nonlinear_newton.pdf

Nonlinear equations:

Newton’s method

MATH2070: Numerical Methods in Scientific Computing I

Location: http://people.sc.fsu.edu/∼jburkardt/classes/math2070 2019/nonlinear newton/nonlinear newton.pdf

Even in zero visibility, a skier knows which way is downhill!

Rootfinding (Newton version)

Given f(x), with derivative f

0

(x), and a starting point x, estimate a root x

∗

for which f(x

1 Newton’s linear model for f(x)

Noting that f(x

∗

) = 0, and dropping the O(x

of the root:

f(x)

0

(x)

Thus, to estimate the root of a function, we try to find a starting point that we hope is already close, and

we must be prepared to evaluate the derivative of the function as part of the iteration.

We hope that, once the iteration is close to the root, the norm of the function will decrease on every step,

in a single starting value x, and along with the function f(x) we now must include the derivative function

1