Change tolerances for error estimates:

Relax error tolerances for stopping:

Make estimated relative distance to the root the main criterion for stopping:

can be used to help speed convergence to higher-order roots:

EvaluationMonitor can be used to keep track of function evaluations used:

Specify the Jacobian for a "black-box" function:

Without a specified Jacobian, extra evaluations are used to compute finite differences:

If you just know the sparse form, specifying the sparse pattern template saves evaluations:

Limit or increase the number of steps taken:

The default number of iterations is 100:

Eventually the algorithm stalls out since this mollifier function has all derivatives 0 at

:

Find a root for

using different methods:

The default (Newton's) method:

Brent's root-bracketing method requiring two initial conditions bracketing the root:

Secant method, starting with two initial conditions:

Monitor when iterative steps have been taken:

Show the steps on a contour plot of

:

Show steps (red) and evaluations (green). A step may require several evaluations:

Find a root using 100-digit precision arithmetic:

Find the root starting with machine precision and adaptively working up to precision 100: