Nonlinear Equations
The problem of finding a solution to a system of n nonlinear equations, F(x) = 0, may be stated as follows:
given F: Rn → Rn, find x* (an element of Rn) such that F(x*) = 0
For example:
x* = [0, 3] or x* = [3, 0]
Note
A solution to a system of nonlinear equations is not necessarily unique.
The most powerful and successful numerical methods for solving systems of nonlinear equations are loosely based upon a simple two-step iterative method frequently referred to as Newton's method. This method begins with an initial guess and constructs a solution by iteratively approximating the n-dimensional nonlinear system of equations with an n-by-n linear system of equations.
The first step formulates an n-by-n linear system of equations (Js = – F) where the coefficient array J is the Jacobian (the array of first partial derivatives of F), s is a solution vector, and – F is the negative of the nonlinear system of equations. Both J and – F are evaluated at the current value of the n-element vector x.
J(xk) sk = – F(xk)
The second step uses the solution sk of the linear system as a directional update to the current approximate solution xk of the nonlinear system of equations. The next approximate solution xk+1 is a linear combination of the current approximate solution xk and the directional update sk.
xk+1 = xk + sk
The success of Newton's method relies primarily on providing an initial guess close to a solution of the nonlinear system of equations. In practice this proves to be quite difficult and severely limits the application of this simple two-step method.
IDL provides two algorithms that are designed to overcome the restriction that the initial guess be close to a solution. These algorithms implement a line search which checks, and if necessary modifies, the course of the algorithm at each step ensuring progress toward a solution of the nonlinear system of equations. IDL's NEWTON and BROYDEN functions are among a class of algorithms known as quasi-Newton methods.
The solution of an n-dimensional system of nonlinear equations, F(x) = 0, is often considered a root of that system. As a one-dimensional counterpart to NEWTON and BROYDEN, IDL provides the FX_ROOT and FZ_ROOTS functions.
Routines for Solving Nonlinear Equations
See Nonlinear Equations (in the functional category "Mathematics" (IDL Quick Reference)) for a brief description of IDL routines for solving systems of nonlinear equations. Detailed information is available in the IDL Reference Guide.
