Multivariate Newton’s Method (newtons_method_n)

Newton's method for finding the root of a differentiable, multivariate, vector-valued function.

newtons_method_n

Newton's method for finding the root of a differentiable, multivariate, vector-valued function.

Syntax

x = newtons_method_n(f,J,x0)
x = newtons_method_n(f,J,x0,opts)
[x,k] = newtons_method_n(__)
[x,k,x_all] = newtons_method_n(__)

Description

x = newtons_method_n(f,J,x0) returns the root of a multivariate, vector-valued function specified by the function handle f, where J is the Jacobian of with respect to (i.e. ) and where x0 () is an initial guess of the root.

x = newtons_method_n(f,J,x0,opts) does the same as the syntax above, but allows for the specification of optional solver parameters. opts is a structure with the following fields:

• k_max → maximum number of iterations (defaults to 200)
• return_all → returns estimates at all iteration if set to true (defaults to false)
• TOL → tolerance (defaults to )

[x,k] = newtons_method_n(__) also returns the number of iterations (k) performed of Newton's method.

[x,k,x_all] = newtons_method_n(__) does the same as the previous syntaxes, but also returns an array (x_all) storing the root estimates at each iteration. This syntax requires that opts.return_all be set to true.

Note

• The Numerical Differentiation Toolbox provides functions for approximating the Jacobian.

Examples and Additional Documentation

• See "EXAMPLES.mlx" or the "Examples" tab on the File Exchange page for examples.
• See "Root_Finding_Methods.pdf" (also included with download) for the technical documentation.