=0 (a root of the equation, or a zero of the function f) The problem is called root finding or zero finding Example Existence/uniqueness