Change tolerances for error estimates:

Relax error tolerances for stopping:

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

DampingFactor 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:

Inspect the number of Jacobian evaluations needed by different methods:

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 :

Method options are also explained in Unconstrained Optimization.

Find a root for using different methods:

Define a function that monitors the steps and evaluations used by FindRoot:

The default (Newton's) method:

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

Secant method, starting with two initial conditions:

Select the affine covariant Newton method:

Change tolerances for error estimates:

Relax error tolerances for stopping:

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

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:

For an isomorphism , the inverse is the root of :

An approximate inverse for the exponential function:

It is very close to the built-in Log function:

A "black-box" function giving the period of an oscillation:

Plot its inverse:

Solve a boundary value problem , using a shooting method:

Use points on either side of the root to give bracketing starting values:

Plot the solution:

Solve the boundary-value problem , with n collocation points:

Consider as a first-order system :

Equations for collocation using the trapezoidal rule:

Use 0 as a starting value:

Find a solution for a particular value of ϵ: